Crystal Structure of Intermetallic Compounds

CRYSTAL STRUCTURES OF INTERMET ALLIC COMPOUNDS

Edited Ьу

j. Н. Westbrook Brookline Technologies, Вallston Spa, New York, USA

and

R. L. Fleischer Union College, Schenedady, New York, USA

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Library of Congress Cataloging-in-Publication Data

Crystal structures in intermetallic compounds / edited by J. H. Westbrook and R. L. Fleischer.p. cm.

'This volume is one of four now being published, each of which consists of reprints ofchapters from the 1995 comprehensive two-volume set—Intermetallic compounds,principles and practice . . . selected sets of chapters are collected, each set being on asingle theme . . . reprint volume 1'—Pref.

Includes bibliographical references and index.ISBN 0-471-60880-7 (pbk. : alk. paper)—ISBN 0-471-60814-9 (set)1. Intermetallic compounds—Structure. 2. Crystallography. 3. Alloys. 4. Physical

metallurgy. I. Westbrook, J. H. (Jack Hall), 1924- II. Fleischer, R. L. (Robert Louis),1930- III. Title: Intermetallic compounds.

TA483.C79 2000620.1 '699—dc21 99-052446

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0 471 60880 7ISBN 0 471 60814 9 (set)

Typeset by Dobbie Typesetting Ltd, Tavistock, DevonPrinted and bound in Great Britain by Antony Rowe, Chippenham, WiltshireThis book is printed on acid-free paper responsibly manufactured from sustainable forestry,in which at least two trees are planted for each one used for paper production.

Dedication

To the memory of

John Herbert Hollomon1919-1985

Wise, vigorous, effective advocate of the relevance and value of scientific research inindustry.

His strong belief in the synergetic interaction of Principles and Practice in the field ofmetallurgy impelled him to assemble an innovative, diverse staff at General Electric,and to inspire independent exploration that benefited both science and engineering.

Oksana BodakDepartment of Inorganic Chemistry,L'viv State University,Lomonosova str. 6, 290005,L'viv 5, Ukraine

Jo L. C. Daams,Philips Research Laboratories,Prof. Holstlaan 4, 5656AA,Eindhoven,The Netherlands

Evgen GladyshevskiiDepartment of Inorganic Chemistry,L'viv State University,Lomonosov Str. 6,290005, L'viv 5, Ukraine

JUrgen HauckInstitut fiir Festkorperforschung,KFA Forschungszentrum,Postfach 1913, D-52425 Jiilich,Germany

Contributors

Kenneth F. KeltonDepartment of Physics,Campus Box 1105Washington University,St Louis, MO 63310-4899, USA

Carl C. KochDepartment of Materials Scienceand Engineering, North CarolinaState University, Raleigh,NC 27695, USA

Klaus MikaInstitute fiir Festkorperforschung,KFA Forschungszentrum,Postfach 1913, D-52425 Julich,Germany

Michael V. NevittDepartment of Physics andAstronomy, Clemson University,231 Kinard Laboratory,Clemson,SC 29634, USA

Erwin E. HeIInerInstitute for Mineralogy,Philipps University, Lahnberge,D-35043 Marburg, Germany

Erwin PartheDepartment of Inorganic Chemistry,Universite de Geneve,Quai Erenest Ansermet 30,CH-1211, Geneva 4, Switzerland andInstitute for Mineralogy andCrystallography, Universitat Wien,Geozentrum, Althanstrasse 14 A-1090,Vienna, Austria

De hoc, multi noscunt multa,omnes aliquid nemo satis,

(Concerning this, many know much,each a little, none enough.)

—anonymous Latin epigram

David G. PettiforDepartment of Materials,University of Oxford,Parks Road, Oxford0X1 3PH,UK

Walter L. Roth1552 Baker Avenue,Schenectady,NY 12309, USA

Roland SchwartzInstitute for Mineralogy,Philipps University,Lahnberge, D-35043 Marburg,Germany

Pierre VillarsIntermetallic Phases Data Bank(IPDB) and MaterialsPhases Data System (MPDS),PO Box 1, CH-6354 Vitznau,Switzerland

Preface to the 1995 Edition

Intermetallic compounds were last comprehensively reviewed in 1967 in a volume that was edited by one of us(JHW). At that time the field was described as of special interest because it was undergoing 'exponentialproliferation'. That trend continues to the present. The number of intermetallic entries in the Permuterm SubjectIndex shows a doubling period of less than nine years, having reached roughly 1800 entries per year in 1993. Apartfrom scholarly interest, intermetallics have now become of substantial commercial significance; for some, suchas Ni3Al, world-wide use is in the 1000s of tons; for others, for example IH-V semiconducting compounds,although the quantities employed are not in tonnage numbers, their value as vital components of electronic circuitsis in the billions of dollars.

From the 1967 book we remind the reader that 'The first published paper dealing with intermetallic compoundsappeared in 1839, and more than sixty years elapsed before . . . the first review paper by Neville in 1900. However,new results were then appearing so rapidly that fifteen years later two books were printed, devoted exclusivelyto this subject, one by Desch in England and one by Giua and Giua in Italy'. More recently, conference volumesthat deal exclusively with intermetallics but typically only within specific, limited sub-topical subject areas havebecome common. The scope of the present work is as broad as that of its 1967 predecessor. However, the increasedvolume of activity in intermetallics and the increased significance of their applications have necessitated an expansionfrom the 27 chapters of the earlier work to the 75 chapters of the present treatise.

First, what are intermetallic compounds? Generally, such a compound is a structure in which the two or moremetal constituents are in relatively fixed abundance ratios and are usually ordered on two or more sublattices,each with its own distinct population of atoms. Often substantial or complete disorder may obtain, as a resultof low ordering energy or the intervention of some external agency, for example extreme cooling rates, radiation,etc. Deviations from precise stoichiometry are frequently permitted on one or both sides of the nominal ideal atomicratios, necessitating a partial disorder. Here we include as intermetallic compounds all metal-metal compounds,both ordered and disordered, binary and multicomponent. Even the metal-metal aspect of the definition is oftenrelaxed by including some metal-metalloid compounds, such as silicides, tellurides, and semiconductors. We believethis inclusion is appropriate since the phenomenology of many such compounds is nearly identical to metal-metalones, and they provide useful examples of principles, properties, and practices.

The burgeoning literature on intermetallics and the lack of a comprehensive single source of up-to-date descriptionsof where we are, what we need to know, and what we can do with intermetallics created the incentive for thepresent pair of volumes. This work was planned to provide state-of-the-art assessments of theory, experiment,and practice that will form a solid base for workers who wish to know more than their own particular area. Eachauthor was asked to set forth the principles of his or her subject in terms that are meaningful to scientists andengineers who are not specialists in the author's field, and then to progress to include knowledge that workersin their own areas would wish to have. Concluding sections of most chapters give the authors' critical assessmentof the state of their subject and of where they believe further effort is merited.

This work is divided into two volumes in order that each be of manageable size. The first, on the theme Principles,is directed at the science of intermetallics—how do we understand their formation, structure and properties? ThePractice volume considers commercial production and engineering applications of intermetallic compounds. Thereader who browses carefully will recognize that the immediacy of the practice described ranges from hoped-foruse, to beginnings of use, to actual commercial application—depending on the specific subject. Some of the hoped-foruses are fated never to be realized, but the authors have aimed to reveal what the obstacles are so that the readermay make his or her own assessment (and possibly provide a solution!).

We conferred carefully with many people in order to identify authorities for each subject; having recruitedcontributors for the project, we then strove to assist them in achieving clarity and thoroughness from outline todraft to final manuscript. The contributors cooperated superbly, and we thank them for their hard work and highachievement. We sought experts wherever they were to be found, and our international set of nearly 100 authorsturned out to be almost equally divided between the United States and 14 other countries. Manuscripts have infact come from all inhabited continents.

We planned this work as an aid to both scientists and engineers. It can serve as a base for those who wish toknow about intermetallics as an area in which to begin research. Equally it is a resource to workers who are alreadyactive in the field and need, or wish, to expand their knowledge of related science or practical technology. Weexpect that many chapters are appropriate source matter for special topic or seminar courses at the advancedundergraduate and various graduate school levels. It is hoped that passage of the next 25 years will reveal someinfluence of this treatise on the further development of this field.

As an assist to readers we have provided in the following pages a consolidated acronym list and somecrystallographic tables. Nomenclature for crystal structure types is often complex, and some of the authors haveintroduced their own. Generally we have asked authors to include both of two commonly used types of symbolsas they introduce structures. The two-part table following this preface lists many of the common types—byStrukturbericht symbol, prototype name (termed a structure type), and Pearson symbol. Strukturbericht symbolsare only partly significant and systematic: A's are not compound structures but consist of a single lattice of atoms(except for A15!); B's are equiatomic ordered structures; C s have 2-to-l atomic abundance ratios, DO's 3-to-l.Structure type compounds are the specific ones used to designate a particular structure. Thus B2 compounds arealso referred to as CsCl compounds. Many structures are better known to metallurgists and mineralogists by namesother than the formula of the structure type chosen by crystallographers, e.g. Laves, fluorite, Heusler, etc. Suchnames have been added in selected cases. The Pearson symbols tell the crystal symmetry and the number of atomsper unit cell. Thus, B2, CsCl has a primitive (P) cubic (c) structure with 2 atoms per cell and hence the Pearsonsymbol (cP2). The Pearson designation is informative, but it is not necessarily unique. Although there is onlyone cP2 structure, Villars and CaIvert list two cP4s, three cF12s and twenty-two hP9s. Thus to be definitive, boththe structure type and the Pearson symbol need to be given, or the Pearson and the Strukturbericht symbol.

The index in each volume includes the subjects in both volumes of this work, in order that the reader may beable to locate any subject that is addressed. Although the purpose of such combined indices is not to induce theowner of a single volume to purchase the other, it possibly may help to reduce the barrier to such action.

We have benefited from outstanding secretarial help during the three years of this project, first by Phillis Liu,then Constance Remscheid at General Electric, finally Mary Carey at Rensselaer Polytechnic Institute. We appreciatethe hospitality of the General Electric Research and Development Center during the inception and middle periodof preparing these volumes. Assembling the final product has been eased for us by the continuing efforts andcheerful good counsel at John Wiley of Jonathan Agbenyega, Irene Cooper, Philip Hastings, Vanessa Lutmanand Cliff Morgan.

J. H. WESTBROOK, Ballston Spa, New YorkR. L. FLEISCHER, Schenectady, New York

Upon these considerations, we have been induced to undertake the present extensive work, the purposeof which is to instruct rather than to amuse; in which nothing will be omitted that is elegant or great;but the principal regard will be shown to what is necessary and useful.

—Isaac Ware, 1756

Preface to theReprint Volumes from

Intermetallic Compounds: Principles and Practice

This volume is one of four now being published, each of which consists of reprints of chapters from the1995 comprehensive two-volume set Intermetallic Compounds: Principles and Practice. In the presentvolumes selected sets of chapters are collected, each set being on a single theme. In this format readerswho are interested in a particular aspect of intermetallic compounds can have a less weighty volumespecific to their subject; a volume that can be produced more economically than the full, original 1900-page set; and that includes a modest updating of the subject matter.

The subjects in most cases are taken from one or more chapter groupings of the original Volume 1 or2: Hence reprint volume 1, Crystal Structures of Intermetallic Compounds, contains the ten chaptersfrom the original work under the heading Crystal Structures; reprint volume 2, Basic MechanicalProperties and Lattice Defects of Intermetallic Compounds, contains three from Property Funda-mentals, four chapters from Defect Structures, and two from Kinetics and Phase Transformations;reprint volume 3, Structural Applications of Intermetallic Compounds contains the thirteen chapters thatwere under that same topic; and finally reprint volume 4, Magnetic, Electrical, and Optical Propertiesand Applications of Intermetallic Compounds, contains two chapters from the section on PropertyFundamentals, seven from Electromagnetic Applications and one from Miscellaneous. Although eachchapter is reprinted nearly intact (only typographic and factual errors corrected), the author or authorswere given the option of adding a brief addendum in order to add whatever new perspective has arisenover the intervening few years. Some have chosen to do so; some have not, either in the preferred casebecause they felt none was needed or because the four-month window of opportunity they were givento satisfy our and the publisher's desire for promptness did not fit their work schedule. Corrections tothe original chapters that were so lengthy that they would upset the original pagination are to be foundin the addenda at the end of each relevant chapter.

Where an addendum is particularly relevant to a portion of the original chapter being reproduced, amargin mark (*) alerts the reader to refer to the added pages at the end of the chapter. Cross-referencesto other chapters relate to the original 1995 two-volume work, the tables of contents of which appear atthe end of this volume.

JHWRLF

Acronyms

ID two-dimensional BH buried heterostructure3D three-dimensional BIS bremsstrahlung isochromat6D six-dimensional spektroskopie

BM Bowles-Mackenzie (theory ofACAR angular correlation of annihilation martensitic transformation)

radiation BSCCO bismuth-strontium-calcium-copperACPAR angular correlation of positron oxide

annihilation radiation BSE back-scattered electronsAE atomic environment BT Bhatia-Thornton (partial structureAES Auger electron spectroscopy factor for liquid alloys)AET atomic environment type BW Bragg-Williams (theory of ordering)AIM argon induction melting BZ Brillouin zoneALCHEMI atom location by channeling

enhanced microanalysis CAM c-axis modulatedALE atomic layer epitaxy CANDU Canadian deuterium-uraniumAM air mass (power reactor)AMT Advanced Materials Technology, CAP consolidated under atmospheric

Inc. pressureAN atomic number CAT computer-assisted tomographyAP atom probe CBLM cluster Bethe lattice methodAP atomic property CC cluster centerAPB antiphase boundary CCD charge-coupled deviceAPD antiphase domain CCGSE concentric-circle grating surface-APD avalanche photodetector emitting (laser)APE atomic property expression CCIC cabled conductor in conduitAPW augmented plane wave CCMAI crystal chemical model of atomicAR antireflection interactionsARIPES angle-resolved inverse photoemission c.c.p. cubic close-packed

spectroscopy CCT continuous cooling transformationARPES angle-resolved photoemission CD compact disc

spectroscopy CD climb dislocationASA atomic-sphere approximation CEBAF continuous electron-beam acceleratorASW augmented spherical wave facility

CEF crystalline electric fieldBC bond charge CERN Centre Europeenne Rechercheb.c.c. body-centered cubic NucleaireBCS Bardeen-Cooper-Schrieffer (theory CFT concentration-functional theory

of superconductivity) CMC ceramic-matrix compositeb.c.t. body-centered tetragonal CN coordination number

CO cubo-octahedron ESR electroslag refinedCP coordination polyhedron ETP electrolytic tough pitch (copper)CPA coherent-potential approximation EXAFS extended X-ray absorption fine structureCRSS critical resolved shear stressCS chemisorption f.c.c. face-centered cubicCSF complex stacking fault f.c.t. face-centered tetragonalCSL coincidence-site lattice FENIX Fusion Engineering InternationalCSRO chemical short-range order Experimental Magnet FacilityCT chisel toughness FET field effect transistorCTE coefficient of thermal expansion FIM field ion microscopyCVD chemical vapor deposition FLAPW full-potential linearized augmentedCVM cluster variation method plane waveCW cold worked FLASTO full-potential linearized augmentedCW concentration wave Slater-type orbitalCW continuous wave FLMTO full-potential linearized muffin-tinCWM Connolly-Williams method (theory orbital

of phase transformations) FOM figure of meritFP Fabry-Perot (laser)

D-A donor-acceptor FT phase transformationDB diffusion bonding FZ floating zoneDBTT ductile-brittle transition temperatureDC direct chill (casting) GB gain x bandwidth (product)DC direct current GB grain boundaryDCA direct configurational averaging GFT glass-forming tendencyDF density functional GGA generalized gradient approximationDFB distributed feedback GITT galvanostatic intermittent titrationDFT density-functional theory techniqueDH double heterojunction GPM generalized perturbation methodd.h.c.p. double hexagonal close-packed GRPA generalized random-phasedHvA de Haas-van Alphen (effect) approximationDLZR directional levitation zone melting GS ground stateDOS density of states GT Goody-Thomas (electronegativity)DPA displacement per atomDPC demonstration poloidal coil HB horizontal BridgmanDRP dense random packing HBT heterojunction bipolar transistorDS directional solidification HCF high-cycle fatigueDSC displacement shift complete h.c.p. hexagonal close-packed

HEMT high-electron-mobility transistore/a electron/atom (ratio) HIP hot isostatic pressingEAM embedded-atom method HPT heteroj unction phototransmitterEBPVD electron beam physical vapor HR high resolution

deposition HREM high-resolution electron microscopyECI effective cluster interaction HRTEM high-resolution transmission electronECM embedded-cluster method microscopyEDC electro-optic directional coupler HSCT high-speed civil transportEDM electrodischarge machining HTS high-temperature superconductorEDX energy-dispersive X-ray HVEM high-voltage electron microscopy

(spectroscopy) HVTEM high-voltage transmission electronEELS electron energy-loss spectroscopy microscopyEMF electromotive forceEPI effective pair interaction IAE irregular atomic environmentESF extrinsic stacking fault IAET irregular atomic environment type

IC integrated circuit LO longitudinal optical (wave)IC investment cast LPCVD low-pressure chemical vaporIDOS integrated density of states depositionIEM interstitial-electron model LPE liquid-phase epitaxyIGC Intermagnetics General LPPS low-pressure plasma sprayingIHPTET integrated high-performance turbine LPS long-period superstructure

engine technology LRO long-range orderILS invariant line strain LSDA local spin-density approximationIMC intermetallic compound LSI large-scale integrationIMC intermetallic matrix compositeIMC inverse Monte Carlo (method) /ASR muon spin relaxationIPM independent-particle method MA mechanical alloying

(approximation) MAPW modified augmented plane waveIPS invariant plan strain MB Martinov-BasanovIQC icosahedral quasicrystal (electronegativity)IR infrared MBE molecular beam epitaxyISF intrinsic stacking fault MBT metal-base transistorIT (positive) inner tetrahedron MC Monte CarloITER International Thermonuclear MCS Monte Carlo simulation

Experimental Reactor MD molecular dynamicsIV intermediate valence MEE migration-enhanced epitaxyTT71^ • .. n u a. A A . A MESFET metal Schottky field-effect transistorJFET junction field-effect transistor x*Trrv *„• ^ - ^ + rr T ,

MFTF Mirror Fusion Test FacilityKKR Korringa-Kohn-Rostoker (bond- MISFET metal-insulator-semiconductor field

calculation method) effect transistorKSV Khantha-Cserti-Vitek (deformation MJR McDonald jelly roll (superconducting

model) cable construction)KTP potassium titanyl phosphate MLR multi-layer reflectorKW Kear-Wilsdorf (dislocation locking MMC metal-matrix composite

mechanism) MN Mendeleev numberMO magneto-optical

LA longitudinal acoustic (wave) MOCVD metal-organic chemical vaporLAPW linearized augmented plane wave depositionLASTO linearized augmented Slater-type MOS metal-oxide-semiconductor

orbital MOSFET metal-oxide-semiconductor fieldLCAO linear combination of atomic orbitals effect transistorLCF low-cycle fatigue MOVPE metal-organic vapor phase epitaxyLCT large coil task MQW multiple quantum wellLCW Lock-Crisp-West (radiation MRI magnetic resonance imaging

analysis) MRSS maximum resolved shear stressLD laser diode MRT orthodontic NiTi alloyLDA local-density approximation MT muffin tinLEC liquid-encapsulated Czochralski MTD martensitic transformation diagramLED light-emitting diode MVA million volt-amperesLEED Low-energy electron diffractionLEISS low-energy ion scattering NASP National AeroSpace Plane

spectroscopy NET Next European Torus (fusion device)LHC Large Hadron Collider NHE normal hydrogen electrodeLKKR Layered KKR (structure calculation) NMI National Maglev InitiativeLME liquid metal embrittlement NMR nuclear magnetic resonanceLMTO linearized muffin-tin orbital NN nearest neighborLNT liquid nitrogen temperature NNH nearest-neighbor histogram

NNN next nearest neighbor RDS rate-dermining stepNOR negative OR (logic operator) RE rare earth (metal)NSR notch/strength ratio RF radiofrequency

OAZ oxidation-affected zone R H E reversible hydrogen electrode

ODR oxygen dissolution reaction R ™ n g l?~ 1 O n m 0 ^ d , v v ÔDS oxide dispersion-strengthened R K K Y Ruderman-Kittel-Kasuya-YoshidaOEIC optoelectronic integrated circuit ( e l e c t r o n i n t e r a c t i o n s )OH octahedron r m s - root mean squareORNL Oak Ridge National Laboratory R R R residual resistivity ratioOT (negative) outer tetrahedron R ^ rap.dly sohd.fiedOTMC orthorhomic Ti-matrix composites R S P r a p l d s o h d l f i c a t i o n processing

RSS resolved shear stressPAS positron annihilation spectrosxopy RT room temperaturePBC periodic bond chain RUS resonance ultrasound spectroscopyPBT permeable-base transistorPCM phase-change material SAD selected-area diffractionPCT pressure-composition-temperature SAED selected-area electron diffractionPD phase diagram SAGBO stress-assisted grain-boundary oxidationPDF pair distribution function SAM-APD separate absorption and multiplicationPDOS phonon density of states avalanche photodetectorPFC planar flow casting s.c. simple cubicPH Pearson's Handbook SC semiconductorPHACOMP phase computation SCE standard colomel electrodePKA primary knock-on atom SCH separate confinement heterostructuresPL photoluminescence SDC specific damping capacityPM powder metallurgy SDW spin-density wavePMTC phenomenological martensite SEM scanning electron microscopy

transformation concept SESF superlattice extrinsic stacking faultPN periodic number SF stacking faultpnpn type of photothyristor SG spin glassPPDF partial pair distribution function SHS self-propagating high-temperaturePPM path-probability method synthesisPPV Paidar-Pope-Vitek (Ll2 hardening SI/VLSI semi-insulating very large-scale

model) integrationPS Pearson symbol SIA self-interstitial atomPT phase transformation SIC self-interaction correlationPTMC phenomenological theory of SIM stress-induced martensite

martensite crystallography SIMS secondary-ion mass spectrometryPVD physical vapor deposition SIS superconductor-insulator-PZT lead zirconate titanate (ceramic) superconductor

SISF superlattice intrinsic stacking faultQC quasicrystal(line) SIT static inductance transistorQCSE quantum confined Stark effect SM semimetalQFD quantum formation diagram SMA second-moment approximationQN quantum number SMA shape-memory alloyQSD quantum structural diagram SME shape-memory effectQW quantum well SPF superplastic forming

SQUID superconducting quantumRBS Rutherford back scattering interference deviceRC ribbon comminution SRO short-range orderRCS replacement-collision sequence SSAR solid-state amorphizing reactionRDF radial distribution function SSD structural stability diagram

SSF superlattice stacking fault ULSI ultra large-scale integrationSTA Atlas of Crystal Structure Types USW ultrasonic waveSTEM scanning transmission electron UTS ultimate tensile strength

microscopy UV ultravioletSTM scanning tunneling microscopySV Sodani-Vitole change of Paidar et VAR vacuum arc refined

al. model VCSEL vertical-cavity surface-emittinglaser

TA transverse acoustic (wave) VEC valence-electron concentrationTB tight binding VGF vertical gradient freezingTCP topologically close-packed VHF very high frequencyTD thoria dispersion VIM vacuum induction meltingTDFS temperature dependence of flow VLS vapor-liquid-solid

stress VLSI very large-scale integrationTE thermoelectric VPE vapor phase epitaxyTE transverse electric (field) VPS vacuum plasma sprayingTEC thermoelectric cooler VUV vacuum ultravioletTEG thermoelectric generatorTEM transmission electron microscopy WB weak beamTEP triethylphosphene WGPD waveguide photodetectorTGW Teatum-Gschneidner-Waber WLR Wechsler-Lieberman-Read (theory

(atomic radius) of martensitic transformation)TIP thermally induced porosity WS Wigner-Seitz (cell)TK Takeuchi-Kuramoto (dislocation WSS Winterbon-Sigmund-Sanders

locking mechanism) (model of irradiation damage)TM transition metal wt.ppm weight parts per millionTM transverse magnetic (field)TMA titanium-molybdenum-aluminum XC exchange-correlation

(alloy) XD™ exothermic dispersion (synthesisTO transverse optical (wave) process)TPA two-photon absorption XIM X-ray inspection moduleTSRO topological short-range ordering XPS X-ray photoelectron spectroscopyTT truncated tetrahedron XRD X-ray diffractionTTS tubular tin source XUV extreme ultravioletTTT time-temperature-transformation

YAG yttrium aluminum garnetUHF ultra-high frequencyUHV ultra-high vacuum ZIF zero insertion force

Crystal Structure Nomenclature*Arranged Alphabetically by Pearson-Symbol Designation

Pearsonsymbol

cFAcFS

cFll

cF\6

cF24

cF32cFSlcF56

cF6ScFSQcF112

CF116

ClIc/16c/28c/32c/40

cISl

cISAcISSc/76c/80c/96c/162c/>lcPlcP4

cPS

Prototype

Struktur-bericht

designation

CuC (diamond)

NaCl (rock salt)ZnS (sphalerite)CaF2 (fluorite)

MgAgAsAlCu2Mn (Heusler)

BiF3 (AlFe3)NaTlAuBe3

SiO2 (0 cristobalite)Cu2Mg (Laves)

CuPt3

UB12

Al2MgO4 (spinel)Co3S4

Co9S8

Sb2O3 (senarmontite)Fe3W3C (if carbide)

NaZn13

Cr23C6

Mn23Th6, Cu16Mg6Si7 (G-phase)W

CoUTh3P4

CoAs3 (skutterudite)Ge7Ir3

Pu2C3

Cu5Zn, (7 brass)Fe3Zn10 (7 brass)

Sb2Tl7cxMn (x-phase)

Cu15Si4Mn2O3

AlLi3N2

Mg32(Al,Zn)49

aPoCsCl

AuCu3

ReO3

AlFe3C (perovskite)CaTiO3 (perovskite)

Fe4N

AlAA£1£3ClChLl1DO3£32CIS.C9CISL\.Dl,Hl1Dl2DS9DS4E93D23DS4D%.AlB.

Dl1DO2

D%,DSC

DB2

DS1

Z,22

>112DS6

DS1

E9d

DS€

Ah

BlLl2

DO9

LV2

Elx

LY

Spacegroup

Fm 3mFd\mFmImFAZmFmImFAZmFm\mFm 3mFdImFAZmFdImFdZmFmZcFmZmFdZmFdZmFmZmFdZmFdZmFmZcFmZmFmZmImZm/2,3/43rfImZ

ImZmIAZd/43mImZmImZmIAZmIAZdIaZIaZImZ

PmZmPmZmPmZmPmZmPmZmPmZmPAZm

Pearsonsymbol

cP6cPlcPS

CPU

cPIOcPZ6cPZ9cPSlhPlhPl

hPZ

hPA

hPS

HP6

hPS

HP9

hPIOhPll

Prototype

Ag2OCaB6

Cr3Si (/3W)FeSi

Cu3VS4 (sulvanite)FeS2 (pyrite)

NiSbS (ullmanite)/3Mn

BaHgn

Mg2Zn11

Cu9Al4 (7 brass)HgSn6-10

MgWCAlB2

CdI2

Fe2NLiZn2

7SeotLaBN

C (graphite)NiAs

ZnS (wurtzitc)La2O3

Ni2Al3

CaCu5

CoSnCu2TeHgS

MoS2

Ni2InNa3AsNi3SnTiAsCrSi2Fe2P

rAgZnSiO2 (high quartz)

Pt2Sn3

CuSMgZn2 (Laves)

SiO2 (3 tridymite)

Struktur-bericht

designation

CZDl1

AlS£20Hl4

ClFO1

AlZD2e

DSe

DS,AiAZBh

CZlC6

L1Z

ckASAZ'Bk

A9BS1

BADS2

DS13

Dld

£35

ch£9ClBS2

Z)O18

DO19

B1

CAOC22Bb

CSDSb

£18C14ClO

Spacegroup

PnZmPmZmPmZnP2,3

P43mPaZPl1ZP4J2PmZmPmZ

PAZmP6/mmmP63/mmc

P6mlP6/mmm

PZmIP63/mmcP63/mmc

PZ1IlP63/mmcP63/mmcP63/mmcP63/mmc

P^mcPZmIPZmI

P6/mmmP6/mmmP6/mmm

PZ1IlP63/mmcP63mmcP63/mmcP63/mmcP63/mmc

Pe2IlP61m

PZP6211

Pf^/mmcP63/mmcP63/mmcP^/mmc

continued

•Adapted (with additions and corrections) from ASM Handbook, Vol. 3, 10th ed, ASM International, Materials Park, OH.

Arranged Alphabetically by Pearson-Symbol Designation (continued)

Pearsonsymbol

hP 14hP 16

hPIS

HP20

hP24

hP2ShR\

hR2hR3hR4hR5

HR 6

HR 7

HR10/i/*12/z/*13/z/?15/i/*20hR26

hR 32mC6mCSmC\2mCl4mC\6mP\2

mP20mP22mP24mP32

mP64oC4oCS

oC\2oC16oC20oC24oC2SoF24oF40oF4SoF12oF 128o/12of 14o/20o/28oP4oP6

oPS

Prototype

W2B5

Mn5Si3

Ni3TiAl4C4Si

Al8FeMg3Si6

Mg2NiFe3Th7

Th7S12

Cu3PMgNi2 (Laves)

Co2Al5

aHg/3PoaAsaSm

NaCrS2

Bi2Te3

Ni3S2

CaSi2

NiS (millerite)Al4C3

Mo2B5

OrAl2O3 (corundum)BaPb3

Fe7W6 (//-phase)B4C

HoAl3

Cr5Al8

CuPtAuTe2 (calaverite)

CuO (tenorite)ThC2

6Ni3Sn4

FeKS2

AgAuTe4 (sylvanite)ZrO2

As2S3

Co2Al9

FeAsSAsS (realgar)

0SeaSeaU

CaSiaGaCrB

I2

P (black)ZrSi2

BRe3

PdSn4

PdSn2

Al6MnTiSi2

Mn4BCuMg2

GeS2

aSSiS2

Ta3B4

Al4UGa2Mg5

AuCdFeS2 (marcasite)

CaCl2

aNpJjNiSi

Struktur-bericht

designation

*>*k

D8 8

DO24

E9A

E%

caDlO2

DS k

DO21

C36DS11

AlOAiAlC19FSx

C33

C12B13Dlx

DSjDS1

DS5DKDS10

Li1C34526C1

Dla

FSaElb

C43D5,DSd

EO1

B1

A1

Ak

A 20Bc

AllB33A 14AllC49EKDlc

SiC54

/116C42Dlb

Dlb

DS t

B19CISC35Ac

Bd

Spacegroup

P63 /mmcPd1ZmCmP63/mmc

P63mcP62mP6222P63mc/>63/mP63cm

P63/mmcP6/mmc

R3mR 3mR 3mR 3mR 3mR 3mR 32R 3mR 3mR 3mR 3mR 3cR3mR3mR3mR3mR3m

R 3mCVmC2/cC2/cC2/mC2/cP2/cP2x/cP2x/cP2x/cP2x/cP2x/cP2x/cP2x/cCmcmCmmcCmcaCmcmCmcaCmcaCmcmCmcmAba2Aba2CmcmFdddFdddFdddFdd2FdddIbam

Im mmImmaIbamPmmaPnnmPnnmPnmaPbnm

Pearsonsymbol

oPS

oPl2

oP 16

oP20

oP24

oP40tI2

tI4tI6

US//10

//12

//14//16

//18//26//28//32

tP2tP4

tP6

/PlO

/P16/P20tP30

/P40

/P50

Prototype

/3Cu3TiFeBGeSSnS

MnPTiB

Co2Si, NiSiTi (E-phase)Co2SiHgCl2

Al3NiAsMn3

BaS3

CdSbCuS2Sb (wolfsbergite)

Fe3C (cementite)Cr3C2

Sb2S3

Sb2O3 (valentinite)AuTe2 (krennerite)CuFe2S3 (cubanite)

TiO2 (brookite)Cr7C3

aPaIn

/3SnCaC2

MoSi2

ThH2

Al3TiAl4BaMoNi4

Al2CuThSi2

Al2CdS4

Al3ZrCuFeS2 (chalcopyrite)Cu2FeSnS4 (stannite)

Ir3SiMoBSiU3

TlSeFe8N

Mn12ThMnU6

Cr5B3

Ni3PW5Si3

6CuTi0Np

AuCuCuTi3

7CuTiPbOPb3Sr

PtSCu2SbPbFCl

TiO2 (rutile)Pb4PtSi2U3

PdSB4Th/3U

oCrFeAl7Cu2Fe

Zn3P2

yB

Struktur-bericht

designation

DO0

B21£16B29B31

&3C37C28DO20

D0d

DO 1 7

ADO11

DS10

DS,

&

DlO1

AS

cuaC I l 6

DO22

Dl3

DKC16Cf

E3DO23

El{

H2,DO;

kB37Dl1

D2b

D2C

Di1

DO,DKL2.A,

L6a

BUBlO

BVC38EO,CADld

DS.BUDh

Di,E%DS9

Spacegroup

PmmnPnmaPnmaPmcnPnmaPnmaPnmaPbnmPmnbPnmaPmmnP42xmPbcaPnmaPnmaPnmaPnmaPccn

Pma2PnmaPbcaPnma

I4/mmmI4/mmmI4x/amdI4/mmmI4/mmmI4/mmmI4/mmmI4/mmm

I4/mI4/mcmI4x/amd

/4I4/mmm

I42d/42m

I4/mcmI4x/amdI4/mcmI4/mcmI4/mmmI4/mmmI4/mcmI4/mcm

/4/4/mcm

P4/mmmP42.2

P4/mmmP4/mmmP4/nmmP4/nmmP4\mmmP42/mmcP4/nmmP4/nmmP42/mnmP4/nbmP4/mbmP42/m

P4/mbmP42/mnmP42/mnmP4/mncP42/nmcP42/nnm

Arranged Alphabetically by Strukturbericht Designation

Struktur-berichtdesignation

Aa

Ab

AcAdAj\AhA1AkA1AlAlA3A3'AAASA6AlASA9AlOAllAllA 13AlAAlSA 16AllA 20BaBbBcBdBe£,( = £33)

kBjBkB1BnBlBlB3BABS1BS2B9BlOBIlB13£16B17£18£19

Prototype

aPa

<*Np/3Np

HgSn6-10

7BaPo/SPoaSejSSeCuWMgaLa

C (diamond)/3SnIn

aAs7Se

C (graphite)aHgaGa

aMn (x-phase)0Mn

I2Cr3Si 03-W)

aSP (black)

aUCoU

fAgZnCaSitjNiSiCdSbCrBMoBWCTiAsBN

AsS (realgar)TiB

NaCl (rock salt)CsCl

ZnS (sphalerite)ZnS (wurtzite)

NiAsNi2In

HgS (cinnabar)PbO

yCuTiNiS (millerite)

GeSPtS (cooperite)CuS (rovelite)

AuCd

Pearsonsymbol

tiltP30oPStPAhPltPSOcPlhRl

mP6AmP32cFAcTlhPlhPAcFStIAtil

hRlhP3hPAhRloCScISScPIOoCScPS

oFllSoCSoCAc/16hP9oCSoPSoP 16oCS/716HPlhPShPA

mP32oPScFScPlcFShPAhPAhP6hP6tPAtPAhR6oPStPA

hPlloPA

Spacegroup

IA/mmmPA2/mnm

PnmaPAlxI

P6/mmmPA2/nnm

Pm3mR3mPl1ZcPl1ZcFm3mIm 3m

P63ZmmcP63Zmmc

FdImIAxZamdIAZmmm

R3mP3xll

P63ZmmcR 3mCmca/43mP4,32CmcaPm3nFdddCmcaCmcm/2,3P3

CmmcPbnmPbcaCmcm

IAxZamdP6ml

P63ZmmcPd^Zmmc

PlxZcPnmaFm 3mPm 3mFA3mP63mc

P63mmcP6^mmc

P3.llPAZnmmPAZnmm

R 3mPnma

PA2/mmcP6)Zmmc

Pmma


£20£26£27£29£31£32£33( = £ /)£34£35£37cacbccce

%ckClChClC3CAC6ClCSC9ClOen,en.C12C14C15ClSbC16CISC19CIlCIlC23CISC31C33C3AC3SC36C37C38C40C42C43C44C46C49CSADO,D0c

Prototype

FeSiCuO (tenorite)

FeBSnSMnPNaTlCrBPdS

CoSnTlSe

Mg2NiCuMg2ThSi2PdSn2ThC2Cu2TeLiZn2

CaF2 (fluorite)MgAgAs

FeS2 (pyrite)Ag2O

TiO2 (rutile)CdI2MoS2

SiO2 (high quartz)SiO2 (/3 cristobalite)SiO2 (/3 tridymite)

CaC2MoSi2CaSi2

AuBe5Al2Cu

FeS2 (marcasite)aSm

TiO2 (brookite)Fe2P

Co2Si, NiSiTi (E-phase)HgCl2AlB2

Bi2Te3AuTe2 (calaverite)

CaCl2MgNi2 (Laves)

Co2SiCu2SbCrSi2SiS2ZrO2GeS2

AuTe2 (krennerite)ZrSi2TiSi2

/3Cu3TiSiU3

Pearsonsymbol

cPSmCSoPSoPSoPScFl6oCStP 16hP6f/16HP 18oF48till

oCIAmCllhP6hP3cFllcFllCPUcP6tP6hP3hP6hP9cFIAhPlltietI6

hR6HP IlcFIAcFIAtilloP6HR3oPIAhP9oPlloPllhP3hRSmC6oP6hPlAoPlltP6hP9o/12

mP12oFlloPIAoClloFIAoPStll6

Spacegroup

/>2,3ClZcPnmaPmcnPnmaFd3mCmcmPA2Zm

P6ZmmmIAZmcmP6211Fddd

lAxZamdAbalClZc

P6ZmmmPbjmmc

Fm3mFA3mPaJ

Pn3mPA2/mnm

PJmIP63Zmmc

Pb2IlFd3m

P63ZmmcIAZmmmIAZmmm

R 3mP6jZmmc

Fd3mFA3m

IAZmcmPnnmR 3mPbcaP61mPnmaPmnb

PdZmmmR3mClZmPnnm

P6y/mmcPbnm

PAZnmmPb2IllbamPlxZcFddlPma2CmcmFddd

PmmnIAZmcm

continued

Arranged Alphabetically by Strukturbericht Designation (continued)


D0c'DO,DO,DO1

DO3

DO9

DOn

DO1 7

DO1 9

DO2 0

DO21

DO2 2

DO2 3

DO2 4

D KDXb

DXC

DXd

DKDl7

™.Dl 3

DlbD2CD2dD2eDl1Dl1DlhDlxDl3DSaDSbDSC

DS,D5fD5,D52D53DS,DS,DS9DSXO

DSn

DSx,Dla

oibDlxDl1Dl3D*aDSbD8C

Prototype

Ir3SiAsMn3

Ni3PCoAs3 (skutterudite)

BiF3, AlFe3

ReO3

Fe3C (cementite)BaS3

Na3AsNi3SnAl3NiCu3PAl3TiAl3ZrNi3Ti

MoNi4

Al4UPdSn4

Pb4PtB4ThMn4BB4C

Al4BaMn12ThMnU6

CaCu5

BaHg11

UB12

Fe8NAl6MnCaB6

NaZn13

Si2U3

Pt2Sn3

Pu2C3

Ni3S2

As2S3

CxAl2O3 (corundum)La2O3

Mn2O3

Sb2O3 (senarmontite)Sb2S3

Zn3P2

Cr3C2

Sb2O3 (valentinite)Ni2Al3

6Ni3Sn4

Ta3B4

Al4C3

Co3S4

Th3P4

Mn23Th6, Cu16Mg6Si7 (G-phase)oCrFe

Mg2Zn11

Pearsonsymbol

/716oP 16tillcmcFX6cP4

0PX60PX6hPShPSoP\6hP14US/716

hP\6tl 10ollOoCIOtPXOtPIOoFAOhR\StI\0tI16

tnshP6cP36cFSltixs

oCISCPl

cFUl/PlOhP\0cIAOhRS

mPIOhRlQhPSc/80cFSOoPIOtPAOoPIOoPIOhPS

mC 14o/14hRlcFS6dlS

cF\\6tP30cP39

Spacegroup

IA/mcmPmmn

/4ImI

Fm 3mPm 3mPnmaPAlxm

P63/mmcP63/mmc

PnmaP63cm

IA/mmmIA/mmmP63/mmc

lA/mImmaAbal

PA/nbmPA/mbm

FdddR 3m

IA/mmmJA/mmmIA/mcm

P6/mmmPmJmFm3m

IA/mmmCmcmPm 3mFmZc

PA/mbmP63/mmc

I43dR31

PlxZc/?3c

P3m\Ia3

Fd3mPnma

P42/nmcPnmaPccn

P3m\CVmImmmR 3mFd3m143d

Fm3mP41/mnm

Pm3


D%d

D\DS1DSgDKD8,D%kDS1DS1nDSxDS2DS3DS4DS5DS6D88D89DS10DS nDlO1DlO2EO1EO1EKEhEXxEl1E3E%E%E9CE%E%E%E%FO1FSaFSxFS6HXxHl4Hl6LY

Ll0Ll0(M)Ll1Ll2Ll2'

n;LlxLl2

LyL60

Prototype

Co2Al9Mg32(Al,Zn)49

Ge7Ir3Ga2Mg5

W2B5Mo2B5Th7S12Cr5B3W5Si3

Fe3Zn10)Cu5Zn8J 7 brassCu9Al4]Cr23C6

Fe7W6 0*-phase)Cu15Si4Mn5Si3Co9S8Cr5Al8Co2Al5Cr7C3Fe3Th7PbFClFeAsS

MgCuAl2AgAuTe4 (sylvanite)

CuFeS2 (chalcopyrite)CaTiO3 (perovskite)

Al2CdS4Al7Cu2Fe

Al8FeMg3Si6Mn3Al9SiAlLi3N2

CuFe2S3 (cubanite)Fe3W3C (r> carbide)

Al4C4SiNiSbS (ullmanite)

FeKS2NaCrS2

CuS2Sb (wolfsbergite)Al2MgO4 (spinel)

Cu3VS4 (sulvanite)Cu2FeSnS4

Fe4NCuPt3AuCu

AuCuIICuPt

AuCu3AlFe3C (perovskite)

<5CuTiThH2

AlCu2Mn (Heusler)Sb2Tl7Fe2NCuTi3

Pearsonsymbol

mPllc/162c/40oIlShPX4hRlhPIOtI31tI31cISlcISlcPSlcFXX6HRX3c/76HPX6cF6ShR16hPISoP40hPIO

tP6mP140CX6mPXltIX6cPS/714/P40hPXShP16c/96oP24cF112HP XScPXlmCX6hR40PX6cFS6cPS/716cP5cF32tP4o/40hR32cP4cP5tP2/76

c Fl 6c/54hP3tP4

Spacegroup

Pl1ZcImJ

Im 3mIbam

P63/mmcR 3m

Pd3ZmI4ZmcfTiI4ZmcmIm3m743m/>43mFm 3mR 3m143d

Pd3ZmcmFm3mR 3m

P63ZmmcPnmaP63mc

P4/nmmPlxZcCmcmPlZc141d

Pm3m14

P4ZmncP61m

P63ZmmcIaI

PnmaFd3mP63mcP1X3ClZcR 3mPnmaFd3mP43m742mPm3mFmbc

P4/mmmImmaR3m

PmbmPm3m

P4/mmmI4/mmmFmbmImhm

P6JmmcP4/mmm

vii This page has been reformatted by Knovel to provide easier navigation.

Contents

Contributors ........................................................................................................ ix

Preface to the 1995 Edition ................................................................................ xi

Preface to Reprint Volumes ................................................................................ xiii

Acronyms ........................................................................................................... xv

Crystal Structure Nomenclature .......................................................................... xxi

1. Factors Governing Crystal Structures ..................................................... 1 1.1 Introduction .......................................................................................................... 1 1.2 Strategy to Find the Factors Governing Crystal Structure ................................. 2 1.3 Compound-Formation Diagrams ........................................................................ 6 1.4 Regularities in Intermetallic Compounds ............................................................ 10 1.5 Nine Quantitative Principles ................................................................................ 38 1.6 Quantitative Relations Between Crystal Structures and Physical Properties

of Intermetallic Compounds .................................................................................. 38 1.7 Conclusion ........................................................................................................... 42 1.8 Appendix ............................................................................................................. 46 1.9 References .......................................................................................................... 48

2. Close-Packed Structures .......................................................................... 51 2.1 Introduction .......................................................................................................... 51 2.2 Stacking Sequences ........................................................................................... 51 2.3 Alloy Formation ................................................................................................... 56 2.4 M xNy Structure Map of a Single Tetragonal Layer .............................................. 56 2.5 Ordering of Atoms in Hexagonal Layers ............................................................. 59 2.6 Hexagonal Close-Packed Structures .................................................................. 61 2.7 Cubic Close-Packed Structures .......................................................................... 61 2.8 Ordered Structures of Complex Close-Packed Alloys ....................................... 66

viii Contents

This page has been reformatted by Knovel to provide easier navigation.

2.9 Structures with Identical Powder Patterns .......................................................... 66 2.10 Homologous Series of Structures ....................................................................... 67 2.11 Symmetry of Ordered Phases ............................................................................ 68 2.12 Ising Model .......................................................................................................... 68 2.13 Occupation of Octahedral or Tetrahedral Interstices ......................................... 70 2.14 Ordered Cubic Close-Packed Interstitial Alloys .................................................. 73 2.15 Ordered Hexagonal Close-Packed Interstitial Alloys ......................................... 76 2.16 Complex Close-Packed Interstitial Alloys ........................................................... 77 2.17 Disordered Alloys ................................................................................................ 78 2.18 Ordered Ternary and Quaternary Compounds .................................................. 79 2.19 Notation ............................................................................................................... 80 2.20 References .......................................................................................................... 80

3. Body-Centered Cubic Derivative Structures ........................................... 83 3.1 Introduction to the Definition of Symbols ............................................................ 83 3.2 The I Framework ................................................................................................. 91 3.3 Frameworks of the I Family with Polyhedra Allocated Around the I Points ....... 99 3.4 Nets in Orthohexagonal Arrangements .............................................................. 110 3.5 Summary ............................................................................................................. 115 3.6 References .......................................................................................................... 115

4. Wurtzite and Sphalerite Structures .......................................................... 117 4.1 Introduction .......................................................................................................... 117 4.2 Definition and Classification of Adamantane Structures .................................... 117 4.3 The ZnS Stacking Variants ................................................................................. 118 4.4 Valence-Electron Rules for Adamantane-Structure Compounds ...................... 119 4.5 Compositions of Adamantane-Structure Compounds ........................................ 122 4.6 Ordered Adamantane-Structure Types .............................................................. 126 4.7 Additional Experimental Rules for Adamantane-Structure Compounds ............ 130 4.8 Concluding Remarks ........................................................................................... 132 4.9 Acknowledgements ............................................................................................. 132 4.10 Appendix ............................................................................................................. 132 4.11 References .......................................................................................................... 134 Addendum ................................................................................................................... 136

Contents ix


5. Atomic Environments in Some Related Intermetallic Structure Types .......................................................................................................... 139 5.1 Introduction .......................................................................................................... 139 5.2 The Atomic-Environment Approach .................................................................... 140 5.3 Observed Atomic Environments ......................................................................... 143 5.4 Concluding Remarks ........................................................................................... 157 5.5 Acknowledgements ............................................................................................. 158 5.6 References .......................................................................................................... 158

6. Some Important Structures of Fixed Stoichiometry ............................... 161 6.1 Introduction .......................................................................................................... 161 6.2 MoSi 2-Type Phases ............................................................................................ 162 6.3 CuAl 2-Type Phases ............................................................................................. 167 6.4 NiTi 2-Type and Fe3W3C-Type Phases ................................................................ 169 6.5 Mn 23 Th6-Type Phases ........................................................................................ 172 6.6 NaZn 13-Type Phases ........................................................................................... 173 6.7 Fe 3C-Type Phases .............................................................................................. 174 6.8 Th 3P4-Type Phases ............................................................................................. 175 6.9 Summary ............................................................................................................. 176 6.10 References .......................................................................................................... 176

7. Topologically Close-Packed Structures .................................................. 179 7.1 Introduction .......................................................................................................... 179 7.2 Close Packing of Atoms with Equal and Nearly Equal Sizes ............................. 182 7.3 Close Packing of Atoms with Unequal Sizes ...................................................... 184 7.4 Summary ............................................................................................................. 190 7.5 References .......................................................................................................... 193

8. Structure Mapping ..................................................................................... 195 8.1 Introduction .......................................................................................................... 195 8.2 Binary Structure Maps ......................................................................................... 196 8.3 Ternary Structure Maps ...................................................................................... 210 8.4 Conclusion ........................................................................................................... 213 8.5 References .......................................................................................................... 213

x Contents


9. Magnetic Structures .................................................................................. 215 9.1 Introduction .......................................................................................................... 215 9.2 Exchange Interactions and Magnetic Structure .................................................. 215 9.3 Magnetic Structures ............................................................................................ 217 9.4 Remarks .............................................................................................................. 226 9.5 References .......................................................................................................... 227

10. Quasicrystals and Related Structures ..................................................... 229 10.1 Introduction ........................................................................................................ 229 10.2 The Icosahedral Phase ..................................................................................... 230 10.3 Decagonal Phase .............................................................................................. 241 10.4 Crystalline Approximants .................................................................................. 247 10.5 Structural Similarities to Liquids and Glasses .................................................. 255 10.6 Real-Space Structures ...................................................................................... 256 10.7 Electronic Properties ......................................................................................... 260 10.8 Concluding Remarks ......................................................................................... 261 10.9 Acknowledgements ........................................................................................... 262 10.10 References ........................................................................................................ 262 10.11 Further Reading ................................................................................................ 267

Index .................................................................................................................. 269

Chapter 1

Factors Governing Crystal Structures

Pierre VillarsIntermetallic Phases Data Bank (IPDB) and Materials Phases Data System (MPDS)9

Postal Box 1, CH-6354 Vitznau, Switzerland

1. Introduction

The fundamentals of the constitution of an alloyingsystem are determined by the crystal structure of itsintermetallic compounds and its phase diagram.Knowing these fundamentals enables scientists to solvemany problems in materials science, and therefore it isimportant to have easy access to the experimentallydetermined data, especially as the experimental workto determine such information is very time- and cost-intensive.

Recently, scientists working in this field have had theadvantage of access to comprehensive, up-to-datehandbooks for crystal structures as well as for phasediagrams. Pearson's Handbook of CrystallographicData for Intermetallic Phases\ second edition (Villarsand Calvert, 1991) contains critically evaluatedcrystallographic data for over 25 000 distinctly differentintermetallic compounds (over 50 000 binary, ternary,etc., entries) covering the world literature from 1913 to1989. The Atlas of Crystal Structures for IntermetallicPhases (Daams etal., 1991), a companion of Pearson'sHandbook, contains for most of the intermetalliccompounds in Pearson's Handbook graphicalrepresentations of the crystal structures. The handbookBinary Alloy Phase Diagrams, second edition (Massalskiet fir/., 1991) contains about 4000 phase diagrams, mostof them critically evaluated phase diagrams. In 1994ASM International will publish the Handbook of TernaryAlloy Phase Diagrams (Villars et al., 1993), whichcomprehensively covers the world literature from 1900to 1989 and will contain information for over 8800ternary systems including over 15 000 isothermal

sections, liquidus and solidus projections, as well asvertical sections. For the literature up to the year1977, there also exists the very comprehensiveMulticomponent Alloy, Constitution Bibliography(Prince, 1956, 1978, 1981).

Looking at the research activities in the last 20 yearswith respect to the number of systems investigated,grouped according to binary, ternary, and quaternarysystems, a shift in the ongoing research from binary toternary and quaternary systems is clearly seen. Theresearch field of multinary systems is a huge field forpotential novel materials with optimized known physicalproperties (e.g. tP68 BFe14Nd2-type permanent magnetsand, in the inorganic field, the new high-Tc super-conductors) as well as 'new' physical properties, as therecent past has proven with the quasicrystals (seeChapter 20 by Kelton in this volume). All this wasachieved by moving from the binary to the ternary andquaternary systems, but this trend confronts us withmany additional 'difficult-to-handle' problems. Wehave to be aware that in the future most research willbe done on multinary systems; therefore, in this chapterwe will first discuss the binary intermetallic compounds(systems), also called binaries, and whenever possibleextend the discussion to ternary intermetallic com-pounds (systems), called ternaries. We will try to showthe main problems by extending to ternaries ourknowledge gained by investigation of binaries andpropose ideas which, according to the author, have ahigh probability of success in the quantitative extensionto multinaries of regularities found in binaries. Theexperimental variables we have are the selection of thechemical elements, their possible combinations, their

Crystal Structures of Intermetallic Compounds. Edited by J. H. Westbrook and R. L. Fleischer ©1995, 2000 John Wiley & Sons Ltd

concentrations, the temperature, and the pressure. Ofenormous consequence from the practical point of viewis the variety of combination possibilities of the chemicalelements and the increase in the possible number ofcombinations on going from binaries to multinariesaccompanied by a much larger available concentrationrange. Unfortunately, the number of available experi-mentally determined data is in inverse proportion tothese opportunities.

We have a relatively robust database for binaries, buta sparse database for ternaries, and almost no data forquaternaries. This is quantitatively demonstrated bylooking at the relevant numbers of binary, ternary, andquaternary systems. In the binary systems, taking 100chemical elements into account, there exist (100x99)/(1x2) = 4950 binary systems, a large number, but still,with united, coordinated international research efforts,all these systems could be experimentally investigatedby the end of this century. Therefore, one wouldinevitably have found among these the most interestingand economically important binary materials. Massalskiet al. (1991) and Villars and Calvert (1991) containinformation on about 4000 systems, so already we havea very robust database with about 80% of the possiblesystems fully or partly investigated experimentally. Withthe ternary systems the situation looks completelydifferent. There exist (100 x 99 x 98)/(l x 2 x 3) = 161 700systems. Villars and Calvert (1991) and Villars et al.(1994) contain experimentally determined data for onlyabout 8800 systems, most of which have only been partlyinvestigated. So here we have a sparse database withonly about 5% of the potentially available systemshaving been partly investigated. In addition, in orderto establish structures and phase relationships, one hasto prepare and investigate at least 10 times more samplesper system in the ternary case compared to the binarycase. In the quaternary case with (100 x 99 x 98 x 97)/(1 x2x3x4) = 3 921 225 potential systems, less than0.1% have been partly investigated. Without having verygood guidelines, it is a hopeless situation to searchsystematically for novel materials with an adequatesuccess rate in multinary systems. Therefore, the onlypracticable way to go is, with the help of the robustbinary database, to find regularities, such as laws, rules,principles, factors, tendencies, and patterns valid withinthe binaries, and then to extend these to the ternariesand quaternaries in such a way that the binaries andmultinaries can be treated together; otherwise theregularities would be based on too few data sets andtherefore would not be trustworthy. In addition, theregularities should show an accuracy clearly aboverandomness to be of practical use (these can easily

be checked with the available experimentally determineddata).

To increase the efficiency in the successful search for'new' intermetallic compounds, the main effortsshould go toward creating an internationally accessibleinformation-prediction system incorporating all data-bases (experimentally determined facts) as well asgenerally valid principles and the 'highest-quality'regularities. The problems involved in such a projecthave been reviewed by Westbrook (1993). The com-bination of the experience and intuition of theexperimentalist together with easy access to the dataalready experimentally determined in the form of up-to-date handbooks as well as access to the envisionedinformation-prediction system would very much helpto coordinate world research activities. Furthermore,it would reduce the number of unwanted duplicationsas well as increase the probability of investigating firstthe most promising systems and not, as in the past,provide us with the systems in a random statisticalsequence. Otherwise we will have to wait until the year5500 for the next 140 generations of scientists toinvestigate the ternary systems, assuming an activity ratesimilar to that in this century.

2. Strategy to Find the Factors GoverningCrystal Structure

When we talk about factors governing crystal structureswe intend to reduce these to atomic property expressions(APEs) of the constituent chemical elements, so that theexisting experimental data can be systematized. Forthe purpose of this chapter, we consider that the onlyregularities of real practical value are those able tosystematize a large group of data with an accuracy inthe range of at least 95%. For the not yet experimentallyinvestigated systems, one goes in exactly the reverse way,assuming the validity of the regularity found for a well-defined group of data. The more data considered, themore trustworthy is a prediction based on suchregularities. To make predictions one starts fromthe chemical elements, their concentrations, and thetabulated atomic properties (APs) of the elements, andcalculates the APEs for a system of interest. Therefore,APs are only of practical interest where they are knownfor most chemical elements with adequate accuracy.From the APEs, in the context of the consideredregularity, one can predict how the systems of interestmost probably (with an accuracy of at least 95%) willbehave. Then scientists can decide which systems arepromising to investigate experimentally.

In principle, it would be sufficient to use as input onlythe atomic numbers (ANs) and the compositions of theintermetallic compounds of the systems under con-sideration. Slater (1956) once made a comment asfollows: 'I don't understand why you metallurgists areso busy in working out experimentally the constitution[crystal structure and phase diagram] of multinarysystems. We know the structure of the atoms [needingonly the AN], we have the laws of quantum mechanics,and we have electronic calculation machines, which cansolve the pertinent equation rather quickly!' Some 35years later Chelikowsky (1991) writes in an excellentreview the following: 'Although the interactions inintermetallic compounds are well understood, it is notan easy task to evaluate the total energy of solids, evenat absolute zero. As the energy of an isolated atom isin the order of about 106eV, but the cohesive energy

only in the order of about 1-10 eV/atom, one must havea method that is accurate to one part on 106, or better.'The other fact that greatly complicates evaluating thecohesive energy by theoretical methods is the numberof particles involved. Given that a macroscopic solidmay contain 1023 nuclei and electrons, it is impossibleto determine the total energy of the crystal structurewithout some approximations. Within the last 15 years,two advances have made it possible to predict thecohesive energy of solids by numerical solutions ofthe quantum-mechanical equations of motion, e.g. theSchrodinger equation: the invention of high-speedcomputers and the device of one-electron potentials,which greatly simplifies many-body interactions.

The accuracy of these computations is usually not atthe same level as experiment. Nonetheless, it is nowpossible, for chemical elements and simple intermetallic

Figure 1. An atomic property, AP, versus a function of atomic number, AN, plot of the pseudopotential radii after Zunger,R* , representing an example of the size factor. In such diagrams the chemical elements are ordered by increasing group numberand each group is ordered by increasing quantum number. This special atomic-number (AN) scale was used instead of a linearlyincreasing atomic-number scale because it gives units of comparable size for distance between the atoms within any group aswell as along any period. The scale chosen for the comparison, of course, does not affect the number of factors (groups), onlythe appearance of the patterns changes

compounds, to predict whether a given crystal structureis the most stable one at 0 K and 1 atm. We have to stressthat the crystal structures (and from Pearson's Handbook(Villars and Calvert, 1991) we know there are at least2750 different ones) have to be given as input for first-principles calculations, and this for each potential inter-metallic compound. Assuming the potential intermetalliccompound crystallizes in one of the 2750 known crystalstructures and knowing its nominal stoichiometry—AB,AB2, etc.—it may still, in some cases, require a fewhundred first-principles calculations for each potentialintermetallic compound; even with high-speedcomputers, this is not yet workable. But the largerproblem is that the differences of the cohesive energiesof those few hundred calculations will be so small thatthe accuracy would have to be in the range of one partin 109 to determine the most stable crystal structure.The complexity of the above-mentioned problem showsthat one cannot expect that within the next decade theconstitution (crystal structure and phase diagram) ofmultinary systems will be calculated from first principles.

Meanwhile it is therefore sensible to adopt semi-empirical approaches based on the experimentallyknown data to search for the most reliable regularities.As at the moment it is impossible to start only from theatomic property AN, it is obvious to try to find outwhich other APs of the chemical elements are needed

to describe the alloying behavior. Villars (1983)conducted a survey of 53 different APs as a functionof the AN (182 variables in all when the differentmethods of determination are taken into account), andit was found that there were only five main groups, herecalled factors. The results are best seen in AP versusAN plots, as shown in Figure 1 for the pseudopotentialradii jR +̂p after Zunger (1981) for s, p, and d elements.The f elements have been left out because in most casesonly incomplete data are available. In all but a fewcases (19 out of 182) very regular symmetric patternswere obtained in such plots; this means a regular behaviorwithin a group with increasing quantum number (QN)as well as along a period with increasing AN.

In these diagrams the chemical elements are orderedby increasing group number and each group isordered by increasing quantum number. This special'AN' scale was used instead of a linearly increasing ANscale because it gives units of comparable size fordistance between atoms within any group as well asalong any period. The scale chosen for comparison doesnot affect the number of factors (groups), of course;only the appearance of the patterns changes. It shouldbe stressed that the equivalence of APs belonging to thesame group is of a qualitative, not a quantitative nature.Adherence to one or another of the five factors is veryobvious. Table 1 summarizes the idealized characteristics

Table 1. Idealized characteristics of the 'patterns' in the atomic property (AP) versus atomic number (AN) plots of the fivefactor (groups)

Factors (groups)

Size factor

Atomic-numberfactor

Cohesive-energyfactor

Electrochemicalfactor

Valence-electronfactor

Long periods Groups

First Second Third

Straight lines with negative slopes and a slight maximumaround the Ni and Cu group elements

Straight lines with positive slopes

s and d elementsLine with twomaxima at theV group and theCo group

p elements

Line with a clearmaximum atthe V group

Line with a clearmaximum atthe Mn group

s elements d elements p elements



Straight lineswith negativeslopes

The first twoand the lasttwo are irregular;the rest arestraight lineswith positiveslopes

Straightlines withnegativeslopes

Straight lines with negative slopes

s and d elementsLines with a maximum around V to Ni group

p elementsStraight lines with positive slopes

Straight lines with positive slopes and adiscontinuity at Cu

Straight lineswith negativeslopes

The first twoand the last twoare irregular;the rest arestraight lineswith positiveslopes

Straightlines withnegativeslopes

For every group, horizontal straight lines

Table 2. Atomic properties (APs) of the chemical elements grouped according to the five factors (groups)

Factors (groups)

Size factor

Atomic-numberfactor

Cohesive-energy factor

Electrochemical factor

Valence-electron factor

Element property determined experimentally or derived from a model

Classical crossing points of the self-consistently screened, non-local atomicpseudopotentials

Radius of maximum radial electron density for outer orbitals fromHermann-Skillman calculations0

Renormalized orbital radiusRadius calculated by Hartree-Fock-Slater methodIonic radiusSoftness parameterAtomic volumeMetallic radiusCovalent radiusReduced thermodynamic potential at 298 KElectrochemical weight equivalentEntropy of solid elements at 298 KDensity

Atomic numberAtomic weightPrincipal quantum numberAtomic electron scattering factorBond energy of deep-lying electronsSpecific heatWavelength of K and L seriesMaximum number of electrons in the solid elementMelting pointBoiling pointHeat of fusionHeat of vaporizationHeat of sublimationEnergy for atomization of 1 mol of the solid element at 0 KBulk modulusYoung's modulusTorsion modulusCompression modulusCrystal lattice energyDimer dissociation energySurface tensionLiquid-solid interfacial energyEnthalpy of formation of monovacanciesCohesive energySolubility parameterCompressibility modulusLinear coefficient of thermal expansion at 273 K

ElectronegativityChemical potential (after Miedema)*/^3 (after Miedema)*s electron binding energys-p parameterPositron annihilation rateElectron affinityHardnessNormal electrode potentialFirst ionization potentialTerm value (after Herman and Skillman)"

Number of valence electrons (corresponding to group number)Number of vacancies or holes in the d bands above the Fermi level

Number ofdata sets

15

6633263

3422274654

34433

323

422

13

23533

"Herman and Skillman (1963).Me Boer et al. (1988).

Figure 2. Plot for the chemical elements of the relationshipof two atomic properties, APs, belonging to the cohesive-energyfactor (group), namely cohesive energy and melting point T(Chelikowsky, 1979)

of those five factors in AP versus AN diagrams, andTable 2 lists the 53 APs grouped by those five factors.Figure 2 shows two APs belonging to the same factorplotted against each other, the cohesive energy versusmelting point T of the chemical elements. As a firstapproximation a linear dependence may be seen to exist.

The five factors (groups) are:

• Size factor• Atomic-number factor• Cohesive-energy factor• Electrochemical factor• Valence-electron factor

In Table 3 we have given, in periodic table rep-resentation for the chemical elements, an AP for eachfactor for which accurate and complete data areavailable as well as (and this is very important forpractical use) values that are independent of theconstituent chemical elements of a compound. Theseare: pseudopotential radius after Zunger (1981), /?f+p;atomic number, AN; melting point, T; electronegativityafter Martynov and Batsanov (1980), XM&B; and groupnumber = number of valence electrons, V. It is clearlyshown in Table 3 that there exist no overall tendenciesbetween those five factors, so they represent the mostindependent factors. In recent years some other APshave come to the attention of the author like, forexample, average electron distances in the structuresof the individual elements (Schubert, 1990), the

energodynamic potential (Volchenkova, 1989), andvibrational frequencies and dissociation energies ofhomonuclear diatomic molecules (Suffczynski, 1987),etc.; however, all of them could be assigned to one ofthe five factors chosen.

In Section 4 we will always check to determine towhich factor(s) the AP and respectively the APE belongthat are successful in organizing large groups ofexperimentally determined data. At the end we will findout which of those five factors are governing crystalstructures.

3. Compound-Formation Diagrams

A fundamental question that has to be answered firstbefore discussing crystal structures is: * Which systemsform at least one "new" intermetallic compound?'

3.1 Binary System

The most outstanding model for binary systems isMiedema's model, which is excellently explained in theseries Cohesion and Structure, volume 1 (de Boer et al.,1988). Two APs of the constituent chemical elementsenter the description of enthalpies of formation AZ/^:the chemical potential for electronic charge (electro-negativity) XM and the electron density at the boundaryof the Wigner-Seitz atomic cell, «ws. Both APs belongto the electrochemical factor group. Table 2.4 of de Boeret al. (1988) gives the most recent recommended valuesfor X™ and «ws for most chemical elements. The keyexpression for the sign of the enthalpy of formationAH*0* of binary alloys is

înterface a { _ (A^M)2 + (Q//>)(A*^] U)

P and Q are constants for certain groups of systems,e.g. systems containing only transition elements. Theconcept that the energy effect upon alloying is generatedsolely at the contact surfaces between dissimilar Wigner-Seitz atomic cells results consequently in the fact thatA//1^ does not vary with the concentration as long asthe basis atomic cell A remains fully surrounded bydissimilar atomic cells B. For intermetallic compoundsthe elastic mismatch effect will be of no practicalimportance, since only crystal structures that arefavorable for the given constituent chemical elementsizes will be realized. Therefore, relation (1) directlyapplies to intermetallic compounds ABj, if they aresufficiently rich in B such that A atoms are completely

Cohe

sive

ener

gy (

eV/a

tom

)

T(K)

Table 3. Five atomic properties (APs) of the chemical elements, each representing an accurate and complete example of our five factors (groups). Thefirst to fifth bars from left to right within a chemical element box are:

CHEMICAL ELEMENT

for the size factor

for the atomic-number factorfor the cohesive-energy factorfor the electrochemical factor

for the valence-electron factor

pseudopotential radiiafter Zunger (1981)

atomic numbermelting pointelectronegativity after

Martynov and Batsanov (1980)valence-electron number

(ii) AN(iii) T(iv) XM&»

(V) V

surrounded by B neighbors, and therefore the enthalpyof formation AHfoT of the intermetallic compound AB^per mole of A is equal to

A ufor A Trinterface /")\A//AB^ = A/ /AinB (2)

Assuming that Q and P are truly constant for arbitrarychoices of the constituent chemical elementsrlhe^sign of theenthalpy of formation is simply determined by the ratio

W=\AXM\/\Anl^s\ (3)

For W> Q/P the AHfoT value is negative, while in theopposite case the A//for value is positive. The analysisof the sign of the predicted and experimental AHfoT < 0is demonstrated for binary systems including two tran-sition metals (Figure 3). Each symbol in the X™ versus

rtws diagram corresponds to a particular binary system.Since information on phase diagrams is easier to retrievethan numerical values of A//for, in assigning * + ' or ' • 'to the points in this plot, the following criteria are used:' • ' i n the binary systems with one or more intermetalliccompounds that are stable at low temperature,indicating A//toi;< 0; and ' + ' if there is no intermetalliccompound in the binary system or if both terminal solidsolubilities are smaller than 10 at .%, indicating A//for

>0. There are complications in binary systems combiningnon-transition metals with transition metals. If theanalysis is performed in the same way as in Figure 3,the demarcation line deviates from a straight line, andthere appears to be a systematic deviation, which canbe resolved more favourably by recognizing a de-pendence on the number of conduction electrons of thenon-transition metal.

Another semiempirical approach by Villars has alsoproven to be successful and is described in Villars (1985).The demarcation lines (surfaces) are much morecomplicated and consequently less trustworthy;therefore we recommend the use of Miedema's model.Nevertheless in Villars' model one gets, for the casewhere intermetallic compounds are absent in binarysystems, additional information as to the following fourphase diagram types: complete solid solubility, completesolid insolubility, simple eutectic, and simple peritecticsystems. Figure 4 shows a part of a chemical elementversus chemical element plot (Villars et aL, 1989). Eachfield represents a binary system. For the systems whereexperimentally determined phase diagrams show theabsence of intermetallic compounds, ' x ' is given.Where no experimental data are available, predictionsare given indicated by ' \ * (after Miedema) and 7 ' (afterVillars, only given in the cases where Miedema gave noprediction). Figure 4 shows that, upon alloying, the s-s,d3-7-d3"7, f-f, s-d3'7, s-f and d3~7-f chemical elementcombinations almost always result in the absence ofintermetallic compounds; see element designations inTable 3.

3.2 Ternary Systems

Attempts to extend Miedema's model to ternary systemsfailed. The extension of compound-formation maps toternary systems is described in Villars (1986). Inthe ternary systems we face two additional majorproblems: only very few ternary systems are fullydetermined, and for many of them only a part of anisothermal section has been experimentally determined.During an extensive literature search (Villars andCalvert, 1985) we found in 1984 information on 5598

Figure 3. Miedema's two-dimensional compound-formationmap for binary systems containing two transition metals. Forthe meaning of the ' + ' and ' • ' symbols, see text; each symbolrepresents one binary system

ternary systems, which represents only 3.6% of thepossible 161 700 systems. Furthermore, looking at thealloying situation from the constitutional point of view,the situation becomes definitely more problematic.Having in mind the two additional difficulties, whichcannot be avoided, one can only have confidence in aprediction for ternary systems if it is possible to separatethe two groups (compound formation and respectivelyits absence) in one diagram with relatively simpledemarcation lines in a two-dimensional diagram orrespectively demarcation surfaces in a three-dimensionaldiagram. The absence of compound formation isexperimentally established for 550 ternary systems(0.3%). Based on the experimental fact that no ternary

system with compound formation is known whosebinary systems are all of the type that do not formintermetallic compounds, we included all combinations(ternary systems) that are surrounded by the 583experimentally established binary phase diagrams showingno compound formation and ended up with 1602 additionalternary systems where the absence of intermetalliccompounds is expected (assuming the correctness of theabove-mentioned statement). All together we thus know2152 ternary systems in which no 'new* ternary inter-metallic compound is formed (1.3%). To assign a ternarysystem to the compound-forming group, we do not needto have a complete isothermal section, since knowledgeof a single ternary compound with a fully determined or

Figure 4. Part of a chemical element versus chemical element plot containing the experimental data and predictions for compoundformation or respectively its absence in binary systems. For the meaning of the symbols, see text

Ele

men

t A

Element B

Figure 5. Villars' three-dimensional compound-formation plotof the section for VE = 2 for ternary systems. For the meaningof 'AVo' and ' • ' symbols, see text; each symbol representsone ternary system

assigned structure is sufficient. For all such ternarysystems we checked where the nearby binary inter-metallic compounds have the same crystal structure.Where we found this situation we assumed a solidsolubility range between the binary and ternary inter-metallic compounds and therefore excluded thosesystems from the new-compound-formation group. Byadjusting our coordinates used for the binary systems(Villars, 1985) to the ternary systems, we came to thefollowing coordinates:

• the magnitude of Zunger's pseudopotential radiisums,

(|^+p,ABl + |Ai?f+p,Ac| + |A^f+P|Bc|)/3 (4)

• the magnitude of the difference in the number ofvalence electrons,

( |AFA B | + |AKA C | + |AKB C | ) /3 (5)

• and the sum of the ratios of the melting temperaturesin kelvin,

( r A / r B + TA/TC+ TB/Tc)/3 (6)

where TA> TB> Tc. The values for i?sz+p, F, and Tare

given in Table 3.In one three-dimensional diagram we plotted the

2152 systems where the absence of intermetallic com-pounds is observed and 5048 systems where compoundformation is observed. For the user's convenience wemade our plots on 12 sections with various isovalence-electron difference (|AKAB| + |AKAC| + |AKBC|)/3values. Figure 5 shows such a section for (|AKAB|+ IAKAC| + IAFBC|)/3 = 2. The symbol ' A ' stands forno compound formation determined by experiment; 'O'stands for no compound formation by extrapolationfrom experimentally established binary phase diagrams,all showing absence of intermetallic compounds; and' • ' stands for compound-forming systems. With arelatively simple demarcation surface it was possibleto separate these two groups satisfactorily. Figure 6shows the schematic three-dimensional formationdiagram. The separation surface is based on 2152 +5048 ternary systems and is accurate to 94%, whichmight be in the range of experimental accuracy. Itis nicely seen from Figure 6 that the absence of'new' intermetallic compounds is found along thethree coordinates; this means that the magnitude of anyone of the factors, i.e. the pseudopotential radiidifference expression, or the valence-electron differenceexpression, the melting-point ratio expression of theconstituent chemical elements, has to be small. Inthe cases where the values of any two or all threeexpressions are small, we also have absence of inter-metallic compounds.

4. Regularities in Intermetallic Compounds

We have tried to summarize the most outstandingpublications about regularities and gave special pref-erence to the ones which included in their investigationlarge groups of well-defined data sets, in the rangeof hundreds to thousands of data sets. Again, when-ever possible, we discuss first the binaries followedby the ternaries (multinaries) in the subsequentapproaches.

Next Page

1. Introduction

The large variety of metals, ordered and disorderedalloys, and interstitial alloys with close-packed structureshas fascinated many theoreticians and experimentalistsand inspired them to write articles and books about thissubject. A mathematical treatment for rigid-spherepacking, for example, was given by Patterson and Kasper(1959, 1967), while Schubert (1964, 1967) and Ogawa(1974) investigated long-period ordered alloys. The largenumber of crystal structures was classified by Wyckoff(1964), Pearson (1967, 1972), and Villars and Calvert(1986). The close-packed structures with coordinationnumber (CN) 12 of each metal atom were representedon structure maps with metal atomic radii, electro-negativity, and number of electrons as parameters(Villars and Hulliger, 1987; Villars et al.9 1989; Chapter18 by Pettifor in this volume). In our present andprevious investigations (Hauck et al., 1988a, 1989) wehave extended the numerical analysis of the environmentfrom the 12 nearest neighbors to metal atoms at largerdistances using the mathematical formulas of Pattersonand Kasper (1959, 1967). All structures from thedifferent compilations, from the simplest to the long-period ordered structures, can be ordered by thisprocedure in structure maps. These can be comparedwith the structure maps obtained by theoreticians, ase.g. Ducastelle (1991), with a different approach to theanalysis of attractive or repulsive interactions betweenmetal atoms in close-packed structures.

2. Stacking Sequences

When pure metals crystallize, the bonding tendencies

Crystal Structures of Intermetallic Compounds. Edited by J. H. We

of the individual atoms should be identical. The sameenvironment can be expected for each metal atom. Twolimiting situations may be discerned: either the structureis based on a three-dimensional (3D) framework ofdirectional covalent bonds as in gray a-Sn with thediamond structure or it is a packing of spherical metalatoms as in the close-packed structures. Metal atomswith weak bonding behave like tennis balls packed ina basket. The packing consists of coplanar hexagonallayers (Figure 1). Successive layers are stacked withcenters of spheres falling directly over the centers of thetriangular interstices of the two-dimensional (2D) layerbelow. Each layer has two sets of triangular intersticesas outlined for the unit cell in Figure 1. Each of the two

Figure 1. Close-packed hexagonal layers of spheres withposition, e.g. A. The next hexagonal layer can be placed eitheron B or C positions. Hexagonal unit cells with a = b-d(diameter of atoms), c = nj2/3d, n = number of layers

itbrook and R. L. Fleischer ©1995, 2000 John Wiley & Sons Ltd

Chapter 2

Close-Packed Structures

Jtirgen Hauck and Klaus MikaInstitut fur Festkorperforschung, KFA Forschungszentrum, D-52425 Jtilich, Germany

sets forms the same lattice as the spheres themselves.The three different positions that can be occupied bymetal atoms in successive layers are conventionallydenoted as the A, B, and C positions (Figure 1).Different stacking sequences of hexagonal layers, whichcan be characterized by the sequence of positions, e.g.ABACB. . . , give rise to different crystal structures.The part of the volume of the basket that can be filledby the tennis balls reaches 7497b, which is the highestpossible density for packing of identical spherical atoms(Max, 1992). Therefore, these structures are called close-packed (for details see Patterson and Kasper, 1959,1967).

Rigid spheres without long-range interactions forman infinite number of possible stacking sequences.Metallic cobalt can be obtained in this type of packingwith a disordered sequence of A, B, and C positionsafter successive annealing above and below 450 0C(Wyckoff, 1964). At temperatures above 450 0C itprefers the ABCABC . . . packing; at lower temperaturesand after long annealing it crystallizes in ABAB . . .packing. The periodic sequence ABAB . . . of two layers(AB) is called hexagonal close-packed (h.c.p. or h), andthe periodic sequence ABCABC . . . of three layers

(ABC) is cubic close-packed (c.c.p. or c), because ofthe hexagonal or cubic symmetry, respectively. Theh.c.p. and c.c.p. metal structures are usually distinguishedby the prototype elements Mg and Cu and theStrukturbericht or Pearson symbols (Table 1). Forty-eight of the 59 elements shown in Figure 2, the rare-earthand the actinide metals except Eu, U, Np (and the veryunstable elements Pm, Es, Fm, etc.), can be obtained ina close-packed modification (Villars and Calvert, 1986).Some metals can even crystallize in the same structuretype but with different lattice constants, as e.g. the low-and high-pressure forms of Ce with c.c.p. stacking (Villarsand Calvert, 1986). The atomic radius of Ce decreasesdiscontinuously with increasing pressure. Also the noblegases with very weak van der Waals forces crystallize inh.c.p. (He) or c.c.p. (He, Ne, Ar, Kr, Xe, Rn) structure(Wyckoff, 1964). The structure of solid He depends onthe isotope. Other elements forming molecules, as e.g.O2, F2,12, B12 (Pauling, 1945, 1960; Wyckoff, 1964) orC60 fullerene (David et aL, 1991), crystallize in close-packed structures with rotational disorder of the moleculesoccurring at increased temperatures.

A few metals crystallize also in more complex close-packed structures with a maximum of nine layers

Table 1. Stacking sequences of «<9 hexagonal layers with conventional or stacking symbols and number M' of metal atomswith different coordination T1, prototype with Pearson and Strukturbericht symbols, space group of undistorted structures, andpopulation (maximum values from different compilations). The Hg and In structures at n = 3 are rhombohedral and tetragonalvariants of the Cu structure

Layersn

23

456a6b7a7b7c8a8b8c8d8e8f9a9b9c9d9e9f9g

Conventional symbols

ABABC

ABACABCACABCBABABCACBABCACACABCBCACABCABACABABACACABCBABABABCBACABABCACACBABCACABCABCABACBABABCBCACABCACACACABCBCACACABCABACACABCACBCACABCBCABACABCABCBAC

Stacking symbols

hh or (h)2ccc or (c)3

chch or (ch)2ccchhhchchhhcchcc or (hcc)2ccchhhhcchhchhcccchchchhhchhh or (chhh)̂hchchhhhhchcchchhcchhhccccchhccchccchccc or (hccc)2chhchhchh or (chh)3ccchhhhhhcchhchhhhcccchchhhccchchchhcchhcchchccccchcch

Af

11

22323332334332334544

Prototype(symbols)

Mg (hP2/A3)Cu (CF4/A1)Hg (hRl/AlO)In (tI2/A6)La (hP4)

MxN,MxN,

MxN,MxN,

Sm (hR3)

Spacegroup

P63/mmcFm3mR3mI4/mmmP63/mmcP3mlP6m2P63/mmcP3mlP3mlP3mlP63/mmcP6m2P3mlP6m2P3mlP63/mmcR3mP3mlP3mlP3mlP3mlP3mlP3ml

Population

271369

52833

32

(Table 1). These structures can be described un-ambiguously by the stacking symbols h and c introducedby Pauling and Jagodzinski (Pearson, 1972; Mardix,1990). This notation describes a layer as h or c type

according to whether the layers on each side of it arethe same type, such as B in ABA, or of different types,such as B in ABC. The two-layer h.c.p. sequence ABgets the symbol hh; the three-layer c.c.p. sequence ABC

Ato

mic

rad

ius

(pm

)E

lect

ron

egat

ivit

y (V

)

Transition metals Main group elements

Figure 2. Electronegativity <f>* (de Boer et al., 1988) and radii (Schubert, 1967) of la-8a main group and Ib-IOb transitionmetals with cubic ( • ) or hexagonal ( A ) close-packed crystal structures. Some metals (o) are not observed in a close-packedmodification

has the symbol ccc. The lanthanides La to Sm, Gd, andthe actinides Am to Cf crystallize in the ABAC orchch = (ch)2 stacking (This stacking is identical toABCB or hchc stacking by variation of the origin.) Sm,Gd, and the heavier lanthanides Tb to Ho, Tm, Lu cancrystallize also in the ABABCBCAC or chhchhchh =(chh)3 stacking. A comparison of the stacking symbolsand population data in Table 1 shows the preferenceof the elemental metals for simple sequences hh, ccc,(ch)2 or (chh)3. Some more complex sequences arefound in alloys MJM^ (Table 1), which are discussed inSection 8.

The hh hexagonal close-packed and the ccc cubicclose-packed structures are the only structure types witha single environment for close and distant neighbors ofall atoms (Af= 1). These structures have been observedin 673 binary and ternary systems (Villars and Hulliger,1987). The five sequences of layers with two differentenvironments of metal atoms (Af = 2) (Pauling, 1945)have been found in 65 systems (Table 1). The metal

atoms of the other sequences of layers contain three ormore different environments. The tendency to smallnumbers M1' of metal atoms with different environmentcan be described by Pauling's rule of parsimony: thenumber of essentially different kinds of constituents in acrystal tends to be small (Pauling, 1929). The structureswith one or two different environments of metal atomsalso have the highest symmetry.

Atoms of close-packed structures with diameter d=\have T1 = 6 nearest neighbors in the same layer andTx = 6 neighbors in the two adjacent layers (three ineach) at distance d (Figure 3), for a total of 12. Thenumber T2 = 6 of second-nearest neighbors at distancefed is also identical for all close-packed structures.The atoms of the hh structure, however, have T3 = 2third-nearest neighbors at distance J873tf and 18 fourth-nearest neighbors at distance fed; while atoms ofccc packing have 24 third-nearest neighbors at distancefed (Figure 3, Table 2). The coordination polyhedraof T1, T2, and T3 neighbors of the h.c.p. structure

/7 Layer

Nearest Nextnearest

Thirdnearest(c.c.p.)

Thirdnearest(h.c.p.)

Figure 3. Metal atoms of five hexagonal layers /i, /i± 1, and n±2. There are 12 nearest, six next-nearest, and 24 third-nearestmetal atoms at distances d, fed, and fed in ABC layers of cc .p . structure; and 12 nearest, six next-nearest, and two third-nearestmetal atoms at d, fed, and J$73d in AB layers of h.c.p. structure

correspond to CN 12', CN 6 (octahedron), and CN 2,and the coordination polyhedra of the c.c.p. lattice toCN 12 (cubooctahedron), CN 6, and CN 24", as shownin Chapter 11 by Villars in this volume. The metal atomsof the experimentally observed sequences (ch)2, ccchh,(hcc)2, (chhh)2, and (chh)3 have two different valuesfor T3, T4 and higher coordination shells. The morecomplex structures contain three or more metal atomswith different environments. The averaged T3 and T4

coordination values of these structures depend onthe fraction, / , of h layers as given by T3 = 2/ andT4 = 24-6f.

The number M' of metal atoms with different setsof T1 values is often identical to the number ofWyckoff positions, the number of symmetricallydifferent atom positions. For example, the numbers ofWyckoff positions of the Mg, Cu, Sm, and La structuresare identical to Af1", but for the « = 8b, 9b (Af1" =3)structures (Table 1) these numbers are larger than M'(with values of 5 for example).

The different stacking sequences with n layers in A,B, or C positions can be described by hexagonal unitcells with the lattice parameters a = b = d (diameter ofatoms) and c = n\2/3d (Figure 3). The smallest unit cellof ccc packing containing only one metal atom is rhom-bohedral with lattice constants a = d and a = 60°.

Usually, the ccc structure is described as the face-centered cubic (f.c.c.) lattice containing four metalatoms with lattice constants ac = Jld. The f.c.c. latticecan also be considered as a stacking of tetragonal layersin the K and L positions (Figure 4). The c-axis of thehexagonal cell corresponds to one of the four spacediagonals of the cubic cell. The only periodic 3D packingof spheres having close-packed hexagonal layers in four

Figure 4. Packing of tetragonal layers at K and L positionsin face-centered cubic metal structure and relation to c.c.p.stacking of hexagonal layers in ABC position

inclined directions is ccc. All other sequences are main-tained in one direction only.

A comparison of the close-packed structures showsthat the interactions between metal atoms, e.g. in theccc Cu structure, must be strong enough to stabilize theatoms of the third layer at the proper position. The Smatoms of chhchhchh structure must be stabilized in theninth layer.

An interaction between metal atoms with directionalbonding can also be deduced from the distortion ofsome crystal structures. The c/a ratios of the latticeconstants of the hh hexagonal close-packed structuresof Ca and Sr are almost ideal with c/a = J8/3 = 1.63.The c/a ratio is slightly decreased to 1.56 for most h.c.p.metals and increased to c/a = 1.86 and 1.89 for Zn andCd, respectively (Pauling, 1945, 1960; Wyckoff, 1964;Laves, 1967). The distance between layers is alsoincreased in cubic close-packed Hg, Po, and Te withthe rhombohedral angle a = 71 -103 ° instead of a = 60°(Villars and Calvert, 1986).

The distance between two identical tetragonal layersof ccc packing, which is ac = Jld for cubic Cu, can beincreased up to 1.66d in In, Ga, La or Ce or decreased to1.33d in Pu or 1 .Od in W. Tungsten has a body-centeredcubic structure, which will be discussed by Hellner andSchwarz in Chapter 13 in this volume. The 74% densityof the close-packed structures, however, is reduced byhexagonal or tetragonal distortions, e.g. to 68% for thebody-centered cubic structure (Laves, 1967; Pearson,1972). These strongly distorted structures are consideredas different structure types in some compilations (Hoand Douglas, 1968). This is one reason that thepopulation numbers of different structure types are onlyrough values for comparison. We have taken themaximum values from the compilations of Wyckoff(1964), Pearson (1972) or Villars and Calvert (1986)of experimentally determined crystal structures. The

Table 2. Self-coordination numbers T1= 7™** of pure metalsin hexagonal and tetragonal plane, h.c.p., c.c.p., and interstitialatoms 1° at octahedral sites of the h.c.p. metal lattice, andR^/d1 values in parentheses; /?, = radius of /th shell, d-diameter of spheres

i

123456789

10

7™ (R]Zd1)

Hex.plane

6 (1)6 (3)6 (4)

12 (7)6 (9)6(12)

12 (13)6(16)

12 (19)12 (21)

Tetr.plane

4 (1)4 (2)4 (4)8 (5)4 (8)4 (9)8(10)8(13)4(16)8(17)

h.c.p.

12(1)6(2)2 (8/3)

18(3)12 (11/3)6(4)

12(5)12 (17/3)6(6)6 (19/3)

c.c.p.

12 (1)6 (2)

24 (3)12 (4)24 (5)8 (6)

48 (7)6 (8)

36 (9)24 (10)

1° (h.c.p.)

2 (2/3)6(1)

12 (5/3)2 (8/3)6(3)

24(11/3)6(4)

12 (14/3)12 (17/3)2(6)

undistorted structures are compared in the present chapterfor the analysis of the self-coordination numbers T1.

3. Alloy Formation

The atoms M and N, e.g. of a binary alloy MxN ,̂ havedifferent sizes and a different tendency to exchangeelectrons, which gives rise to different bonding. Theelectron density and the radii of the M and N atomscan be varied by charge transfer between the M and Natoms. The two parameters do not vary independentlyfor most metals (Figure 2). The electronegativity valuesof the elements can be related to the electron work-function and indicate the tendency of a metal toexchange electrons (Pauling, 1945, 1960; de Boer et aL,1988). The electronegativity increases with increasingnumber of electrons to a maximum for the 8b- 10b tran-sition elements. Those elements with an almost filledd shell try to accept electrons to complete the d shellwith 10 electrons, while elements with few outerelectrons like the Ia and 2a alkali and alkaline-earth or the Ib or 2b transition metals with lowerelectronegativity are ready to donate these electrons. TheIa and 2a metals are then stabilized in the sphericals2p6 shell of the noble gases, the Ib and 2b transitionmetals in the d10 shell with decreased radii. The electro-negativity increases again for the 4a-7a elements becauseof the tendency to attain the stable s2p6 octet of thenoble gases. The radii of metal atoms decrease withincreasing number of electrons to a minimum for thestable d10 electron configuration (Schubert, 1967). Alsothe radii of the lanthanides La to Lu are decreasedslightly as indicated by the arrow in Figure 2 as a resultof the same effect of the filling of the inner 4f-electronshell. The relation between radii and electronegativityindicates the tendency to form spherical metal atomsin alloys. Some metals like Mo, which do not form close-packed structures because of directional bonding of theouter electrons, form close-packed structures in alloys,as e.g. cubic close-packed MoNi4 or MoPt2 (Sections7 and 10). The directional interactions between metalatoms, however, can even be increased on the formationof an alloy. The CuPd or FeTi alloys, for example,crystallize in an ordered body-centered cubic structurein spite of the close-packed structures of the com-ponents. Other pairs of metals like Cu and Rh, withthe c.c.p. structure, do not form compounds at all butsegregate as two limited solid solutions.

The interactions between the M metal atoms inMJNTy alloys can be attractive, as e.g. with segregation,or repulsive. Both interactions can be distinguished by

an analysis of the self-coordination numbers T1 of Matoms with M atoms. Analysis of the different ways ofsphere packing showed the same number Tx = 12 andT2 = 6 of nearest and next-nearest M atoms, respectively.These numbers are reduced on the formation of orderedMxNy alloys and can be used to obtain structure maps(Hauck et aL, 1988a, 1989; Mika and Hauck, 1990).

The derivation of structure maps will be outlined forthe simple examples of the tetragonal and the hexagonallattice nets of single metal-atom layers. The singletetragonal and hexagonal 2D layers of M atoms (Figures1 and 4) are the basic units for describing the 3D close-packed metal structures (Beattie, 1967; Lima-de-Fariaand Figueiredo, 1969; Beck, 1969).

4. MxNy Structure Map of a SingleTetragonal Layer

The metal atoms M and N of MxN^ alloys can beordered in a large number of different structures. Thestructures are characterized by the self-coordinationnumbers of the metal atoms (T1, T2, T3) to obtain asubdivision of Pettifor's structure map (see Chapter 18in this volume) for close-packed alloys with T1 =CN =12. The numerical procedure to obtain a structuremap with a correlation between the different structureswill be outlined for a single tetragonal layer occupiedby M and N atoms (Figure 4). The procedure containsthe following steps:

(1) A large variety of structures is obtained by asystematic variation of the unit cell (Figure 5(a)).

(2) The metal positions are occupied by M atoms to amaximum percentage of 50% at y/x=\. Thestructures at higher M content are identical byexchange of the M and N atoms.

(3) The different crystal structures can be characterizedby the coordination of the M atoms with other Matoms in the first, second, and third coordinationshells (T1, T2, T3) and the ratio (y/x) of N and Matoms, e.g. the notation 204; 1 is used for the struc-ture with T1 = I metal atoms M at distance d, noM atoms in the second coordination shell, T3 = AMatoms at distance Id (Figure 5(b)) and y/x= I.jTnax=7™x= J W x ^ 4 ^Q t h e m a x i m u m seJf_

coordination numbers of the tetragonal layer.The coordination numbers of each shell are averagedfor structures that have differently coordinated Matoms, e.g. the 323, 442, and 323 coordinationnumbers of the three types of M atoms in the firstsketch in Figure 5(d) are averaged in the 3.3 2.7 2.7; 1structure.

Figure 6. T1,. T2 structural map of tetragonal single-layerstructures Tx T2 T3; 1 of Figure 5

(4) The crystal structures characterized by the co-ordination numbers T1T2 T3 and a fixed compositiony/x can be plotted as single points in a Tx-T1-T3

coordinate system or as projection in the Tx-T2

plane as outlined for Tx T2T3;! structures in Figure6. All structures are found to fall within a triangle,with the three structures 444;(1), 204; 1, and 044; 1,which have a single environment of M atoms, at thecorners. The structures 322;1 and 123; 1 have oneand three environments, respectively. The structure444;(1), with the composition given by (1) inparentheses, is used to indicate that the structurecan only be obtained in the limit of very large unitcells, because the N atoms must also have thisenvironment.

(5) The structures shown on the structure map can beconsidered as combinations of the variously shadedstructural units u, v, x, and y (Figures 5(c) and (d)).The structures on the left side of Figures 6 and 7are obtained by the combination of u and v units,structures on the right side by a combination of xand y units. Points within the triangle are obtainedby the combination of all the subunits (Figure 5(e)).

Figure 5. Tetragonal single layers for equal numbers of M and N atoms: (a) different unit cells; (b) occupation with M atoms(large circles) and characterization by T1 T2 T3 self-coordination numbers and composition y/x; (c) different subunits forconstruction of different structures (d,e)

(6) Structures T1T2 T3; y/x with different y/x values canbe included in the same structure map (Figure 7) byusing Cowley's short-range order parameter a,(Ducastelle, 1991), which can vary within- 1 â,-< 1. The a, values can be obtained from thecoordination numbers Tf4 and Tf* of M and Natoms or from the Tf and y/x values (Hauck et al.,1988a):

J-N _ ynnax _ ^jnnax _ JM)x/yymax _ yM , yN _ ymax

d.T™* =Tf- (77"» - Tf)x/y

The coordination numbers Tf and T™ of M and Natoms are identical at the composition y/x= 1 butdifferent at other compositions. Structures with

different compositions can be obtained by using thesame structural subunits. The area mapped out bythe maximum range of CK1, a2 values is different atdifferent compositions, as outlined for yZx = 2 andy/x =3 in Figure 7.

(7) The Ot1 values are zero for a random distribution ofM and N atoms, because the mean value of Tf +T™ equals Tf**, which corresponds, for example toT1 = T2= T3 = 2 at composition y/x= 1 (Figure 7).Very small a, values are expected for alloys withvery weak interactions and in particular at hightemperatures. Positive at values are obtained forattractive interactions of M atoms, i.e. for clusterformation or segregation. The 202;2 and 322;2structures of Figure 7 consist of single and double

Figure 7. a,, a2 structural map of tetragonal single-layer structures T1 T2T^yZx and surface structures /, Xl2

A t t r a c t i o n of M atomsClus te rs

Disordered

a l l o y

Repulsion of M atomsSegregat ion

rows of M atoms, respectively. The size of the Mand N clusters is increased to complete segregationof M and N atoms in the 444;(1) structure. Negative(X1 values indicate repulsive interactions between Matoms, e.g. Coulomb repulsion.

(8) Two-dimensional tetragonal structures are observed,for example, for gas molecules adsorbed on the(100) surfaces of c.c.p. metals (MacLaren et aL,1987). The occupation of the surface of metal atomswith gas molecules depends on the size of the gasmolecules, the equilibrium gas pressure, and theinteraction between the molecules. The unit cell isusually described by the length of the two sidesZ1 Xl2 as outlined in Figure 7. There are differentstructures with identical unit cells for occupationof two or more positions, as can be seen by the two2x3 structures 103;2 and 022;2.

5. Ordering of Atoms in Hexagonal Layers

Each metal atom within a hexagonal layer has T^ = 6nearest, r™ax = 6 next-nearest, and r™ax = 6 third-nearest neighbors at distances d, ^d, and 2d, respectively(Figure 3). The structure map (Figure 8) contains sixstructures with a single environment at the corners ofan irregular pentagon: 666;(1), 422;1, 202;2, 226;1,006;3, and 060;2. The 226;1 and 006;3 structures haveidentical Ot1 values. The crystal structures existing alongthe different borderlines can be obtained by com-binations of the structural elements in the same way aswas done for the tetragonal plane structures. The cross-hatched area of the pentagon with the limiting structures202;2 and 060;2 cannot be obtained at the compositiony/x- 1, because the borderlines depend on y/x (item(6) of Section 4).

Figure 8. ax, a2 structural map of hexagonal single-layer structures Tx T2T2;y/x and surface structures /, Xl2

Segregation'

Figure 10. a,, a2 structure map of ordered h.c.p. structures

S e g r e g a t i o n

Figure 9. Five hexagonal close-packed structures MxN,: 1262;(1) Mg, 602;la LiRh, 462;1 AuCd, 042;3 TiCu3, and 062;3 SnNi3;and four theoretical structures at the border of the structure map in Figure 10. Projection of four layers ABA' B' with M atoms atA, B, A' or B' positions. Nearest-neighbor A and B positions are connected by full lines; unit cells are indicated by broken lines

Positions

The two-dimensional structures of adsorbed gasmolecules with the unit cells 1 x 1 , 1 x 3 , and J3 x J3 canbe compared to the analogous structures 1 x 1 , 1 x 2 ,and J2 x J2 of the tetragonal plane. The cells 1 x 2 and1 x p are different unit cells of the 226;1 structure.

6. Hexagonal Close-Packed Structures

The hexagonal close-packed alloys consist of twohexagonal layers in A and B positions with identicalenvironment of 12 nearest, six next-nearest and twothird-nearest neighbors at distances d, Jld, and Js/3d,respectively (Figure 3). The maximum Cx1-(X2 range ofordered h.c.p. MJN,, alloy structures is within a tetragonwith the 12 62;(1), 602;l, 442;1, 462;1, and 062;3structures at the corners (Figures 9 and 10). The 530;land 930;l structures are also at the corners in the three-dimensional «!-«2-^3 space (Mika and Hauck, 1990).Several structures with a single environment of metalatoms are inside the structure map. The 930;l structurewith two layers of M atoms followed by two layers ofN atoms can be considered as partially segregated sheetsof M and N atoms. The M atoms of the 602; Ia and602; Ib structures are clustered in layers and chains,respectively. The M atoms of the remaining structures

of Figure 9 are as far apart as possible. The 462; 1 and06 2; 3 structures with identical a, values correspond tothe AuCd and SnNi3 structures. The other experi-mental structures are found at lower a2 values.

The architecture of the different structures built upfrom structural subunits is more complicated than forthose from single tetragonal or hexagonal layers. Theconstruction of MN and MN3 structures by the com-bination of u, v, x, and y units is shown in Figure 11.The SnCu3 structure, for example, can be obtained bythe combination of these units in the sequence u4vu4v.The theoretical structures at the left border of Figure 10consist of alternating layers of M and N atoms. Theother structures have layers with identical compositionand environment of the M atoms in these layers (Table3). Therefore, these structures can also be described bythe type and the stacking sequence of hexagonal layers.

7. Cubic Close-Packed Structures

Each metal atom of the c.c.p. structure has r™ax = 12,T™ax = 6, and 7 ^ = 24 for the first-, second-, andthird-nearest neighbors at distances d, fid, and fid(Figure 3). The maximum T1 T2T3 values are reducedin MxNy alloys where y/x^ 1 (Table 4). The structure

Figure 11. Architecture of h.c.p. MN and MN3 structures at the right border of the structural map (Figure 10), and ZrAu4,Sb3Cu10, Cd12Au42 (CdAu35) and Cd26Au72 (CdAu28) structures with two hexagonal layers. M atoms ( • ) and N atoms (•) offirst layer, only M atoms (o) of second layer, u, v, x, y structural subunits for puzzle assembly of crystal structure architecture.Second- and third-nearest neighbors of single-layer M atoms are connected by broken lines for pattern recognition

Table 3. Hexagonal close-packed alloys MJ<ly characterized by the self-coordination numbers T1 of M atoms in the h.c.p.structure and in each hexagonal plane, number M of metal atoms with different sets of h.c.p. T1 values, and maximumpopulation values from different compilations

T1T2T3-JZx

h.c.p.

1262;(1)930;l602;la602;lb53O;la,b462;1442;1622;153O;ld,e73O;la,b220;2a,b222;20.9 3.7 2;2.8202;3042;30 4.4 2;3O52;3062;3042;3.3142;3.5022;4000;5

hex. plane

666;(1)666;(1)666;(1)226;1226;1226;1242;1242;1242;1422;1060;2060;20 2.32.8;2.8226;1022;30 1.6 2.8;3014;3006;3022;3.3O13;3.5040;4060;2

M'

122222222222

12437525653

Prototype (symbols)

Mg (hP2/A3)

LiRh (hP2)

AuCd (OP4/B19)

TaPt2 (oC12)Cd26Au72 (hP98)SbAg3 (oP4)TiCu3 (oP8)SnCu3 (oC80)CdAu3 (oC32)SnNi3 (hP8/D019)Sb3Cu10 (hP26)Cd12Au42 (hP54)ZrAu4 (oP20)SnAu5 (hR6)

Space group

P63/mmcP3mlP6m2PmmnC2/m; Fdd2PmmaPnmaPbcmP2/c; C2/cP2/m; P2/mC2/c; P6,22CmcmP63/mmcPmm2PmmnCmcmCmcmP63/mmcP63/mP63/mcmPnmaR3

Population

271

1

14

111

2512

862131

Table 4. Cubic close-packed alloys MxNy characterized by the self-coordination numbers T1 of M atoms in c.c.p. structure andin tetragonal planes (only occupied planes perpendicular to the direction with highest symmetry are included; in Ga3Pt5 twodifferent planes exist), number M' of metal atoms with different sets of T values, and maximum population values

T1T2T3JZx

c.c.p.

12624;(1)6012;la6012;lb6212;la,b4416;14 5.6 9.6;1468;1

546;1.25356;1.52.7 4.7 5.3;1.722 12;2a248;2254;2a254;2b1310;2.50.8 48;2.60.7 3.3 9.3;2.7204;3048;3054;3

060;3

028;4000;7020;8

tetr. plane

444;(1)204;l204;l204;l044;l044;l444;(1)

121;1.253 3 3;0.25004;3, 044;l202;2044;l044;l044;l110;2.5044;l0.7 1.3 0;2.7204;l044;l044;l

044;l

000;4004;3000;8

M[

1222262

34423434

106333

2

233

Prototype (symbols)

Cu (CF4/A1)CuPt (hR2/Ll,)CuPt (cF32/Ll3)

UPb (tI8)CuAu II (oI40)CuAu (tP4/Ll0)NaHg (oC16)V4Zn5 (til8)Ti2Ga3 (tPIO)Ga3Pt5 (oC16)MoPt2 (oI6)ZrSi2 (OC12/C49)ZrGa2 (oC12)HfGa2 (tI24)Mn2Au5 (mC14)Nb5Ga13 (oC36)Mo3Al8 (mC22)CuPt3 (oC8)TiAl3 (tI8/D022)ZrAl3 (til6/DO23)CdAu3 II (til6)AuCu3 (cP4/Ll2)SiU3 (tI16/D0c)SrPb3 (tP4)MoNi4 (tI10/Dla)GeCa7 (cF32)TiPt8 (til8)

Space group

Fm3mR3mFd3mPmmn, P42/mmc14/amdImmaP4/mmmCmcmI4/mmmP4/mCmmmImmmCmcmCmmm14,/amdC2/mCmmmC2/mCmmmI4/mmm14/mmmI4mmPm3mI4/mcmP4/mmmI4/mFm3m14/mmm

Population

36911

21

9381158

30281111

4372

3865

151623

Figure 12. a,, a2 structure map of ordered c.c.p. structures

map contains 28 different structures with a singleenvironment of M atoms (Hauck et aL, 1988a). The12 624;(1), 6 0 12;la,b, 44 16;1, and468;l structures arethe limiting structures (Figures 12 and 13). The 60 12;laand b structures are homometric structures—structureswith identical T1 values, but different symmetry, as willbe discussed in some detail in Section 9. The 060;3AuCu3 structure and the 468; 1 CuAu structure haveidentical a, values. These structures can be comparedwith the h.c.p. 062;3 SnNi3 and 462; 1 AuCd structuresfor the same T1 and T2 values (Figure 10). The 4416; 1UPb and the 048;3 TiAl3 structures correspond tothe h.c.p. 442; 1 and 042;3 TiCu3 structures. PtVcrystallizes in the CuAu or AuCd structures (Villars andHulliger, 1987). The 4416;1 UPb structure is identicalto the NbP structure (Villars and Calvert, 1986). Thearchitecture of structures along the 12 6 24;(l)-60 12; Iaboundary is analogous to that for the h.c.p. structures.The M and N atoms are completely segregated in

12 6 24;(1) with decreasing thickness of hexagonal layersprogressing along the boundary (first row in Figure 13).Close-packed planes of M and N atoms alternate in the6012;la CuPt structure as in the h.c.p. 602;la LiRhstructure. The c.c.p. structures at low Of1, includingmost experimental structures, can be assembled fromtetragonal layers rather than from hexagonal layersalone as is the case of h.c.p. structures.

The limiting structures at the right border 060;3,048;3, 468;1, and 4416;1 can be split into structuralsubunits u, v, x, and y, which can be combined like theparts of a puzzle to obtain the crystal structures foundat the right border (Figure 14). The u ' , v', x', and y'subunits are obtained from the u, v, x, and y subunitsby a variation of the origin. The ZrAl3 structure isobtained from u and u' units and the CuAu II structurefrom a combination of the x and y structural subunits.The MN2 structures ZrGa2, HfGa2, and ZrSi2, and theNb5Ga13 structure are obtained by combination of u,


Figure 13. T1T2T^yZx cx.p. structures MxN^: 12 6 24;(1) Cu, 60 12;1 CuPt, 4416;1 UPb, 468;1 CuAu, and 060;3 AuCu3;and theoretical structures at the border of the structure map (Figure 12). Occupation of hexagonal layers ABC. . . by M atomsin the first row and of four tetragonal layers KLK'L f in the other structures. Two unit cells of 6012;la structure are showrin the second row

Figure 14. Architecture of c.c.p. MN3, MN, and MN2 structures from u, u ' , v, v ' , x, x ' , y, y' subunits and structures withtwo tetragonal layers with M atoms at z = 0 ( • ) and z= 1/2 (o). Positions of N atoms at z = 0 (•) for pattern recognition

Table 5. Characterization of structures with a single environ-ment of M atoms

Tetr. layers Hex. layers

CuAu 044;l, 444;(1) 226;1UPb 044;l, 220;l 242;1AuCu3 044;l, 444;(1) 006;3TiAl3 Q44;l, 444;(1), 020;3 022;3

v, x, and y units (Figure 14). Other structures, likeGa3Pt5, Mo3Al8, TiPt8, V4Zn5, Ti2Ga3, and Mn2Au5

(Figure 14) and the structures from homologous series(Section 10), MoNi4 and MoPt2, can be considered asa packing of two ordered tetragonal layers. The tetra-gonal MxNy layers perpendicular to the drawing planeare separated by layers of N atoms in Ga3Pt5 andNb5Ga13, and in all MN3 and MN2 structures. Most ofthe experimental structures are at boundary lines for thegiven composition. The only exception in Table 4 isV4Zn5, which lies at the boundary line of the structuremap belonging to the hexagonal layer (Figure 8).

The structures with a single environment of M atomscan be characterized by tetragonal or hexagonal layers

with small unit cells, as shown in Table 5. The structureswith Af>2 (Table 4), as e.g. the CuAu II or ZrAl3

structure at the right boundary of the structure map(Figure 12), consist of u,v,x,y structure subunits, whichare connected by 044; 1 tetragonal planes. The otherlayers of the CuAu-UPb and AuCu3-TiAl3 parentstructures do not fit together. The structural subunitsof many structures on the right side of the structure map(Figure 12) are connected by204;l or044;l tetragonalplanes (Table 4). Most (theoretical) structures on theleft side of the structure map are connected by 666;(1)hexagonal planes.

The CK1, OL1 values of the different layers of a structureare sometimes similar to the Of1, a2 values of the struc-ture itself (Figures 7, 8, and 12) and are located in thesame area of the structure map. An example is the 204; 1tetragonal and 226;1 hexagonal layer of the 60 12;IaCuPt structure. In other structures, as e.g. 22 12;2aMoPt2, the 020;2 and 202;2 tetragonal layers or the202;2 and 060;2 hexagonal layers are at different loca-tions of the structure maps (Figures 7 and 8), probablybecause of the directional bonding of Mo atoms.

Table 6. Complex close-packed ordered alloys M ^ with the same hexagonal planes as h.c.p. or c.c.p. structures, and maximumpopulation values

Layers

236b9a

234

2346b7c9a

10121415

236b

12

2346a9a

10

Stacking symbol

hhCCC

(hcc)2

(chh)3

hhCCC

(Ch)2

hhCCC

(Ch)2

(hcc)2

cccchch(chh)3

(hhchc)2

(hhcc)3

(hhhchhc)2

(hchcc)3

hhCCC

(hcc)2

(hhcc)3

hhCCC

(Ch)2

hchchh(chh)3

(hhchc)2

Hex. PlaneT1T2T^yZx

226;1226;1226;1226;1

060;2060;2060;2

006;3006;3006;3006;3006;3006;3006;3006;3006;3006;3

022;3022;3022;3022;3

014;3014;3014;3014;3014;3014;3

Prototype (symbols)

AuCd (OP4/B19)CuAu (tP4/Ll0)IrTa (oP12)LiSn(Nb2Rh3) (mP6)

SnAu5 (hR6)OO8;5WAl5 (hP12)

SnNi3 (hP8/D019)AuCu3 (cP4/Ll2)TiNi3 (1^16/DO24)PuAl3, VCo3 (hP24)Ti(Pt089Ni011), (hP28)BaPb3 (hR12)7-Ta(Pd067Rh033), (hP40)PuGa3 (hR16)Ba(Pb0 8TlO2)3(hP56)HoAl3 (hR20)

TiCu3 (oP8)TiAl3 (tI8/D022)0-NbPd3 (oP24)/3-NbPt3 (mP48)

CdAu3 (oC32)ZrAl3 (til6/DO23)MgAu3 (oC64)CdAu28 (oC96)"CdAu2 7 (mC576)MgAu, (0CI6O)

Space group

PmmaP4/mmmPmmaP2/m

R3P3,12P6322

P63/mmcPm3mP63/mmcP63/mmcP3mlR3mP^/mracR3mP^/mmcR3m

PmmnI4/mmmPmmnP2/m

CmcmI4/mmmCmcmAmm2C2/mCmcm

Population

149342

1

3

86386

1271

26141

15

254024

171111

"An alternative structure to this (h P98) is given in Table 3.

8. Ordered Structures of ComplexClose-Packed Alloys

The complex close-packed metals (Table 1) are similarto h.c.p. stacking with only one direction of packingof the hexagonal layers. AU structures of Table 1 containa threefold axis in the c direction of the stacked layers.The metal atoms of the complex structures of Table 1already have two or more different environments.The values of Tx and T2, however, are identical forall complex close-packed metals. The «i-a2 struc-ture maps are the same as for h.c.p. and c.c.p.alloys. The architecture of the borderline structuresresembles those of h.c.p. alloys containing hexagonallayers of M atoms. These structures can be dividedinto five groups corresponding to different Cx1, a2

values and different hexagonal layers (Table 6) (Beck1969; Zhao et al., 1991). The h.c.p. CdAu3 structure,

for example, with ax- -0.33, U2 = OJS9 is graduallyaltered to the c.c.p. ZrAl3 structure with differentstacking of 014;3 hexagonal layers by variation of theAu content or partial substitution of Cd by In (Schubert,1964) (Table 6). The c.c.p. 008;5 structurecorresponding to h.c.p. SnAu5 and (ch)2 WAl5 is onlyobserved in the interstitial alloy V6C5 (see Section 14).

9. Structures with Identical Powder Patterns

Some c.c.p. and h.c.p. structures are homometric withidentical coordination T1 of all atoms for all / (Figure15) and some other structures differ only in higher co-ordination spheres (Hauck et al., 1988a). The homometricstructures cannot be distinguished by powder diffractionmethods, if the lattice is undistorted (Patterson, 1944).Other structures are enantiomorphous, i.e. the crystal

Figure 15. Examples of homometric pairs of structures with identical self-coordination numbers T1: 6212;la,b, 6012;la,b,22 12;2a,b, and 224;3a,b are c.c.p.; 220;2a,b, and 5 3O;la,b are h.c.p. The positions of M atoms are given by the projectionheight. The different M positions of homometric pairs are encircled. N atoms at z = 0 (•)

structures contain right- or left-hand screw axes thatcannot be distinguished by powder diffraction. Thehomometric, but non-enantiomorphous, structures arecharacterized by adding a suffix a, b, c, . . . to theircoordination descriptions. The h.c.p. 530;Ia,b struc-tures (Figure 15) are homometric, but differ from thehomometric 5 30;Id,e structures in the fourth and highercoordination shells. The h.c.p. 602;la,b structures(Figure 9) differ from the fourth coordination onward.The c.c.p. 2212;2c structure is different from the homo-metric 2212;2a,b structures (Figure 15) in the sixth andhigher coordination shells. The crystal structures a, b,c, . . . with identical positions on the structure maps(Figures 10 and 12) are stabilized by about the sameamount of lattice energy and are sometimes coexisting, ase.g. the 008;5b,e,f V6C5 stacking variants (Section 14).

There are also homometric structures of close-packedmetal atoms with a different stacking sequence butidentical T1 values (Mardix, 1990), e.g.

hcchchcc h h c h c h hand hcchchchhcchhhc

which are two different structures containing the sameunit cell with 15 layers and the same percentage (53.3%) ofh layers in the same space group P3ml. Other structuresof Table 1, e.g. 9a and 9c, or 9d and 9e, have identical T1

for /= 1-8 but differ in T9 and higher coordination shells.

10. Homologous Series of Structures

A second type of special structures exhibits identical a,values but at different composition y/x. These structureshave identical or closely related unit cells and formhomologous series of structures that are filled upsuccessively by M atoms until T™* is reached (Table 7).Each M and each N atom of these structures must havethe same set of numbers T1T2T3, respectively.The structure with the lowest M content y/x = r = k

Figure 16. Examples of homologous crystal structures with identical a, values at different concentrations y/x: M atoms at z = 0(•) and z = 0.5 (o). Positions of N atoms (•) at z = Q for pattern recognition

Table 7. Self-coordination numbers of some homologous serieswith different r and k values (see text)

r

222333344444

k

210321043210

T1 T2 T^y/x

c.c.p.

2212;2aMoPt2,b7418;0.5a Pt2Mo,b12 6 24;0 Cu060;3 AuCu3468;1 CuAu8616;0.33 Cu3Au12 6 24;0 Cu028;4 MoNi433 12;1.56416;0.679520;0.25 Ni4Mo12 6 24;0 Cu

h.c.p.

220;2a,b741;0.5a,b12 62;0Mg062;3 SnNi3462;1 AuCd862;0.33 Ni3Sn12 6 2;0 Mg

(TfT$T?,r) is filled up with M atoms in steps ofk = 0, 1, . . . , r (Hauck et al., 1988a):

T1(Ic) = 77ax - (77 a x - Tf)Mr

y/x = M(r+\-k)

The homologous series of structures also contain struc-tures with M and N atoms interchanged, as e.g. AuCu3

and Cu3Au (Table 7). There are three homologousseries for c.c.p. and two series for h.c.p. with r=2, 3, and4, respectively (Table 7). The r = 3 series for the c.c.p.060; 3 AuCu3 and 468;1 CuAu structures correspondsto the h.c.p. 062;3 SnNi3 and 462;1 AuCd structuresat identical Ot1, a2 values (Figures 10 and 12). The c.c.p.2212;2 structures (Figure 16) can be compared with theh.c.p. 220;2 (Figure 15) or 222;2 TaPt2 (Hauck et al.,1988a) structures. The pairs of closely related c.c.p.structures 048;3 TiAl3 and 4416;1 UPb (Figure 14)and the corresponding h.c.p. 042;3 TiCu3 and 442; 1structures (Figure 11) are not homologous because ofdifferent a2 values (Figures 10 and 12). The TiPt8 andV4Zn5 structures with different a, values even crystallizein the same space group at different y/x and can beconsidered as one structure type (Villars and Calvert,1986), if the V4Zn5 structure is written as Zn(V4Zn4).

11. Symmetry of Ordered Phases

The metal atoms of the h.c.p. and c.c.p. lattices havea single environment with a high point symmetry forthe metal atoms (Hahn, 1983; Villars and Calvert, 1986):

4 - 2— 3 — for the Cu atoms with c.c.p. structurem m

6m2 for the Mg atoms with h.c.p. structure

The site symmetry is reduced by distortion (d), e.g. ofthe Cu lattice, or by different stacking (s), as e.g. the(ch)2 La structure, or by formation of ordered struc-tures (o) (Figure 17). Structures with a single environ-ment of all M and N atoms, as e.g. AuCu3 or CuAu,are usually highly symmetric. Structures containing Matoms with different environments are built fromdifferent structural subunits, as e.g. ZrAl3, CuAu II orZrGa2 (Figure 14). The symmetry of these structures isusually lower because of the different symmetry elementsof the structural subunits. The populations of the dif-ferent structure types obey Pauling's rule of parsimony:the number of essentially different kinds of constituentsin a crystal tends to be small (Pauling, 1929). Structuretypes with a single environment of metal atoms M andN (M' = 2 in Table 3) with high symmetry, as in thecase of the h.c.p. alloys with AuCd or SnNi3 structureor the c.c.p. alloys with CuAu, AuCu3, MoNi4 orMoPt2 structure (Table 4), are observed frequently,whereas structures containing metal atoms with differentenvironments, as e.g. CuAu II, ZrGa2 or Mo3Al8, arerare. The fact that highly symmetric structures can onlybe constructed from M and N atoms with a singleenvironment shows that the frequent observation of highsymmetry (Laves, 1967) is a consequence of Pauling'srule for small numbers of constituents.

In most cases, the experimentally observed symmetryis identical with the symmetry determined for the undis-torted lattice. This can be used to find a structural modelof the ordered structure from electron diffraction patterns.Alloys with small single-crystal domains of orderedstructures can be investigated by electron diffraction todetermine the unit cell and the symmetry from thediffraction pattern. These values can be compared withthe list of theoretical structures to calculate the powderpattern for X-ray or neutron diffraction.

The symmetry of a few ordered alloys, as e.g. NaHg,CdAu3 II, SiU3 or SrPb3 (Table 4), is further decreasedowing to distortion similar to that in tetragonal In orrhombohedral Hg compared to cubic Cu (Table 1).

12. Ising Model

Crystal structures of ordered alloys MxN,, have beencalculated for central pairwise interactions V1 (/= 1, 2,3, . . .) between nearest-, next-nearest-, third-nearest-neighbor, etc., metal atoms within the Ising model (see

Order of group

Figure 17. Crystallographic point-group symmetry of M atomsin different ordered alloys MJ^,. The order of the pointsymmetry of M atoms is reduced by ordering (o), by stacking(s), e.g. of La atoms in chch layer sequence, or by distortion (d)

Chapter 2 by Turchi in this volume). The resultsobtained for the single tetragonal layer (Kaburagi, 1978),the single hexagonal layer (Kudo and Katsura, 1976;Hiraga and Hirabayashi, 1977), h.c.p. (Kudo andKatsura, 1976) and c.c.p. (Kanamori and Kakehashi,1977) alloys can be compared with the structures in thestructure maps. The present notation T1 T2 T3;y/x forthe self-coordination numbers T1 and concentrationy/x is similar to the notation of the structures used byKanamori and Kakehashi (1977). The interaction para-meters F1 and F2 between nearest and next-nearestneighbors were varied in the Ising model calculationsand the structures obtained by different proceduresplotted in a K1-K2 coordinate system (de Novion andLandesman, 1985). These plots can be compared withthe Otx-Oi2 structural maps of Figures 7, 8, 10, and 12.The parameters a, seem to be almost proportional to V1.Only simple structures are obtained by variation of the V1

and V2 interaction parameters. The complex structureswith different environments of M atoms are only foundby the additional variation of F3, F4, etc. (Ducastelle,

1991). Structures at different positions on the borderlineof the structure map, e.g. CuAu II compared to CuAu,are stabilized by different F3, F4, etc., which cause aslightly different F2 and a2 (Figure 12).

The interaction parameters F, = 0 for a statistical(random) distribution of M and N atoms correspondwith (X1 = O. The a, values are positive for attractive inter-actions between M atoms and negative for repulsiveinteractions. The structures can be classified within fourfields by consideration of F1, F2 only: (I) F1, F2 <0;(II) F1 <0 , F2 >0; (III) F1 >0 , F2 <0; and (IV) F1, F2

> 0. These areas are subdivided by the diagonals intoIa, Ib, etc. (Figure 18). Most experimental structuresare in (I) or (II), with a repulsive interaction F1

between nearest-neighbor M atoms. Structures in Iliaand IV, with attractive interactions F1 between nearest-neighbor M atoms, already contain multiple layers ofM and N atoms as a first step toward segregation.Structures with covalent bonding between M atoms arein Ia and IHb. The structures of Mb contain clusters,e.g. sheets or rows of M atoms.

povalent bondingClusters

Ionic bonding

Random


Coulomb repulsion

Figure 18. The different areas of the a,, ct2 structure map of close-packed alloys as derived from the Ising model

The M atoms of structures in lib are as far apart aspossible, which can be correlated with repulsive inter-actions V1 between nearest M atoms, e.g. Coulombrepulsion. These structures have the highest Madelungfactors (Hauck et al.9 1988a). The crystal structures withcovalent and ionic bonding are separated by the lineOtx = OL2. The M atoms of ionic compounds with low Mcontent are expected at T1 = T2 = O because of therepulsive interactions. The relation OL1 = Ot2, however, isalso valid for covalent compounds with the relationT1ZT2= 7^3V 7^**, with preferential occupation of theinner coordination shells T1 and T2. The Ct1 = a2 line ofdilute alloys intersects the borderline 4a j + ot2 + 1 = 0 ofthe h.c.p. and c.c.p. structure maps at the concentrationy/x= 5. Therefore, ionic and covalent compounds withy/x^5 and T1 = T2 = O should be on the line Cx1 = Cx2.Ordered ionic compounds with y/x < 5 are expectedwith increased T2, ordered covalent compounds withincreased T1.

The structures with a maximum interaction are at theouter borderlines. Structures with composition y/x= 1,2, and 5 are on the borderline 4(X1 + cx2 + 1 = 0 in (I,Ha); structures with y/x= 1 and 3 are on 3Cx1 + 1 = 0 in(lib); and structures with all compositions are on2otx = ct2+\ in (III, IVb). Structures with differentcompositions are in the field, e.g. structures withy/x = 2 are on Aa1 + 1 = 0 in (II) (Figure 12) because of

geometric restrictions of the close-packed metals.Structures with y/x =2 are less frequent than structureswith y/x= 1 or 3 (Table 4). They can be considered asa combination of structural subunits of compositiony/x= 1 and 3, as was outlined before. Other structuresinside the Cx1-(X2 field like c.c.p. 6212;la,b have thesame composition as structures at the borderline, e.g.60 12;la,b, 52 14;1 or 8212;1 (Figure 12). These structuresare supposed to be less favorable than structures at theborderline with maximum interactions between Matoms.

13. Occupation of Octahedral or TetrahedralInterstices

Interstitial atoms I = H, C, N, O, . . . can occupytetrahedral or octahedral interstices of the close-packedmetal structures. There are one octahedral 1° and twotetrahedral F atoms per metal atom M for the com-position MI0I2̂ at a complete occupation (Figure 19).Small spheres with radii less than 22.5% or 41.497b ofthe metal atom radii can be inserted in the voids betweenfour or six metal atoms, respectively. The 140Io densityof the metal atom sphere packing is increased atcomplete occupation of all the tetrahedral and octa-hedral interstices by IJ0Zo and 5.3%, respectively, to a

Figure 19. Position of interstitial atoms 1° and I1 in octahedral and tetrahedral sites of the ABC c.c.p. or AB h.c.p. M lattice.Only 1/2 or 1/3 (for 6-Ni3C) of the octahedral sites 1° at identical positions in c direction of h.c.p. M lattice are occupied atz/y = 2 or 3, respectively

total density of 81%. A maximum radius of 59% ofthe metal atom radii is considered for 1° atoms byHagg's rule for interstitial alloys with increased metalatom distances (Toth, 1971).

The octahedral interstitial atoms 1° of the c.c.p.metal structure are at successive positions ABC, similarto the metal atoms, but shifted by c/2 (Figure 19). Thetetrahedral interstitial atoms V9 with the twofold con-centration of the I0 atoms, form a repeated ABCABCsequence with distance c/6 between layers of interstitialatoms.

The octahedral atoms 1° of the h.c.p. metal structurewith an AB sequence of metal layers are all on Clayers with distance c/2 between two layers (Figure 19).

The tetrahedral sites are at A and B positions with thesequence BAAB and distance c/4 between layers ofinterstitial atoms I1. The tetrahedral atoms I1 close toa metal atom are in positions analogous to those foroctahedral sites. The h.c.p. metal atom at a B position,for example, contains two adjacent layers of I1 at Apositions and two layers 1° at C positions, whereas theV and 1° atoms close to a B atom of the c.c.p. M latticeare at the different A and C positions in each case.

The spatial distributions of the six octahedral sites1-6 and the eight tetrahedral sites a-h close to ac.c.p. and h.c.p. metal atom are shown in Figure 20.The nearest-neighbor V atoms of the h.c.p. structure,e.g. at c and g positions, are only Jl/6 d apart compared

Figure 20. Positions 1-6 of octahedral and a-h of tetrahedral sites in the neighborhood of a c.c.p. or h.c.p, M atom. Someneighbors of the self-coordination numbers T°. and T) of octahedral and tetrahedral interstices are shown

Coo

rdin

atio

n nu

mbe

r

Lattice constant, <7c(nm)

Figure 21. Coordination of hydrogen atoms in c.c.p. M structures with cubic lattice constants ac

with JT/2d in c.c.p. (Figure 20) (d= diameter of Matoms). The shortest distance between the I0 atoms ofthe h.c.p. structure, e.g. at positions 1 and 4, is j2/3dcompared to d in the c.c.p. case. The second h.c.p.coordination shells T\ and T\ are at the same distanceas the first c.c.p. coordination shells T\ and T°.

The small hydrogen atoms can occupy both tetra-hedral and octahedral interstices (Figure 21). Thetetrahedral sites are occupied completely, e.g. in VH2,and partially, e.g. in ZrH1 6. The metal lattice of thesehydrides is c.c.p. instead of h.c.p. (M = Ti, Zr, Hf, rareearths) or b.c.c. (M = V, Nb) for the pure metals. Thepreference for a c.c.p. instead of a h.c.p. metal latticecan be explained by the short distance between nearesttetrahedral sites in the h.c.p. structure (Hauck, 1983).The occupation of octahedral sites in MH with c.c.p.of the metal lattice (where M = Li, Na) instead of ah.c.p. structure is also preferred because of the largerdistance between H atoms. Both sites are occupied inMPI2 = MH3 with c.c.p. of the big rare-earth metalsM = La to Nd, Yb and with h.c.p. of the small rare-earth

metals M = Y, Nd, Sm, and Gd to Tm, Lu. NdH3 isc.c.p. at r>350 0C and h.c.p. at lower temperatures.The short distances between the nearest interstitial sitesof the h.c.p. lattice are increased by a hexagonaldistortion from c/a = 1.63 to ~1.80 (Villars and Calvert,1986) (see also Chapter 21 by Schlapbach et al. inVolume 2).

The preference of H atoms for tetrahedral oroctahedral interstices has been explained by the differentsize of the interstitial sites (Westlake, 1983) or thedifferent mechanisms of MH bonding (Hauck, 1983).The first type of model considers a minimum holesize of 40 pm (or a critical metal atom radius of139 pm) and a minimum distance of 210 pm be-tween H atoms to be necessary. The larger octahedralsite should be occupied in metals with small atomicradii, the smaller tetrahedral site in metals with largeatomic radii.

The second type of model is based on a chargetransfer between H and metal atoms with differentelectronegativities 0*:

Met

al a

tom

rad

ius,

r (n

m)

Tetrahedral sites

Octahedral sites

Electronegativity,<£* (V)

Figure 22. Hydrogen occupancy of octahedral sites in metals with small radii and high electronegativity, otherwise tetrahedral sites

M5-H6+ for metals with high <£*,e.g. M = Cr, Mn, Fe, Co, Ni, Pd

M6+H6- for metals with low </>*,e.g. M = Li, Na, Ca, Sc, Y, Ti

The H5+ atoms prefer the octahedral site with abonding to the transition-metal d electrons. The H5~occupies octahedral interstices of metals with no delectrons, otherwise tetrahedral interstices. Both sitescan be occupied in vanadium depending on temperature,hydrogen concentration, and isotope (Hauck, 1983).

The two models give the same prediction for mostmetals because of the correlation between small metalatom radii and high electronegativity (Figure 22). Thecritical radius of metal atoms, 139 pm, corresponds withthe cubic lattice constant of 393 pm and can explain thevariation from octahedral coordination in PdD0 7 andLiH to the tetrahedral coordination in VH2, etc.(Figure 21). Octahedral sites, however, are also occupiedin NaH, KH, RbH or CsH, which have larger latticeconstants. In rare-earth hydrides the smaller tetrahedralsites are occupied first, while octahedral sites areoccupied at H/M ^1.95. This behavior can be explainedby the second type of model with H atoms as a sensitiveprobe for d electrons. The d electrons of the rare-earth

metals are depleted with increasing H content because ofthe charge transfer to H6~. The loss of d electronsgives rise to a metal-semiconductor transition, differentmagnetic properties, and smaller lattice constants.

The other interstitial atoms I = C, N, O, . . . are toobig for the occupation of tetrahedral sites. They occupyoctahedral sites in the close-packed M structures (seeTables 8 to 10) with composition M2I ,̂ 0.5 ^yZ z^l.Most compounds are non-stoichiometric. Orderedphases with yZz = 0.5, 0.63, 0.75, 0.83, and 0.88 occurat low temperatures (Toth, 1971).

14. Ordered Cubic Close-Packed Interstitial Alloys

The interstitial atoms I = H, C, N, . . . can occupy theoctahedral sites of the c.c.p. M structure at the com-position M7Iy with 0.5 ^y Zz^ 1. The remaining (z-y)octahedral sites are empty. The larger number of metalatoms and interstitial atoms can be neglected in adescription of the ordered structures, if the positionsof the smaller number of vacant sites • are consideredin the formula M 2 D^, x+y = z. The positions of theM and I atoms must only be considered for the effectof lattice distortion caused by vacancy ordering and for

Table 8. Cubic close-packed interstitial alloys M2DxI^ with I = H, C, N and vacancies • at octahedral sites. Some antitypestructures uJAy\v I = O, S, Se, F, Cl, I, with M atoms at octahedral interstices of close-packed I atoms are also included.Number Af of M, I, and • with different sets of T1 values (all minority components have only one set of T1 values), andmaximum population values are also shown. For tetragonal planes the same holds as in Table 4

TJ1T^yZx

c.c.p.

12 6 24;(1)

6012;la6012;lb4416;1

408;1.67606;23010;2a3010;2b3010;2c2212;2a204;3208;3048;306O;3107;4028;4OO8;5aOO8;5b008;5eOO8;5f006;7

tetr. plane

444;(1)

204;l204;l044;l

202;2102;2103;2102;2202;2204;l004;3044;l044;l000;4000;4002;5000;5002;5

000;7

M

2

333

4355444444,34355555

Prototype (symbols)

NaCl (CF8/B1)NiO (hR2)CoO, GeSb (tI4)Gd2C, a-NaFeO2 (hR3,4)Ti2C (cF48)Pd2D, Ti2N (til2)

Ti8C5 (hR13)In2S3 (hP5)TiCl3, Li2SnO3 (mC32,48)AlCl3 (mC16)Sc067S (oF80)D Ti2 n O2 (oI8)Tm075Se, OsCl4 (oC 14,10)HfI4 (mC40)Nb4N3, SnF4 (tI14,10)D Nb3 D O3, Nb4C3 (cP6,7)UCl5 (mP24)UF5, PdD08, Ti08O (tI12,18)Lu083S (0F88)V6C5 III (mC44)V6C5I, Sc083S (mC22)V6C5 II, Li5ReO6 (hP33,36)V8C7 (cP60)

Space group

Fm3mR3mI4/mmmR3mFd3m14,/amd

R3mP3mlC2/cC2/mFdddImmmCmmmC2/cI4/mmmPm3mP2,/cI4/mFdddC2/cC2/mP3,12P4332

Population

67744542

116

101121221211221

the determination of the space group. The sublatticeof vacancies plus interstitial atoms D xly can be com-pared with the lattice of ordered MxN7 alloys (Haucket al., 1988b). The vacancies and I atoms exhibitthe same ABC layer sequence as the metal atoms ofc.c.p. M (Figure 19). Therefore, the distribution ofvacancies of the n xly sublattice can be described bythe same coordination numbers T1, T2, T3 and con-centration y/x as for ordered c.c.p. alloys MxN^ (Table8). The carbides Ti2C, Gd2C, Ti8C5, V6C5, and V8C7

are located in area Ia of the structural map, in-dicating covalent-type bonding, while the nitridesTi2N, Nb4N3 and the hydrides Pd2H, PdD 0 8 arelocated at low a{, indicating Coulomb-type inter-actions (Figure 23).

Two other types of interstitial alloys can be comparedif the composition is formulated in a similar way to thatused for the Mz • xly compounds with the conventionthat z>y >x: DxMÎ2 with I = O, S, Se, F, Cl, I as in

Ti0 8O, LiIc83S, Tm0 75Se, SnF4, AlCl3 or HfI4 (Table 8);and n ^M7 • xOy as in D Ti2 n O2 or D Nb3 D O3 withvacancies in both sublattices.

The I = O, S, Se, F, Cl or I atoms are bigger thanthe M atoms. Therefore, the M atoms fill part of theoctahedral interstices of the close-packed I lattice.The structures are the antitype of M2DxI^ (Lima-de-Faria et al., 1990). The sublattices of M andI = O atoms of NbO and TiO are only partly filled. Thevacancies of the M and the I sublattices form the samestructure: 2212;2 for TiO and 060;3 for NbO.

Most interstitial alloys are metallic; some aresuperconducting, like PdD, TiC, NbN, TiO, NbOor Sc083S (Toth, 1971; Moodenbaugh et al., 1978).The halides are non-metallic. Other non-metalliccompounds that can be compared with the antitypestructure DxM7I2 are the ternary oxides MxN7I2, e.g.Li5ReO6, Li2SnO3 or Qf-NaFeO2 (Table 8), with Re,Sn or Fe atoms at the vacancy positions.

Figure 23. a,, a2 structure map of c.c.p. interstitial alloys M2DxI^. Some halides aJsAylz with metal atoms at octahedralinterstices of c.c.p. I lattice (antitype structure) are also included

Ti 2 C, Gd2C, CdCl2. NaFeO2 ( fac ia l )

Pauling's electrovalence ru le

TiO (trans+cis)

Segrega t ion

Pd2H, Ti2N (meridional)

T iC l 3 , A l C l 3 , Sc0-67S, Li2SnO3 ic/s)

The compositional parameters x, y, and z = x+y ofthe non-metallic compounds cannot vary independently.The electroneutrality rule requires compensation of thenegative charge of the oxygen or halide atoms. Thecharge g(l) = - 2 or - 1, respectively, of these anionsshould be compensated by the electro valence #,/«, ofthe P1 neighboring metal atoms with coordinationnumber nt and charge q{ as defined by Pauling (1929,1945):

The I = O atoms of the N^MxI2 compounds, e.g.Li5ReO6, Li2SnO3, and Cx-NaFeO2, are coordinated bypM = 6x/z = 1, 2 or 3, respectively, M atoms (M = Re7+ ,Sn4+, and Fe3+) and/?N = 6y/z = 5, 4 or 3, respectively,N atoms (N = Li+ or Na+ atoms), which compensatethe - 2 charge of the oxygen atoms according toPauling's electrovalence rule. The structures of thesecompounds correspond with the V6C5, TiCl3, andGd2C structures (Table 8) and are at the borderline ofthe (I, Ha) field of the structure map (Figure 23) withthe correlations 4Cx1 + a2 + 1 = O and Qf1 + a3 = O betweenOf1, OL1, and a3 values (Hauck et al., 1989).

The metal atoms of the interstitial alloys M2I^, I = H,C, N, are surrounded by p = 6y/z I atoms: /7 = 3 aty/z = 0.5, e.g. in TiC05 or TiN05; /? = 4 at y/z = 0.61;and p = 5 at y/z = 0.83, e.g. in V6C5. The three I atomsof MI0 5 are in the neighboring positions 1, 2, 3

"Some atoms of DTi2DO2, etc., have a cis configuration.

(Figure 20) of facial configuration, as in TiC0 5 orGdC05 , or further apart at 1, 2, 5 in meridionalconfiguration, as in PdH0 5 or TiN0 5 (Table 9) (Hauck,1981). The self-coordination number T1 of the 6012;IbTiC05 structure is increased compared with the 4416;1TiN0 5 structure, because of the close distance betweenI atoms in facial configuration. The M atoms of thefacial configuration can be shifted to the C atoms to

Figure 24. Self-coordination numbers Tx T2T3 of interstitial atoms in covalent (upper row) or ionic (lower row) compounds.An interstitial atom is placed in the middle of eight face-centered cubic cells with lattice constant ac. The numbers z give theposition of neighboring interstitial atoms at height zac/2. Nearest neighbors are connected by bold lines. The broken lines indicatethe unit cells. Ti2C and V8C7 are cubic with lattice constant ao = 2ac

Table 9. Interstitial alloys M., • J.y with single configurationof tetrahedral or octahedral sites (Figure 20)

y/z

c.c.p. 0.50.50.670.67

0.750.83122.53

h.c.p. 0.50.5

0.670.670.8313

M4I,

Ti2C, Gd2CTi2N, Pd2HLi2SnO3, AlCl3

D Ti2 D O2, Li2ZrO3,Na2PtO3

Fe3O4

V6C5, Li5ReO6

TiCLaH2

LaH2-5

LaH3, CeD3

Ta2CCo2C, S-Nb2C,

f,e-Fe2NCr2S3

CK-Al2O3

Cr5S66'-NbNHoD3

Configuration

1-3 (facial)1, 2, 5 (meridional)1-4 (cis)1, 2, 5, 6 (trans)"

1-3, h1-51-6a-h1, 2, 5, a-h1-6, a-h

1-32-4

1-42-51-51-61-6, a-h

decrease the M-C bonds by covalent bonding. Areduction of all Ti-N bonds with nitrogen atoms onmeridional configuration 1, 2, 5 is not possible. The Natoms of the 4416;1 TiN05 structure, however, arefurther apart and are stabilized by a 1.9% increasedMadelung factor for the Coulomb interactions.

The other compounds with y/zÔ.5, 0.67 or 0.83,as e.g. Ti8C5, V8C7 or PdD08, contain two differentconfigurations of M atoms (Hauck et al., 1988b). InTi8C5, for example, with/? = 6>/z = 3.75, two Ti atomshave three C atoms, and six Ti atoms four C atoms inthe same configuration as the compounds Ti2C andAlCl3 (Table 9). Ti8C5 is located in the vicinity of Ti2Cand AlCl3 of the structure map (Figure 23). Thestructures close to the borderline with only two differentconfigurations obey Pauling's rule of parsimony(Pauling, 1929). The structures at or close to theborderline 3^1 + 1=0, e.g. 060;3 and 468;1, which areobserved as ordered alloys, are unfavorable for inter-stitial compounds because of two or more configurations.The Nb atoms, for example, of 060;3 Nb4C3 aresurrounded either by four or six C atoms. The numberof configurations, however, is reduced to a singleconfiguration in NbO ( • Nb3 a O3) and in ternaryinterstitial alloys AlFe3C and FeNiN (Section 18).

The T1 and T2 values in the T1 T2 T3;y/x notation ofinterstitial compounds describe the connection of octa-hedra by corners and edges, respectively. The 107;4

UCl5 structure contains UCl6 octahedra connected by acorner to (UC15)2, while the UF6 octahedra of the 028;4UF5 structure are connected by two edges in a one-dimensional row. The CTi6 octahedra of the 6012;IbTi2C structure are linked by all corners, the NTi6octahedra of 4416; 1 TiN05 structure by four cornersand four edges. The vacancies of 008;5 V6C5 • or theRe atoms of 008;5 Li5ReO6 form isolated octahedra.

The covalent and ionic compounds can be comparedby two series of interstitial compounds MzaJ.y, theTi2C series with covalent bonding and the Ti2N serieswith ionic bonding (Figure 24).

The vacancies of the 4416;1 Ti2N structure are filledup with interstitial atoms in such a way that they areas far apart as possible. These structures with 50, 33,25, 20, and 12.5% vacancies, shown in the lower rowof Figure 24, have a minimum of T1 coordination. Thevacancies of the 6012;Ib Ti2C structure in the upperstructural series are successively occupied at increasedT1 and T2 = O. The increased Tx coordination numbersof this series allow covalent bonding to interstitial atomsin neighboring facial or cis configuration. The Madelungfactors of these structures are decreased by 1-3% (Figure25). This difference of the lattice energy must be com-pensated by covalent bonding to stabilize the structure.The n V6 octahedra of V6C5 • or ReO6 octahedra ofLi5ReO6 are isolated at the composition y/x^5. Thevacancies or Re7+ atoms are as far apart as possiblefor a maximum Coulomb energy. Covalent bonding isalso favorable at this composition, because of thepossibility of decreasing all Re-O bonds of ReO6 or in-creasing all a -V distances of the • V6 octahedra forcovalent V-C bonding. Therefore, the 008;5 structure isfavorable for both systems. This structure lies at theintersection of the Aa1 + a2 + 1 = O boundary line of thestructure map (Figure 23) and the a{ = a2 line for dilutedalloys, as e.g. V8C7 with T1 = T2 = O. The OO8;5a-istructures are different stacking variants, which canoccur in the same sample, like V6C5 I, II, and III, oras disordered structures. Some structures like 008;5e,fare homometric, analogous to 6012;la,b Ti2C andGd2C (see Section 9 and Figure 15) (Parthe and Yvon,1970; Hauck et al., 1988b).

15. Ordered Hexagonal Close-PackedInterstitial Alloys

The octahedral interstitial sites of the h.c.p. metals areall on C positions between each layer of metal atoms(Figure 19). The interstitial sites form a simple hexagonallattice with a distance c/2 = Jl/3d between neighboring

y/x

Figure 25. Madelung factors MF7JyTx versus concentrationy/x for the undistorted covalent and ionic compounds ofFigure 24

Covalent

Ionic

Figure 26. ot2i a5 structure map of h.c.p. Mza^Ly interstitialalloys, corresponding to a,, oe2 of hexagonal plane (Figure 8)

layers. The distance between two layers is only 82% ofthe shortest M-M distance (Table 2) in undistortedstructures. This can explain why interstitial atompositions at identical projection sites are not occupiedin adjacent layers because of the size of interstitial atomsor the Coulomb repulsion between these atoms. There-fore, the ordered distribution of I atoms in h.c.p. metalscan be treated as single hexagonal layers (Hiraga andHirabayashi, 1977). The coordinates of the structuremap (Figure 26) are identical with the Ot2-Ot5

coordinates of the three-dimensional case (Table 2).The formulas of these interstitial compounds are

identical to those for c.c.p. interstitial alloys:

Mz a Jiy, x+y=zt for the small interstitial atoms I = N,C

DxMÎ2, x+y = z, the antitype for bigger atoms I = O,I

Most ordered compounds have a composition z/y = 2because of the alternating empty and occupied sites inthe c direction (Figure 19). The individual hexagonallayers are of different composition as outlined in Table10. Frequently, a phase transition to the W2C structureis observed at high temperatures with a disordereddistribution of C atoms (Toth, 1971). The e-Ni3C ande-Fe3N structures contain 060;2 hexagonal layers (Table10) with N or C atoms in different positions. Thesestructures were observed in ZrOx at A:=0.30 and 0.34,

respectively, and more complex stacking variants atintermediate compositions (Hirabayashi et al.9 1974).Complete occupation of all octahedral sites is obtainedin 6'-NbN and the NiAs antitype structure (Table 10)at the distortion of c/a ~1.9 (Villars and Calvert, 1986).

Recently new compounds Na96In97Z2, Z = Ni, Pd orPt with fullerene-like cages were obtained in the NiAsstructure (Sevov and Corbett, 1993). In74 fullerenes,which are connected by pentagonal faces, form a h.c.p.structure. Distinct M60( = In48Na12) polyhedra occupythe octahedral interstices. The In74 and M60 cages arecentered by In10Z clusters.

16. Complex Close-Packed Interstitial Alloys

The octahedral positions in h.c.p. alloys with stackingsymbol hh are identical, e.g. at CC (Figure 19), anddifferent for c.c.p. alloys with stacking symbol ccc, e.g.at ABC. The octahedral positions in complex close-packed alloys are identical in the h layers and differentin each c layer. The sequence CCBB, for example, isobtained for the four-layer structure chch and CABBBfor the five-layer structure ccchh. Octahedral sites ofneighboring layers are vacant ( • ) at identical projectionsites, e.g.

and

because of the short distance between these octahedralsites. Therefore, the concentration of interstitial atomsis reduced (Parthe and Yvon, 1970).

The neighborhoods of M atoms in c and h layerscorrespond to c.c.p. and h.c.p. in Figure 20. They can

Hex. planeT1T2T3IyZx

666;(1)

666;(1)666;(1)666;(1)422;1242;1

226;1226;1

060;2060;2060;2disordered

Prototype(symbols)

5'-NbN (hP4/BS1)

NiAs (hP4/B8,)Cx-Ta2C (hP3)CdI2 (hP3/C6)£-Nb2C (oP12)T-Fe2N (oP12)

Co2C (oP6)CaCl2 (oP6)

e-Fe2N (hP9)C-Fe3N (hP8)e-Ni3C (hR8)W2C (hP4/L'3)

Spacegroup

P63/mmc

P63/mmcP3mlP3mlPnmaPbcn

PnnmPnnm

P31mP6322R3cP63/mmc

Population

1

1544

5214

21

9429

Table 10. Hexagonal close-packed interstitial alloys withdifferent occupation of hexagonal planes of interstitial atoms,and maximum population values

be converted to each other by rotation of the lower halfby 60°. The complex close packing can be influencedby small variations of concentration or temperature insome compounds (Pearson, 1967, 1972). Some inter-stitial alloys occur in both ordered and disorderedforms in complex close-packed M structures, as e.g.MoC and Mo3C2 in (hcc)2. The structures of Sn4P3 andTa3MnN4 are observed in (ch)2, Ta2VC2 and Tb2C in(chh)3, Ti3SiC2 in (chhh)2, and T-V4C3 in (hhcc)3

(Pearson, 1967, 1972; Parthe and Yvon, 1970).

17. Disordered Alloys

Many alloys form extensive substitutional solid solutions,in particular alloys of metals with similar radii andelectronegativity (Figure 2). Some alloys, as e.g. AuCuand AuCu3, are disordered at high temperatures and

become ordered at lower temperatures. The same appliesto interstitial alloys, as e.g. V6C5 or Ti2C, which havea disordered distribution of C atoms at high tem-peratures but an ordering of the C atoms (and vacancies)at lower temperatures (see Chapter 23 by de Novion inthis volume). The X-ray or neutron diffraction patternsexhibit diffuse intensity, which is concentrated on linesor surfaces in reciprocal space. The diffuse,intensity canbe analyzed for the short-range order parameters a,. Theexperimental values (Schweika, 1985; de Novion andLandesman, 1985) are usually between «, = 0 for a ran-dom distribution of metal or C atoms with no diffuseintensity and the values of the ordered alloys (Figure27), and should shift to the values of the ordered alloys asthe order-disorder transition temperature is approached.The relations 4^1 + a2 + 1 = 0 and ^1 + a3 = 0 were ob-tained for the al9 a2, a3 short-range order parametersof oxides like LiFeO2 and carbides from the shape of

Figure 27. Experimental short-range order parameters a,, a2 of disordered alloys MxN^ or MzoxCy carbides within the c.c.p.a,, OL2 structure map (Figures 12 and 23)


the diffuse intensity patterns in reciprocal space (Sauvageet aL, 1974; Ducastelle, 1991). These relations can beexplained by a disordered arrangement of carbon atomsin configurations allowed by Pauling's electrovalencerule (Table 9) or by the disordered T1 configurations6012;l, 3010;2, 2212;2, 008;5 or 4416;1 (Figure 24)(Hauck, 1985). There are many stacking variants ofthese configurations (Hauck et al., 1988b).

18. Ordered Ternary and Quaternary Compounds

Few ternary and quaternary ordered alloys with close-packed metal structures are known (Figure 28). Alloyswith the composition M^Nj,, X1 ^y, can be describedby the self-coordination numbers T1 of the minoritycomponents M^. There are three sets of T1 values forternary and six sets for quaternary alloys. The Ot19Ot1

values are identical for the 468;1 or 060;3 derivativestructures or in the same area of the structure map, ase.g. for D4Co5Ge7. The T1 values of all M .̂ can beaveraged according to the frequency of their occurrence.These values can be compared with the self-coordinationnumbers of alloys with different sets of T1 values,which was discussed in Section 2 for close-packed metalswith different stacking sequences. All metal atoms forn = 2 and 3 (h.c.p. and c.c.p. structures) have the sameenvironment (Af=I) (Table 1). The n = 4, 5, 6b, 8a,and 9a layer stackings of Table 1 contain Af = 2 metalatoms with different T1 values, the other structuresthree, four or five different sets of T1 values (Table 1).Therefore, the structures with two different T1 at theratio y/x can be formulated as pseudobinary alloys MN(n = 4, 8a), MN2 (/* = 6b, 9a), and MN4 (n = 5). Otherstacking sequences are pseudoternary, e.g. M1M2N3

Figure 28. Ternary and quaternary compounds with Tx T2T3\y/x of different components with composition y/x. Onlypositions of minority elements are plotted, full and opensymbols for elements at projection height z = 0 and 0.5,respectively, or as multiples of ao/4 in cubic Co5Ge7

(/? = 6a), M1M2N4 (n = 7), M1M2N5 (« = 8b), M1M2N2

(A7 = 8c,e,f), M1M2N4 (n = 9b), and M1M2N4 (n = 9c).The population of the different structure types decreaseswith increasing Af. Ordered binary and ternary alloys,however, have not been observed with these structures.The MN, MN2, and MN4 binary alloys, for example,favor other structures in the h.c.p. or c.c.p. structures(Tables 3 and 4). Most of the observed binary h.c.p.and c.c.p. MJN,, compounds have two sets of T1 values,which are identical at concentration y/x-1. The orderedc.c.p. interstitial alloys Mzu^y are ternary compoundssimilar to the isomorphous ternary oxides NaFeO2,Li2SnO3, etc., with consideration of the vacancypositions • and with three sets of T1 values for M, I,and n in most compounds (Table 8). The vacancies ofsome compounds can be filled, as e.g. in Ti2CH. Thebinary oxides TiO ( • Ti2 • O2) and NbO ( • Nb3 • O3) arepseudoquaternary with two identical sets of T1 (Table8). These compounds can be compared with FeNiN awith identical T1 values for the metal atoms, thenitrogen atom, and the vacancy (Figure 28).The different metal atoms of these structures have asingle configuration of interstitial atoms according toPauling's rule of parsimony (Pauling, 1929).

Other alloys forming solid solutions, like the binaryCrNi2 or the ternary CrNiFe2 5, are listed in the Custructure type (Villars and Calvert, 1986) with a singleenvironment of metal atoms, though the short-rangeorder parameters are different from at- O (Figure 27).Metals with similar radii and electronegativities can alsosubstitute for one component of an ordered binary alloy.Such ternary alloys have been investigated frequently tovary the structure of metals of binary alloys in small steps:

(i) The stacking sequences 7c and 14 were obtainedby variation of the Ti-Pt-Ni and Ba-Pb-Tlcompositions (Tables 1 and 6).

(ii) Structures with a different sequence of structuralsubunits were obtained, e.g. in (Zr3Al)(Al3Si5)(Schubert, 1964) with yvv' xuu' y' v' vx' u' u com-pared to ZrSi2 (Figure 14) with yvv'y'u'u.

(iii) Homologous structures with identical environmentof all atoms at different compositions (Table 7) canbe filled up with different atoms. The Pd and Auatoms, e.g. of PdAuCu2 (Figure 28), have the060;3 coordination of the AuCu3 structure. BothPd and Au atoms together and the two Cu atomstogether exhibit the 468; 1 configuration of theCuAu structure.

(iv) Some alloys, as e.g. RhZr3 (AuCu3 structure),become unstable on hydrogen loading and segregateor become amorphous (Bowman, 1988).

(v) Other structures, which are unstable as pure alloys,can be stabilized by interstitial atoms, e.g. AlFe3

with the AuCu3 structure in AlFe3C (Figure 28).Among the 139 known compounds with thisstructure (Villars and Calvert, 1986) there are manyalloys that are not stable without the presence ofinterstitial atoms. The atoms of the AlFe3Cstructure have the same positions as the atoms ofthe CaO3Ti perovskite structure with oxygen atomsat the Fe positions. The sublattice of the metalatoms Ca and Ti in CaTiO3, however, is body-centered with oxygen atoms at the octahedralinterstices.

(vi) The crystal structures of some alloys can bevaried by insertion of interstitial atoms. The c.c.p.ZrAl3 structure, e.g. of MnPd3 (Flanagan andCraft, 1992), changes to the AuCu3 structurein MnPd3H • 3, which is isomorphous withAlFe3Cn3 (Figure 28).

If, however, the properties of the three or morecomponents are too different or the concentrationof the substituting component is too high, thenthe alloys become unstable and favor structureswithout close packing. As we pointed out at thebeginning of this chapter, close packing can be expectedfor identical atoms or at least similar atoms withweak directional bonding and is not likely to be foundin a child's drawer containing a collection of differentlysized balls.

19. Notation

A, B, C positions of metal atoms in hexagonallayers

K, L positions of metal atoms in tetragonallayers

n number of hexagonal layersn = 2 layers in A,B positions of h.c.p. or h

hexagonal close-packed structuren = 3 layers in A,B,C positions of c.c.p. or

c cubic close-packed structured diameter of atoms = distance between

the nearest M atomsa = dt c=nj2/3d hexagonal lattice constants of struc-

tures with n layers/ fraction of h layers in complex close-

packed structuresM^Ny, r=y/x^l alloy composition with minority

component MTj self-coordination numbers for / = 1,

2, . . . coordination shell

CN coordination numberT*** maximum self-coordination number

in pure metalsTx T2 T3;y/x characterization of crystal structures

by the M atom environment, whereT1 is the number of /th nearest neigh-bors and y/x the ratio of thecomponents

ah - l ^ a / ^ 1 Cowley's short-range order para-meter, CL1T?**= T1-(Tf^-T1)Zn theCx1, Oi2 parameters are used for struc-tural maps

± V1 repulsive ( - ) or attractive (+) inter-actions between M atoms, e.g.Coulomb repulsion or covalentbonding

M number of M and N atoms withdifferent environments (= differentT1 values)

20. References

Beattie, H. J. Jr (1967). In Intermetallic Compounds (ed. J. H.Westbrook). Wiley, New York, p. 144.

Beck, P. A. (1969). In Advances in X-ray Analysis, Vol. 12(eds C. S. Barrett, J. B. Newkirk, and G. R. Mallett).Plenum, New York, p. 1.

Bowman, R. C. Jr (1988). In Hydrogen Storage Materials (ed.R. G. Barnes). Mater. Sci. Forum, 31, 197.

David, W. I. F., Ibberson, R. M., Matthewman, J. C ,Prassidec, K., Dennis, T. J. S., Hare, J. P., Kroto, H. W.,Taylor, R., and Walton, D. R. M. (1991). Nature, 353, 147.

de Boer, F. R., Boom, R., Mattens, W. C. M., Miedema,A. R., and Niessen, A. K. (1988). Cohesion in Metals:Transition Metal Alloys (Cohesion and Structure, Vol. \)(eds F. R. de Boer and D. G. Pettifor). North-Holland,Amsterdam,

de Novion, C. H., and Landesman, J. P. (1985). Pure Appl.Chem., 57, 1391.

Ducastelle, F. (1991). Order and Phase Stability in Alloys(Cohesion and Structure, VoI 3) (eds F. R. de Boer andD. G. Pettifor). North-Holland, Amsterdam.

Flanagan, T. B., Craft, A. P., Niki, Y., Baba, K., andSakamoto, Y. (1992). /. Alloys and Compounds, 184, 69.

Hahn, Th. (ed.) (1983). International Tables for Crystallography,Vol. A. Reidel, Dordrecht.

Hauck, J. (1981). /. Less-Common Met., 11, 99.Hauck, J. (1983). J. Less-Common Met., 94, 123.Hauck, J. (1985). /. Less-Common Met., 105, 283.Hauck, J., Henkel, D., and Mika, K. (1988a). Z. Phys., B71,

187.Hauck, J., Henkel, D., and Mika, K. (1988b). Z. Kristallogr.,

182, 297.

Hauck, J., Henkel, D., and Mika, K. (1989). Int. J. ModernPhys., B3, 1425.

Hirabayashi, M., Yamaguchi, S., Asano, H., and Hiraga, K.(1974). In Order-Disorder Transformations in Alloys (ed.H. Warlimont). Springer, Berlin, p. 266.

Hiraga, K., and Hirabayashi, M. (1977). J. Physique, 38, C7,224.

Ho, S.-M., and Douglas, B. E. (1968). /. Chem. Educ, 45,474.Kaburagi, M. (1978). J. Phys. Soc. Japan, 44, 54.Kanamori, J., and Kakehashi, Y. (1977). /. Physique, 38, C7,

274.Kudo, T., and Katsura, S. (1976). Prog. Theor. Phys., 56, 435.Laves, F. (1967). In Intermetallic Compounds (ed. J. H.

Westbrook). Wiley, New York, p. 129.Lima-de-Faria, J., and Figueiredo, M. O. (1969). Z.

Kristallogr., 130, 41, 54.Lima-de-Faria, J., Hellner, E., Liebau, F., Makovicky, E.,

and Parthe, E. (1990). Acta Crystallogr., A46, 1.MacLaren, J. M., Pendry, J. B., Rous, P. J., Saldin, P. V.,

Samorjai, G. A., van Hove, M. A., and Vredensky, D. D.(1987). Surface Crystallographic Information Service: AHandbook of Surface Structures. Reidel, Dordrecht.

Mardix, S. (1990). Acta Crystallogr., A46, 133.Max,N. (1992). Nature, 355, 115.Mika, K., and Hauck, J. (1990). Acta Crystallogr., A46

(Suppl.), C-269.Moodenbaugh, A. R., Johnston, D. C , Viswanathan, R.,

Shelton, R. N., DeLong, L. E., and Fertig, W. A. (1978).J. Low Temp. Phys., 33, 175.

Ogawa, S. (1974). In Order-Disorder Transformations inAlloys (ed. H. Warlimont). Springer, Berlin, p. 240.

Parthe, E., and Yvon, K. (1970). Acta Crystallogr., B26, 153.Patterson, A. L. (1944). Phys. Rev., 65, 195.Patterson, A. L. and Kasper, J. S. (1959, 1967). In International

Tables for X-Ray Crystallography, Vol. II (eds J. S. Kasperand K. Lonsdale). Kynoch Press, Birmingham, p. 342.

Pauling, L. (1929). / . Am. Chem. Soc, 51, 1010.Pauling, L. (1945, 1960). The Nature of the Chemical Bond.

Cornell University Press, Ithaca, NY.Pearson, W. B. (1967). A Handbook of Lattice Spacings and

Structures of Metals and Alloys, Vol. 2. Pergamon,London.

Pearson, W. B. (1972). The Crystal Chemistry and Physics ofMetals and Alloys. Wiley, New York.

Sauvage, M., Parthe, E., and Yelon, W. B. (1974). ActaCrystallogr., A30, 597.

Schubert, K. (1964). Kristallstrukturen zweikomponentigerPhasen {Crystal Structures of Binary Phases). Springer,Berlin.

Schubert, K. (1967). In Intermetallic Compounds (ed. J. H.Westbrook). Wiley, New York, p. 100.

Schweika, W. (1985). Nahordnung und Wechselwirkungen inder Legierung Ni089 Cr011. Jiil Report 2004, Forschungs-zentrum Julich.

Sevov, S. C , and Corbett, J. D. (1993). Science, 262, 880.Toth, L. E. (1971). Transition Metal Carbides and Nitrides.

Academic Press, New York.Villars, P., and Calvert, L. D. (1986). Pearson's Handbook

of Crystallographic Data for Intermetallic Phases, VoIs1-3. American Society for Metals, Metals Park, OH.

Villars, P., and Hulliger, F. (1987). J. Less-Common Met.,132, 289.

Villars, P., Mathis, K., and Hulliger, F. (1989). In TheStructures of Binary Compounds {Cohesion and Structure,Vol. 2) (eds F. R. de Boer and D. G. Pettifor). North-Holland, Amsterdam, p. 1.

Westlake, D. G. (1983). / . Less-Common Met., 91, 1.Wyckoff, R. W. G. (1964). Crystal Structures, VoIs 1-3.

Interscience, New York. (Reprinted 1982 by Krieger,Malabar.)

Zhao, J.-T., Gelato, L., and Parthe, E. (1991). ActaCrystallogr., C47, 479.

This chapter was originally published in 1995 as Chapter 12 in IntermetallicCompounds, Vol. 1: Principles, edited by J. H. Westbrook and R. L. Fleischer.

1. Introduction to the Definition of Symbols

There is a great variety of body-centered structures(I family) among intermetallic compounds, which allowsone to classify them into main types and subtypes, andwhich includes as well compounds where single pointsof the complex are replaced by coordination polyhedra.It was Hermann (1960) who developed a nomenclaturefor lattice complexes, which is useful in deriving all latticecomplexes in space groups by subgroup relations,saying: This terminology has proved satisfactory forbrief descriptions of space groups and crystal structuresand may be used for the systematic treatment of anumber of crystal-geometric problems which have beenraised by Niggli, Weissenberg, Laves and the author.The practical application to such problems is to be thesubject of further publications.' And Hermann, as theauthor of Strukturbericht volumes 1 and 2, knew whathe was talking about; but sadly his life was too shortto realize his ideas. Twelve years later we could publishthe NBS Monograph 134 Space Groups and LatticeComplexes (Fischer et al., 1973).

An important step had already been taken bypublishing the first Internationale Tabellen zurBestimmung von Kristallstrukturen (1935) edited byHermann in cooperation with Astbury, Bradley, Jones,Lonsdale, Mauguin, Wyckoff, and others. In the firsttables comparisons of different notations of crystalclass (point groups) and space groups have been givenby Schoenflies, Rinne et al. and Hermann-Mauguin(complete orthohexagonal), including the names ofthe general face forms by Schoenflies, Miers andGroth.

The category crystal class (point group*) was statedwith a scheme showing all subgroup relationships andthe names for the crystal forms and their special formsincluding limiting forms (Figure 1). The diagram givesa survey of the subgroup relations between variouscrystal classes. In this figure every connection line meansthat the crystal class at its lower end is a subgroup ofthat at the top. Double lines mean that the upper classcontains the lower as subgroups in two non-equivalentpositions. In the complete Hermann-Mauguinnomenclature, the subgroup relations are more clearlyrecognized than in the abbreviated forms. In the actualdescriptions, every crystal class is given its proper Millerindices (hkt) for special and general forms, and a letteras a short reference. The face symmetry belonging toeach form is also given and the representation in astereographic projection; representations of the crystallo-graphic point groups 222 = D2, 4m2 = D241, 312/m = D3d,and 43m = Td in the orthorhombic, tetragonal,rhombohedral, and cubic systems are given in Table 1.The general face forms are rhombic disphenoid or tetra-hedron, tetragonal scalenohedron, ditrigonal trapezo-hedron, and a hexakistetrahedron. In both cases D2d

and Td the special form is a tetrahedron, in the firstcase a tetragonal tetrahedron. The duals of the faceforms are the coordination polyhedra, for which asymbolism was developed by Donnay et al. (1964)(Table 2). In the chosen examples 4d 1, 8d 1, 12d 1, and

*A crystallographic point group is a group that maps a pointlattice onto itself. Consequently, rotations and rotoinversionsare restricted to_the well-known 32 crystal cases formed by 1,2, 3, 4, 6, and 1, 2( = m), 3, 4, 6.

Chapter 3

Body-Centered Cubic Derivative Structures

Erwin E, Hellner and Roland SchwarzInstitute for Mineralogy, Philipps University, Marburg, Germany


24d 1 are the general face forms, which characterize thecrystal classes; the names of the faces are used by Grothas the names of the crystal classes. The other parts withpoint symmetry . . 2, . . m, etc., are reduced, specialforms.

The term iattice complex' in three-dimensional spacewas introduced by Niggli (1919) in his book GeometrischeKristallographie des Diskontinuums: 4As alreadymentioned, the complex of equivalent points in a spacesystem is comparable in every way with the complex ofequivalent faces in a crystal class. In the normalcrystallography each face form has a special symbol.Names, as cube, trapezohedron, scalenohedron, pedium,pinacoid etc., are given to simplify the description. Wewill call an equivalent point complex just a (simple)lattice complex and try to find simplifying symbolisms,which are useful for special reasons . . . .'

Several attempts have been made by Niggli. In thefirst Internationale Tabellen (1935) Hermann has achapter on lattice complexes, either with no degree offreedom or with one degree (according to Niggli),invariant or univariant lattice complexes, tabulated andall the space groups in which they can be realized('occur'). In the present table two point positions areallocated to the same lattice complex if they can betransformed into one another by any rotations orchanges of scale, provided the corresponding crystalsystem and the directions of its axes are not alteredthereby.

Just as all crystal forms of a particular type may befound in different point groups, the same latticecomplex may occur in different space groups. Forexample, the complex P 'cubic primitive lattice' maybe generated, for instance, in Pm3m la,b, Fm3m 8c,

Orde

r k of

gro

up

Figure 1 Maximal subgroups and minimal supergroups of the three-dimensional crystallographic point groups. Full lines indicatemaximal subgroups; double or triple full lines mean that there are two or three maximal normal subgroups with the same symbol.Broken lines refer to sets of maximal conjugate subgroups. The group orders are given on the left. Full Hermann-Mauguinsymbols are used

and Ia3 8a,b with site symmetry m3m, 43m, and .3.realized by Cl in CsCl, and F in CaF2.

The body-centered cube (I) may be generated byspace-group operators Im3m a, Fd3c a, and Ia3d a,as realized in W metal and by Al in grossulariteCa3Al2(SiO4J3. By this criterion (space-group genera-tors) the Wyckoff positions of all space groups (1731entries in the space-group tables, 1128 types of Wyckoffsets) are uniquely assigned to 402 lattice complexes.

If a lattice complex can be generated in different space-group types, one of them has the highest site symmetry,and is called the characteristic space group or standardrepresentation of this lattice complex. All the other spacegroups are called subgroups of the space groups of thecharacteristic type.

The number of degrees of freedom of a latticecomplex is the same as that of its Wyckoff position,i.e. the number of coordinate parameters that can vary

Table 1 The three-dimensional crystallographic point groups D2, D2d, D3d, and Td (see International Tables for Crystallography,1983, table 10.2.2)

Orthorhombic system

2 2 2

4

2

2

2

d

C

b

a

1

. . 2

. 2 .

2 . .

Symmetry of special projections

Tetragonal system

4m2

8

4

4

1

D2d

d

C

b

a

1

. m .

. . 2

2mm.


Rhombic disphenoid or rhombic tetrahedronRhombic tetrahedron (u)

Rhombic prismRectangle through origin



Pinacoid or parallelohedronLine segment through origin (q)

Pinacoid or parallelohedronLine segment through origin (m)

Pinacoid or parallelohedronLine segment through origin (i)

Along [ 1 0 0 ]2mm

Along [0 1 0]2mm

Along [0 0 1]2mm

I

Tetragonal scalenohedronTetragonal tetrahedron cut off by pinacoid (I)

Ditetragonal prismTruncated square through origin

Tetragonal dipyramidTetragonal prism

Tetragonal disphenoid or tetragonal tetrahedronTetragonal tetrahedron Q)

Tetragonal prismSquare through origin

Tetragonal prismSquare through origin (h)

Pinacoid or parallelohedronLine segment through origin (e)

Along [0 0 1]4mm

Along [10 0]m

Along [110 ]2mm

(hkl) (hkl) (hkl) (hkl)

(hkO) (hkO) (hkO) (hkO)

(hOl) (hOl) (hOT) (hOT)

(OkI) (OkI) (OkT) (OkT)

(0 0 1) (0 0 T)

(oio) (olo)

(ioo) (Too)

(hkl) (hkl) (khl) (khl)(hkl) (hkl) (khl) (khl)(hkO) (hkO) (khO) (khO)(hkO) (hkO) (khO) (khO)(hhl) (hhl) (hhl) (h h I)(hhl) (hhl) (hhl) (hhl)(hOl) (hOl) (OhI) (OhI)

(10 0) (TOO) (OTO) (0 10)

(i 10) (TTo) (iTo) (TiO)

(ooi) (oo T)

(continued)

Table 1 (continued)

Trigo3 Im

3 l im

war/ sy:D3d

stem

Hexagonal axes12

6

6

6

d

C

b

a

1

. . m

. . 2

3 . m

Ditrigonal scalenohedron or hexagonalscalenohedronTrigonal antiprism slicedoff by pinacoid (I)

Dihexagonal prismTruncated hexagon through

origin

Hexagonal dipyramidHexagonal prism

RhombohedronTrigonal antiprism (k)

Hexagonal prismHexagon through origin

Hexagonal prismHexagon through origin (i)

Pinacoid or parallelohedronLine segment through origin (e)

(hkil)(khil)(hkil)(k h i I)(h k i 0)(khiO)(hkiO)(khiO)(hOhl)(Oh hi)(hOhl)(Oh hi)

(hh2hl)(hh2hl)(112 0)(1120)(10 10)(10 10)(0 0 0 1)

(ihkl)(hikl)(ihkl)(h ikl)(i hkO)(hikO)(ihkO)(hikO)(hhOl)(hhOl)(hhOl)(hhOl)

(2hhhl)(h2hhl)(2 1 1 0 )(0 2 10)( 1 1 0 0)( 1 1 0 0)( 0 0 0 1 )

(kihl)(i khl)(kihl)(i khl)(k i h 0)(ikhO)(kihO)(ikhO)(Oh hi)(hOhl)(Oh hi)(hOhl)

(h2hhl)(2hhhl)( 1 2 10)(2 1 1 0 )(0 1 1 0 )(0 1 1 0 )


Cubic system

4 3 m Td

24 d 1

12 c . . m

6 b 2 . mm

4 a . 3 m

Along [0 0 1]6mm

Hexatetrahedron or hexakistetra-hedronCube truncated by twotetrahedra Q)

C Tetrahexahedron or tetra- ikishexahedronOctahedron truncated by cube

Trigonotritetrahedron or tristetra-hedron (for \h\<\l\)Tetrahedron truncated by tetra-hedron (i) (for \x\ < \z\)

Tetragonotritetrahedron or delto-hedron or deltoid-dodecahedron(for |A|>|/|)Cube and two tetrahedra (i)

^ (for\x\>\z\) JRhombododecahedron

Cubooctahedron

Cube or hexahedronOctahedron (f)

TetrahedronTetrahedron (e) or

Symmetry of special projectionsAlong [00 1]

4mm

Along [10 0]2mm

(hkl) (Jill)(lhk) (lhk)(klh) (klh)

(OkI) (OkI)(10 k) (10 k)(klO) (klO)

>(hhl) (Ji hi)(lhh) (lhh)(hlh) (hlh)

( 1 1 0 ) ( T l O )(0 11) (0 1 1 )( 1 0 1 ) ( 1 0 1 )( 1 0 0) ( 1 0 0 )(0 10) (0 1 0)(0 0 1) ( 0 0 1)( 1 1 1 ) ( 1 1 1 )(1 1 1 ) ( U 1)

Along [ 1 1 1 ]3m

Along [2 1 0]

(Ji kT) (hkT)(lhk) (lhk)(klh) (klh)

(OkT) (OkT)(10k) (10k)(klO) (klO)

(Ji hi) (hhT)(lhh) (lhh)(hlh) (hlh)

( T l O ) ( l T O )( 0 1 1 ) ( 0 1 1 )( 1 0 1 ) ( 1 0 1 )

(TiT) (iTT)( i n ) ( i n )

Along [ 1 1 0 ]m

/ w/y(khl) (JcJiI) (kJiT) (khT)(lkh) (lkh) (lkh) (I k h)(hkl) (hlk) (hlk) (hlk)

(kOl) (kOl) (ArOO ( * 0 / )(/ArO) (/A:0) (/ArO) (/A:0)(0/Ar) ( 0 / * ) (01k) (01k)

independently within the Wyckoff position. Accordingto the number of degrees of freedom, a lattice complex iscalled invariant, univariant, bivariant, or trivariant. Forinvariant lattice complexes Hermann (1960) proposedcapital letters to symbolize the geometrical arrangements:

P, I ,F, C, A letters are chosen in connection with the Bravaisnotation

J from the American toy 'Jackstones' or 'Jacks'(three orthogonal arms with a commoncentral intersection)

D for 'diamond' arrangementT shortest connections of 'tetrahedra'W in connection with the 'JS1W-AlS' typeS with the point symmetry 4 (Schoenflies S)V with the point symmetry 222 (Schoenflies V)+ Y planar trigonal surrounding (self-coordination)

The symbols of invariant lattice complexes are listed inTable 3. Note that, if invariant complexes allow pairs ofenantiomorphic point configurations, these are distin-guished by the sign + or - in front of the symbols.

Table 2 Symbols for coordination polyhedra and polygons (developed by Donnay et al. (1964) and expanded by publicationsin Physics Data, changed by Lima-de-Faria et al. (1990)) sorted alphabetically. Differing symbols are set bold

Donnay et al. (1964), expanded bypublications in Physics Data

(Ic) or (11)(2c)(21)(3c)(3D(4c)(41) or (4s)(4py)(4t)(4u)(5by)

(6a)(61)(6o)(6r)

(7py)

(8a)

(8by)(8c)(8cu)(8d)(8r)

(9hco)(1Or)(12a)(12aco)(12co)(12d)(12i)(12p)(12tt)(12a2z)(15ttp)(12tt4t)(16thr)(18cohc)(24co)(24oc)(24scu)

Lima-de-Fariaet al. (1990)

[H][2n][21][3n][31][4n][41] or [4s][4y][4t]

[5by][51][6ap]

[6o][6p][7by]

[6plc][6p2c][8ap][8acb][8by]

[8cb][8do][8p][6p3c]

[12aco][12co]

[12i][12p][12tt][14FK][15FK][16FK]

Single pointTwo atoms not collinear with the central pointTwo atoms collinear with the central pointTriangle not coplanar with the central pointTriangle planar with the central pointQuadrangle not coplanar with the central pointQuadrangle coplanar with the central pointTrigonal pyramidTetrahedronApproximate quadrangle, four atoms nearly coplanarTrigonal bipyramidPentagon coplanar with the central pointTrigonal antiprismHexagon coplanar with the central pointOctahedronTrigonal prismPentagonal bipyramidHexagonal pyramidMonocapped trigonal prismBicapped trigonal prismTetragonal antiprismAnticubeHexagonal bipyramidOctagon not coplanar with the central pointCubeDodecahedron with triangular facesTetragonal prismTricapped trigonal prismHalved cubooctahedronPentagonal prism not coplanar with the central pointHexagonal antiprism not coplanar with the central pointAnticubooctahedron, twinned cubooctahedronCubooctahedron with triangular and quadrangular facesCubooctahedron with square facesIcosahedronHexagonal prismTruncated tetrahedron with triangular and hexagonal facesFrank-Kasper polyhedron with 14 verticesTruncated trigonal prism, Frank-Kasper polyhedron with 15 verticesFrank-Kasper polyhedron with 16 verticesTetragonal hexagon prismCubooctahedron, hexacappedCubooctahedron with triangular and octagonal facesCubooctahedron with quadrangular and hexagonal facesSnub-cube

If the point configuration is shifted by the vector(\\\)> (Hi) o r (Hi) f r o m t n e standard setting, thelattice complex symbol is followed by one, two or threeprimes on the capital letters, e.g. F ' , F" or F"'. Thestar * stands for a combination of two complexes, whichcan be mapped onto each other by T or (\\ \)- The

invariant cubic lattice complexes are drawn inFigure 2. Symbols of the site sets and coordinatesof the corresponding points are listed in Table 4. Asite set is symbolized by a string of numbers andsmall letters. The product of the numbers gives thenumber of equipoints in the site set, whereas the letters

Table 3 Symbols of invariant lattice complexes in their characteristic Wyckoff position

supply information about the degrees and directions offreedom. For characteristic Wyckoff positions, thedescriptive symbols of lattice complexes with degreesof freedom consist, in general, of four consecutive parts:shift vector, distribution symmetry, central part, andsite-set symbol. Either of the first two parts may beabsent. The central part contains one, or more than one,symbol of an invariant lattice complex (see Fischeret al.9 1973).

Two examples now follow: (1) I4xxx I43mcxxx)designates site sets composed of tetrahedra in parallelorientations replacing the points of a cubic body-centeredlattice; the vertices of these tetrahedra are located onbody diagonals. (2) . . 2I4xxx (Pn3mexxx) representsthe lattice complex for which, in contrast to example(1), the tetrahedra around 000 and yj-y differ inorientation; they are related by a twofold axis (. . 2)( = orientation symmetry) parallel to a face diagonal.

Figure 2 The 16 invariant lattice complexes in their standard settings. Numbers are given in eighths of projection axes. 0.4 means0 and 4 = 4/8= 1/2

Table 4 Symbols of site sets and coordinates of correspondingpoints

Site-setsymbol

Ix2yz

IXX

\xx2 y zIXX

\xx2yz\xxxIxxx3yz

\ylylx

\z\z2x\z2x2y

\z2xxU2y\z4xx\z2xy

\xy\xz\xyz2x2x2y2x2y2z

2x2z2x2yz

2xx2xx2y2xx2y2z

2xx2z2xx2yz

2xx2xxx2y2y2z

2z2xy2xy2z

2xy2xz2y

2xxz2yz2yz2x

2xyz3x3x2y

3x2y2z

3x2z

3x2yz

Coordinatesystem

othcOthtthhC

C

moOmothcOOtOtOmothmoOothcOtOOOth

tttttthremoOmothcmothmomomotoOamohh

h

h

h

Coordinates of site-setpoints

JCOO

xyz; xyzxyz; x-y,y,zxxOxyz; yxzxxOxyz; yxzXXX

xyzQOyOxyO; xyO0OzxOz;xOzxyz; xyz; xyz; xyzxxz; xxzOy z; Oy zxxz; xxz; xxz; xxzxyz; xyzxyOxOzxyz± (A: O O)±(xyO; xyO)±(xyz; xyz; xyz; xyz)± (xOz; xOz)xyz; xyz; xyz;xyzxyz; xyz; x-y, y, z;

y-x,y, z±(xxO)±(xyO; yxO)±(xyz; xyz; yxz; yxz)±(xxz; xxz)xyz; xyz; yxz; yxz±(xxO)+ (xxx)±(OyO)±(0yz;0yz)±(00z)±(xyO)+ (xyz; xyz)±(xOz)±(xyz; xyz)+ (xxz)±(Oyz)±(xyz; xyz)±(xyz)JtOO; OJCO; XXOxyO; yxO; y, x-y, 0;

x-y, y, 0; x, y-x, 0;y-x, x, 0

xyz; xyz; yxz; yxz;J, x-y, z; y, x-y, z;x-y, y, z; x-y,y, z; y-x, x, z; y-x,x, z; x, y-x, z; x,y-x, z

x Oz; x Oz; 0xz;0xz;xxz; xxz

xyz; yxz; y, x-y, z;x-y, y, z; x, y-x, z;y-x, x, z

Site-setsymbol

3JCJC

3xx2y

3xx2y2z

3xx2z

3xx2yz

3z3z2y

3xy3xy2z

3yz4x4x2 y

4x2y2z

4x2z4x2yz

4xx4xx2z

4xxx4xy4xy2z

4xz4xz2y

4xxz4xxz2y

4xyz6x

6x2y

6x2y2z

6x2z

6x2yz

Coordinatesystem

rehh

h

h

reh

rrhh

rtctt

tt

tctC

tttt

tt

tchh

h

h

h


JCJCOQx x 0; x, 2 x, 0; 2 3c, 3c, 0xyO; yxO;y, x-y, 0;

y-x, y, 0; x, x-y, 0;y-x, x, 0

xyz; xyz; yxz; yxz; y,x-y, z; y, x-y, z;y-x, y, z; y-x, y, z;x, x-y, z; x, x-y, z;y-x, x, z; y-x, x, z

xxz; xxz; x, 2x, z; x, 2x,z; 2 x, x, z; 2 x, x, z

xyzQ; yxzQxyz; yxz; y, x-y, z; y-x,

y, z; y-x, x, z; x, x-y, z0OzQ0yzQ;y0zQxyO; y, x-y, 0; y-x, x, 0xyz; xyz; y, x-y, z; y,

x-y, z; y-x, x, z; y-x,X, Z

OyzQ± ( ^ 0 0 ; OJCO)±(jcÔ; JCÔ; .yjcO; ^3c0)±(xyz; xyz; xyz; xyz;

yxz; yxz; yxz; yxz)±(JCOZ; xOz; Oxz; Oxz)xyz; yxz; xyz; yxz;

yxz; xyz; yxz; xyz±(JCJCO; JCJCO)±(xxz; xxz; xxz; xxz)xxx; 3c3cJCQ± ( J C ^ O ; yxO)±(xyz; xyz; yxz; yxz)xOz; xOz; Oxz; Oxzxyz; xyz; xyz; xyz;

yxz; yxz; yxz; yxzxxz; xxz; x~x~z; xxzxyz; yxz; xyz; yxz;

xyz; yxz; xyz; yxzxyz; xyz; yxz; yxz±(JCOO; JCJCO; OJCO)±(X-)

7O; J;JCO; JC, x-y, 0;x-y, JC, 0; y, y-x,0; y-x, y, 0)

±(xy z; xy z; yxz; yxz;JC, JC-J>, z; JC, x-y, z;x-y, x, z; x-y, JC, Z; y,y-x, z; y, y-x, z; y-x,y,z;y-x,y,z)

±(JC0z; xOz; xxz; xxz;Oxz; Oxz)

xyz; xyz; yxz; yxz; JC,x-y, z; x, y-x, z; x-y,x, z; y-x, x, z; y, y-x, z;y, x-y, z; y-x, y, z;x-y, y, z

(continued)

attribute of a framework is the forming of proper voidssuitable for occupation by the other atoms of the crystalstructure. Most frameworks are defined using latticecomplexes and polyhedra (Fischer and Koch, 1974).

A symbolism for coordination polyhedra wasdeveloped by Donnay et al. (1964) as the dual of thecrystal forms. It begins with the multiplicity of thecomplex followed by a symbol for the coordination: 1stands for planar surrounding, y for pyramidal, p forprismatic, a for antiprismatic, by for bipyramidal, t fortetrahedral, o for octahedral surrounding, etc. TheInternational Union of Crystallography, Commissionon Crystallographic Nomenclature, Subcommittee onthe Nomenclature of Inorganic Structure Types (Lima-de-Faria et al., 1990) proposed changes for somesymbols. Table 2 shows symbols for coordinationpolyhedra as given by Donnay et al. (1964) andexpanded by publications in Physics Data (see Hellnerand Pearson, 1986) and as given by Lima-de-Faria etal. (1990).

For a crystal-chemical description of a crystalstructure one needs to know the geometrical behaviorof the homogeneous Wyckoff positions as they varywith the atomic parameters lattice axes relations and theangle between the axes and the sizes of the atoms orclusters. Discussions of sphere packing conditions forthe tetragonal and cubic positions are given by Fischer(1971, 1973, 1974, 1991, 1993). But if more than twopositions form the framework, one again needs somehelp by group-subgroup relations.

There is no unique way to describe crystal structures.We will use lattice complexes, frameworks, polyhedra,and nets (and combinations thereof) to describe b.c.c.derivative structures.

2. The I Framework

The I lattice complex consists of all body-centered cubiclattices. It corresponds to the sphere packing type 8/4/cl(Fischer, 1973), i.e. 8 is the number of nearest neighbors,4 the number of the smallest mesh to reach the nearestneighbor, and cl is an enumeration of nets with the samecoordination number (CN). Only one kind of voidexists, i.e. flattened tetrahedra around a W* configur-ation. The maximal radius of the sphere, which fills upthe voids, is very small (Hellner et al., 1981).

Elemental representatives of the I structure (namedthe W A2 (cI2) type) are the alkali metals, the alkaline-earth metals, and some transition metals like Cr, /3-Ti,V, and elements below them in the periodic table. Theinclusion especially of the alkali atoms in this group,

In consequence, the listing of space groups with theirlattice complexes seems to be generally easy and goesautomatically: see Table 5. Multiples of invariant latticecomplexes are denoted by nanbnc, where the subscriptnumbers indicate that the cell is repeated na times alongaxis a, etc. So I113 gives an I complex tripled along c.The product nanbnc gives the order of the new cell.

A framework is an arrangement of atoms or ions thatare in contact with each other. The most essential

Table 4 (continued)

Site-setsymbol

6xx

6xx2z

6z6z2x6Z2x2y

6z2xx

6z2xx2y

6z4x

6z4x2y

6z4xx

6z2xy

6z4xy

6xy

6xy2z

6xz6xz2y

6xxz6xxz2y

6xxz6xxz2y

6xyz

8xxx\2xx

Coordinatesystem

rehh

C

C

C

C

C

C

C

C

C

C

h

h

hh

rrhh

reh

C

c


±(x3c0Q)±(xxO; x, 2 x, 0; 2x, 0)±(xxz\ xxz; X1 2 x, z\ x,

2 Xy Z; 2 Xy Xy Z'y 2 Xy Xy Z)

±(00zQ)±(x0zQ'yX0zQ)±(xyzQ; xyzQ; xyzQ]

xyzQ)xxzQ; xxzQ; xxzQ;

xxzQxyzQ; yxzQ\ xyzQ;

yxzQ; xyzQ; yxzQ;xyzQ; yxzQ

+ (xQzQ; xOzQ; OxzQ;OxzQ)

±(xyzQ; xyzQ; xyzQ;xyzQ; yxzQ; yxzQ;yxzQ;yxzQ)

±(xxzQ; xxzQ; xxzQ;xxzQ)

xyzQ; xyzQ; xyzQ;xyzQ

xyzQ xyzQ; yxzQ;yxzQ; xyzQ; xyzQ;yxzQ; yxzQ

+ (xyQ'y yy y-Xy 0; x-y, x,0)

±(xyz'y xy z; >>, y-Xy z\ yyy-Xy z\ x-yy Xy z; x-yy

XyZ)±(x0z\ Oxz; xxz)±(xyz; yxz; x-yy yy z; y,

X-yy Zy Xy y- Xy Z'y y- XyXyZ)

±(xxzQ)±(xyzQ;yxzQ)± (X X Z\ Xy 2 Xy Z'y 2 Xy Xy Z)

±(xyz; yxz; xy x-yt z;x-y, Xy z\ y-Xy 3c, z\ x,y-Xy z)

±(xyzQ)+ (xyz; yt x-yt z; y-xy xy

Z)+ (XXJC; x x x Q)±(jtx0Q; xxOQ)

Table 5 Assignments of Wyckoff positions to Wyckoff sets and lattice complexes in some space groups. Characteristic Wyckoffsets are marked by asterisks

229 Im3m

961 I6z4x2y *Im3m 1

48 k I6z4xx *Im3m k

48 j I6z4x *Im3mj

48 i -j~j~J-4..P26xx *Im3m i

24 h I12xx *Im3mh

24 g .3.J*4x *Im3mg

16 f I8xxx *Im3m f

12 e I6z *Im3me

12 d W* *Im3md

8 c } { { P 2 Pm3ma

6 b J* *Im3m b

2 a I *Im3m a

230 Ia3d

96 h 4a..Y**3xx"2yz *Ia3d h

48 g 4a..Y**3xx *Ia3d g

48 f .3.S*2z *Ia3df

32 e 4..Y**2xxx *Ia3de

24 d S* *Ia3d d

24 c V* *Ia3d c

16 b Y** *Ia3db

16 a I2 Im3ma

155 R32 (hexagonal axes)18 f R3x2yz *R32 f

9e 00yR3x *R32 d

9d R3x

6 c R2z R3m c

3 b 0OyR R3ma

3a R

155 R32 (rhombohedral axes)

6 f P3 2yz *R32 f

3 e yyyP3xx *R32 d

3d P3x*

2 c P2xxx R3m c

1 b y y y P R3m a

1 a P

217 I43m

48 h I6z2xx2y *I43m h

24 g I6z2xx *I43mg

24 f .3.J*4x Im3mg

12 e I6z Im3me

12 d W* Im3md

8 c I4xxx *I43m c

6 b J* Im3m b

2 a I Im3m a

220 I43d

48 e .3dS4xyz *I43d e

24 d .3.S2z *I43dd

16 c 4..I2Y**lxxx *I43dc

12 b 'S *I43da

12 a S

121 I42m

16 j I4xxx2y *I42mj

8 i I4xxz *I42m i

8 h 0y0(C2z)c P4/mmm g

8g 00yI4x I4/mmmi

8f I4x

4 e I2x I4/mmm e

4 d 0 y | € c P4/mmm a

4 c Oy 0Cc P4/mmm a

2 b 00y I I4/mmma

2a I

122

16e

8d

8c

4b

4aI42d.2.vD4xyz

4..vTFclxvD2z

00y VDVD

*I42d e

*l42d d

14,/amd e

I4/amd a

23 12228 k I2x2y2z *I222 k

4j 0y0I2z Immme

4i I2x

4 h y00I2y

4g I2y

4f 00Î2x

4e I2x

2 d OyOI Immm a

2 c 00y I

2 b y 0 0 1

2a I

24 12,2^1

8d x°T-2CcBblx2yz *I21212l d

4 c 0xT-2.BbAalz Immae

4 b | |02. .A aC c ly

4 a ioi..2CcBblx

with only one s electron in the valence-band shell, hasforced Pauling (1940) to propose the unsynchronizedresonance of bonds between positive ions and covalentlybonded clusters of two, three or four atoms, to explainthe large electrical conductivity. The pseudopotentialapproach to band structure has been of interest tounderstand better the electronic properties in solids andits relation to the total energy of simple sp-bondedmetals as given by Heine et al. (1970), including thefittings to experimental data (Cohen and Heine, 1970).Further, the orbital-dependent radii proposed by Simonand Bloch (1973) have been used by Zunger (1980) todemonstrate a structural separation plot for intermetalliccompounds (see also Chapter 11 by Villars in thisvolume). In spite of these developments in theory andapplications, there is no experimental information fromphotoelecton spectroscopy of metals with which tocompare magnetic models for the alkali and noblemetals from diatoms to the solid state (Malrieu et al.,1984) including the valence-electron excitations in thealkali metals (vom Felde et al., 1989) using electronenergy-loss spectroscopy (EELS) (see also Chapter 6 bySingh in this volume).

Structures with a framework description can becategorized by splitting and order. Splittings for orders1, 8, 27, and 216 are discussed in the following sections.Table 6 gives a comprehensive listing of b.c.c. derivativestructures that can be described by I frameworks(Hellner and Sowa, 1985).

2.1 Splittings of the I lattice complex

A lattice complex is represented by its characteristicWyckoff position in the space-group type that allowsthe highest site symmetry. In subgroups, the latticecomplex is split in two or more Wyckoff positions.Some splittings of the I lattice complex are shown inTable 7.

The splitting of the I complex in Pm3m to P + P 'produces the CsCl B2 (cP2) type, the heteroplanarcompound for the coordination number 8 with the idealradius quotient ^q = ̂ cation/ânion = 0-91 a nd a lowerlimit of flq = 0.73; followed by NaCl Bl (cF8) type(F + F') for CN 6 with Rq = 0.414; followed by the ZnS(zinc-blende) B3 (cF8) type (F+ F") for CN 4 withRq = 0.225. A lattice-energy calculation has beenproposed by Born (1923). Madelung (1918) has madea proposal for a simple summation of the energy terms.Some of the 300 intermetallic representatives of the Bltypes statistically occupy both positions and havemetallic band structures, but there exists an interestingheteropolarity in the intermetallic compounds Cs+Au"

and Rb+Au- with /?Au_ =2.00 A (Knecht et al., 1978);see Figure 3.

Hume-Rothery (1926) realized the importance of theratio of the total number of valence electrons to the totalnumber of atoms, the valence-electron concentration(VEC), in controlling the composition limits of metallicphases such as the disordered b.c.c. /3-brass structuresin CuZn, AgZn, AuZn, and AgCd with VEC = 1.5 butalso Cu3Al with VEC = 6/4 =1.5 and in Cu5Sn withVEC = 9/6 =1.5; the /3-manganese structure has thesame VEC= 1.5 as Ag3Al, Cu5Si, and CoZn2.

Jones (1934) has interpreted this in terms of the Blochtheory of metals, as inferred from experimental workon the Fermi surfaces.

2.1.1 The 8th Order of the I Complex

The 8th order of the I complex itself, named I222, canbe seen in the space group Ia3d in the Wyckoffposition 16a, point symmetry 3 000 and in Fd3c in16a and 23 000: the first one is occupied in Ga4Ni3, thelatter one by 16 Al in voltaite, a hydrous aluminum ironsulfate mineral.

As shown in Figure 4 from the I222m3m (8th order)via P222n3m to Fd3m one reaches the splitting productD+ D' =I222, which describes the NaTl structure. Zintland Dullenkopf (1932) and Zintl (1939) described thistype first and discussed the electron transfer from Nato Tl to get sp3 hybridization, as in the elements ofmain group IV, C, Si, Ge, Sn. For the isotype I—IIcompounds, like LiZn or LiCd, a statistical occupationof the sp3 hybridization was assumed to explain theirparamagnetic behavior. Furthermore, the solubility ofthese compounds in liquid ammonia (NH3) shows thepresence of negative anions.

From I222m3m via P222m3m to Fm3m one reachesthe splitting product F + F' + P222 = I222, which describesAlMnCu2, the Heusler structure.

The subgroup F43m of Fm3m permits all four Fsto exist; therefore F +F ' +F" +F ;" = I222 is occupied inCuTiHg2. If one subtracts an F from I222 one gets asubclass (I222-F) that has anti-CaF2 or Mg2Sn as atype compound.

The next splitting is found via I222m3m to P222m3mto Im3m to Pm3m; P + P ' + J + J' + P2'22 = I222 for thecompound Fe13Ge3. In the space group 12^ an 8pointer on the body-diagonal is realized with one degreeof freedom, namely 8a 3 xxx in the compound CoU withJCCO = 0.294, ^ = 0.0347 respectively. Therefore thedescription may be allowed to idealize the para-meters to A:= 1/4 and x = 0, and P2xxx + P2xxx«I2xxx,which is the type CoU.

Table 6 Frameworks of the I family and allocation of observed structure types thereto. Filled-up and defect frameworks areincluded herein. Numbers of known isotypic substances are those given in the compilation of Villars and Calvert (1985)

Substance

Order 1W

/3-Ag2S

Pa

/3-CuZn (CsCl)

HgMn

NaCl

ZnS (sphalerite)

Cu2O (cuprite)

FeS2

InS

a-Po

Order 2

CuTi

CaGaN

Order 3

AlCu2(MoSi2type) c/a~ 2.9

MoPt2

C2IrU2, Mn3N2

Al4Ba

Order 4

Pd5Ti3

Mo3U22

0-Np

Order 5

Al3Os2

Order 6

AlAu2

Order 7Cu4Ti3

Order 8NaTl

AlCu2Mn(Heusler alloy),

BiLi3 (metallic)

HgNa

SiU3

Li2ZnSbCuSi2Zr,

AsCuSiZrGaPt3

Fe13Ge3

BiF3 (non-metallic)

Symbol

cI2

cI20

tI2

cP2

tP2

cF4

cF8

cP6

oP6

oP8

cPl

tP4

tP6

tI6

oI6

tnotno

tP8

tP8

tC8

UlO

oP12

tI14

cF16

cF16

oC16

tI16

cF16

tC16

tP16

cP16

cF16

Spacegroup

Im3m

Im3m

I4/mmm

Pm3m

P4/mmm

Fm3m

F43m

Pn3m

Pnnm

Pnnm

Pm3m

P4/nmm

P4/nmm

I4/mmm

Immm

I4/mmm

I4/mmm

P4/mmm

P4/nbm

C42.2

I4/mmm

Pnma

14/mmm

Fd3m

Fm3m

Cmcm

I4/mcm

F43m

C4/amm

P4/mbm

Pm3m

Fm3m

Wyckoff

2a

2a, 6b, 12d

2a

Ia, Ib

Ia, Id

4a, 4b

4a, 4c

2a, 4b

2a, 4g

4g2

Ia

2c, 2c

2c2

2a, 4e

2a, 4g

2a, 4e2

2a, 4d, 4e

la,b, 2g, 2h2

2a, 2c, 4f

4a, 4c

2a, 4e, 4e

4c3

2a, 4e3

8a, 8b

4a, 4b, 8c

4c2, 8g

4a, 4b, 8h

4a, 4b, 4c, 4d

4a, 4b, 4c2

4e, 4f, 4g, 4h

la,b, 3c,d, 8g

4b, 8c

Framework

[I]

[i]

[i] = [P + P ' l

[ I ] - [ P - H P ' ]

F + F'

I = F + F"

[ I ] + F "

[n..0y0I2xy] +1

n..0y0I2y + n.. I2xy

[ I - P ' ] = P

[I112] = [0l0CIlz + 0 l | C I l z ]

[OHi 1 1 2 Iz ]+0-HrCiIz

[I113] = [I + 0 0 | l 2 z ]

[I131Iy][I113]+I2z

[I113] + 0 H c 1 1 2

[I114] = [P114+TTTP114][Ii,4] = [C + 0 | 0 C + P2'21]I221Iz

I115

1.2,HTWXZ

I117

I2 2 2=[D + D ' ]

I222= [F + F ' + P'222]

/ 3 l 0 \OO|llxy A / ( 8 )M(8)= 1 3 0

i \ 0 0 1 /OOTI222

I222= [F + F ' + F " + F'"]

I222=[P222 +P222IZ]

00|l222lzx

I222 = [ P222 + P222 ]

= [P'2 2 2 + P + J + P ' + J ][ I 2 2 2 - F ] + F = [P2'22 + F ' ] + F

Number ofsubstances

241

1

3

401

7

-700

120

3

44

1

9

15

4

84

8

7

462

1

1

3

4

4

1

34

204

8

5

34

8

2

Substance

C (diamond)NaCl (metallic)/3-Hg4PtMg2SnAsAgMgThH2

AsLiMnAsPd5Tl

Order 9Te4Ti5

Order 27Tl7Sb2

a-MnCu5Zn8

T-Al4Cu9

Ag2Hg3

Order 64In3Li13

P d 4 - J e

CoMnSb

Sc11Ir4

Mn23Th6

Bi4Cu4Mn3

Ga4Ni3

Order 216Li22Pb5

Li22Si5

Cu41Sn11

Mg6Pd

Na6Tl

SmnCd45

Symbol

cF8cF8ClIOcF12cF12tF12tC12tC14

tI18

cI54

cI58

c!52

cP52cI52

cF128

cF120

cF120cF120

cF116

cF88

CI112

cF432

CF416

cF396

cF408

cF448

Spacegroup

Fd3mFm3mIm3mFm3mF43mF4/mmmC4/ammC4/mmm

I4/m

ImSmI43mI43m

P43m123

Fd3m

F43m

Fd3mFm3m

Fm3m

Fm3m

Ia3d

F23

F43m

F43m

F43m

F43m

Wyckoff

8a4a, 4b2a, 8c4a, 8c4a, 4c, 4d4a, 8d4a,b,c2a,c,d, 8i

2a, 8h, 8h

2a, I2e, I6f, 24h2a, 8c, 24g2

8c2, 12e, 24g

4e4, 6f, 6g, I2i2

8c2, 12d, 24f

8a,b, 16c,d, 32e, 48f

4a,c, 16e4, 24f, 24g

8a, 16c,d, 32e, 48f4a,b, 24d,e, 32f*

4b, 24d,e, 32f2

8c, 24d, 24e, 32f

16a, 48f,g

4a-d, 16e8,(24f,g)2, 48h4

16e8, 24f2,g2, 48h4

4d, 16e5, 24f*,g, 48h5

16e6, 24f2,g, 48h5

4a,b,c,d, 16e6,24f,g, 48h6

Framework

[ I 2 2 2 -D ' ] =D

[1222-Pi2I = [F + ? ' ][I222-J*] = [1 + Pi2]U 2 2 2 - F ' ] = [F+P2 '22][ I 2 2 2 -F ' ] = [F + F" + F'"][ I 2 2 2 -F ' ] = metrically distorted cell[ I 2 2 2 -F" Iz][I2 2 2-OO|C]

I33.

I333 = [ W*(4tc) + I(«8cu,6o) ]W*(4tc) + l(#12tt,4t )W*(4tc) + l(4t+,4t ,60) or ~ [I333- I] =

«l(io4t + 4t 6ol2co)* [I3 3 3-I] =I(io4t + 4t 6ol2co)- [I3 3 3-H = [W*(4tc>] +I(io4tf4t -60)

I444= [D + D' + JJ^ + T + T' +F^22X]

= [P444+F?22][ I 4 4 4 -D ' ] = [J222 + F + J2'22 + F"+TS( +

T ' + T " + T'"][ I 4 4 4 -D ' ] = [J222xs, + F '̂22x + T + T / + D]

[I444-P222] = [F + F'+J222 + F(6O)+ F'(8cu) + F(8cu)]

[I4 4 4-F,P'2 2 2] = [F(8cu) + F'(8cu)

+ J2'22x + J222 + F ' ]

[I444-P222.F'(8CU)] =

[P222 + J222 + J '222 + F(8cu)]

[I444-Y**] = H222+ J2*22 + (Pi44-Y**)]

I666 = t W?22(4tc) + I222(#io4t+ 4t ,60) ]

* [I666-I222] = [W2*(4tc)] +I222(io4t + 4t 60)= I222 [ io4t+ 4 t ' 6012co ]

[l666-((I222-F'"),F(4t + ),F'(4t + ),F'"(4t + ),F'"(60))] +F'"(12tt) or [W*2(4tc)] +P2(4r,6o) + F"(4t ,4t + ,6o) + F'"(«12tt,4t )

[I666-(I222,F(4t+),F'"(4t+),F'"(6o))] +F'"(12tt) or[W*2(4tc) ] + D"(4t ,4t+ ,60) 4- F(4t ,60)+ F"'(12tt,4t)

[l666-(F'(4t + ),F"(4t + ),F'(6o),F"(6o))]+ F'(12tt),F"(12tt) or W*2[4tc] +D'"(«8c,6o)+ D"(#12tt,4t)

Number ofsubstances

9665

41129942

49

12

1

60

1791

1

2

4

158

4

3

5

2

3

1

19

2.7.2 The 27th Order

The 27th order of the I complex is named I333 with thetransformation matrix

/300X030

\003 /

The number of points increases from 2 to 54: but thiscannot be realized by a one-point configuration in a

space group with only generators and translators. Itmust be a combination of lattice complexes with specialparameters and interpenetration. But this configurationhas already been recognized for Tl7Sb2 in Struktur-bericht volume 1 (1931) by Ewald and Hermann as'Substitutionsiiberstruktur des A2-Typs\ Figure 5 showsthe three main lattice complexes I(6o), I(8c), andW*(4tc), plus two Tl atoms in I. The numbers indicatethe heights in the projection direction a2 in n/6.Besides the fact that only this one representative(Tl7Sb2) is known for I333 but with vacancies at the 2pointer I (in 000 and j j y , the complex I(8c) splitsinto I(4t+) and I(4t~) with different distances to theorigin and a z parameter for the W*(4t) complex.Subclasses of I333 include y-brass, Cu9Al4, and Fe3Zn10.Representatives of the 7-brass structure (cI52) with theVEC = 21/13« 1.62 are the intermetallic compoundsCu5Zn8, Ag5Zn8, Au5Zn8, Cu9Al4, Cu31Sn8, Cu31Si8,Co5Zn21, Pt5Zn21, Rh5Zn21, and Na31Pb8. They can beregarded as a subclass of the I333 type. Another splittingis realized in the a-Mn structure (or \ phase) W*(4t) +1,I12tt, I4t, which has 58 atoms per cell. Al12Mg17 andRe24Ti5 are said to have the same a-Mn structure (cI58).


A structure type with more than 155 representatives,named Mn23Th6 cF116 Fm3m, represents an I444 withvacancies of the CaF2 type (I444-FjP222).

The Mg6Cu16Si7 type has been shown by Nagorsonand Witte (1953) to be isotypic with Mn23Th6. ButSc11Ir4 cF120 Fm3m has an I444-P2 type. (Structuresin this group as well as those in Section 3.7 are alsodiscussed in Chapter 17 by Nevitt and Koch in thisvolume).

Oxid

atio

n nu

mbe

r

Au(metctUic)

Binding energy (eV)

Figure 3 Binding energy of the Au(4f7/2) level in compoundswith different oxidation states

Table 7 Splittings of I Bravais lattices in different orders

Order

1

8

27

64

216

Space group

Im3mPm3m

Ia3d (r)Fd3c (a)Im3mFd3mFm3mF43mP43m

Im3m

Fd3mFm3m

F43m

Splitting multiplicity

21, 1

16162, 6, 88, 88, 4, 44, 4, 4, 41, 3, 1, 3 ,4 ,4

2, 12, 16, 24

8, 8, 48, 16, 16, 324, 4, 24, 8, 24, 32, 32

4x4, 8x16, 4x24, 4x48

Framework symbol

I = P + P'

*2

I22= [I+ J*+ P2']

I2=[D + D']I2 = P2' + F + F'I2=[F + F '+F" + F"']I2=[P + J + P ' + J ' + F " + F ' " ]

I3 = [ W*(4tc) +1 + I(6o) + I(8cu) ]

I4= [D + D' + JJx + T + T' +F1J]I4= [F+ F' +J2+ P2 + J2 x+ F2+ F2 ']

I6=[W*(4tc) + I2(«4t + ,4r,6o)]

Figure 4 Splitting of point configurations belonging to invariant cubic lattice complexes or limiting forms in class-equivalent subgroups of Im3m

Figure 5 Lattice complexes I(6o), I(8c), and W*(4tc) (12e, 16f and 24h are atom positions in the Wyckoff notation)

Ideal:

Tl7Sb2:


If one doubles the length of the axes in the cubic systemfor Tl7Sb2, Cu5Zn8, and the a-Mn type, one arrives atthe Li22Si5, Cu41Sn11, and Na6Tl types of 216th order,correlations that were found by our symbolism. Notethat the VEC «1.635 for Cu41Sn11 is in the Hume-Rothery range, because the next higher VEC = 7/4 = 1.75belongs to the hexagonal e structure, i.e. CuZn3 in theCu-Zn binary system.

Table 8 is a slightly modified version of a table givenby Fornasini et aL (1978). It is supplemented by theatomic parameters of Mg6Pd (Samson, 1972) andNa6Tl (Samson and Hansen, 1972). The three basicstructures Tl7Sb2, a-Mn, and Cu5Zn8 contain only onetype of nested polyhedra each. The two sets around 000and YTT within the corresponding unit cells arerelated by translations to each other. In the secondcolumn the corresponding cluster description is listed.All point configurations are either face-centered latticesor they form polyhedra around these lattices. Thefollowing symbols are used:

• CC cluster center(12tt) TT truncated tetrahedron(4t + ) IT positive (inner) tetrahedron(4t~) OT negative (outer) tetrahedron(6o) OH octahedron(12co) CO cubooctahedron

The structures of Tl7Sb2, a-Mn, and Cu5Zn8 are listedin the right part of Table 8, the corresponding descrip-tions with symbols of lattice complexes and coordinationpolyhedra are added in the last column. I(8c) symbolizesisolated cubes around an I configuration and W*(4tc)means corner-connected tetrahedra around a configur-ation of the cubic invariant lattice complex W*.

The different intermetallic phases included in Table 8differ from each other by the occupation or non-occupation of some of the kinds of point configurationslisted. Configurations W*(4tc) or, with respect to thecrystal structures with enlarged unit cells, the corres-ponding configurations F(12co) + F'(12co) + F"(12co)+ F'"(12co) = Wf22(4tc), however, are occupied in everycase. Therefore, a more careful discussion of these con-figurations seems appropriate.

In summary, within these eight crystal structures, thelarge voids around I or I2 are filled in several differentways. With the aid of the symbols given above, thecrystal structures are described in Table 6.

These symbols show very clearly the common featureof all these crystal structures, namely the three-dimensional connected frameworks W*(4tc) as

illustrated in Figure 6. For a description of thegeometrical and physical properties, this symbolismseems more adequate and useful than the clusterdescription (Hellner and Koch, 1981a,b).

In this context it may be mentioned that in Ir3Ge7

the W*(4tc) lattice complex collapses to a W* only, sothat the description may more simply be I(8c), I(6o),W*; this description has been compared with the idealTl7Sb2 in I333 and Cu5Zn8 7-brass by Hellner and Koch(1980).

3. Frameworks of the I Family with PolyhedraAllocated Around the i Points

The following frameworks do not correspond to a latticecomplex. They are defined with the help of polyhedralocated at the equipoints of the lattice complex, centeredor not. Table 9 lists structures that can be described bysuch frameworks.

3.1 The I(4t) Framework

Such a framework may be described as a set of parallel-oriented tetrahedra, the centers of which form anI lattice. The tetrahedra do not share vertices butthey are connected by additional shortest distances (cf.Figure 7). The parameter Jt= 3/16 is specialized withrespect to sphere packings, but not with respect toDirichlet domains.

3.2 The I(6o) Framework

An I(6o) framework as shown in Figure 8 consists oftwo configurations P(6o) and P ' (60) that are shiftedrelative to each other by ( { } } ) . All distances betweenpoints of the P(6o) and P ' (60) configurations are longerthan the shortest distances within the framework, i.e.the framework does not correspond to a sphere packingbut to interpenetrating sphere packings of type 5/3/c3(Fischer, 1973).

No crystal structure with a nearly ideal I(6o) frame-work has been found. In all cases the x parameter ofthe framework atoms is close to 1/4. In the structureof SF6 cI14 Im3m the F-F distances within a fluorineoctahedron are shorter than the distances between atomsfrom different polyhedra because of the S-F bondings.

3.3 The D(6o) Framework

A D(6o) framework consists of ideal octahedra, thecenters of which form a D configuration, which is a

Table 8 The atomic parameters for five crystal structures of the 216th and three of the 27th orders. For Li22Si5 point position 48(h)xyz, the x and y coordinatesare averaged to facilitate the comparison. For the basic structures the list of coordinates is extended to the array of eight unit cells corresponding to theenlarged unit cells of the further structures

27th order216th order

I(12tt)

I(4t+) I(8c)I(4 t ) 1(*C)

I(6o)J*(21)W*(4tc)

Cu5Sn8

I43mj = 8.878

Zn 0.0545 x 2Cu 0.9140 X 2Cu 0.1779 X 2Zn 0.1564 x 2

0.0183x2

Zn 0.3045 x 2Cu 0.1640 x 2Cu 0.4279 x 2Zn 0.4064 x 2

0.2683x2

Zn 0.5545 X 2Cu 0.4140 X 2Cu 0.6779 X 2Zn 0.6564 X 2

0.5183x2

Zn 0.8045 x 2Cu 0.6640 x 2Cu 0.9279 X 2Zn 0.9064 x 2

0.7683x2

Tl7Sb2

Im3m<7=11.618

Tl

Tl 0.0852 x 2Tl 0.9148 x 2Sb 0.1569x2Tl 0.1749x2

0.0000

Tl

Tl 0.3352 x 2Ti 0.1648 x 2Sb 0.4069 X 2Ti 0.4249 X 2

0.0000

Tl

Tl 0.5852 X 2Tl 0.4148 x 2Sb 0.6569 x 2Tl 0.6749 x 2

0.0000

Tl

Tl 0.8352 x 2Tl 0.6648 X 2Sb 0.9069 X 2Tl 0.9249 x 2

0.0000

a-MnI43m0 = 8.911

MnMn 0.0448 X 2

0.1410x2

Mn 0.9089 x 2

Mn 0.1785 x 20.0173x2

MnMn 0.2948 X 2

0.3910x2

Mn 0.1589x2

Mn 0.4285 X 20.2673 x 2

MnMn 0.5448 X 2

0.6410x2

Mn 0.4089 X 2

Mn 0.6785 x 20.5173x2

MnMn 0.7948 x 2

0.8910x2

Mn 0.6589 x 2

Mn 0.9285 x 20.7673x2

Cu41Sn11

F43m0=17.98

Cu 0.0573Sn 0.9113Cu 0.1763Cu 0.1562

0.0186

Cu 0.3005Cu 0.1657Cu 0.4241Sn 0.4084

0.2680

Cu 0.5504Cu 0.4166Cu 0.6765Cu 0.6465

0.5278

Cu 0.8062Cu 0.6664Sn 0.9309Cu 0.9087

0.7631

Li22Si5

F230=18.75

Li

Li 0.080Si 0.911Li 0.167Li 0.164

0.005

Li

Li 0.330Li 0.167Si 0.428Li 0.413

0.250

Li

Li 0.583Li 0.420Si 0.678Li 0.662

0.497

Li

Li 0.833Si 0.665Li 0.917Li 0.917

0.750

Na6TlF43m0 = 24.154

Tl 0.90136Na 0.10858Na 0.14228

0.03346

Na 0.30267Tl 0.16755Na 0.43185Na 0.41016

0.26791

Na 0.55549Na 0.41542Tl 0.66484Na 0.65395

0.52856

Na 0.801910.90985

Na 0.67234

Na 0.943960.76942

Mg6PdF43m0 = 20.108

Pd 0.90038Mg 0.10720Mg 0.14355

0.03398

Mg 0.30250Pd 0.16790Mg 0.43518Mg 0.40605

0.27391

Pd 0.40653Mg 0.61866Mg 0.65097

0.52812

MgMg 0.79984

0.90720

Mg 0.66868

Mg 0.943450.76760

Sm11Cd45

F43m0O = 21.699

Sm

Cd 0.0834Cd 0.9126Cd 0.1573Sm 0.1735

0.0142

SmCd 0.2958

0.3904

Cd 0.1636

Cd 0.43770.2627

CdCd 0.5455

0.6403

Sm 0.4059

C 0.67280.5123

Cd

Cd 0.8297Sm 0.6618Cd 0.9105Cd 0.9161

0.7637

Clustersymbol

Cluster AA CC 4(a) 000A TT 48(h)ju:z x

ZA IT \6(e)xxx xA OT \6(e)xxx xA OH 24(f)*00 xA CO 48(h)xx:z x

Z

Cluster BB CC 4(C)IIIB TT 4S(h)xxz x

ZB IT 16(e)jax xB OT 16(e)xx* xA OH24(g)x|T xA CO 48(h)jurz x

Z

Cluster CC CC 4(b)|||C TT 48(h)xxx x

ZC IT \6(e)xxx xC OT \6(e)xxx xC OH24(f>yi xC CO 48(h)ja:z x

Z

Cluster DD CC 4(<1)HTD TT 4S(K)XXZ x

ZD IT \6(e)xxz xD OT \6(e)xxx xD OH 24(g)xff xD CO 48(h)jtxx: x

Z

Symbol ofthe pointconfigur-ation

FF(12tt)

F(4t+)F (4 t )F(6o)F(12co)

F"F"(12tt)

F"(4t + )F"(4r)F"(6o)F"(12co)

F'F'(12tt)

F'(4t+)F'(4f)F'(6o)F'(12co)

F'"(12tt)

F'"(4t + )F '"(4t)F'"(6o)F'"(12co)

subgroup of I2. The axes of these octahedra areoriented parallel to the unit-cell edges and their verticesare connected by additonal shortest distances (cf.Figure 9). The corresponding sphere packing type is8/3/c3 (Fischer, 1973).

In a D(6o) framework the ideal small octahedraaround T are corner-connected. Each D octahedronshares faces with four T octahedra and four distortedD' (4t) octahedra; each T octahedron shares faces withtwo D octahedra and six D' (4t) octahedra. The edges of

Figure 6 Frameworks of crystal structures of the 27th order, (a) Framework W* [4tc] as part of the crystal structure of Tl7Sb2:symmetry Im3m 24(h) xxO, AT = 0.3497, z coordinates in /J/100. (b) Framework W*[4tc] as part of the crystal structure ofa-Mn: symmetry I43m 24(g) xxz,_x = 0.3571, z = 0.0346, z coordinates in /i/lOO. (c) Framework W* [4tc] as part of the crystalstructure of Cu5Zn8: symmetry I43m 24(g) xxz, x = 0.3128, z = 0.0366, z coordinates in «/100

Table 9 Frameworks of the I family with polyhedra allocated around the i points and observed structure types thereto. Numbersof known isotypic substances are those given in the compilation of Villars and Calvert (1985)

(continued)

Substance

BFe

MoNi4

GeMg2S4 (olivine)

PTi3

GeK

SF6

Re7Si6U4

TiO2 (rutile)

ErGa4V2

MoNi4

FeS2 (marcasite)

FeSb2

CFe2

RbNiCrF6

NiTi2

Fe6W6C

Fe3W3C

(X-RhBi4B2CoW2

B4Ta3

BFe

B2IrMo2

Al11La3

B2CoW2

Ir3Sn7

46H2O-8Cl

N2

Nd5Mo3O16

Tl7Sb2

HfSiTe (Cu2Sb

type)

W

Te4Ti5

Cu4Nb5Si4

(filled-up Te4Ti5)

Na6Tl

Symbol

0P8

tllO

oP28

tP32

cP64

cI14

cI34

tP6

tI14

tllO

0P6

0P6

0P6

cF72

cF96

cF104

cFI12

cl 120

oIlO

oI14

oP8

oP20

oI28

oIlO

cI40

cP54

cP64

cP96

cI54

tP6

cI2

tI18

tI26

cF408

Spacegroup

Pnma

I4/m

Pnma

P42/n

P43n

Im3m

Im3m

P42/mnm

I4/mmm

I4/m

Pnnm

Pnn2

Pnnm

Fd3m

Fd3m

Fd3m

Fd3m

Ia3d

Immm

Immm

Pnma

Pnnm

Immm

Immm

Im3m

Pm3n

Pm3n

Pn3n

Im3m

P4/nmm

Im3m

I4/m

I4/m

F43m

Wyckoff

4c2

2a, 8h

4a, 4c4, 8d

8g4

8e2, 24i2

2a, 12e

a, c, d, e

2a, 4f

2a, 4d, 8h

2a, 8h

2a, 4g

2a, 4c

2a, 4g

b, c, f

d, e, f

a, d, e, f

c, d, e, f

c, h

2a, 4f, 4h

2c, 4g\ 4h

4c2

4g5

2a, 2c, 4g, 4j, 812

2a, 4f, 4h

12d, 12e, 16f

2a, 6c,d, 16i, 24k

16i, 481

8c, 12d,e, 16f, 48i

2a, 12e, 16f, 24h

2a, 2c2

2a

2a, 8h2

2a, 8h3

48h4

FrameworkNumber ofsubstances

1.2,411(21 oo )(«6r31varcon)[010]

[| |0I(41) + I]

T.2, (00-yI2xylz(«4t) + B121, { { y F l x z )

[I(4t) + 00|l(4t) + T T T P 2 2 . ( 2 1 ) + 0 2 - I c n ^ t 2 6 ) ]

[I(4t + ) + I (4 r )+ . .n I(12tt) + W'(4t + )]

!(•6o) = I(6o) + I

[P2'22 + W* + I(6o)]+I

!(•6oc)

00|l(6o2c) +Oi^C 1 1 2

TT0I(6o2c)

n. !(•6ovarcon.)

n.. I(«6o)

!(•6ovarcon.)

D(6o) + D',T

[D(6o) + D ' ( 4 t ) + T ' ]

[D(6o) + D'(4t~) + T ' ] 4-D

[D(6o) + D ' ( 4 t ) + T ' ] +T

..dI222(6o) + V*

0l|l2y(«6r41) le )4c) + I

|00I2y(#6r2x31412e212c oo )

1.2,0111(21 oo )(«6r31varcon)[010]

n.. 0i0I(21(ll)(«6r3cvarcon.)

(0ill(612e) + 0||B12Ily(51 l e ( l c) ^ ) + I131

[I(8cu) + W* + J*(21)]

[I(8cu) + J*(41) + W] +1,W'

[I(8cu)8t + W'(8a)st]

[I(8cu) + J*(8a)] +P2'22,W*,I(6o)

[ I(8cu) + W*(4tc) + I(6o) +1 ]

[0|ll(8alzlzv 'a rcon .)]

!(•12dvarcon.)

I(#12co2x41)

!(•12co2x4l) + etc.

[I2(12co) ] + etc. = W2*(4tc) + etc.

106

16

14

26

6

3

18

17

16

44

4

5

60

3

54

1

15

17

106

9

15

15

23

1

193

241

12

3

1

the D' octahedra are 66% larger than the shortestdistances within the framework. Each D' octahedronis surrounded by four T' antiprisms and four D'(4t)octahedra. AT ' antiprism has two common faces withD' octahedra and six common faces with D'(4t)octahedra. As a consequence, each D' (4t) octahedronshares faces with one D octahedron, one D' octahedron,three T octahedra, and three T' antiprisms.

The 'centers' of the distorted D' (4t) octahedra lie inthe interior of the large D' octahedra. The shortestdistances between these centers are much smaller thanthe sum of the radii for two D' (4t) voids calculated withthe aid of the distances to the framework atoms.Therefore, it is not possible to locate atoms at the'centers' of the D' (4t) voids, because such atoms wouldbuild up isolated tetrahedra without contact with any

Table 9 (continued)

Substance

Mg6Pd

Sn11Cu41

8th order 7-brass

Sm11Cd45

Na6Al5-3-Si10-4O32

Pt8Cd41

H3PW12O40-6H2O

Al12W

CoAs3

Fe4LaP12

H3U

Pr3Rh4Sn13

NaZn13

Al11La3

Fe4Si2Zr

Cu4Si4Zr3

Ni5Si3Y

a-Mn

Ge38P8Br8

GeK

Qf-SeCl4

ThMn12

Mg32(Al1Zn)49

Ge38P8Br8

46H2O-8Cl

Pr3Rh4Sn13

B4CeCo4

B4Co4Nd

BaCuO2

Na30Al30Si66O192

-98H2O (zeolite

ZK-5)

Symbol

CF396

cF416

cF448

cI60

cI392

cP160

cI26

cI32

cI34

cP32

cP40

cF112

oI28

tP14

oI22

oP36

cI58

cP54

cP64

cP160

tI26

cl 162

cP54

cP54

cP40

tP18

tP18

cI402

CI304

Spacegroup

F43m

F43m

F43m

Im3m

I43m

Pn3m

Im3

Im3

Im3

Pm3n

Pm3n

Fm3c

Immm

P42/mnm

Immm

Pnma

I43m

P43n

P43n

P43n

I4/mmm

Im3

P43n

Pm3n

Pm3n

P42/nmc

P42/n

Im3m

Im3m

Wyckoff

48h4

48h4

48h4

8c, 12d, 16f, 24h

8c4, 12e2, 24f, 24g9, 48h2

2a, 6d, 8e, 24h, k4, 481

2a, 24g

8c, 24g

2a, 8c, 24g

2a, 6c, 24k

2a, 6d, 8e, 24k

8a, 8b, 96i

2a, 2c, 4g, 4j, 812

2b, 4g, 81

2d, 4e, 4f, 4h, 8n

4c9

2a, 8c, 24g2

2a, 6c,d, 8e2, 24i

8e2, 24i2

8e2, 24i6

2a, 8f, 8i, 8j

2a, 12e2, 16f, 24g3, 48h

2a, 6c,d, 8e2, 24i

2a, 6c,d, 16i, 24k

2a, 6d, 8e, 24k

2b, 8g, 8g

2b, 8g, 8g

2a, 12d,e3, 16f, 24h2,

48i, 48j2, k3

16f, 48i,j,k2, 961

Framework

[I2(12co)] +etc. = W2*(4tc) + etc.

[ I2(12co) ] + etc. = W2*(4tc) + etc.

= I2(io(4t+)(4r)(6o)(12co))

[I2(12co) ] + etc. = W2*(4tc) + etc.

[I(12co) + P2'22}+I(8cu)st,Ws*

[ [I(12co) + J*(12co) + P2'22(12co)] +etc.]

[I(#4t),(12co)(#6o)5c] + J*(41,8r)st,J*

[I(12i) + I]

[I(12i) + P2'22]

[1(12I) + P ^ ] + I

..n I(12i) + I, W

[..n I(12i) + P2 '22]+I, W'

..cP222(12i) + P222,P2'22

!(•12r2x6Ia41) + 0 0 | l + 0 |0B l 2 1 ly

n.. 00|l(#12r2x6M2c61)

0|0I(#12r2><61a612e 00 )(100)

T.21 | | | l2 1 1lxz(#12r61)v a r .c o n .

[ ! (•12t t ,4r) + W*(4tc)]

[..n I(12tt) + I(4t+) + I ( 4 r ) + W ' ] +1,W

[..n I(12tt) + I(4t + ) + I ( 4 r ) + W'(4t + ) ]

[I(12tt) + I ( 4 r ) + W(12tt) + W(4t + ) ] +

I(4t+),W(4t-)

I(#16thr41)

!(•20Pd(#12tt)varcon.)

. .nI(#20pd) + W,W'

[..nI(20pd) + W ] + I , W

1(2Op(I86)+ 1,W

!(•24scd)

00|l(«24gon2 x 8 1 8 x 4 1)

[I(24cod) + [Pi44 - I(8cu) ] + W* + J*(8r)st

+ J*(21) + P2'22(6o)] +etc.

I(48cod(«4tc)) + I(8cu)

Number ofsubstances

2

2

19

1

6

19

40

5

59

47

15

6

26

7

60

6

132

10

47

31

26

Figure 7 I(4t) framework: (a) point configuration with lines indicating the tetrahedra around I; (b) I(4t) tetrahedron around} y } ; (c) J* dodecahedron (8d) with 8 vertices and 12 faces around Oyy; (d) W* tetrahedron around O-^T- Z coordinates givenin fir/100

Figure 8 I(6o) framework: (a) point configuration with lines corresponding to sphere contacts, octahedra around I; (b) W*tetrahedron around -y°T'» (C) W+(^0) tetrahedron around 00.32320.3232; (d) trigonal antiprism around P2', { { { . z coordinatesgiven in a/\00

Figure 9 D(6o) framework: (a) point configuration with lines indicating the octahedra around D; (b) D' octahedron aroundy j y ; (c) T octahedron around YTT> (d) trigonal antiprism around T', { { { ; (e) D'(4t) octahedron 'around' 0.4375 0.43750.4375; (f) hexacapped cubooctahedron (18cohc) around D', y y y , combined from a D' octahedron, four T' antiprisms, andfour D'(4t) octahedra. z coordinates given in «/100

other atom of the crystal structure. For the descriptionof some crystal structures it may be more adequate,therefore, to construct large coordination polyhedraaround D' made up from the first and from the secondcoordination shells. The difference of the distances is15%. The resulting polyhedra with 18 vertices may bedescribed as hexacapped cubooctahedra (18cohc). Eachof them is a combination of the original D' octahedron,four D'(4t) octahedra, and four T' antiprisms. Theselarge polyhedra overlap the T' antiprisms.

Several crystal structures of different type show theD(6o) framework shown in Figure 9.

The crystal structures of Fe3W3C and Fe6W6C arevery similar. They may be described by a basicframework D(6o) of tungsten atoms, the T' and D'(4t)voids of which are occupied by iron atoms. The carbonatoms are located in the T and the D octahedra,respectively. The iron atoms by themselves, however,build up a second heterogeneous basic framework withsmaller sphere radii. Taking this into account, the struc-tures may be described as [ D(6o) + (T' + D' (4t)) ] + Tand [D(6o) + (T' + D'(4t))] +D, respectively.

In NiTi2 three quarters of the Ti atoms form a basicframework D(6o). The other Ti atoms occupy the T ;

voids, which allow radii almost equal to the radii of theframework atoms. The smaller Ni atoms form a D'(4t)arrangement (see also Chapter 17 by Nevitt and Kochin this volume).

3.4 The ..dI2(6o) Framework

The ..dI2(6o) framework of Figure 10 is related to theI(6o) framework in a similar way as the a..P2(6o)framework to the P(6o) framework: an undistorted I(6o)framework with 96 points per unit cell and doubledtranslation periods can be built up in the general positionof Ia3d. Without losing shortest distances, the octa-hedra may be tilted in such a way that additional shortestdistances are produced. Octahedra of different orien-tations can be mapped onto each other by glidereflections according to diagonal d glide planes. Thisis indicated by the orientation-symmetry part ..d of theframework symbol. The sphere packing type correspon-ding to such a framework is 6/3/c3 8. The respectiveDirichlet domains are not especially simple, but theyshow some small faces that may be regarded as singlepoints or lines for the derivation of the voids. From thisview, six kinds of voids remain within a ..dI2(6o)framework: ideal octahedra around I2; slightlyelongated tetrahedra around S*; trigonal prisms aroundY** and tetragonal antiprisms around V*, both slightlydeformed; highly distorted octahedra with twofold

symmetry; and highly distorted tetrahedra in a generalposition.

With the high number of symmetrically differentshort distances within a ..dI2(6o) framework, smallchanges of the coordinates permit adaptation torelatively different radii of atoms located in the voids.This explains the large variety of crystal structures withthe garnet type, which contain ..dI2(6o) frameworks.In addition some further structure types with such aframework exist. These other structure types differ withrespect to the connections of the occupied voids (Hellneret al., 1979).

The bismuth atoms of a-RbBi4 may be regarded asa basic framework with the tetragonal antiprismaticvoids occupied by rhodium atoms. In this structure thecorners of the antiprisms have nearly equal distancesfrom the centers.

3.5 The 1(12i) Framework

An ideal 1(12i) framework, as illustrated in Figure 11,consists of parallel-oriented icosahedra with centersforming an I lattice. The vertices of these icosahedraare connected by additional shortest distances. Thecorresponding sphere packing type is 9/3/c3. Each pointof such an ideal framework has nine nearest neighborsand one additional neighbor with a slightly largerdistance (6%). The coordination number of an 1(12i)framework should therefore be regarded as 10. Withvery similar coordinates, sphere packings of anothertype (6/3/clO) may be obtained. Compared with 9/3/c3,four contacts per sphere are lost, but the additional tenthshort distance of 9/3/c3 gives rise to the sixth spherepacking contact on 6/3/clO.

The aluminum atoms of WAl12 form a slightlydistorted 1(12i) basic framework with the icosahedralvoids centered by the tungsten atoms.

In skutterudite, CoAs3, the arsenic atoms form ahighly distorted 1(12i) configuration (y = 0.3431,Z = 0.1503) with the octahedral voids occupied by cobaltatoms. The cobalt and arsenic atoms together formdistorted pentagonal dodecahedra (20pd) around I.These dodecahedra share those eight vertices which referto the cobalt atoms.

3.6 The ..nl(12i) Framework

A ..nl(12i) framework consists of ideal icosahedra withcenters at I. The icosahedra around the points 00 0 andy y y have opposite orientations but their axes areparallel to each other. These icosahedra can be mappedonto each other by a glide reflection along a diagonal

Figure 10 The ..dI2(6o) framework: (a) point configuration with lines indicating the octahedra around I2, only polyhedra inthe lower half of the unit cell are shown; (b) S* tetrahedron around 0 ̂ T ; (C) [P4'-Y**]XX(21) tetrahedron around 0.9197 0.14310.0339; (d) trigonal prism around Y**, { } { ; (e) W2'xx octahedron around 0.2971 0.9529 0.1250; (f) tetragonal antiprism aroundV*, 0-|-g-. z coordinates given in a/100

Figure 11 I(12i) framework: (a) point configuration with lines indicating the icosahedra around I; (b) four-sided pyramid aroundI(12i), 0 0.1575 0.4123; (c) P2' octahedron around { { { ; (d) trigonal prism around J* (21), 0.3981 Oy.z coordinates given ina/100

Figure 12 The ..nl(12i) framework: (a) point configuration with lines indicating the icosahedra around I; (b) W tetrahedronaround \\0; (c) P2' octahedron around \ \ \ \ (d) 12-coordinated polyhedron around W , | 0 y . z coordinates given in a/100

Figure 13 The ..cP2(12i) framework: (a) point configuration with lines indicating the icosahedra around P2, only polyhedra withcenters at z = 0 are shown; (b) tetrahedron around 96(i) Oy Z1 0 0.0752 0.1968; (c) J2' tetrahedron around ^ 0 0; (d) P2' snub-cube (24scu) around T T T - Z coordinates given in a/100

glide plane. The vertices of the icosahedra are connectedby additional shortest distances (cf. Figure 12). Thecorresponding sphere packing type is 7/3/cl.

The crystal structure of UH3 contains a distortedframework of this type built up by the hydrogen atoms(y = 0.155, z = 0.31). The uranium atoms fill up all 12coordinated voids within the framework. In AuZn3 thezinc atoms form a very similar configuration as abasic framework (y = 0.16, z = 0.30). The gold atomsoccupy analogous positions as the uranium atoms inUH3.

3.7 The ..CP2Cl2i) Framework

A member of the ..cP2(*12i) family is the structureNaZn13 in Fm3c. A framework ..cP2(12i) contains idealicosahedra in two orientations (cf. Figure 13). The axesof all icosahedra are parallel to one another. Polyhedraof different orientations can be mapped onto each otherby glide reflections along diagonal glide planes accordingto the orientation-symmetry symbol ..c given in frontof the framework symbol; furthermore besides twosmall tetrahedra, one finds a large snub-cube, which isfilled by large alkali, alkaline-earth, or rare-earthelements.

4. Nets in Orthohexagonal Arrangements

In [1 1 0)-type planes, the b.c.c. structure forms 36 netsof atoms consisting of isosceles triangles with angles ofapproximately 55°, 55°, and 70° (Figure 14). Eachtriangle thus has one unique corner and it is convenientto indicate this by chemical component in consideringthe ordered CsCl structure, since in (1 1 0) planes the

triangles of the CsCl structure have two corners (55°)occupied by one component and one (70°) by theother. Indeed, it is seen that the 36 net is made upof two interpenetrating 44 rectangular nets. In theb.c.c. or CsCl structures, successive nets in [110]directions are stacked one above the other, so that theunique corners of the isosceles triangles of one net centerthe midpoint of the edge opposite the unique corner ofthe triangles in the nets above and below it, as shownin Figure 14, thus giving two stacking positions A°and B°.

In this way, b.c.c. structures and distorted b.c.c.structures can be described by nets. A more generalapproach is possible by orthohexagonal arrangementsas defined in Hellner et al. (1992b) and Pearson (1993):A hexagonal unit cell can be described on orthorhombicaxes in one of three possible orientations according tothe matrices:

M:?M1!)-M5i)given in the International Tables for X-Ray Crystal-lography (1952, pp. 19, 21). Such an orthohexagonalcell {a, b, c) has dimensions, for example, a-aQ,b = J3ao, c = c0, where a0 and C0 are the dimensions ofthe primitive hexagonal cell. Atomic arrangements areprimitive (P) in the hexagonal cell and centred (A, B,C or F) in the orthorhombic cell.

Previously Hellner (1986) had loosely used the symbol'hC to refer to the approximately orthohexagonalatomic arrangement in the few structures with tetragonalor orthorhombic symmetry in which it was observed.Our present study of orthorhombic structures revealsthat the phenomenon is much more widespread than

Figure 14 CsCl (or b.c.c. with one component atom) (1 1 0) plane of atoms, indicating triangle angles and two equivalent stackingsites A° and B°

previously imagined, so that it is necessary to adoptmore precise symbols with which to describe it. Hencewe now adopt a matrix notation (|M| =2) to describethe centered orthohexagonal arrangement indicating itby the superscript symobol 'o' (cf. VD, VT). This wefollow by a, b, or c to indicate the unique axis of thetetragonal, orthorhombic, or monoclinic structure onthe plane normal to which (i.e. 1 0 0), (0 1 0) or (0 0 1)respectively) the unit-cell edges (i.e. b/c, a/c, a/brespectively) have the approximate ratio J3 or 1/J3required for an orthohexagonal description. Thus wehave the three matrices

/ 1 0 0 \ / 1 0 M / 1 1 0 \oa = 0 1 1 ) ob=[ 0 10 and oc = 1 1 0

\ 0 T l / V l O l / \ 0 0 1 /

which can be applied to the primitive hexagonal latticecomplexes or point configuration descriptors; e.g. oaH,obH, OCH, oaG, obG, 0CG, oaN, obN, OCN, oahD, ochT,OC±Q, etc. It is only necessary to consider one orien-tation of each of these orthohexagonal arrangements,and we choose that corresponding to O2 above.Although it should be obvious, perhaps it has to bestated that, in order to give a framework descriptionin orthohexagonal terms, not only does the ratio of twounit-cell edges have to be approximately 1/J3, but alsothe arrangement of the atoms on the plane defined bythe ratio has to be appropriate; it does not sufficeto have one condition alone satisfied. Thus, forexample, in the SiTi oP8 structure, although the atomicarrangement on the (0 1 0) plane may correspond to afunction of obH, this description should not be used,as c/a = 1/1.375 is far removed from the value 1/J3

In Table 10 we give the primitive hexagonal latticecomplexes H, G, and N in their standard settings andthe corresponding point configurations OCH, OCG, andOCN in their standard settings in orthorhombic andmonoclinic space groups when a/b = ̂ (or 1/J3), butit should be noted that here OCN is crystallographicallyheterogeneous.

We can now apply this notation to describe an

approximately orthohexagonal arrangement of atomswhen it is found on [1 0 0)-type planes of structures ofany class, tetragonal, orthorhombic, monoclinic oranorthic, since the matrices oa, ob, and oc apply onlyto the transformation of primitive hexagonal toorthohexagonal, the hexagonal cell being taken witheither a, b, or c unique for future comparisons.However, we also find approximately orthohexagonalarrangements to occur sometimes on (1 1 0]-type planesin crystals of these classes; indeed, it is the most frequentoccurrence of orthohexagonal arrangements in thetetragonal class. Since we have no mechanism forstructural description on (1 1 0]-type planes, such cellshave to be reset so that the [1 1 Oj-type planes become(1 0 0J-type planes.

In the tetragonal class this resetting can be indicatedby specifying a space-group change, e.g. P-+C, I->F,C->P22, etc., but in the orthorhombic and lower classesthe appropriate space-group changes have not beenworked out. Hence the resetting has to be indicated byspecifying the matrix change (generally |M| =2, sub-sequently referred to as M(2) for ease in tabulating) thatit involves. This matrix change must not be confusedwith that of primitive hexagonal to orthohexagonal,since it is quite independent thereof.

The reset tetragonal cells have axes a, b9 c appropriateto the space group quoted, but in the lower classesthe reset cells have new axes a' =b' = (a2 + 62)1/2,a' =cf =(a2 + c2)l/2 or b' +c' = (l?1 + c2y/2 and the ratiosc/a1, b/a' or a/b' equal J3 or its reciprocal, or anintegral multiple of either in the case of a superstructureof the orthohexagonal cell.

Another reason for our change to matrix notation isthat the previous use of the prefix h as in hC to referto a centered orthohexagonal arrangement of atoms canbe confused with its use in hD and hT where it signifiesdiamond, D, and tetrahedra, T, point configurationdescriptions of these cubic lattice complexes when theyoccur in the primitive hexagonal cell. This confusionis now removed, and indeed we can, for example,describe ochD without ambiguity.

Table 10 Standard settings of H, G, and N in the primitive hexagonal cell and of OCH, OCG, and OCN point configurations inorthorhombic and monoclinic cells with b/a = h

36

63

6363

Primitive hexagonal settingin P6/mmm

HGN

Ia2c3f

Orthorhombic settingin Cmmm,

b/a = $o c H

OCQ

o c N

2aAiyy = \2b, 4e

Monoclinic setting in C2/m(b unique),

b/a = $y (3 = 90°

o c H

OCQ

o c N

2a4g, y = r2b, 4e

In the descriptions of tetragonal cells with ortho-hexagonal arrangements, complications do arise never-theless, since projections down A and B are identical.When the orthohexagonal description occurs on (1 0 0)and (010) planes, we choose (01 0) so that thematrix symbol is obH, etc. When it occurs on the (110)plane and a centered tetragonal setting is required,

we choose (0 1 0) of the centered tetragonal cell sothat again the matrix symbol in the centered cell isobH, etc. However, because projections down a andb are identical, * non-cry stallographic' features mayoccur in the point configurations used to describethe structure. Although these may require splitting ofWyckof f sites so that half the atoms occur in one point

Heights nh 2

Heights n/8

Figure 15 Diagram of the AlAu2 oP12 structure with idealized metric a/2 = b and idealized atom positions showing the

/ 1 0 l \M(I) = I 0 1 0 I cell, the framework (2f)yy^(obH211+obG211) as a superstructure of the MoSi2 tI6 structure

U O l /

Table 11 Orthorhombic and tetragonal structures exhibiting centered pseudohexagonal atomic arrangements on (1 10 }-type planesand requiring a centered orthorhombic or tetragonal cell for description. Note 1/^3=0.577

Substance Symbol Axial ratioSpacegroup Wyckoff

Centeredcell Framework

Number ofsubstances

Superstructures

CaCr2Al10 tP52 ^ =0.565 P4/mmm 2a,c, 8g,h,i2,j2 tC104 °bT2^0H261ly + >0G 2 4 1 lx

+ TTON241Iy)

1

Edge-centered stacking

Pa

Al3Os2

MoSi2

MoPt2

ReSi2

AlAu2

Al4Ba

BaNiSn3

Be2CaGe2

tI2

tllO

tI6

oI6

oI6

oP12

tllO

tllO

tPIO

— =0.582aj2

— - =0.646°5aj2

i=lJ34

< =1.731

y+b2

- =1.895*J(a/2)2 + b2

a~2bc

— =1.739a]2

— =1.604ajl

-^= =1.745a]2

I4/mmm

I4/mmm

I4/mmm

Immm

Immm

Pnma

I4/mmm

I4/mmm

P4/nmm

2a

2a, 4e2

2a, 4e

2a, 4g

2a, 4i

4c3

2a, 4d,e

2a3, 4b

2a,b,c3

tF4

tF20

tF12

M(I)

M(I)

M(I)

M(A)

tF20

tF20

tC20

(2f)obH

(100obHlz

(2f)ob(H + G)

/ 1 0 l \(2f)oc(H + G), Af(2)= 0 10

V l O l /

/ i T o \(2f)ob'(H + G), M(2) = [ 1 1 0

\ 0 0 1 /

( 2 0 ™ ° b ' ( H + G)2n

/ i 2 ° \a = c, M(2)= T 1 °

V l O l /

(40olroob '(H +G)211

/ 1 2 0 \A/(4)= 1 2 0

V l O l /

( 2 0 ( T T T + TTT)°bH(«4t r e oo )(001)

+ (2f)obH or(4d)(000 + y00)obH

- (2f)iOOobU) + (2f)iO0obH2z)

(4d)(000 + |O0)o bHlz

+ (20(>iiobHlz

(20ob(i-irH + (ooo + joo)2

H(«4t4e oo )) or(001)

(4d)(000 + y0°)° b H - (2f)TTTobH)

+ (20TTT°b H2z

3

4

84

8

1

4

462

6

10

i

(continued)

configuration and half in another; this seems unavoid-able!

AlAu2 oP12 with c/[(a/2)2 + b2]U2= 1.895 is aninteresting example of an orthorhombic structure that,with idealized a/2 = b and idealized atom positions,can be set in a cell with \M\ =2 and can be describedwith an orthohexagonal framework: (2f)TTir(obH211

+ obG2n); see Figure 15. This clearly shows it to be a

superstructure of the MoSi2 tI6 structure, which in thecentred tF12 setting is described as (2f)(obH + obG). Itcould be argued that the above ratio of 1.895 isconsiderably larger than the ideal ratio 1.732 = J3;however, many substances with MoSi2 structure alsohave axial ratios (c/a) considerably larger than theideal value of c/a = 2.45 of MoSi2 in I setting andrequired for an ideal orthohexagonal cell. Hence, slight

'Note, in centered cell, 5a/c= 1.548 <J3.Îdealized atom positions.c Actual atom positions.^Note this filled-up MoSi2 structure is like a MoSi2 structure phase such as AuZr2 with c/a« 3.5 instead of 2.5.

Table 11 {continued)

Substance

C2IrU2

CoNb4Si

Al2Cu

BaMg2Sn2

Ga5Mg2

GaU

Ru2Sn3

Symbol

tllO

tP12

tI12

tP20

tI28

oC32

tP20

Axial ratio

— =2.537*a$

-^z =0.577a$

-^- = 0.568

4= =1.750x2a]2

# + * =1.751b

-^= =0.568x2a\2

Spacegroup

I4/mmm

P4/mcc

I4/mcm

P4/nmm

I4/mmm

Cmcm

P4c2

Wyckoff

2a, 4e2

2a,c, 8m

4a, 8h

2a,b,c6, 4f

4e, 8h, I6n

4a,c, 8e,f,g

2b,c, 4f,i, 8j

Centeredcell

tF20

tC24

tF24

tC40

tF56

M(2)

tC40

FrameworkNumber ofsubstances

(2f)ob(H + (000 + 0Oy)G)

(2f)ob((i00 + i<4-)Hly

+ (00^ + OOf)H + -HOG)

(20ob( (OOi + 0(>i)H + (xOO +

Ôy)Hly + i | 0 G ) or (2f)ob

, i 2C(O(^- +OOf)H(«4t2e oo

(100)

+ (|O0 + ^ ) H I y )

(2f) o b ( l i J H 1 1 2 + (OOO + J O O ) H 1 1 2

( • 4 ^ oo ))(001)

(2f)ob (G + (OTO + (>TT)H 1 y

+ (0OT + OOT)HIZ + ( T T T +

}^)H2xyz)

(2f)(0Oy + T<>T)oc'N + (2f)

6

19

101

3

1

1

(i^+irl+M+fH-r'Hixz

+ TTTl2I2Iy + TTTl2,22xzly)

/ i o T \M(2)= 0 10

V l O l /

(4d)[obH112 + ( o bG 1 1 2 - i0iP(21))] 3

latitude in bending 'rules' may give benefits in showingstructural relationships. MoSi2 structures are also dis-cussed in Chapter 17 by Nevitt and Koch in thisvolume.

Orthohexagonal atomic arrangements may exhibitone of the three possible cases of edge-centered stacking(f), (q), (d), which we have already had occasion todescribe in conjunction with the earlier symbols hC andhG (e.g. Hellner et al., 1990). With 36 nets there areonly three possible variants of edge-centered stacking:(f) which involves two stacking sites A0, B0; (q) threesites B0, C0, D0; and (d) four sites A0, B0, C0, D0

(Hellner et al., 1992b, ch. 2). We recognize this fact byhenceforth referring to (2f), (3q), and (4d) in order tobring the nomenclature in parallel with that for medianstacking, where we no longer refer to hexagonal andcubic close packing, respectively, as (h) and (c),but as (2h) and (3c). Thus we have achieved logicalconformity with other stacking descriptions such as, forexample, (4h), (6h), etc., which are observed in realstructures.

Structures also occur in which the stacking isintermediate between the edge-centered and medianpositions. An example of this is found in the TaPt2

structure (oC12, 4c and 8g of Cmcm). There is asuperstructure of the centered pseudohexagonal cellalong a, and since b:a/3 = 1.708, the angles of thetriangles of the 36 nets formed by atoms on the (0 0 1)planes scarcely differ from 60° (see e.g. Hellner andPearson, 1984). The cell contains two 36 layers stackedalong [00 1], but the sites are certainly intermediatebetween edge-centered and median. In these circum-stances, it appears that the best description that can begivenis(2fh)0l|ocH311ly.

Table 11 lists orthorhombic and tetragonal structuresexhibiting centered pseudohexagonal descriptions.

5. Summary

The I structures are described with the 2 pointer I andits voids as a sphere packing type; the higher orders givethe chance to describe them by splitting procedures, asin the 8th, 27th, 64th, and 216th orders. Especially the7-brass structure is described as W*(4t) with large voidsaround I. The compound Tl7Sb2 is an ideal 7-brasswith filled voids. The structures of 216th order arefilled with clusters as in Tl7Sb2 and as in the a-Mnstructure.

Other structures of the I family are frameworks withpolyhedra around each i point, which can partly behandled as homogeneous sphere packings with self-

coordination and different voids. Besides some hetero-geneous cases, a net description follows at the end withorthopseudohexagonal settings and gives relationsamong a lot of hexagonal, rhombohedral, and lower-symmetry crystal classes.

If further information about the behavior in lower-symmetry space groups is wanted, see Hellner andPearson (1986, 1987, 1988) and Hellner et al. (1992b,1993) in Physics Data 16-5, 16-6, 16-7, 16-9, 16-10. Inaddition to the hexagonal and rhombohedral structuresin Physics Data 16-8 (Hellner et al., 1990), there isanother survey in Physics Data 16-4 (Hellner et al.,1992a).

6. References

Born, M. (1923). Atomtheorie desfesten Zustandes. Leipzig,Berlin.

Cohen, M. L., and Heine, V. (1970). Solid State Phys., IA, 37.Donnay, J. D. H., Hellner, E., and Niggli, A. (1964) Z.

Kristallogr., 120, 369.Fischer, W. (1971). Z. Kristallogr., 133, 18.Fischer, W. (1973). Z. Kristallogr., 138, 129.Fischer, W. (1974). Z. Kristallogr., 140, 50.Fischer, W. (1991). Z. Kristallogr., 194, 67, 87.Fischer, W. (1993). Z. Kristallogr., 205, 9.Fischer, W., and Koch, E. (1974). Z. Kristallogr., 139, 266.Fischer, W., Burzlaff, H., Hellner, E., and Donnay, J. D. H.

(1973). Space Groups and Lattice Complexes (NationalBureau of Standards, Monograph 134). US Departmentof Commerce.

Fornasini, M. L., Chabot, B., and Parthe, E. (1978) ActaCrystallogr., B34, 2093.

Heine, C , Heine, V., and Weaire, D. (1970). Solid State Phys.,24, 1, 249.

Hellner, E. (1986). Z. Kristallogr., 175, 227.Hellner, E., and Koch, E. (1980). Can. J. Chem., 58, 708.Hellner, E., and Koch, E. (1981a). Acta Crystallogr., A37, 1.Hellner, E., and Koch, E., (1981b). Acta Crystallogr., B38, 376.Hellner, E., and Pearson, W. B. (1984). Z. Kristallogr., 168,

255.Hellner, E., and Pearson, W. B. (1986). Frameworks for 'Inter-

metallic Phases' with Structures in Space Groups of the4-Stem ofU/mmm {Physics Data 16-5). FIZ 4, Karlsruhe.

Hellner, E., and Pearson, W. B. (1987). Structure TypeDescriptions for Intermetallic Phases in the Space GroupsI4/mmm and I42/mcm and their Subgroups (PhysicsData 16-6). FIZ 4, Karlsruhe.

Hellner, E., and Pearson, W. B. (1988). Structure TypeDescriptions for Intermetallic Phases in the Space Groups14,/amd and 14,/acd and their Subgroups (Physics Data16-7). FIZ 4, Karlsruhe.

Hellner, E., and Sowa, H. (1985) The Cubic Structure TypesDescribed in Their Space Group with the Aid ofFrameworks (Physics Data 16-3). FIZ 4, Karlsruhe.

Hellner, E., Gerlich, R., Koch, E., and Fischer, W. (1979).The Oxygen Framework in Garnet and its Occurrence inthe Structures OfNa3A12Li3F12, Ca3A12(OH)12, RhBi4 andHg3TeO6 (Physics Data 16-1). FIZ 4, Karlsruhe.

Hellner, E., Koch, E., and Reinhardt, A. (1981). TheHomogeneous Frameworks of the Cubic CrystalStructures (Physics Data 16-2). FIZ 4, Karlsruhe.

Hellner, E., Schwarz, R., and Pearson, W. B. (1990) StructureType Descriptions for Intermetallic Phases in theHexagonal and Rhombohedral Systems (Physics Data16-8). FIZ 4, Karlsruhe.

Hellner, E., Schwarz, R., and Pearson, W. B. (1992a).Introduction to an Inorganic Crystal Chemistry II (PhysicsData 16-4). FIZ 4, Karlsruhe.

Hellner, E., Schwarz, R., and Pearson, W. B. (1992b).Structure Type Descriptions for Intermetallic Phases inthe Orthorhombic System (Physics Data 16-9). FIZ 4,Karlsruhe.

Hellner, E., Schwarz, R., and Pearson, W. B. (1993).Structure Type Descriptions for Intermetallic Phases inthe Monoclinic System (Physics Data 16-10). FIZ 4,Karlsruhe.

Hermann, C. (1960). Z. Kristallogr., 113, 142.Hume-Rothery, W. (1926). The Structure of Metals and Alloys.

Institute of Metals, London.International Tables for Crystallography (1983). Vol. A (ed T.

Hahn). Reidel, Dordrecht.International Tables for X-Ray Crystallography (1952). Vol. 1

(eds N. F. M. Henry and K. Lonsdale). Kynoch Press,Birmingham.

Internationale Tabellen zur Bestimmung von Kristallstrukturen(1935). Vol. 1 (ed. C. Hermann). Gebriider Borntrager,Berlin.

Jones, H. (1934). Proc. R. Soc, A144, 225.Knecht, J., Fischer, R., Overhoff, H., and Hensel, F. (1978).

/ . Chem. Soc, Chem. Commun., 905.Lima-de-Faria, J., Hellner, E., Liebau, F., Makovicky, E.,

and Parthe, E. (1990). Acta Crystallogr., A46, 1.Madelung, O. (1918). Phys. Z., 19, 528.Malrieu, J. P., Maynau, D., and Daudey, J. P. (1984). Phys.

Rev., B30, 1817.Nagorson, G., and Witte, H. (1953). Z. Anorg. Chem., 217,144.Niggli, P. (1919). Geometrische Kristallographie des

Diskontinuums. Leipzig.Pauling, L. (1940). The Nature of the Chemical Bond. Cornell

University Press, Ithaca, NY.Pearson, W. B. (1993). Z. Kristallogr., 204, 239.Samson, S., (1972). Acta Crystallogr., B28, 936.Samson, S., and Hansen, D. A. (1972). Acta Crystallogr.,

B28, 930.Simon, G., and Bloch, A. N. (1973). Phys. Rev., B7,

2754.Villars, P. , and Calvert, L. D. (1985). Pearson's Handbook

of Crystallographic Data for Intermetallic Phases, Vol. 1.American Society for Metals, Cleveland, OH.

vom Felde, A., Sprosser-Prou, J., and Fink, J. (1989). Phys.Rev., B40, 10181.

Zintl, E. (1939). Angew. Chem., 52, 1.Zintl, E., and Dullenkopf, W. (1932). Z. Phys. Chem., B16, 183.Zunger, A. (1980). Phys. Rev., B22, 649.

This chapter was originally published in 1995 as Chapter 13 in IntermetallicCompounds, Vol. 1: Principles, edited by J. H. Westbrook and R. L. Fleischer.

1. Introduction

Some 25 years ago when the same author wrote thechapter on 'Wurtzite and sphalerite type structures'for the first edition of Intermetallic Compounds(Parthe, 1967), these kinds of compounds, particularlyIII-V compounds such as InSb and GaAs, were thecenter of attention of solid-state physicists and chemists,who were excited by the new discoveries in semiconductorphysics and the resulting new devices that revolutionizedso many facets of our daily life. This was perhaps thereason why a chapter on wurtzite and sphalerite-relatedstructures was included in the book, although thesecompounds are not intermetallics in the usual sense.

Today the center of action has shifted to other kindsof compounds such as the high-temperature oxidesuperconductors. These days, new wurtzite andsphalerite-related structures do not appear so often inthe literature. However, the reader will be treated here toone of the few successful attempts to correlate structuralfeatures with composition. In particular, it will be shownthat two parameters, to be calculated from the com-position and the valence-electron contribution of theparticipating elements, can be used not only to pinpointpossible errors in the composition or structure ofreported compounds, but also to predict probablestructural features of unknown compounds.

2. Definition and Classification ofAdamantane Structures

The topics of this chapter are the wurtzite and

sphalerite structures and all the structures related tothem as stacking and/or vacancy and/or substitutionalvariants. For these structures the summary termadamantane structures is often used.*

Adamantane structures, obtained only with normal-valence compounds, form a subdivision of thetetrahedral structures. The cations and anions occupythe Zn and S sites, respectively, of the wurtzite orsphalerite structure (or another stacking variant).Every atom on a cation (Zn) site is tetrahedrallycoordinated by four anions. Conversely, every atom onan anion (S) site is tetrahedrally coordinated by four(or fewer in the case of a defect adamantane structure)cations.

The origin of the coining of the term * adamantanestructure' stems from the close geometrical relation-ship between the two diamond and the two ZnSmodifications. As shown in Figure 1, the wurtzite andthe sphalerite structures are ordered substitutionalvariants of the hexagonal diamond and the cubicdiamond structure, respectively. The rare hexagonaldiamond, also called lonsdaleite (Frondel and Marvin,1967), is a stacking variant of the common cubicdiamond.

The adamantane-structure compounds are con-veniently classified according to three aspects:

*In the earlier literature can be found the term 'adamantine'structures (Spencer et al.% 1962); however, von Schnering (1989)pointed out that, since there are only single bonds present, itis more appropriate to denote these structures as 'adamantane'structures.

Chapter 4

Wurtzite and Sphalerite Structures

Erwin PartheLaboratoire de Cristallographie aux Rayons X, Universite de Geneve, 24 Quai Ernest Ansermet,

CH-1211, Geneva 4, Switzerlandand

Institute for Mineralogy and Crystallography, University of Vienna, Geozentrum, Althanstrasse 14,A-1090 Vienna, Austria


• The stacking of the ZnS base structure, as in wurtziteor as in sphalerite or as in a more complicated ZnSpolytype.

• The percentage of vacant cation (Zn) sites, or,expressed differently, the overall cation-to-anionratio. One distinguishes between normal adamantane-structure compounds with composition CnAn, whereall Zn and S sites are fully occupied, and defectadamantane-structure compounds with compositionCn _p OpAn, where some of the available Zn sites arenot occupied ( • stands for a vacancy). Examples ofa normal and a corresponding defect adamantane-structure compound are ZnS (wurtzite) and a' -Ga2S3

(where one of three cation sites remains unoccupied).• The substitutions on the cation (or anion) sites, leading

to ternary or multinary adamantane structures. Anexample of a series of ordered substitutionalderivatives of zinc blende is ZnS (sphalerite) -*Cu3SbS4 (famatinite) -> Cu2FeSnS4 (stannite).

We shall demonstrate below that the number of Znsite vacancies in a defect adamantane structure andthe kind of atoms that may substitute on the Zn sitesare not arbitrary, but depend on the valence-electronnumber of the participating atoms.

3. The ZnS Stacking Variants

Some 200ZnS polytypes are known, all having

hexagonal unit cells with base planes of the same sizebut with different cell heights. The S atom partialstructures of all polytypes are close-packed with the Znatoms occupying in regular fashion one of the two kindsof tetrahedral interstices. Five of these ZnS polytypesare presented in Figure 2. Instead of drawing thecomplete hexagonal or triple-hexagonal cells and allthe atoms contained in them, as was done in Figure 1,only the [1120} planes of the cells and the arrangementof the atoms in these planes are shown in Figure 2.

The ZnS polytypes can be considered as an inter-growth of common identical slabs, which are stackedin a different way. Cutting the structures by slicing theS atoms along the horizontal planes, indicated by brokenlines, one obtains the common identical S172-Zn-S172

slabs. As indicated on the upper left of Figure 2, thereexist two possibilities of how two of these slabs can beintergrown. They differ by a rotation of the upper slabby 180° around the hexagonal axis. These twopossibilities of slab stacking give rise to the formationof the different ZnS polytypes.

Assuming that ah and ch are the hexagonal unit-cellconstants of a polytype, then the height of one slab isgiven by

rf=ah(2/3)1/2 = 0.8165ah (1)

This value corresponds to the height of a tetrahedronwith edge length ah. For a polytype built up of N

Figure 1. Structure of hexagonal diamond compared with that of wurtzite and of cubic diamond compared with sphalerite. Forsphalerite the non-conventional triple-hexagonal cell is also shown, which is preferred when one wants to compare sphalerite withthe other ZnS stacking variants. Small filled circles indicate Zn and large open circles C or S atoms

Diamond Zinc sulfide

Hexagonal diamond

and wurtzite

Cubic diamondand sphalerite

intergrown slabs, the ch/ah ratio of the hexagonal ortriple-hexagonal unit cell is given by

ch/ah = N(2/3)l/2 = 0.8165JV (2)

For the particular case of the ZnS polytypes, thenumerical values are: ah-3.825 A9 d-3A23A and ch

= 3.123iVA.Polytypes of great variety are found with binary

normal adamantane-structure compounds, particularlywith ZnS and SiC. In a table published by Mardix(1986) there are listed 196 properly identified ZnSpolytypes. Mardix also advances an approach toexplain the formation of the polytypes as a thermo-dynamic phenomenon taking place with the help of,but also regulated by restrictions imposed by, dis-locations. These ideas have at present no directrelevance for ternary or multicomponent normaland defect adamantane-structure compounds, becauseas base structures only the common wurtzite (ZnS 2H)and sphalerite (ZnS 3C) structures occur here. Oneexception only has been reported. A modification ofCu2ZnGeS4 (Moodie and Whitfield, 1986) crystallizeswith a structure that can be considered as a sub-stitutional variant of ZnS 12R polytype (right-hand sideof Figure 2).

Different notation schemes have been developed tocharacterize the stacking of the slabs. Four different poly-type notations are listed below each drawing in Figure 2.Besides the well-known ABC notation, there are in com-mon use the Jagodzinski-Wyckoff and the Zhdanovnotations. From the last two one can also derive theproper space group for a polytype with a given kind ofstacking. The details are discussed in the appendix(Section 10.1).

4. Valence-Electron Rules for Adamantane-structure Compounds

The adamantane structures form a subdivision of thetetrahedral structures. The latter are formed by iono-covalent compounds for which two valence-electron rulescan be formulated, one referring to the limiting ionic andthe other to the limiting covalent bonding state. We shallfirst discuss the more general rules for tetrahedralstructures and then the modified rules for adamantanestructures.

4.1 Rules for Tetrahedral-Structure CompoundsThe valence-electron rules for tetrahedral structures havebeen amply described in the literature (Parthe, 1972,

Figure 2. Atom arrangements in the {1120} planes of five ZnS polytypes, interpreted as an intergrowth of a common slab. Foreach polytype are given the Ramsdell, ABC, Jagodzinski-Wyckoff, and Zhdanov notations, and the space group. Small filledcircles represent Zn atoms and large open circles S atoms

ZnS 2HwurtziteAB(h)2

(D 2

P63mc

ZnS 3CsphaleriteABC(C)3

OO

F-43m

ZnS 4H

ABAC(hc)2

(2)2P63mc

ZnS 6H

ABCACB(hcc)2

O) 2

P63mc

ZnS 12R

ABACBCBACACB(hhcc)3

(1.3)3R3m

* 1990, 1992, 1996) and we can therefore restrictourselves to a short repetition.

4.1.1 Ionic Bonding State

In the limiting ionic bonding state the valence electronsof the cations are transferred to the anions in order forthem to obtain a stable octet configuration ns2np6. Inthe general case where the cations do not transfer alltheir valence electrons—they are used for bonds betweenthemselves or for non-bonding orbitals on the cations—or where the anions, due to shared covalent bonds betweenthemselves, do not need as many electrons to completetheir octet shells, one can formulate the followingvalence-electron rule, known as the generalized 8-Nrule:

VECA = 8 + CC/(n/m) - AA for CmAn (3)

VECA, the partial valence-electron concentration inrespect to the anion, is defined as

VECA = (mec + neA)/n for CmAw (4)

and can be calculated alone from the composition ofthe compound and the valence-electron contributionof the participating elements (eCj eA). The right-handpart of (3) refers to structural features such as AA,the average number of anion-anion bonds per anion,and CC, the average number of cation-cation bondsper cation, and/or the electrons used for lone electronpairs.

Depending on the value of VECA one distinguishesbetween polyanionic-, and normal-, and polycationic-valence compounds.

• If VECA<8: polyanionic-valence compounds withAA >0.

• If VECA = 8: normal-valence compounds with AA =CC = O.

• If VECA>8: polycationic-valence compounds withC O O .

4.1.2 Covalent Bonding State

In the limiting covalent bonding state all atoms formfour tetrahedral sp3 hybridized bonding orbitals, whichoverlap with the sp3 orbitals of the neighboring atoms.To form an sp3 hybrid each atom needs four valenceelectrons. Each orbit not used for bonding obtains asecond electron and becomes a non-bonding orbital.One can formulate the following valence-electron rule,known as the tetrahedral-structure equation:

VEC = 4 + 7VNBO forCÂ, (5)

VEC, the total valence-electron concentration, is definedas

VEC = (mec + neA)/(m + n) for CmA/J (6)

and can be calculated from the composition of the com-pound and the valence-electron contribution of theparticipating elements. The right-hand part of (5) con-tains the parameter NNBOi the average number of non-bonding orbitals ( = lone electron pairs) per atom. Theselone electron pairs one can 'see' indirectly in a structureby the absence of an expected tetrahedral atom neighbor.

Depending on the value of VEC one distinguishesbetween normal and defect tetrahedral-structurecompounds.

• If VEC < 4: a tetrahedral structure cannot be formed.• If VEC = 4: normal tetrahedral-structure compound

with iVNBO = O.• If VEC > 4: defect tetrahedral-structure compound

with NNBO>0.

4.2 The Adamantane-Structure Equation

As stated above, the adamantane structures form a sub-division of the tetrahedral structures. In the general caseof a defect tetrahedral structure, only N^130, the averagenumber of non-bonding orbitals per atom, is known,but not the relative positions of the non-bondingorbitals. In defect adamantane structures the non-bonding orbitals are grouped in quadruplets. Four non-bonding orbitals are positioned on the four anionssurrounding a vacant Zn site. One can use in thechemical formula the symbol • , which indicates avacant cation site and at the same time a quadrupletof non-bonding orbitals. Denoting the composition ofa defect adamantane-structure compound (as abeginning) as C ^ ^ D ^ with p cation sites beingunoccupied, the value of NNBO can be expressed by

NNBO = 4p/[(m-p) + n)] for Q n ^ n p A n (7)

Inserting (7) into (5), making use of (6), and multiplyingby [(/w—p) + /2], one obtains the following modifiedexpression for the tetrahedral-structure equation:

(m -p)ec + neA = 4[(m -p) + n] +4/7

for C n ^ D 1 A 1 (8)

Since adamantane-structure compounds are normal-valence compounds, one can also make use of (3)

in the special form where CC = AA = 0, which can bewritten as

(m -p)ec + neA = 8« for Cm_p upAn (9)

Combining (8) with (9), one obtains as a result, that

m = n (10)

The total number of cation sites (occupied and non-occupied) is thus equal to the number of anion sites.The compositions of adamantane-structure compoundsare thus CnA^ or Cn^DpAn , for normal and defectcompounds, respectively. Inserting (10) into (7) and (5)gives

N̂BO = 4/7/[(«-/?) + /i] = VEC-4 for C n ^DA n

(H)

and finally

p/n = 2-8/VEC for CnÛpAn (12)

This is the adamantane-structure equation (Parthe, 1987).Depending on the value of VEC one can distinguish

between normal and defect adamantane-structurecompounds.

• If VEC < 4: an adamantane structure cannot be formed• If VEC = 4: normal adamantane-structure compound

with composition CnAn.• If VEC > 4: defect adamantane-structure compound

with composition C^^nÂ^.

Occasionally the adamantane-structure equation isexpressed in a different form. Rewriting (12) as

[{In -/7)VEC] /In = 4 (13)

and substituting VEC by

VEC = [ (n -p)ec + neA] Z[(n-p) + n]forCn_pnpAn (14)

leads to another form of the adamantane-structureequation:

[(n-p)ec+p0 + mA]/[(n-p)+p + n]=4

for C n ^ n 1 A 1 ( }

where (n -p) is the number of occupied Zn sites, p thenumber of unoccupied Zn sites, and n the number ofS sites.

The ratio of the number of valence electrons of allatoms divided by the number of all atoms and vacanciesis always 4 (Pamplin, 1960), or, expressed differently,provided one counts the vacancies as 'zero-valent'atoms, the value of 4VEC is always 4.

4.3 The Upper VEC Boundary

Adamantane structures cannot exist with compoundswhere VEC < 4 because there are not enough valenceelectrons available for each atom to form four sp3

hybrid orbitals. However, there also exists an experi-mental upper VEC limit above which an adamantanestructure does not form. With the VEC value increasingabove 4, more and more cation sites remain unoccupiedand consequently the average number of anion-cationbonds per anion, labelled NBA _> c , decreases. The valueof NBA _ c can be expressed by

NBA _ c = 4 - 4(p/n) = 4 [ (8/VEC) - 1 ] (16)

Simple solutions of (12) and (16) together with knownexamples are given in Table 1.

The experimentally observed upper VEC limit occursat 4.923 where three out of eight cation sites areunoccupied and each anion has, on average, only 2.5tetrahedral bonds to surrounding cations. This VEClimit results from studies by Radaoutsan (1964) on thesolubility of GeSe2 (VEC = 5.333), a defect tetrahedralbut not an adamantane structure) in Ga2 • Se3 (VEC =4.80, a defect adamantane structure), which terminatesat Ga4Ge • 3Se8 (VEC = 4.923).

For compounds with larger VEC values, so manycation sites would be unoccupied that the averagenumber of anion-cation bonds extending from an anionwould be less than 2.5; or, expressed differently, an anion

Table 1. Simple solutions of the adamantane-structureequation for different values of VEC, p/n, and NBA^Ctogether with known examples of compounds with adamantanestructures

Formula

C11A11

C7DA8

C5DA6

C3DA4

C2DA3

C5D3A8

'C D A2'

VEC

4

4.267

4.364

4.57,

4.8

4.923

5.333

p/n

0

1/81/61/4

1/3

3/81/2

N B A . C

4

3.53.333

3

2.667

2.52

Examples

Cu2ZnGeS4 [122464],LiSiNO [1456]

Hg5Ga2Te8 [2532o68]Hg3In2Te6 [2332o66]CdGa2Se4 [ 2 3 2 D 6 J ,Cd3(AsI3) [ 2 3 D ( 5 7 3 ) ]

Ga2Se3 [3 2 a6 3 ] ,Si2N2O [ 4 2 D 5 2 6 ]

(Ga4Ge)Se8 [(3,4)0,6,)Impossible

Figure 3. The layer structure of red HgI2, which can beclassified as a defect tetrahedral but not as defect adamantanestructure (Parthe, 1990)

would have, on average more than 1.5 lone electron pairsattached to it. Owing to the mutual repulsion of the loneelectron pairs, the few remaining anion-cation bondsoften show a greater deviation from the ideal tetrahedralangle of 109.4°. Under these unfavorable circumstances,a wurtzite or sphalerite-related structure with a close-packed anion partial structure does not then form.Compounds with VEC>4.923 may, however, crystallizewith a defect tetrahedral structure.

One example is red HgI2 with VECA = 8 and VEC =5.333, shown in Figure 3. One recognizes slabs cut fromthe sphalerite structure; however, these slabs are notconnected by covalent A-C-A bonds and are verticallyand horizontally displaced with respect to each other.The layer structure of HgI2 can thus not be consideredas a sphalerite structure with cation vacancies.Nevertheless, it is a defect tetrahedral structure withNNBO - 4/3 (two non-bonding orbitals on each iodineatom).

5. Compositions of Adamantane-StructureCompounds

The compositions of adamantane-structure compoundsare controlled by the valence-electron rules given aboveand the valence-electron contribution of the elements.

Table 2. Elements found in adamantane-structure compoundsand their valence-electron contribution

The valence-electron numbers of the elements foundin adamantane-structure compounds are listed in Table2. They correspond (except for transition elements) totheir group number in the periodic table of the elements.In the case of transition elements, where the valence-electron contribution can vary, magnetic measurementsmay permit the distinction between different valencestates, such as between Fe2+ and Fe3 + .

To denote the compositions of adamantane-structurecompounds, we shall not use the usual symbols of thechemical elements, but instead use numbers for them,which correspond to the number of valence electronsgiven in Table 2.

5.1 The Two Methods to CalculatePossible Compositions

There exist two methods to calculate the compositionsof multicomponent adamantane-structure com-pounds: the cross-substitution method and the algebraicmethod.

5.7./ The Cross-Substitution Method

The compositions of possible adamantane-structurecompounds can be derived from the composition of anelement with four valence electrons, such as diamond,by the repeated procedure called 'splitting.' This is thereplacement of a pair of atoms of one kind by twodifferent ones, one to the left and one to the right inthe periodic table of elements, while keeping constantthe ratio of number of valence electrons to number ofatoms. The cross-substitution diagram or splittingdiagram, shown in Table 3, is only one of the manypossibilities. One can derive compositions not only ofnormal adamantane-structure but also of defectadamantane-structure compounds. In the latter case,one has to extend the list of valence-electron numbersby 0, an element with zero valence electrons beingconsidered equivalent to a cation site vacancy • in thestructure.

5.1.2 The Algebraic Method

This method is best suited for a systematic search ofall possible element combinations for a given generaladamantane-structure composition. One looks forsolutions of the adamantane-structure equation bysystematically changing the valence-electron values ofthe atoms. In Table 5 a demonstration of the applicationof this method is given.

Number ofvalence electrons

1234567

B groupelements

Cu, AgZn, CdB, Al, Ga, InC, Si, Ge, SnN, P, As, SbO, S, Se, Te, PoCl, Br, I

A and T groupelements

LiBe, Mg, Mn, Fe, NiFe

5.2 Binary Normal and Defect Adamantane-Structure Compounds

One first calculates the possible compositions of binarynormal-valence compounds with VECA=8 and thengroups them according to their VEC value as shown inTable 4. Adamantane structures can occur only for 4 ̂VEC<4.923.

There exist four possible compositions for binarynormal adamantane-structure compounds:

17, 26, 35, 44 (17)

and two for binary defect adamantane-structurecompounds:

43a54, 32a63 (18)

5.3 Ternary Normal Adamantane-StructureCompounds

5.3.1 'Two-Cation' Compounds withComposition Cn_xC'xAn

The adamantane-structure equation (12) has for p/n = 0the simple form

[(n-x)ec + xec,+neA]/2n = 4 for Cn_xC;An (19)

which can be rewritten as

x/n = (S-ec-eA)/(ec,-ec) (20)

One calculates the values of x/n for all possible differentvalues of ec<ec, <eA. The results of the calculation are

Table 4. Binary normal-valence compound compositionsordered according to their VEC value. Possible adamantane-structure compositions are printed in bold type

VEC 2.667 3.2 4 4.57, 4.8 5.333

eA = l 17 272

eA = 6 126 26 3 2 n 6 3 462

eA = 5 2352 35 4 3 D 5 4

eA = 4 44

presented in Table 5. Of interest are the five solutionsof x/n where 0< x/n < 1, which are printed in bold type.They correspond to the five possible compositions of'two-cation' compounds:

14253, 2452, 1362, 12463, 13564 (21)

5.3.2 'Two-Anion' Compoundswith Composition CnAn_yAy

The method described above can be modified tocalculate y/n values. One finds thus the five possiblecompositions of 'two-anion' compounds:

33427, 3246, 2257, 23472, 24373 (22)

5.4 Multicomponent Adamantane-StructureCompounds

The compositions of possible quaternary (and morecomponents) normal adamantane-structure compoundscan be obtained by adding and subtracting the formulaelisted in (17) and (21) or (22). The compositions ofternary (and more components) defect adamantane-structure compounds can be calculated by including also

Table 3. Example of a cross-substitution or splitting diagram for the calculation of the compositions of adamantane-structurecompounds together with known examples of normal and defect adamantane-structure compounds

Valence electron numbers of the atoms0 1 2 3 4 5 6 7

Compo-sitions

Examples

the formulae of (18). The method is demonstrated by thefollowing three examples, a quaternary normal and twodefect ternary structures, respectively:

• LiSiNO [1456]

+ 2 6+142 53

- 2 4 52

14 56

• Ag2Hg n I4 [ I 22n7 4]

- I 4 74

+ 32 n 63

+ 2 6- 1 2 3 2 _64_

I22n 74

• Ga4Ge D3Se8 [344n368]

+ 3 6 n 3 69

+ 124 63

- 1 2 3 2 JU_344n3 68

The formulae (17), (18) and (21) or (22) can be addedin any arbitrary proportion such as, for example:

(26)1_x + (136 2 ) ^ l x 2 1 _A6i + x

or

( 3 5 ^ + ( 3 , D 62)x^3l+xn ,5^xG3x

where 0 ^ J C < 1 . As a result the compounds withquaternary normal and ternary defect adamantanestructures are line compounds, their homogeneity rangein the corresponding quaternary or ternary phasediagram consisting of a composition line.

Examples for quaternary line compounds of composition1X2I-X 3x6i+x a r e t n e complete series of solid solutionsZnS-CuGaS2 or CdTe-CuInTe2 (Goryunova, 1965).Examples for ternary line compounds of composition31+A:nx51_A:63x are the complete series of solid solutionsGaAs-Ga2Se3 or InAs-In2Te3 (Goryunova, 1965).

In Figure 4 are presented the composition lines forternary adamantane structures in the 1-3-6, 2-3-6, and3-4—6 ternary phase diagrams. One plots in a ternarydiagram the line connecting points with VECA = 8 andthe lines corresponding to VEC = 4 and VEC = 4.923.Adamantane structures are found on the line forVECA = 8 between the limits 4 < VEC^4.923. In most ofthese line compounds the cations and vacancies occupythe Zn sites in random fashion. With certain compositions,however, ordered structures may occur (as for examplewith 232 n 64 CdGa2Se4 and CdGa2S4).

To obtain a quick idea if an adamantane structurewould be possible and what its homogeneity range mightbe, one can make use of the master diagram (Parthe and

Figure 4. The homogeneity ranges of ternary defect adamantane-structure compounds in the 1-3-6, 2-3-6, and 3-4-6 phase diagrams.The linear composition ranges are indicated with thick lines. Compositions where ordered adamantane structures occur are writtenwith bigger numbers

Table 5. Values of x/n according to equation (20) for the calculation of the possible compositions of ternary normal adamantane-structure compounds of the 'two-cation' type

ec = 2ec = 3

eA = 4

ec,=2

3/1

ec,=3

3/22/1

ec,=2

2/1 2/21/1

2/31/2O

ec,=2

1/1

eA = 6

ec,=3

1/2O

<0

ec,=4

1/3O

<0

ec, = 5

1/4O

<0

O<0

Paufler, 1991) shown in Figure 5. The valence-electronnumbers of the cations are given on the bottom line,those of the anions on the top line. By connectingappropriate points one obtains the triangle (in mostcases, non equilateral) that corresponds to the ternarysystem of interest. Adamantane-structure compoundscan be found only on the short thick line for whichVECA = 8 and 4^VEC^4.923 . The followingexamples may demonstrate the use of the masterdiagram.

• System 2-3-7: An adamantane structure is notpossible in this ternary system.

• System 1-3-5: An adamantane structure is possibleonly for a compound 35 that has no homogeneityrange.

• System 2-4-6: A 'two-cation' adamantane structureis possible between 26 and 2243n368 (VEC = 4.923,the upper limit of the short thick line where/>//! = 3/8).

• System 3-4-5: A 'two-cation' adamantane structuremight be possible on the tie line from 35 to 43 a 54.

• System 2-5-7: A 'two-anion' compound can occuron the composition line between 2257 (VEC = 4) and2 5 D 3 57 7 (VEC = 4.923). The only known examplesfor compounds on this composition line are Cd3PI3

and Cd3AsI3 with disordered defect wurtzite structure(Rebbah et al., 1981), and Zn3PI3 and Zn3AsI3 withdisordered defect sphalerite structure (Suchow et aL,1963).

Given that adamantane-structure compounds must liealong the short thick line of Figure 5, several classesof multicomponent solid solutions meet the require-ments. The more elements with different valenceelectrons are involved, the more complicated becomesthe shape of the possible homogeneity region (points -+lines -• planes-^tetrahedra...). For an example, see theresults of the system InAs-ZnGeAs2-CuInSe2-Cu2GeSe3, studied by Pamplin and Hasoon (1985).Only a few of the possible multicomponent solidsolutions have been investigated in detail. However, testshave been made with selected single multicomponentcompositions. For example, the seven-componentcompound Cu115Zn23Ga75Ge8As16Se30Br4 whereVEC = 4 and VECA = 8 crystallizes with a disorderedzinc-blende structure (Parthe, 1972).

It should be emphasized that the valence-electron rulesrelate to the theoretically possible composition limitsof adamantane-structure compounds. Depending on thesystem of interest, there may occur other competing,thermodynamically very stable compounds, as a

Figure 5. A master diagram to localize the possible homogeneity ranges of adamantane-structure compounds (modified afterParthe and Paufler, 1991)

Cations

Anions

consequence of which the otherwise expectedadamantane-structure compound either does not appearor its homogeneity range does not reach the lowerand/or upper VEC limit. As discussed below, thetendency to form adamantane-structure compoundsdecreases if the elements are from high periods and thecompounds become more metallic. It should also beremarked that the adamantane-structure compoundswith their covalent bonds have, in the solid state, verysmall diffusion rates and the annealing time needed toobtain equilibrium may be extremely long (six monthsfor certain compounds, after Woolley (1964)). Thereare known examples in the literature where the originallypublished solubility limits were later found to be toosmall because the time used for homogenizing thespecimens had been too short.

5.5 Tolerated Deviations from the ExpectedVEC and VECA Values

Minimal deviations of the compositions from theexpected theoretical VEC and VECA values in therange of AVEC/VEC = AVECA/VECA == 10~4 can betolerated without a destabilization of the crystalstructure. The excess or deficiency of valence electronsprovokes pronounced changes in the electrical and/oroptical properties of the compounds. This has beenconfirmed, for example, for CuInSe2 [1362], whereannealing either under maximum or under minimum Sepressure allows the conductivity to switch from p typeto n type (Masse and Redjai, 1986).

The deviations from the ideal VEC value are verysmall indeed. The studies of the homogeneity range ofGaP [35] by Jordan et al. (1974) serve as an example.At 10000C the composition of the Ga-rich side of thehomogeneity range is Ga05001P04999 (VEC = 3.9998)and, close to the melting point, Gd05002P04998

(VEC = 3.9996), while the composition of the P-rich sideis Ga05000P05000 (VEC = 4) for both temperatures. Thismeans that AVEC/VEC has values of 0.5 x 10~4 and1.0 XlO 4 , respectively. In a more recent study ofGaAs (Sajovec et al., 1990), it was found that themaximum surplus of native interstitial defects overvacancy defects is about 5 x 10"5 defects per structuresite.

For compounds where larger deviations in com-position had been initially reported, it was shown latereither that the composition stated was not correct orthat the structure was not really an adamantanestructure. An example is the compound 4Cu2SnSe4*with disordered sphalerite structure (Bok and deWit,1963). Since VEC = 4.286 and VECA = 7.5, a compound

with this composition should not crystallize with anadamantane structure. It appears very probable that4Cu2SnSe4' is in reality identical with Cu2SnSe3 [12463]where VEC = 4 and VECA = 8 (Rivet et al., 1970).Assuming that the latter compound has a homogeneityrange, expressed by Cu2Sn1+ ̂ Se3+2y [1241+J, oy63 + 2y =0 2*63)1 +y + (32 n 63)y - (1362)2>,; 0 ̂ y ^4.5 ] , a possibleadamantane-structure composition, closest to thatreported by Bok and de Wit, would be Cu2Sn15Se4

U 2*1.5°0.564] with VEC = 4.267 and VECA = 8.The only adamantane-structure compound that does

not obey the rules in the normal way is the recentlyreported NiSi3P4 with famatinite Cu3SbS4 type, anordered substitutional variant of the sphalerite type(Il'nitskaya et al., 1991). The formation of anadamantane structure can be understood if one assumesthat the Ni atoms do not contribute any valenceelectrons to the tetrahedral sp3 hybrid bonds betweenthe atoms. This 43054 compound should be a topic offurther studies.

6. Ordered Adamantane-Structure Types

A list of completely determined adamantane-structuretypes, based on wurtzite and sphalerite, is given in theappendix (Section 10.2). Here we only discuss andpresent drawings of the most common ordered structuretypes.

6.1 Binary Adamantane-Structure Types

6.1.1 Binary Normal Adamantane-StructureTypes

Binary normal adamantane-structure compounds withcompositions as given by (17) adopt the wurtzite or thesphalerite-type structure (see Figure 1) or a stackingvariant. The ideal c/a ratio of a wurtzite-type structurehas according to (2) the value of 2(2/3)1/2= 1.633. Theactual measured c/a values range from 1.6O0 (for AlN)to 1.656 (for BP). Fleet (1976) has shown that wurtzite-type compounds are stable relative to the sphaleritetype when c/a< 1.633, but unstable when c/a> 1.633.

The question of the relative stability of the wurtziteand the sphalerite type is a recurring one, which is stillwaiting for an answer. O'Keeffe and Hyde (1978)pointed out that, in compounds which crystallize onlywith the wurtzite type (but not the sphalerite type), thecation-cation distance is close to the sum of the non-bonded radii and concluded that the existence of only awurtzite-type structure was due to this fact.

6.1.2 Binary Defect Adamantane-StructureTypes

As examples of types of binary defect adamantane-structure compounds, we show in Figure 6 schematicdrawings of the structures of two 32 • 63 compounds,

a '-Ga2Se3 and /3-Ga2Se3 (Ollitrault-Fichet et aL, 1980;Ghemard et al., 1983). These structures can be describedby a stacking of identical defect-ZnS-type slabs withordered Zn site vacancies, that is, every third Zn sitenot being occupied. In the upper left corner of Figure6 is shown a segment of such a defect slab where, for

Figure 6. Two ordered binary defect adamantane-structure compounds with composition 32n63. The defect ZnS slab, the stackingpositions, and the slab stacking in oc'-Ga2S3, and /2-Ga2Se3, the first related to wurtzite and the second to sphalerite (Parthe, 1987;modified after Ollitrault-Fichet et al.t 1980; reproduced by permission of Academic Press, Orlando, FL)

Structuresrelated towurtzite

Structuresrelated tosphalerite

Wurtzite Cu3AsS4

enargite

CuFeS2

chalcopyriteSphalerite Cu3SbS4

famatinite

Figure 7. Six ternary normal adamantane-structure types. Cations are represented by small circles (modified after Parthe, 1987)

Li2SiO3

or LiSi2N3

/MMaFeO2

or BeSiN2

C C A 2

1362, 2452

C 2 CA 3

12463, 14253

C 3 CA 4

13564

Cu2GeSe3

reasons of simplicity, the upper anion layer has been leftout. The possible stacking positions of the defect-ZnS-type slabs are indicated in the diagram on the lower leftof Figure 6. The corners of the hexagons correspond tothe cation sites in the A" slab, which are occupied, whilethe squares with A" inscribed correspond to cation sitesin the A" slab, which are not occupied. The lettersdenoting the stacking positions, A, B, and C, areunprimed, primed or double-primed to characterize thesix relative positions of the cation site vacancy. Thedrawings on the right indicate the stacking of thesedefect slabs in a '-Ga2S3 and /3-Ga2Se3. The primes anddouble primes can be ignored if one simply wants toknow the overall stacking, which is AB in a'-Ga2S3

and ABC in 0-Ga2Se3. The ZnS base structures arethus wurtzite and zinc blende, respectively.

Adamantane-structure compounds of composition32n63 that are more metallic (having elements fromhigher periods), such as In2Te3, have disordered struc-tures, which means that vacancies and cations occupythe available cation sites at random.

6.2 Ternary 'Two-Cation9 NormalAdamantane-Structure Types

The five possible compositions given by (21) can beseparated into three groups according to the ratio C:C'of the two kinds of cations:

CC A2, C2C A3, C3C'A4

For each group, wurtzite- and sphalerite-relatedordered ternary structure types are known (Parthe andGarin, 1971) and are shown in Figure 7. In principle,the same structure types could be adopted by ternaryordered 'two-anion' normal adamantane-structure

compounds, for which the possible compositions aregiven by (22); however, no such compounds are known.

Pauling's electrostatic valence rule serves as a firstguide in understanding the order of the atoms andvacancies on the structure sites of wurtzite andsphalerite. It postulates the preferred occurrence ofan atom arrangement where the sum of the bondstrengths (emanating from the surrounding cations)agrees with the formal charge of the anion.

As an example one may consider the /3-NaFeO2

structure, shown in the upper left part of Figure 7.Pauling's postulate is satisfied with the atom arrange-ment shown. It is possible to imagine an arrangementwith a different atom order requiring a much smallerunit cell where all cations represented by small opencircles are at z = O and all cations indicated with smallfilled circles at z = 1/2. This hypothetical structure willnot be found in Nature because it violates Pauling's rule.

Pauling's postulate is, however, not infallible. Forcertain compositions Pauling's rule cannot be satisfiedexactly, as for example the Li2SiO3 and Cu2GeSe3

types, shown in the middle part of Figure 7. In thesecases a selective shortening or elongation of certaintetrahedral bonds may achieve a better effective valencebalance. For more details, particularly with reference towurtzite-derivative structures, see Baur and McLarnan(1982).

A modification of Pauling's rule by the bond-valenceconcept (O'Keeffe, 1989) allows the calculation ofstructural details. As an example, we consider thesupposedly isotypic structures of Li2SiO3 and LiSi2N3.A detailed study of these two structures shows that thetetrahedra are deformed, but in Li2SiO2 and LiSi2N3 indifferent ways (for example, three short and one longLi-O, but one short and three long Si-N distances).These differences in the distances within a given

Figure 8. Three ternary ordered defect adamantane-structure types of composition C 2C • A4, which are substitutional andvacancy derivatives of the sphalerite structure. Cations are represented by small circles, of which one-third are empty and two-thirds filled (Parthe, 1972)

tetrahedron are to be expected according to the bond-valence concept. The two structures are thus not isotypicin a rigorous sense, but constitute two branches of acommon hypothetical structure type with undeformedtetrahedra.

6.3 Ternary Defect Adamantane-Structure Types

Ternary defect adamantane-structure compounds are,in general, line compounds. However, if the ratio ofvacant to occupied cation sites has been specified, onlydiscrete compositions are possible.

6.3.1 Structure Types of 'Two-Cation'Compounds with Composition C2C ^ A4

Since/7/AZ = 1/4 these structure types can occur only withcompounds for which VEC = 4.57j. Using the algebraicmethod, one finds three possible compositions forC2C' • A4 compounds:

In Figure 8 three C2C' • A4 structure types are shownthat are vacancy and substitutional variants of thesphalerite type and where Pauling's electrostatic valencerule is satisfied. The most common structure type is thethiogallate CdGa2S4 (or /3-Ag2HgI4) type, which isfound not only with I22n74 and 232n64 but alsowith 224 n 64 compounds such as Hg2GeSe4 andHg2SnSe4 (Motrya et al., 1986). Also a wurtzite-relatedtype exists, which has been found with /3-ZnAl2S4 HT.

If the small circle in the center of the drawing of the/3-Cu2HgI4 cell is filled, one has obtained a drawing ofthe structure of AgIn5Se8, an ordered 135n268

compound, which can thus be considered as an orderedcation substitution variant of /3-Cu2HgI4.

6.3.2 Structure Type of (Two-Aniony

Compound with Composition C2OA2A '

For p/n=\/3 and VEC = 4.8 three compositions arepossible:

Structuresrelated towurtzite

Cu2CdGeS4

wurtzstanniteHeight

1/2

Structuresrelated tosphalerite

Cu2FeSnS4

stanniteCu2(Zn1Fe)SnS4

kesterite

Figure 9. Cation ordering in two wurtzite- and two sphalerite-related quaternary normal adamantane-structure types ofcomposition C2CCA4. Projection is along the shortest axis

Na2ZnSiO4

An ordered ternary structure type based on wurtzite hasbeen reported for Si2N2O.

6.4 Quaternary 'Three-Cation' NormalAdamantane-Structure Types of

Composition C2CCA4

The 'three-cation' normal adamantane-structure com-pounds are in general also line compounds. However,if the composition has been specified as C 2 CCA 4 ,there exist only four solutions of the adamantane-structure equation, which correspond to the compositions:

122464, 122364, 232454, 134254

The theoretically possible ordered structured typeshave been studied by Parthe et al. (1969), McLarnanand Baur (1982), and Baur and McLarnan (1982). Fourstructure types, which have been determined by X-raydiffraction studies, are presented in Figure 9 in aschematic way. Two are derived from wurtzite and twofrom sphalerite. Pauling's electrostatic valence rule issatisfied by all of them.

Most of the experimental studies were centered on the122464 compounds (Schafer and Nitsche, 1974, 1977;Guen et al., 1979; Lamarche et al., 1991). Some of thecompounds of this family seem to be promising materialfor nonlinear optics. Only scarce information is availableon the possible 134254 compound CuGaGe2P4. 122364

compounds, such as AgCd2InTe4 (Parthe et al., 1969),and 232454 compounds, such as ZnIn2GeAs4 (Pamplinand Hasoon, 1985), exist and are known to havesphalerite (or wurtzite) diffraction patterns. It appears thatordered quaternary structure types do not occur withthese compounds. ZnIn2GeAs4 is just one point onthe tie line between InAs and ZnGeAs2, which forma complete series of solid solutions. However, sincethe formation of adamantane-structure compounds,as stated above, is not free of kinetic problems,it is doubtful that the reported results correspond inall cases to equilibrium conditions. Thus one cannotstate with certainty that an ordered atom arrangementnever forms. In addition, not powder but single-crystal diffraction experiments would be required forthe study of the structure details.

6.5 Example for an Ordered 'Two-Cation' and'Two-Anion' Normal Adamantane-Structure

Type with Composition CCAA '

For this composition there exist five possibilities:

1456, (2356), 2347, (1357), (1267)

Of particular interest are the two compositions withoutsurrounding parentheses, which do not correspond toa mixture of binary normal adamantane structures. Anordered wurtzite-based quaternary structure type isknown for a-LiSiNO. Compounds with sphaleritestructure such as CuGePS are known, but they arenot ordered (see Pamplin, 1987).

6.6 Quaternary 'Three-Cation9 DefectAdamantane-Structure Compounds of

Composition CCCuA4

For this general formula there exist only two particularcompositions that agree with the adamantane-structureequation:

134 n 64, 125 n 64

The composition 345 • 54 is not included here since werestrict ourselves to solutions where the atoms that serveas cations have fewer valence electrons than the atoms weconsider as anions. Adamantane-structure compoundsare known only for the composition 134 D64 , such asAgInSnSe4 (Hughes et al., 1980). These compounds arereported to have a partially disordered chalcopyritestructure. According to Pauling's electrostatic valencerule, one should expect an ordered crystal structure thatis a cation ordering variant of the thiogallate CdGa2S4

type. An ordered wurtzite-related defect adamantanestructure compound of composition 125 • 64 is knownfor AgZnPS4.

7. Additional Experimental Rules forAdamantane-Structure Compounds

It has been shown above how the possible compositionsof adamantane-structure compounds can be calculatedusing the adamantane-structure equation. However, thisvalence-electron rule is a necessary but not a sufficientcondition for the occurrence of an adamantane structure.Compounds are known for which the adamantane-structure equation is satisfied, but the observedstructures are not adamantane structures. One simpleexample is NaCl (VEC = 4 and VECA = 8), which doesnot adopt a normal adamantane structure.

To predict whether a compound (with a permittedcomposition) will actually crystallize with an adamantanestructure, one can make use of one of several structureseparation plots. In each plot one finds an area whereadamantane structures should occur and other regions

where the formation of such a structure is unlikely. Weshall restrict ourselves to a presentation of four structureseparation plots (shown in Figure 10), which are allapplicable to binary equiatomic normal-valencecompounds.

Mooser and Pearson (1959) were the first to present sucha plot. They use as ordinate n, the average principalquantum number of the participating elements, and asabscissa | Ax\, the magnitude of the difference of Pauling'selectronegativity values. A Mooser-Pearson nvs. \Ax\plot for AB compounds with VECA = 8 is shown inFigure 10(a). This plot is based on the observation thatthe tendency to form directional bonds (as they occurin tetrahedral structures) decreases with the following:

• An increase of n. With elements of higher periodsa 'dehybridization' or 'metallization' occurs. The dand f orbitals now have an energy that is comparablewith that of the s and p orbitals and the first two mayform non-tetrahedral combinations with the s and porbitals.

• An increase of |AJC|. The electrons now prefer toremain with the strongly electronegative element anddo not participate any more in the formation of sp3

orbitals.

Phillips and Van Vechten (see Phillips, 1981) basetheir plot on two parameters that can be obtained fromthe measurements of the optical absorption spectra of

Figure 10. Structure separation plots for binary equiatomic normal-valence compounds according to (a) Mooser and Pearson,(b) Phillips and Van Vechten, (c) Zunger, and (d) Villars. Points corresponding to adamantane structures are indicated withopen circles, those of non-adamantane structures with filled circles (modified after Parthe, 1987; reproduced by permission ofthe Materials Research Society, Pittsburgh)

(a) Mooser-Pearson (b) Phillips-Van Vechten

Non-adamantane structuresNon-adomontane

AdamantanestructuresAdamontone structures

(c) Zunger (d) Villars

Non-adamantane structures

Non-adamantanestructures

the compounds and of the elements with a diamondstructure. They use as ordinate C, the average ionicenergy gap, and as abscissa Ehi the average homo-polar energy gap. In the C vs. Eh structure separationplot after Phillips and Van Vechten, shown in Figure10(b), the same compounds are considered as in Figure10(a). There is a clear separation between adamantane-and non-adamantane-structure compounds; however,the energy gap of a compound has to be measuredbefore a prediction of its crystal structure can be made.

Zunger (1980, 1981) calculates (density-functional)pseudopotential p- and s-orbital radii of the atoms. Forhis structure separation diagram of AB compounds, heuses as ordinate the difference between the total effectivecore radii of atoms A and B according to

/jAB=|(rA + /A )_ ( / .B+ rB ) | ( 2 3 )

and as abscissa the sum of the 'orbital non-locality* ofthe s and p electrons on each site according to

/jAB=|rA_ /.A| + |rB_rB| ( 2 4 )

A structure separation diagram R*B vs. R^B for binaryequiatomic normal-valence compounds is shown inFigure 10(c).

Villars (1983) uses for his structure stability diagramsthree parameters: R^B, the difference of Zunger'spseudopotential radius sums as defined by (23); |AJC|,the magnitude of the difference of the Martynov-Batsanov electronegativity values of the participatingatoms; and finally the sum of the number of valenceelectrons. Villars' R*B vs. \Ax\ diagram for com-pounds with eight valence electrons is presented inFigure 10(d). A more elaborate method has recentlybeen described by Villars et al. (1989).

8. Concluding Remarks

It has been demonstrated how the adamantane-structureequation can be applied to predict possible adamantane-structure compositions. Using the structure separationplots, one can foresee whether the formation of anadamantane structure is likely or not. Manyadamantane-structure compounds are already known.But there should exist many more; and, since systematicstudies of those with more unusual compositions havebeen made only in part, it would be worth while to dothis work. The adamantane-structure equation willcertainly prove to be a valuable guide to searchingand for what to expect when such experimental

investigations are carried out. However, there alsoremain some theoretical questions which still have notbeen answered satisfactorily, such as the choice of thewurtzite or sphalerite base type. A new approach isneeded here to find a solution to the problem.

9. Acknowledgements

The author thanks Ms Christine Boffi for her help withthe typing of the manuscript and Ms Birgitta Kiinzlerfor help with the preparation of the drawings. Thisstudy was supported by the Swiss National ScienceFoundation under contract 20-28490.90 and the Alfredand Hilde Freissler Stiftung.

10. Appendix

10.1 Stacking Notations and Space Groups ofZnS and SiC Polytypes

10.1.1 Stacking Notations for Polytypes

Ramsdell notation The notation consists of a numbercorresponding to the number of slabs in the hexagonalor triple-hexagonal cell and a letter that indicates thecrystal system (H = hexagonal, R = rhombohedral, thatis trigonal with R Bravais lattice, T or L = trigonal withP Bravais lattice, C = cubic). The Ramsdell notation forthe five polytypes presented in Figure 2 is listed therein the top row of text. This notation is not unambiguoussince, for the case of polytypes with a higher numberof slabs, differently stacked polytypes may have thesame Ramsdell stacking symbol.

ABC notation Starting from the first slab in positionA, one denotes the two other possible stacking positionsby B and C. The slab stacking sequence in the structureis then expressed by a sequence of capital letters. In theZnS and SiC polytypes, as in all close-packed structures,two successive layers cannot have the same stackingposition letters. The notation is unambiguous (exceptfor a permutation of the stacking letters); however, itis too cumbersome for the notation of very complicatedstackings. Examples are shown in Figure 2.

Jagodzinski- Wyckoff notation The notation consistsof a sequence of small letters, h and c, which areassigned to each slab depending on the sidewaysdisplacement of its two neighboring slabs, the one aboveand the one below, h is assigned to each slab where the

two neighboring slabs are displaced sideways in the samedirection, and c to each slab where the two neighboringslabs are displaced sideways in different directions. Theminimal h,c sequence (often by a factor of 2 or 3 shorterthan the A,B,C sequence) is occasionally supplementedby an added subscript number, which, when multipliedby the number of h and c symbols within theparentheses, gives the number of slabs in one hexagonalor triple-hexagonal unit cell. Examples are given in thethird row from the bottom in Figure 2.

Zhdanov notation In this notation one finds asequence of numerals, each of which represents thenumber of repetitions or successions of slabs with agiven sign for the sideways displacement of subsequentslabs, while the succeeding numeral represents thenumber of successions of opposite sign. Examplesare found in the second row from the bottom in Figure2. The Zhdanov stacking symbol can be obtained fromthe Jagodzinski-Wyckoff symbol as follows: TheJagodzinski-Wyckoff stacking symbol is divided intosections. Each section starts with the letter h. Thenumbers of letters in each section corresponds to thenumerals of the Zhdanov stacking symbol.

10.1.2 Space Groups of ZnS and SiC PolytypesThe ZnS polytypes are non-centrosymmetric and have,except for cubic zinc blende, trigonal or hexagonalsymmetry (space groups P3ml, R3m or P63mc). Inorder to find the proper trigonal or hexagonal spacegroup of a ZnS or SiC polytype for which the minimalZhdanov stacking formula kalbmc . . . is known, oneneeds to calculate three values:

• 7V( = a + Z? + c+ . . .), the number of numerals in theminimal period,

• s ( = ak + bl + cm+ . . . ), the normal sum of allnumerals in the minimal period, and

• E ±, the alternative sum of the numerals in theminimal period, i.e. the difference between the numberof positive and negative displacements.

The space group and the Ramsdell symbol of thepolytype can then be obtained from Table 6. Note that

Table 6. The space group and Ramsdell symbol of ZnS andSiC polytypes

RamsdellN E + Space group symbol

Odd - (186) P63mc Is HEven 3«±1 (160) R3m 35 REven 3n (156) P3ml s T

Wurtzite-related types

CAZnS, wurtzite

hP4 (186) P63mc-62

SBl, 78(see Figure 1)

CC A2

/3-NaFe3 + O2 (or BeSiN2)oP16 (33) Pna2,-<74

SR 18, 422; SR 32A, 28(see Figure 7)

C2C A 3

Li2SiO3 (or Cu2SiS3 HT orLiSi2N3)oS24 (36)Cmc2 ] -^VSR 37A, 74; SR 48A,335; David(see Figure 7)

C3C A 4

Cu3AsS4, enargite (orLi3PO4 LT)oP16 (31) P m n 2 , - 6 VSR 35A, 19; SR 32A, 355(see Figure 7)

C2CCA4Cu2CdGeS4, wurtzstannite

(or Cu2CdSiS4)oP16 (31) P m n V & VSR 34A, 47; SR 38A, 54(see Figure 9)

Na2ZnSiO4 (or Li2BeSiO4,liberite)mP16 (7) Pc-tf8

SR 3IA, 230; Joubert(see Figure 9)

0-Li2Co2+SiO4

oP32 (33) Pna2,-a8

SR 45A, 364

Na2MgSiO4

mP32 (7) Pc-a16

SR 48A, 337

Cu4Cu2+Si2S7

mS56 (6)Cc-tf14

SR 49A, 26

CCAA'a-LiSiNO (or ZnAlNO)

oP16 (29) Pca2,-<z4

SR 46A, 137; Hyde

Sphalerite-related types

ZnS, sphaleritecF8 (216) F-43n>caSBl, 76(see Figure 1)

CuFe3 + S2, chalcopyritetI16 (122) l-42d-dbaSBl, 279see (Figure 7)

Cu2GeSe3

ol 12 (44) Imml-cfbaSR37A, 73(see Figure 7) #

Cu3SbS4, famatinitetI16 (121) \-42m-idbaSR 38A, 15(see Figure 7)

Cu2Fe2+SnS4, stanniteti l6 (121) l-42m-idbaSB3, 96(see Figure 9)

Cu2(Zn5Fe2+)SnS4, kesteriteti l6 (82) \-4-gdcbaSR 44A, 56(see Figure 9)

Cu4Ni2+Si2S7

mS28 (5)C2-c6forSR 46A, 64

Table 7. Normal adamantane-structure types: p/n = 0 andVEC = 4

References:David = David et al. (1973)Joubert = Joubert-Bettan et al. (1969)Hyde = Hyde et al. (1981)SB = StrukturberichtSR = Structure Reports

Wurtzite-related types

^"16.67^ 14 D 1.33^32{p/n = 0.0417, VEC = 4.085)

^-1.203^'0.39° 0.407^2(p/n = 0.2035, VEC = 4.453)

C2C DA4

(/?/* = 0.25, VEC = 4.571)/3-ZnAl2S4 HT

oP32-4 (33) Pna21-j8

SR 32A, 13

( C 2 O D A 4

(p/n = 0.25, VEC = 4.571)

C C C a A 4

(p/n = 0.25, VEC = 4.571)AgZnPS4

oP32 (33) Pna2,-a8

SR 52A, 77

C 5 C D2A8

(p/n = 0.25, VEC = 4.571)

C2DA3

(/?/* = 0.333, VEC = 4.8)a ' -Ga2S3 (or Al2Se3)

mS20 (9) C c VSR 19, 404; SR 31A, 17(see Figure 6)

a-Al2S3

hP30 (169) P6,-tf6

Eisenmann

B2O3 II HPoS20 (36) Cmc2,-fc2aSR 33A, 259

C2DA2A'(p/n = 0.333, VEC = 4.8)

Si2NÔoS20 (36) Cmc2x~blaSR29, 244; Sjoberg

Sphalerite-related types

CUuIn16-67Se32

CP63-0.64 (215)P-43m-Pe1CJCa

SR 56A, 51

Cu0J9In, 203Se2

tP16-3.02 (112)P-42c-nfedb

SR 55A, 57

CdIn2Se4

tP7 (111) P-42m-/i/<7SR 19, 416(see Figure 8)

CdGa^S4, thiogallate (or/3-Ag2HgI4)til4 (82) 1-4-gcbaSR 19, 414; SR 4OA, 157(see Figure 8)

0-Cu2HgI4

tI14 (121) I-42m-/tfaSB 2, 50,331(see Figure 8)

(MnIn2)Te4 (or (ZnIn2)Se4)tI14 (121) I-42m-/cfoSR 41A, 79

AgIn5Se8tP14(lll)P-42-n2baSR 50A, 44

0-Ga2Se3

mS20 (9) Cc-a5

SR 50A, 35(see Figure 6)

s, 2s and 35 stand for numbers that can be obtained fromthe equation for s given above.

Example Besides rhombohedral ZnS 12R with stacking(13)3, shown in Figure 2, there exist three more ZnSpolytypes with 12 slabs in the unit cell. Their Zhdanovstacking symbols are (6)2, 93, and 4422. The firstpolytype has space group P63mc, and the two othersP3ml. The Ramsdell notations for these three structuresare ZnS 12H and twice ZnS 12T.

10.2 List of Adamantane-Structure TypesBased on Wurtzite or Sphalerite

Only completely determined crystal structure typesare considered here. For each type is given thename, the Pearson classification symbol, the space-group number, the standardized Hermann-Mauguinspace-group symbol, the Wyckoff sequence, and areference. The Wyckoff sequence is the sequenceof letters of occupied Wyckoff positions inthe standardized structure description (Parthe et al.,1993). Abbreviations used for references: SB,Strukturberichte; SR, Structure Reports. Structuredetails and lists of isotypic compounds published earliercan be found in Parthe (1972). Normal and defectadamantane-structure types are listed in Tables 7 and8, respectively.

11. References

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Eisenmann, B. (1992). Crystal Structure of a-DialuminiumTrisulfide, Al2S3. Z. Kristallogr., 198, 307-308.

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References: Eisenmann = Eisenmann (1992); Sjoberg = Sjoberget al. (1991); SB= + Strukturbericht; SR = Structure*Reports

Table 8. Defect adamantane-structure types: p/n > 0 and VEC > 4

Goryunova, N. A. (1965). The Chemistry of Diamond-likeSemiconductors. Chapman and Hall, London.

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Hughes, O. H., Woolley, J. C , Lopez-Rivera, S. A., andPamplin, B. R. (1980). Quaternary Adamantine Selenidesand Tellurides of the Form I III IV VI4. Solid StateCommun., 35, 573-575.

Hyde, B. G., O'Keeffe, M., and Shavers, C. L. (1981). CationPacking and Molar Volume in Oxides and Nitrides withthe Wurtzite Structure. J. Solid State Chem., 39, 265-267.

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Joubert-Bettan, C A . , Lachenal, R., Bertaut, E. F., andParthe, E. (1969). The Crystal Structures of Na2ZnSiO4,Na2ZnGeO4 and Na2MgGeO4. J. Solid State Chem., 1,1-5.

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This chapter was originally published in 1995 as Chapter 14 under the title 'Wurtzite andZinc-Blende Structures' in Intermetallic Compounds, Vol. 1: Principles, edited by J. H.Westbrook and R. L. Fleischer. The title change is to avoid any confusion for mineralogists,some of whom classify both hexagonal wurtzite and cubic sphalerite as 'zinc blende'.

Addendum

To replace the last two paragraphs on p. 128 and onthe top of p. 129.

For ordered adamantane structures with composi-tion C2CA3 it is not possible to satisfy Pauling'selectrostatic valence rule in an exact manner*. Forcertain anions the sum of the electrostatic bondstrength values emanating from its neighbouringcations may be larger or smaller than its formalcharge. To achieve a better effective valence balancethis is compensated, in agreement with Coulomb's law,

1It may be possible to satisfy Pauling's electrostatic valencerule even for composition C2CA3 if the structure is partiallyordered and has some sites where cations with differentvalence electrons are mixed. An example is found withtetragonal (Cu(Cu^5/9Sn^4/9)3S4 ( = Cu2SnS3) crystallizing withstannite (or famatinite) type where some sites have mixedcation occupation.

by elongating or shortening certain tetrahedral bondsbetween neighbours. The tetrahedra are thus distortedin a particular way. For a quick estimate of thisdistortion derived from bond-strength sums, see Parthe(1996) and for a more detailed calculation based on thebond-valence concept, see O'Keeffe (1989). An agree-ment between the observed and calculated elongationsand shortenings of interatomic distances is a necessary(but not sufficient) condition for the choice of thecorrect super cell in adamantane structures and thiscan serve as a test.

In wurtzite-related orthorhombic Li2SiO3 (Figure 7)and supposedly isotypic LiSi2N3 one expects (and alsofinds experimentally) that the tetrahedra are distorted,but in a different way, i.e. LiO4 tetrahedra have threeshort and one long Li-O bonds, but correspondingSiN4 tetrahedra one short and three long Si-N bonds.The two structures are thus not isotypic in a rigorous

sense, but constitute two branches of a commonhypothetical structure type with undistorted tetra-hedra. For other details on wurtzite-derivativestructures, see Baur and McLarnan (1982).

In sphalerite-related orthorhombic Cu2GeSe3, deter-mined nearly 30 years ago (Parthe and Garin, 1971),the expected tetrahedron distortions are not observed.The reason might be a wrong super cell choice with theresulting reported interatomic distance values corre-sponding to averages over longer and shorterdistances. It appears worthwhile to reinvestigate thestructure of Cu2GeSe3 using modern, up-to-date,single-crystal diffraction methods.

In the new sphalerite-related monoclinic structure ofCu2GeS3 (Chalbaud et al., 1997) the agreementbetween expected and observed elongation and short-ening of cation-anion distances has been verified.

Another C2CA3 compound with a sphalerite-relatedstructure is Cu2SnS3 for which there exist twoseemingly conflicting sets of structure data. Chalbaudet al. (1999) report a fully ordered monoclinic structurewhich is isotypic to Cu2GeS3, discussed above. Thisstructure was determined on a sample which had beenannealed at 600 0C for fully 30 days. However, Chen etal. (1998) observed the tetragonal stannite type withsome sites having mixed cation occupation. Theirsample with composition (Cu(Cu ̂ 5/9Sn_4/9)3S4

( = Cu2SnS3) was annealed at 700 0C for 2 days. Forthe determination of a fully ordered structure it isobviously necessary to make really certain that thesample has been annealed for a sufficiently long time.In these adamantane structure compounds, character-ized by an important percentage of covalent bonding,the reaction rate for atom ordering is low. To assureperfect ordering the annealing time should better beexpressed in units of months instead of days.

Under the subheading C2CA3 in Table 7, add twonew blocks in the column on the right hand side, to beplaced below the four-line block of Cu2GeSe3.

Cu2GeS3

mS24 (9) Cc-a6

Chalbaud et al (1997)

(Cu(Cu^5/9Sn^4/9)3S4 = Cu2SnS3

(partially ordered)tl 16 (121) \-A2m-idbaStannite Cu2Fe2 + SnS4

or Famatinite Cu3SbS4

type Fe or Sb site occupiedby Cu only, others mixedChen et al (1998)

References

Cenzual, K., Gladyshevskii, R., and Parthe, E. (1995). TYPIX1995 Database of Inorganic Structure Types. Database ontwo floppy disks and User's Guide of 48 pages.Frankfurt: Gmelin-Institut fur Anorganische Chemie.

Chalbaud de, L. M., Delgado de, G. D., Delgado, J. M.,Mora, A. E., and Sagredo, V. (1997). Mat. Res. Bull, 32,1371-1376.

Chalbaud de, L. M., Delgado de, G. D., Delgado, J. M.,Cenzual, K., and Sagredo, V. (1999). J. Solid StateChem., submitted for publication.

Chen, X., Wada, H., Sato, A., and Mieno, M. (1998). J. SolidState Chem., 139, 144-151.

Parthe, E. (1996). Elements of Inorganic Structural Chemistry.Selected Efforts to Predict Structural Features. 2ndedition, 170 pages. K. Sutter Parthe Publisher, 49Chemin du Gue, CH-1213 Petit-Lancy, Switzerland.

Wallinda, J. and Jeitschko, W. (1995). J. Solid State Chem.,114, 476-^80.

1. Introduction

In Nature many simple and complicated intermetallicstructures are realized and they are always legitimate.The mathematical models describing those structures arein general correct, but they do not always reveal thenature of the realized structure. This chapter illustratesthe second statement in an analysis of some relatedstructure types by examination of atomic environments.

Interpretation of chemical and/or physical propertiesof intermetallic compounds is mostly impossible withoutany knowledge of the crystal structure of thosecompounds. For example, information about the latticesymmetry is vital for the interpretation of magneticanisotropy and piezo- and/or ferroelectricity; and band-structure calculations are not possible without know-ledge of the complete crystal structure. It is thereforeessential that we have some means for analyzing thepublished data for their validity and that we are ableto check these data against the crystallographicrules as given e.g. in the International Tables forCrystallography (Hahn, 1983).

We have to keep in mind, however, that althoughsome minor displacements of atoms can lead to changesin symmetry and thus in the strict mathematicaldescription of the structure, they do not necessarilyaffect the crystal chemistry of the compound. Thismeans that minor displacements of the atoms do notalways have a large influence on the short-range atomicarrangement in the crystal structure.

With our atomic-environment approach, described inSection 2, it is possible to analyze crystal structures fortheir geometrical short-range atomic arrangements, and

as a result we are able to define the observed atomicenvironments (AE) by their coordination number andtheir shape. While other chapters in this volume discusscrystal-structure families with obvious internal relation-ships, here we will analyse with the AE approach 10 verypopulous intermetallic structure types, as listed in Table1, where their relationships are not so apparent fromthe usual descriptions. In Pearson's (1972) excellentbook on The Crystal Chemistry and Physics of Metalsand Alloys these 10 structure types (and others) areclassified (ch. 9) as structure types dominated bytriangular prismatic arrangements. In our approach weavoid this kind of classification, because it is in ouropinion difficult to see this triangular prismatic motif;instead, we use, as we will explain later on, anotherclassification based on the observed AEs in thesestructures.

The results of this analysis are given in Section 3,where we also compare the observed AEs and demon-strate how they can be used for the detection of relationsbetween structure types. In most of the 10 analyzedstructure types the AEs change as a function of the c/aratio and/or are dependent on the values of the atomiccoordinates (x, y, z) of one (or more) of the atoms in theasymmetrical unit. We show this dependence withthe help of some examples.

The crystallographic data used in this analysis andgiven in Table 2 are collected from the second editionof Pearson's Handbook of Crystallographic Data forIntermetallic Phases (PH) as compiled by Villars andCalvert (1991). In the past many of the 10 structure typesand the compounds crystallizing therein have been thesubject of similar or partly identical studies, e.g.


Chapter 5

Atomic Environments in Some Related IntermetallicStructure Types

Jo L. C. DaamsPhilips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands

Table 1. List of the 10 analyzed structure types

Space- Space- No. of No.Pearson group group point ofsymbol Formula symbol number sets compounds

hP2 CW P6m2 187 2 35hP3 AlB2 P6/mmm 191 2 330hP6 CaIn2 P63/mmc 194 2 149hP6 InNi2 P63/mmc 194 3 154tP6 Cu2Sb P4/nmm 129 3 197oC8 BCr Cmcm 63 2 193oP8 BFe Pnma 62 2 121UlO Al4Ba /4/mmm 139 3 723tI12 Si2Th /4,/amd 141 2 82oP12 Co2Si Pnma 62 3 495

the compounds crystallizing in the tllO Al4Ba structurehave been studied by Pearson and Villars (1984) amongothers. Whenever necessary, we will give references tothese studies, but it is impossible to give in this briefoverview a complete bibliography.

2. The Atomic-Environment Approach

A crystal structure is completely determined by thefollowing data:• Chemical formula• Space group and unit-cell dimensions• Coordinates of the point sets (atomic positions) and

their occupancy.

These characteristics lead to a fairly large number ofdifferent structure types; nearly 2800 types are listed inPH (Villars and Calvert, 1991), which makes it almostimpossible to see connections or even to detect identities.Many of these structure types are depicted in the Atlasof Crystal Structure Types for Intermetallic Phases(Daams et al.t 1991). In this compilation, for eachstructure type, there is given a three-dimensionaldrawing and two projections, and for each point set adrawing of the atomic environment. A similar type ofdata representation has been made by Grin* andGladyshevskii (1989), who give in their book Gallidesthe crystallographic data for all gallides. They also givefor each structure type a complete description with oneor two projections and some drawings of the AEsobserved in the structure.

The aim of the concept, described more extensivelyin previous work (Daams et al.9 1992; Villars andDaams, 1993; Daams and Villars, 1993), and brieflyrecapitulated hereafter, is to define the AEs as clearlyas possible, so that at the end of the analysis we willbe able to group them into distinct atomic environmenttypes (AETs).

We defined an AE using the method of Brunner andSchwarzenbach (1971), where all interatomic distancesbetween an atom and its neighbors are plotted in a next-neighbor histogram (NNH) as is shown in Figure l(a).

Table 2. Crystallographic data for the analyzed structure types

Pearsonsymbol

hP2

hP3

hP6

hP6

tP6

oC8

oP8

tllO

tI12

oP12

Formula

CW

AlB2

CaIn2

InNi2

Cu2Sb

BCr

BFe

Al4Ba

Si2Th

Co2Si

Atom

WCAlB

CaIn

NilIn

Ni2CuICu2SbBCrBFeBaAllA12ThSi

ColCo2

Si

Wyckoff letterMultiplicity

IaIdIa2d2b4f2a2c2d2a2c2c4c4c4c4c2a4d4e4a8e4c4c4c

SiteSymmetry

6m26m2

6/mmm6m26m23m.3m.6m26m24m24mm4mmm2mm2m.m..m.

4/mmm4m24mm4m2

2mm..m..m..m.

X

01/30

1/30

1/30

1/31/300000

0.0360.180

00000

0.0380.1740.702

y

02/30

2/30

2/30

2/32/30

1/21/2

0.4400.1461/41/40

1/2000

1/41/41/4

Z

01/20

1/21/4

0.4550

1/43/40

0.2700.700

1/41/4

0.6100.125

01/4

0.3800

0.41650.7820.4380.389

Occupation

1111

In most cases a clear maximum gap is revealed, as canbe seen in Figure l(a). All atoms to the left of thismaximum gap belong to the AE of the central atom.This rule is called the maximum-gap rule, and the AEof Figure l(b) is constructed according to this rule withthe atoms to the left of the maximum gap. The AEenvironment shown in Figure l(b) is the fully cappedtrigonal prism, which is a trigonal prism (atoms 1 to6) with three atoms (atoms 7 to 9) capping the equatorialfaces and two atoms (atoms 10 and 11) capping thepolar faces of the prism.

In those cases where this rule leads to AEs with atomsbeside the central atom enclosed or to AEs withatoms on one (or more) of the faces of the coordinationpolyhedron, we use the maximum-convex-volume rule.This rule is defined as the maximum volume around onlyone central atom enclosed by convex faces, with all thecoordinating atoms lying at the intersections of at leastthree faces. This rule was also used in those cases whereno clear maximum gap was detectable. In those caseswhere two (or more) equal, or practically equal, maximumgaps were observed, we kept the number of differentAETs in a structure type as small as possible.

The AETs are characterized using codes and labelsas explained in Daams and Villars (1993). The code ofthe idealized AE shown in Figure l(b) is 92 22°3, whichmeans that we have nine atoms (atoms 1 to 9) adjoiningtwo triangles and two squares, and two atoms (atoms10 and 11) adjoining no triangles and three squares.From the work mentioned earlier on the cubic structure

types (Daams et al., 1992), the rhombohedral structuretypes (Daams and Villars, 1993), and the elements(Villars and Daams, 1993), we think that aftercompletion of the analysis of all symmetries we will haveabout 25 basic atomic environment types. Thus 90%of the intermetallic structures, with about 50000compounds, are composed of combinations of theseAETs. It is important to mention that our AETs arenot isolated 'building' units, as crystal structures mayconsist of interpenetrating AETs.

We would also like to stress that there are some othermethods that can lead to the AETs as we have foundthem. However, we think that the method itself is notso important as long as the AETs found are un-ambiguously defined.

Another well-known local-environment constructionis the so-called Voronoi polyhedron construction, alsoknown as the Dirichlet construction or the Wigner-Seitz cell. This construction of the local environmentin a structure generates AEs that look similar to but arenot identical with our AETs. This Voronoi polyhedronis more often used by physicists, while our approach ismore familiar to crystallographers. For example,Pearson used the Dirichlet construction in the analysisof the compounds crystallizing in tllO Al4Ba (Pearson,1985a) and called the obtained local atomic environ-ments the Wirkungsbereich of an atom (see alsoPearson, 1985b).

For reasons we explain later, we think thatmethods which use (instead of crystal-structure-derived

Figure 1. (a) A typical example of an NNH and (b) the AE (coordination polyhedron) belonging to this NNH

Maximumgap

d/Cfmin

n

interatomic distances) the metallic radii of the atomsare less reliable, e.g. Bruzzone et al. (1970), becausethe metallic radii of the atoms, as given by e.g.Pearson (1972) or the so-called Teatum radii (Teatumet a/., 1960), are strongly dependent on the alloyingelement(s). This view was also emphasized by Pearson(1972) and is confirmed by e.g. the work of Merlo(1988) and Miedema and Niessen (1982).

In his book Structure of Substances Aslanov (1989)uses comparable coordination polyhedra (AEs) inhis so-called crystal chemical model of the atomicinteractions (CCMAI). In this model it is assumedthat all atomic interactions are a result of mutualattractions and repulsions of the atoms in the directcoordination spheres of the atoms. In Aslanov andMarkov (1992), and references therein, they appliedtheir model to intermetallic phase structures, and

some of the structure types analyzed with our approachare also described in this study. The small differencesthat appear between our AEs and theirs are a result ofthe differences in defining which atoms belong towhich coordination sphere. While in our approachthis is a straightforward definition, based on the NNH,the definition is sometimes complicated in theirmodel.

Although there are the above-mentioned differencesbetween both approaches, it is quite obvious thatthere is a strong resemblance. It would be achallenging task to find and describe the agree-ment between both methods so that in the future wewill have a number of 'standard building blocks' forintermetallics which can be used for explaining as wellas for predicting new intermetallic structures and theirphysical properties.

Figure 2. The 12 standard AETs as they are realized in the 10 analyzed structure types (idealized forms)

3. Observed Atomic Environments

3.1 General Remarks

Using the rules given above, we have analyzed all 10structure types of Table 1. Our approach leads toconventional atomic-environment types for thesestructures. However, for those compounds crystallizingwith an extreme range in the reported c/a ratio, e.g.hP3 AlB2, we observed an irregular AE (IAE) forcompounds with the highest c/a ratio. We considernormal AETs as environment types that can bevisualized by coordination polyhedra with a convexvolume, and IAETs as environment types that cannotbe described as a convex volume. Our analysis showedthat in the 10 structure types, with 24 point sets andabout 2500 compounds, for a medium c/a ratio, 12different AETs, as shown in Figure 2, are realized;wherever possible the standardized form of the AETsis depicted. In Table 3 we give the labels, explainedin Daams and Villars (1993), the number of times thatthe AE is observed, and their suggested notation; seealso Jensen (1989) and Parthe (1990). Most of theseAETs have been observed in the already mentionedprevious work and many of them belong, or are relatedto, the group of highly symmetric, frequently occurringAETs (basic AETs).

It has to be remarked that sometimes the differencesbetween some of the 12 observed AETs are very small;for example, the AETs with the labels 9-a and 9-b (seeFigure 2) can be regarded as a perfect trigonal prismwith three atoms capping the side faces of the prism (9-a)and a distorted version thereof (9-b). In AET 9-b thethree capping atoms are shifted outwards to a positionin which two of them form a square with two prism

Table 3. List of observed AEs with their suggested description

atoms and consequently the AET (9-b) can be nameda polar-capped square antiprism. A similar resemblancecan be observed for the AETs with the labels 10-c and10-1 (see Figure 2). These minor changes in AE and theconsequences for the crystal chemistry of the compound-families are still under investigation.

In the past 25 years the known compounds crystal-lizing in these 10 structure types have changed frommainly binary (Pearson, 1958, 1967) to almost com-pletely ternary (Villars and Calvert, 1991). In ouranalysis this means that in the six structure types, wherewe have two occupied atom positions in the asymmetricalunit (see Table 2), we must have in ternary compoundstwo different atoms mixing on one position in thestructures. From the geometrical (or chemical) point ofview such mixing seems improbable in many cases,because, for example, the metallic radii of the mixingatoms differ sometimes by as much as 30%. In Section3.5 this sometimes paradoxical behavior of metal atomsin crystal chemistry will be discussed in relation to thestructure types.

In the following we will describe in detail, for twoexamples, hP3 AlB2 and hP6 CaIn2, the AEs as theyare realized.

3.2 The AEs in hP3 AlB2

In hP3 AlB2 we observe different AEs for the twoatoms in the asymmetrical unit as a function of the c/aratio, which ranges in this structure type from 0.597 (forBi2Tl) to 1.083 (for AlB2) to 1.474 (for Ag2Nd). In thisstructure type both atoms are on specific positions withfixed coordinates, so there is no influence on the AEsof a changing atom coordinate.

For a medium c/a ratio, as in hP3 AlB2, we observefor the atom on the Ia position (Al) the full-cappedhexagonal prism as an AE and for the atom on the 2dposition (B) the equatorial-capped trigonal prism.

If we analyze the AEs step-by-step through the NNHof the respective atoms in hP3 AlB2, we see that thefirst 12 neighbors (B) of the Al atom form a hexagonalprism. The next six neighboring atoms (Al) are thecapping atoms of the side faces of the prism, while thenext two atoms (Al) are the polar caps, together leadingto the full-capped hexagonal prism. The NNH of theAl atom is given in Figure 3, while the AE is depictedin Figure 2 with label 20-a. For the B atom we see thatthe first three neighbors (B) form an in-plane trianglearound the central atom. The next six neighboring atomsform a triangular prism on which the first threeneighbors are capping atoms on the side faces; takentogether they form the equatorial-capped trigonal prism.

Label

9-a9-b

10-c10-1

11-a12-b13-b

14-d14-e17-e20-a22-a

Numberobserved

4311

222

22221

Description

Equatorial-capped trigonal prismMonopolar-capped square antiprismPolar-capped square antiprismMonopolar-equatorial-capped trigonal

prismFull-capped trigonal prismCubooctahedronPolar, monoequatorial-capped,

pentagonal prismPolar-capped hexagonal prismPolar-capped anti-cuboctahedronFull-capped pentagonal prismFull-capped hexagonal prismPolar, eight-equatorial-capped,

hexagonal prism

Figure 3. The NNH of the Al atom in hP3 AlB2; the AET is shown in Figure 2 with label 20-a

Figure 4. The NNH of the B atom in hP3 AlB2; the AET is shown in Figure 2 with label 9-a

Figure 5. The NNH of the Tl atom in hP3 Bi2Tl; the AET is shown in Figure 2 with label 14-d

Maximum

gap

Maximum

gap

Maximumgap

The NNH of the B atom is given in Figure 4 and theAE is depicted in Figure 2 with label 9-a.

For the minimum c/a ratio, as in Bi2Tl, we observefor the atom on the Ia position (Tl) as an AE the polar-capped hexagonal prism; see Figure 2 (with CN = 14,label 14-d). The NNH of the Tl atom is given in Figure5 and we see that as a consequence of the different c/aratio that the six in-plane Tl atoms are shifted to theright of the maximum gap and therefore no longerbelong to the AE of the central (Tl) atom. For the atomon the 2d position (Bi) the full-capped trigonal prismis realized as the AE; see Figure 2 (with CN =11, label11-a). The NNH of the Bi atom is given in Figure 6,and here we observe that we have two Bi atoms forming

caps on the triangular faces of the prism. These two Biatoms are from the neighboring unit cells, above andbelow.

For the maximum c/a ratio, as in Ag2Nd, weobserve for the atom on the Ia position (Nd) theequatorial-capped hexagonal prism AE; see Figure 7(with CN = 18, label 18-d); the two Nd atoms from theunit cells above and below belong no longer to the AEof the central (Nd) atom. For the atom on the 2dposition (Ag) the central atom is in this arrangementsurrounded by three Ag atoms forming a triangle; seeFigure 8 (with CN = 3, label 3#b).

With this example we have shown that the AEs in agiven structure type can change dramatically within that

Maximumgap

Figure 6. The NNH of the Bi atom in hP3 Bi2Tl; the AET is shown in Figure 2 with label 11-a

Maximum

gap

Figure 7. (a) The NNH of the Nd atom in hP3 Ag2Nd and (b) the corresponding AET (CN= 18, label 18-d)

d/Cmin

n

n

d/drnn

structure type as a function of the c/a ratio. The CNfor the atom on the Ia position varies from 14(minimum c/a) to 20 (medium c/a) to 18 (maximumc/a), and the CNs for the atom on the 2d position are11, 9 and 3 respectively.

It shows that, although the mathematical modelsdescribing these compounds are identical with all com-pounds assigned to this structure type having the samespace group and same positions occupied, the com-pounds are not the same in terms of the short-rangeatomic arrangement for the respective atoms in thecompounds.

Adding the AE information to the descriptive modelsmeans that hP3 AlB2 can no longer be regarded as a* single structure type', but that it should be recognizedas consisting of different subtypes or, as we will explainlater, of different coordination types. As a consequence,it also shows that it is incorrect to assign a givencompound to a given structure type based only on thestandard mathematical description.

This sometimes dramatic change in short-rangeatomic arrangement has been observed in manystructure types in which the compounds show a widerange in c/a ratio, and this effect has therefore beendescribed by many authors. For example Pearson (1972,1979) already distinguished two groups of compoundscrystallizing in hP3 AlB2 based on the c/a ratio,whereby the compounds with the lowest c/a ratio(c/a < 0.9) mainly consisted of combinations with groupI (alkaline) or group II (alkaline-earth) metals.

In the following example we show some changes inAEs, as a function of changing c/a ratio, which are lessdramatic as compared with the example given above.

3.3 The AEs in hP6 CaIn2

In hP6 CaIn2 we also observe different AEs for thetwo atoms in the asymmetrical unit as a function of thec/a ratio. The c/a ratio ranges in this structure type from1.382 (for CuGaTi) to 1.583 (for CaIn2) to 1.812 (forLaPtSb).

For a medium c/a ratio, as in CaIn2, we observe asan AE for the atom on the 2b position (Ca) a distortedpolar-capped hexagonal prism and for the atom on the4f position (In) the monopolar-equatorial-capped tri-gonal prism.

If we analyze the AEs step-by-step through the NNH(Figures 9(a) and 10(a)) of the respective atoms in hP6CaIn2, we see that the first six neighbors (In) of the Caatom form a trigonal prism. Together with the next sixneighboring atoms (In), these 12 atoms form a distortedhexagonal prism, on which the next two neighboringatoms (Ca) are polar caps; see Figure 9 (b). This AEis distorted because the In atoms do not form a planarsix-membered ring owing to the slight displacement ofthe z coordinate from the ideal position (0.455 <-• 1/2).

For the In atom we see that the first three neighbors(In) form a non-planar triangle with the central atom.Inclusion of the next atom (In) gives a tetrahedron, ofwhich three side faces are capped by the next three atoms

Figure 8. (a) The NNH of the Ag atom in hP3 Ag2Nd and (b) the corresponding AET (CN = 3, label 3#b)

Maximumgap

d/dmin

n

(Ca). Including the next three atoms (Ca) gives thedepicted AE, which shows a triangular prism of Caatoms with three In atoms capping the equatorial facesand an In atom capping one of the triangular faces ofthe trigonal prism (Figure 10(b)).

For the minimum c/a ratio, as in CuGaTi, we observeas an AE for the atom on the 2b position (Ti) the same

distorted polar-capped hexagonal prism as in CaIn2

(see Figure 9), and for the atom on the 4f position(M = Ga, Cu) an elongated form of the full-cappedtrigonal prism (see Figure 11). It is clearly shown in theNNH that the extra M atom is practically in the middleof the maximum gap and it is an arbitrary decision toinclude it in the AE.

Maximumgap

Figure 9. (a) The NNH of the Ca atom in hP6 CaIn2 and (b) the corresponding AET; the idealized form of the AET is shownin Fieure 2 with label 14-d

Maximum

gap

Figure 10. (a) The NNH of the In atom in hP6 CaIn2 and (b) the corresponding AET, which is also shown in Figure 2 withlabel 10-1

For the maximum c/a ratio, as in LaPtSb, we observefor the atom on the 2b position (La) the full-cappedhexagonal prism (see Figure 2, label 20-a) and for theatom on the 4f position (M = Pt, Sb) the monopolar-equatorial-capped trigonal prism (see Figure 2, label10-1). The NNHs of both atoms are shown in Figure12 (La) and Figure 13 (M = Pt,Sb).

3.4 Related Structure Types

The examples given in Section 3.3 demonstrate one ofthe advantages of our approach: it shows directly the

resemblance in the short-range atomic arrangementsbetween atoms in different structure types, and it givesus a tool to classify structure types with the same AEs.Therefore, we have defined a so-called coordinationtype. When in different structure types the same AEsand the same number of AEs are realized, we definethese structure types as belonging to the same co-ordination type. In the group of 10 analyzed structuretypes we have two examples of structure types belongingto the same coordination type (see Table 4). In oC8 BCrand in oP8 BFe the same AEs are realized, as is alsothe case in hP3 AlB2 and in til2 Si2Th.

Figure 12. The NNH of the La atom in hP6 LaPtSb; the AET is shown in Figure 2 with label 20-a

Maximumgap

Figure 11. (a) The NNH of the M atom (M = Cu, Ga) in hP6 CuGaTi and (b) the corresponding AET; the idealized form ofthe AET is shown in Figure 2 with label 11-a

Maximumgap

d/dmm

n

d/dnm

n

If we include the AEs as they are realized in certainstructure types as a function of c/a ratio, we have anincreased number of partly overlapping structures andtherefore of structures belonging to the same co-ordination type. The other realized AEs, in the 10structure types, are depicted in Figure 14, and it isobvious that these AEs are distorted or derived formsof the normal AEs, with atoms sometimes included orexcluded.

The relations between the 10 structure types, if weinclude the boundary compounds, can easily be seen inTable 5.

Relations between the 10 structure types and otherstructure types can be found in the same manner, whenwe compare the realized AETs with those of otherstructure types. As already mentioned, most ofthese structure types are depicted in Daams et al, (1991).A good example of a structure type with many relatedstructure types is hP6 InNi2. The realized AETs (11-aand 14-d) in this structure type are also realized in thestructure types given in Table 6. Some of these relationsare quite obvious, so hP6 Fe2Si and Ni2Si are refined(or assigned) to a space group with a lower symmetry,where the atomic coordinates of one of the atoms are

Pearson

symbol

hP2

tP6oC8oP8hP3tl12oP12hP6hP6

tno

Formula

CW

Cu2Sb

BCr

BFe

AIB2

Si2Th

Co2Si

CaIn2

InNi2AI4Ba

Number of

point-sets

2

3

2

2

2

2

3

2

3

3

Number of

compounds

35

197

193

121

330

82

495

149

154

723

Coordinationnumbers

149/9/129/179/179/209/2010/13/1310/1411/11/149/12/22

Observed AtomicEnvironment Types

o - ^ c p - Q - Q - D c p a j f p a j

m m

Number of

AETs

1

2

2

2

2

2

2

2

- t -

Figure 13. The NNH of the M atom (M = Pt, Sb) in hP6 LaPtSb; the AET is shown in Figure 2 with label 10-1

Table 4. Coordination types for a medium c/a ratio

Maximum

gap

d/Cmin

n

just fractions away from the ideal positions that occurin hP6 InNi2.

For hP6 Ge4Ni7 we have to translate all atomiccoordinates by 0, 0, 1/4 and then we obtain thesame coordinates as in hP6 InNi2. Related to hP3AlB2 is hP3 BaPtSb, and related to tllO Al4Ba aretllO BaNiSn3, tllO H2PdZr2, and tllO C2IrU2. Thecompounds crystallizing in hP3 AlB2 with minimumc/a ratio, having AEs 11-a and 14-d, are relatedto the structure types of hP3 Cd2Ce and hP3 CuZn3,since in these structure types the same AEs arerealized.

In this brief overview it is impossible to givea complete survey of all relations between the 10analysed structure types and other structures. Acomplete analysis of all structure types, as theyare listed in Villars and Calvert (1991), is at thismoment only partly realized, and it will take us

at least another two years to complete this analysis.We estimate that with our approach we can reducethe number of structure types from 2800 differentmathematical descriptions to about 500 differentcoordination types.

Another approach for representing relations betweenespecially complicated structures is the so-called inter-growth concept as proposed by Kripyakevich andGladyshevskii (1972), among others. In this concept,relations between structure types are written as a linearhomogeneous series of intergrowing identical segments,in which these segments are slabs of well-knownstructures. For a series of inhomogeneous linearstructures the concept is described by Grin' et al. (1982),and in a lecture presented at the nineteenth InternationalSchool of Crystallography, Grin' (1992) made anattempt to give a more systematic approach to theconcept. An improvement to this concept could be

Figure 14. The remaining AEs as they are realized in the 10 structure types covering the whole range of c/a ratios

Table 6. List of related structure types for hP6 InNi2

Pearson symbol Formula No. of compounds

hP3 CuZn3 2hP6 Fe2Si 13hP6 Ge4Ni7 1hP6 Hg 1hP6 Ni2Si 1

the use of AEs in the descriptions of the slabs becauseit would show more directly the relations betweenthe combining slabs and the structures in the series.In Chapter 12 by Hauck and Mika in this volume,the close-packed structures and the relations betweenthem are extensively described, and they give aprocedure for ordering them in structure maps.

Table 5. Coordination types for the complete range of c/a ratios

Pearsonsymbol

hP3hP6tMOtnotnotP6tl12tP6tP6oP12tP6tno0C80P80P80P80P8hP3tl12tl12hP6oP12hP6hP60C8hP3hP6hP6hP60C8oP12hP2

Formula

Ag2NdKPZnCu2Se2TICo2LaP2

GdGe2Pt2HfSiTeBaGe2

Cu2SbMnSbZbAgTeTIBi2UAI4BaBCrSiThCeCuBFeDyNiAIB2

Si2ThSi2UAsAuCaCo2SiCaIn2

LaPtSbAIHfBi2TICuGaTiBiIn2

InNi2BaSiPRe2

CW

Parenttype

AIB2

InNi2AI4BaAI4BaAI4BaCu2SbSi2ThCu2SbCu2SbCo2SiCu2SbAl4BaBCrBFeBFeBFeBFeAIB2

Si2ThSi2ThInNi2Co2SiCaln?

CaIn2

BCrAIB2

CaIn2

InNi2InNi2BCrCo2SiCW

Coordinationnumbers

3/183/3/184/8/204/8/225/12/228/8/179/129/9/129/9/129/12/169/12/179/12/229/179/179/179/179/179/209/209/209/9/2010/13/1310/1410/2011/1311/1411/1411/11/1411/11/1411/1513/13/1514/14

c/a ratio

maximal (max)max

I maxmedium (med)

I minimal (min)maxminminminminmedmin

[medminmedmedmaxmedmedmaxmedmedmedmaxmaxminminminminminmaxcomplete range

Observed Atomic Environment Types

3.5 Structure Types with Atoms Mixing onOne Position

In some structure types, as for example hP3 AlB2,when we look at the reported compounds, we canobserve that a great variety of complicated intermetallicshave been described in this rather simple structure. Anelement (Ti), binary, ternary, and even quaternarycompounds have been described in terms of this binarystructure type, with two atomic positions in theasymmetric unit. This means, for example in the ternarycase, that structures have been described where we havetwo or more atoms randomly occupying the same positionin the structure, and, given the published stoichio-metries, also in a variety of ratios in that position. Fromthe geometrical point of view, as in the hard-spheremodel, this is only acceptable if the mixing atoms wouldhave about the same metallic radii; otherwise we havea description of a structure in which an atom is not inclose contact with its direct neighbors.

Solutions for this problem are often given bydescribing these structures by a lower-symmetry spacegroup. For example, the hP3 BaPtSb structure type isdescribed in space group 187 with the Pt and Sb atomson the Id and If positions with atomic coordinates 1/3,2/3, 1/2 and 2/3, 1/3, 1/2, respectively. Although theatoms are now described on different positions, theiractual position in the layer remains the same as wouldbe the case if this compound were described in terms of

hP3 AlB2. This solution is in fact in many cases aneven worse description than the random distributiondescription as in hP3 AlB2, because in the formerdescription the same atom is always on top of itselfgoing from one unit cell to the other. As a result of largedifferences in metallic radii, this would introduceenormous strains in the crystal, and it is improbable thatsuch a crystal can exist.

However, if we assume that these descriptions arecorrect, we must accept that atoms are not hard spheresbut that they have a certain degree of compressibility,which, as already mentioned, is highly dependent on thealloying atom. Since almost all compounds describedin these structure types have a transition-metal and ap element mixing on the same position, both atoms canshow this compressibility. As observed and reportedmany times before, the metallic radii of the p elementsespecially show this behavior. It is therefore probablycorrect to state that their reported radii are only validin the pure-element structures.

For compounds where we have two atoms mixing onthe A position in an AB2 structure, we observe thesame behavior, e.g. in Mo3YB8 we have Mo and Y, with1.40 and 1.81 A as their reported metallic radii,respectively, occupying the same A position. Thenumber of improbable combinations of this sort is,however, much smaller than the number of com-binations for the other (B) position, wherein the biggestdifferences in radii appear only in combinations with

Figure 15. Periodic system representation of the binary 1:2 compounds crystallizing in hP3 AlB2

the rare-earth metals. In most of the reported com-binations there are two early d elements mixing, and theyhave in general much smaller differences in metallicradii.

In hP3 AlB2 there are four quaternary compoundsreported in which we have a rare-earth (RE) metal onthe A position and three elements (Cu, Si, and Zn)mixing on the B position with the compositionRE2CuZnSi2.

Before we consider some relations between the atomicradii and the (shortest) interatomic distance in thestructure, we present some statistical information aboutthe compounds crystallizing in hP3 AlB2. This in-formation can be very helpful for finding errors instructure descriptions.

For the reported binary compounds, crystallizing inthis AB2 type of structure with the ideal 1:2 stoichio-metry, we have visualized the realized combinations ina so-called periodic system representation as shown inFigure 15. The elements observed on the A position aremarked with a light gray shading and the elementsobserved on the B position with a darker gray shading,elements reported for the A as well as the B positionhave both shadings with a diagonal separator. Closerexamination of Figure 15 shows us that we have sixelements that can occupy the A as well as the B position,namely Al, Cu, Ag, Au, Th, and U. Three of theseelements (Al, Ag, and Au) occupy the A position onlyin combination with boron. Th occupies the B positiononly in CuTh2; and, because the other combination,Cu2Th is also reported, it is likely that CuTh2 isincorrect, especially since both compounds are reportedwith the same unit-cell dimensions. U occupies the Bposition only in combination with Ti in TiU2, thiscompound being assigned to the hP3 AlB2 structuretype.

From the 153 ternary compounds crystallizing in hP3AlB2 with various (A,A')B2 or A(B,B')2 stoichiometry,we have about 30 compounds in which random mixingof atoms is reported on the A position, e.g. as inB4CrMo where Cr and Mo occupy the A site. If weanalyze the occurrence of the atoms in this structure incombination with the position of the elements in theperiodic system (see Villars and Girgis, 1982), weobserve that only a limited number of combinations arerealized in this structure type.

In Table 7 we give the complete set of observedternary combinations in hP3 AlB2 for atoms mixingon the B position, their mixing ratios, and theelements reported on the A position. Not all ofthe elements reported on the A position form acompound with the given combination. For example,

in the ternary compounds with Ag and Si, the 1:1combination is only observed for Nd (NdAgSi), whilethe 2:4 combination is observed for Ce, Dy, Er, Eu,La, Nd, and Pr in the compounds RE3Ag2Si4.

For some mixtures we observe that the combinationsreported form an almost complete line compound seriesin the ternary phase diagram, e.g in the combinations ofthe rare-earth metals with Al and Ga. There is, however,so far no evidence from ternary phase diagrams that

Table 7. Binary combinations in hP3 AlB2, for the atomsmixing on the B position (and their stoichiometric ratios) inthe structure

B-positionelements

Ag-SiAl-Ga

Al-GeAl-SiAl-SnAs-PtAu-SiB-BeB-CB-CoBe-GeBe-SiCo-GaCo-GeCo-SiCu-GaCu-Ge

Cu-InCu-Si

Cu-SnFe-GeFe-SiGa-NiGa-SiGe-IrGe-NiGe-PdGe-PtGe-RhGe-ZnIn-NiNi-SbNi-Si

P-PtSi-TiSi-Zn

Ti-Zr

B-positionstoichiometric

ratios

1:1 2:43:1 7:3 5:31:1 4:6 1:31:157:1 13:3 3:17:1 13:3 1:17:1 13:3

29:111:37:1 17:3 5^ M

19:13:13:53:51:31:31:3 2:81:3 1:5 1:91:1 2:4 1:3

1:33:1 1:1 3:52:4 1:31:12:41:3 2:8

39:11:13:13:1 1:18:28:23:11:13:11:11:1 2:4 1:3 2:8

5:2 27:131:11:1

5:1

Elements reported onthe A position

La,Ce,Pr,Nd,Eu,Dy,Er( Na,Sr,Ce,Nd,Sm,Gd,[ Tb,Dy,Ho,Er,Tm

La,Ce,NdSr,La,Ce,NdLa,Ce,NdEuEuTi,Zr,Hf,TaTaGdCa,SrCa,Sr,BaSmCeLa,Ce,Pr,Nd,Sm,Eu,Gd,ThSm,Ho,Lu

[ Sr,Ba,Y,La,Ce,Pr,Nd,Sm,[ Gd,Tb,Dy,Ho,Er,Tm,Lu

La,Ce,Pr,Nd,Sm,Eu( Sr,Ca,Y,La,Ce,Pr,Nd,Sm,[ Gd,Tb,Dy,Ho,Er,Tm

LaLa,NdLa,Ce,Pr,Nd,Sm,Gd,UHoCaNaLa,Ce,Pr,Nd,Sm,EuNdNdNdSrLa,Ce,Pr,Nd,Sm,EuCe,Pr,Nd,Sm,Eu

[ Sr,Ca,La,Ce,Pr,Nd,Sm,I Gd,Dy,Er

Ca,Sr,EuGd

{ Y,Ce,Nd,Sm,Eu,Gd,Tb,I Dy,Ho

U

this line compound series really exists. In the ternaryphase diagram of Al-Ga-La, as collected in Bodak andGladyshevskii (1985), we see that such a line compoundseries exists for LaGa2. ̂ Alx whereby x varies from 0 to1.8 within the hP3 AlB2 structure type. For LaAl2 thereported structure type is cF24 MgCu2 at low tem-peratures; at high temperatures (> 1363 K) there is acompound reported (La2Al5) having the hP3 AlB2

structure with the La and Al atoms mixed on bothpositions.

From the 330 compounds reported in the hP3 AlB2

structure type, at least 10% of these compounds arereported with a stoichiometry differing from the ideal1:2 composition. In compounds with stoichiometryratios of 2:3, 2:5 or 3:5, as e.g. in Ho2Ge3 or Gd3Si5,we have, as mentioned before, atoms mixing on bothpositions or partly occupied positions.

In Table 8 we give the observed ternary combinationsin hP3 AlB2, where the atoms are mixed on the A

position; in the compounds with the stoichiometryAA'B4 the AA' combinations are 1:1 mixed; forthe other combinations the mixing ratios are given in thetable.

In Section 3.6 we show for compounds crystallizingin some of the analyzed structure types the relationbetween the shortest interatomic distance in the structureand the mean radius of the atoms, at the shortestdistance, in that structure.

3.6 Atomic Radii and Interatomic Distances

Pearson (1979) and Villars and Girgis (1982) reporteda linear dependence between the interatomic distancesand the (concentration-weighted) mean radii (dAB vs.R) for compounds crystallizing in various binary inter-metallic structure types. With the help of someexamples, we will show that this is correct for the mainpart of the compounds in a given structure type; but,for some subgroups of compounds within one structuretype, different linear dependences can exist.

For demonstration purposes it is not necessary tomake the rather complicated calculations given in thepaper by Villars and Girgis; instead, we show this lineardependence by plotting the mean atomic radius of theatoms concerned versus half of the interatomic distancebetween these atoms.

In hP3 AlB2 we have two atoms in the asymmetricunit and half of the interatomic distance between those

Figure 16. Plot of dAB vs. R for the binary compoundscrystallizing in hP3 AlB2. Exceptional cases are noted in the text

Table 8. Binary combinations in hP3 AlB2, for the atomsmixing on the A position (and their stoichiometric ratios) inthe structure

A-positionelements

Al-MgCr-MoCr-TaCr-TiCr-VHf-NbHf-TaHf-TiHf-ZrMo-TiMo-ZrNb-TiNb-VNb-ZrNp-PuTa-TiTa-VTa-ZrTi-VTi-ZrAl-NbAl-TaCr-NbMn-MoMo-NbMo-YGd-NbTi-WGd-UU-Y

A-positionstoichiometric

ratios

1:21:32:22:22:23:17:3

10:11:12:3

Elements reportedon the B position

B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B4B6B8B8B8B8B8B20B22

Ga4Ga10

Figure 17. Plot of dAB vs. R for the ternary compoundscrysallizing in hP3 AlB2

atoms is l/2[l/3a2+ l/4c2]172. For the binary com-pounds, the mean radius is the mean of the metallic radiiof the atoms involved; for the ternary compounds, wefirst calculate the mean radius of the atoms mixing on thesame position, weighted for the mixing concentration,and then calculate the mean radius of the atoms in thestructure.

Figure 19. Plot of dAB vs. R for the binary compoundscrystallizing in tP6 Cu2Sb. Exceptional cases are noted in thetext

In Figures 16 and 17 this linear dependence is plottedfor the binary and ternary compounds, respectively,crystallizing in hP3 AlB2. We see that there is almostno difference between the two plots and that there isa 1:1 dependence, meaning that the mean radius of theatoms involved equals the available space in thestructure. In Figure 16 we observe that two compounds,NdAg2 and PrAg2, shown by the open circles, are faraway from the 1:1 dependence, the interatomic distancebetween the rare-earth atoms and the silver atom beingtoo long (0.448 nm, see Figure 7(a)) compared with themean of the metallic radii. This discrepancy is, ofcourse, a direct consequence of the very long c-axis, andit is questionable whether these compounds are in factrealized in this structure type. It is more likely that thesecompounds are incorrectly classified under this structuretype.

The next example shows that there is not always a1:1 dependence, but that this dependence varies fromstructure type to structure type, with each of themcombining their particular groups of atoms. In Figure18 we have plotted this dependence for the binary (2:1)compounds crystallizing in oP12 Co2Si. We see that ingeneral the mean radii of the atoms are more than halfthe shortest interatomic distance, although a linearrelationship still exists. The opposite is observed for thebinary compounds crystallizing in tP6 Cu2Sb, where ingeneral the mean radii are smaller than half the shortest

Figure 18. Plot of dAE vs. R for the binary compoundscrystallizing in oP12 Co2Si

interatomic distance (see Figure 19). The broken linesin Figures 16 to 19 are the result of a least-squaresanalysis.

This contrasting behavior can be understood andexplained if we compare, for two AB2 structure types,

the groups of atoms that are combined in the respectivecompounds. In Figure 20 we show in the periodic systemrepresentation oP12 Co2Si compounds and in Figure21 tP6 Cu2Sb compounds. While in tP6 Cu2Sb therare-earth atoms are mainly on the A position, the

Figure 20. Periodic system representation of the binary 1:2 compounds crystallizing in oP12 Co2Si

Figure 21. Periodic system representation of the binary 1:2 compounds crystallizing in tP6 Cu2Sb

opposite is observed for oP12 Co2Si, where they are onthe B position.

Another remarkable difference is observed when wecompare the combinations that are formed: tP6 Cu2Sbis almost exclusively observed for combinations of rareearths with a p element, while in oP12 Co2Si there isno such preference. This preference can explain thedifferences in the slopes of the lines drawn in Figures 18and 19; in tP6 Cu2Sb the bulk of the compounds havea covalent bonding character, which results in smallerradii, while in oP12 Co2Si we have predominantlymetallic bonding.

From the 72 reported binary tP6 Cu2Sb compounds,with a 2:1 stoichiometry, in only seven combinationsdo we have a p element (As or Sb) on the A positionand a d element (Sc, Cr, Mn, Fe or Cu) on the Bposition. Figure 19 shows that these seven compounds(marked by open circles) are also placed in this plotoutside the main group of combinations, as are threeother combinations, namely EuO2 and GdO2 (markedby an asterisk), and GdS2 (marked by an open square).The two oxides probably have an ionic bondingcharacter, while the third compound, GdS2, is reportedto be a high-temperature as well as a high-pressure phase.

To analyze the reported compounds in oP12 Co2Siin the same manner is much more complex, because inthis structure type there is a much wider variety ofcombinations formed. However, some of the com-binations can give us valuable information about thepossibilities in this structure type. Figure 20 shows thatthere are at least five combinations possible in whicha rare-earth (Eu or Yb) or an alkaline-earth atom (Ca,Sr or Ba) is reported on the A position. If we look upall representatives in which this is realized, we see thatthis combination is only reported for compounds withhydrogen in the B position. Elements reported on theA as well as the B position are Pt, Ge, P, As, Sb, S,and Se. The latter two, S and Se, are on the A positiononly in combination with cesium, and As and Sb onlyin combination with rhodium. For P and Ge there arereported many combinations but e.g. Ge on the Bposition only occurs in combination with Mo or W, bothcompounds stable only at high temperatures.

With these brief examples, we have shown that thecombination of a simple method, e.g. a mean radii vs.interatomic distance plot, and a periodic system arrayshowing all possible combinations gives muchinformation about the validity and the nature of thecompounds crystallizing in a particular structure type.Such knowledge can be very helpful in explaining crystalstructures, and to a certain extent the physical propertiesof old as well as new intermetallic compounds.

Thus this analysis illustrates the thesis set forth inSection 1 that examination of the AEs of the 10seemingly diverse structure types of Table 1 leads to thefollowing kinds of insights:

• Striking resemblance in the short-range atomicarrangements between the structure types and thusbetween compounds crystallizing therein.

• Simple recognition of relations between structures byintroducing coordination types.

• Crystal chemical information available for additionto the standard mathetmatical description.

• Rather simple calculations for checking reportedcrystal-structure data.

4. Concluding Remarks

In this paper we have introduced our atomic-environment approach, and we have shown that theseunambiguously defined AEs can be very useful in crystalchemistry, especially when we know that in intermetallicstructures only a limited number of AEs are realized.From this and earlier work we have good reason tobelieve that within two years, after the completion ofthe analysis of all crystal symmetries, we will recognizeabout 25 standard atomic-environment types. TheseAETs can be used to explain crystal-structure data andto detect relations between structures. It is our firmbelief that, by combining our AET with predictivemodels such as the Miedema model, the Villars structurestability diagrams and the structure maps of Pettifor,these models will gain in reliability as well as inpredictability.

The observed atomic environments in the analyzedstructures suggest a complementary classification method(classification in so-called coordination types), whichwill lead to a considerable decrease in structure typesthat are needed. This classification in coordinationtypes gives, independent of the mathematical descriptionof the structures, geometrical information about theshort-range atomic arrangements in the structures.

In general, highly symmetrical AEs are formed inintermetallic compounds. Unusual AEs are only formedwhen there is a large difference in metallic radii of theatoms forming the compound. This deduction isconfirmed by Aslanov in his paper concerning the crystalchemical model of atomic interactions of intermetallicphases (Aslanov, 1992). Although there are somedifferences between the definitions in the model ofAslanov (CCMAI) and in our approach, especiallyregarding the coordination sphere of an atom, the

resemblance of the AEs observed by both methods isin general so overlapping that a more thoroughinvestigation is necessary to decide whether one methodis preferable to the other.

We are convinced that the value of the intergrowthconcept will be strengthened if in this model AEs areused in describing the intergrowing slabs. By introducingAEs for the atoms in the slabs, the relations betweenthe respective structures and the slabs will be morevisible, and it will also show whether there areimprobable structure transformations.

We have shown that with simple calculations andplots we can visualize the bonding character of thecompounds crystallizing in a structure type. Presentingall the compounds crystallizing in a particular structuretype in a periodic system array gives us valuable infor-mation about the possible combinations (of groups ofatoms) that are, or to some extent can be, realized inthis structure type.

Combining the above methods shows systematicallywhich of the published crystal-structure data are correct.Applying all these methods to new structure data can pre-vent us from publishing those data in a wrong description.

The importance of AEs, when combined with so-called structure stability diagrams (SSD) or quantumstructure diagrams (QSD), is described in Chapter 11by Villars in this volume. In that chapter and also inearlier work (Villars and Hulliger, 1987; Villars et al.,1989), Villars shows that combining the AEs with theSSD improves the reliability of the SSD.

The metallic radii as they are published among othersby Teatum et al. (1960) are not valid in most inter-metallic compounds, probably due to the fact that theassumption that they are correct for CN= 12 is nevermet in most compounds. We have shown that (seeTables 4 and 5) the coordination numbers can vary from3 to 22 in the structure types analyzed. In general weobserved low coordination numbers, CN < 9, for thep elements, CN numbers between 9 and 14 for the delements, and CN> 12 for the s and f elements.

5. Acknowledgements

The author is grateful to Dr P. Villars, H. W.Wondergem, and D. B. de Mooy for their interest inthis work and their critical reading of the manuscript.Special thanks are due to Dr A. R. Miedema who,before his sudden death, enthusiastically supported ourwork on systematics in intermetallic structures, who wasalways available for advice, and who participated innumerous fruitful discussions.

6. References

Aslanov, L. A. (1989). Structure of Substances (in Russian).Moscow State University Press, Moscow.

Aslanov, L. A., and Markov, V. T. (1992). A Crystal ChemicalModel of Atomic Interactions. 6. Intermetallic PhaseStructures. Acta Crystallogr., A48, 281-293.

Bodak, O. I., and Gladyshevskii, E. I. (1985). Handbook ofTernary Systems Containing Rare-Earth Metals (inRussian). L'vov State University Press, L'vov.

Brunner, G. O., and Schwarzenbach, D. (1971). ZurAbgrenzung der Koordinationssphare und Ermittlung derKoordinationszahl in Kristallstrukturen. Z. Kristallogr.,133, 127-133.

Bruzzone, G., Fornasini, M. L., and Merlo, F. (1970). Rare-Earth Intermediate Phases with Zinc. J. Less-CommonMet., 22, 253-264.

Daams, J. L. C , and Villars, P. (1993). Atomic-EnvironmentClassification of the Rhombohedral Intermetallic StructureTypes. Proc. Workshop on Regularities, Classificationsand Predictions of Advanced Materials. J. Alloys andCompounds, 197, 243-269

Daams, J. L. C , Villars, P., and van Vucht, J. H. N. (1991).Atlas of Crystal Structure Types for Intermetallic Phases.ASM International, Materials Park, OH.

Daams, J. L. C , Villars, P., and van Vucht, J. H. N. (1992).Atomic-Environment Classification of the Cubic Inter-metallic Structure Types. /. Alloys and Compounds, 182,1-33.

Grin', Yu. N. (1992). The Intergrowth Concept as a UsefulTool to Interpret and Understand Complicated Inter-metallic Structures. In Modern Perspectives in InorganicCrystal Chemistry (ed. E. Parthe). Kluwer, Dordrecht.

Grin', Yu. N., Yarmoluk, Ya. P., and Gladyshevskii, E. I.(1982). The Crystal Chemistry of Series of InhomogeneousLinear Structures. Sov. Phys. Crystallogr., 27, 413-417.

Grin', Yu. N., and Gladyshevskii, R. E. (1989). Gallides (inRussian). Metallurgiya, Moscow.

Hahn, T. (ed.) (1983). International Tables for Crystallo-graphy, Vol. A., Reidel, Dordrecht.

Jensen, W. B. (1989). Crystal Coordination Formulas.In The Structure of Binary Compounds (eds F. R. de Boerand D. Pettifor). Elsevier, Amsterdam, pp. 105-146.

Kripyakevich, P. I., and Gladyshevskii, E. I. (1972). Homo-logous Series Including New Structure Types of TernarySilicides. Acta Crystallogr., A28, Supplement, s97.

Merlo, F. (1988). Volume Effects in the IntermetallicCompounds Formed by Ca, Sr, Ba, Eu and Yb with OtherElements. J. Phys., F18, 1905-1911.

Miedema, A. R., and Niessen, A. K. (1982). Volume Effectsupon Alloying of Two Transition Metals. Physica, B114,367-374.

Parthe, E. (1990). Elements of Inorganic Structural Chemistry.Sutter Parthe, Petit-Lancy, Switzerland.

Pearson, W. B. (1958). A Handbook of Lattice Spacings andStructures of Metals and Alloys. Pergamon, London.

Pearson, W. B. (1967). A Handbook of Lattice Spacings andStructures of Metals and Alloys, Vol. 2. Pergamon, London.

Pearson, W. B. (1972). The Crystal Chemistry and Physics ofMetals and Alloys. Wiley-Interscience, London.

Pearson, W. B. (1979). The Stability of Metallic Phases andStructures: Phases with the AlB2 and Related Structures.Proc. R. Soc, A365, 523-535.

Pearson, W. B. (1985a). The Cu2Sb and Related Structures.Z. Kristallogr., Ill, 23-39.

Pearson, W. B. (1985b). The Most Populous of all CrystalStructure Types, the Tetragonal BaAl4 Structure. J. SolidState Chem., 56, 278-287

Pearson, W. B., and Villars, P. (1984). Analysis of the Unit-Cell Dimensions of Phases with the BaAl4 (ThCr2Si2)structure I, II and III. /. Less-Common Met. ,97, 119-143.

Teatum, E., Gschneidner, K., and Waber, J. (1960). ReportLA-2345, US Department of Commerce, Washington, DC.

Villars, P., and Calvert, L. D. (1991). Pearson's Handbookof Crystallographic Data for lntermetallic Phases, 2ndedn. ASM International, Materials Park, OH.

Villars, P., and Daams, J. L. C. (1993). Atomic-EnvironmentClassification of the Chemical Elements. Proc. Workshopon Regularities, Classifications and Predictions ofAdvanced Materials. J. Alloys and Compounds, 197,177-196

Villars, P., and Girgis, K. (1982). Regularities in Binarylntermetallic Compounds. Z. Metallk., 73, 455-462.

Villars, P., and Hulliger, F. (1987). Structural-StabilityDomains for Single-Coordination lntermetallic Phases./. Less-Common Met., 132, 289-315.

Villars, P., Mathis, K., and Hulliger, F. (1989). EnvironmentClassification and Structural Stability Maps. In TheStructure of Binary Compounds (eds F. R. de Boer andD. Pettifor). Elsevier, Amsterdam.

This chapter was originally published in 1995 as Chapter 15 in lntermetallic Compounds,Vol. 1: Principles, edited by J. H. Westbrook and R. L. Fleischer.

1. Introduction

Well over 1000 intermetallic compounds have beenreported for the nine varied structure types of fixedstoichiometry treated in this chapter, a number that isa substantial fraction of all the compounds whosestructures have been recently described and tabulated(Villars and Hulliger, 1987; Villars and Calvert, 1991).While their structural features differ widely, the familiesof compounds discussed here exhibit several unifyingthreads as regards the atomic size and the positions ofthe components in the periodic table. Theseconsiderations will be explored throughout the chapter.

Since the first edition of this book (Westbrook, 1967)there has been a substantial increase in the number ofcompounds reported and compiled, so much so that itis no longer practicable to include complete lists of thephases. Villars and Calvert (1991) should be consultedfor such compilations. A major factor in the growthhas been the addition of large numbers of ternarycompounds to earlier lists of primarily binarycompounds.

An important benefit of the greatly enlarged databaseis the refinement of earlier rules and rationales foroccurrence. Some of the earlier qualitative stricturesregarding atomic size and valence have been weakenedor invalidated as gaps in the data have been filled in.Newer, more quantitative, models are being developedfor intermetallic phases (Nevitt, 1967; Villars, 1984;Pettifor, 1986a, b; Rajasekharan and Girgis, 1983;

Miedema et ai, 1975); thus far, however, they have hadonly limited application to the compounds treated inthis chapter, as will be noted.

The compound families treated in this chapter show,for the most part, close adherence to characteristicstoichiometries. The discussion will begin with threeAB2 compound families whose structure types areMoSi2, CuAl2, and NiTi2. The closely related Fe3W3Ctype will be included with the last of the three. Twogroups of compounds of more complex stoichiometrythat are isostructural, or at least quasi-isostructural, willbe treated next. These are the Mn23Th6 type and theCu16Mg6Si7 type. The discussion will then turn to threecompound families possessing dissimilar structures, buthaving in common the dominant involvement byelements of the lanthanide and actinide periods. Thesecompounds belong to the NaZn13, CFe3, and Th3P4

structure types. The last of these is unusually prolific inits occurrence: over 350 compounds have been reported.

Readers unfamiliar with the Wyckoff notation usedin the tables of atomic positions that follow shouldconsult the International Tables for Crystallography(Hahn, 1989) for clarification. Briefly, each Wyckoffnotation, consisting of a number denoting themultiplicity and an arbitrary letter, refers to the positionof a set of atoms in the unit cell that share the samepoint symmetry. The x, y, and z values, used inconjunction with the generalized atomic coordinates forthe specific structure in the International Tables, providethe location of each atom in the unit cell.

Chapter 6

Some Important Structures of Fixed Stoichiometry

Michael V. NevittDepartment of Physics and Astronomy, Clemson University, Clemson, SC 29634, USA

Carl C. KochDepartment of Materials Science and Engineering, North Carolina State University,

Raleigh, NC 27695, USA


2. MoSi2-Type Phases

This family of AB2 compounds has been discussed andanalyzed to a greater extent than the others covered inthis chapter. Its treatment will be somewhat moredetailed, therefore. The compounds have the Pearsonsymbol tI6 (Strukturbericht Cllb); the space group isI4/mmm and space group number 139. The atomicpositions for the prototype MoSi2 (Thomas et al.,1985) are given in Table 1. The cell dimensions area = 0.3203 nm, c = 0.785 nm and axial ratio c/a = 2.45.

Villars and Calvert (1991) list 84 binary or ternarycompounds that crystallize in the MoSi2-type structure.Many of these compounds involve a lanthanide as theA component and noble or group IVA (Ti group) metalsas the B component, but sometimes as the A metal. Ithas long been recognized that these compounds can beclassified into two main groups with respect to their axial

Table 1. Prototype MoSi2

WyckoffAtoms notation Symmetry x y z Occupancy

Mo 2(a) 4/mmm 0 0 0 1.0Si 4(e) 4mm 0 0 0.333 1.0

ratios (Nevitt, 1967; Raynor, 1974); that is, a group oflow axial ratio between 2.29 and 2.60, and a group ofhigh axial ratio between 3.188 and 4.684. More recentlyHellner and Pearson (1984) have divided these groupingsinto four as illustrated in the histogram of Figure 1.These include the large groupings of compounds aroundc/a ratios of either 2.45 or 3.45 as well as smaller groupswith c/a at 3.0 and 4.24. They extended their study ofthe tetragonal distortion of I (body-centered) and F(face-centered) lattice complexes to phases with AB2

stoichiometry with the MoSi2-type structure. Thegeometry of these groupings is related to the relativesizes of the component atoms. From the geometry ofthe structure and the assumption that there aresimultaneous contacts between spherical atoms in theA-A and A-B directions, it may be determined thatthe ideal radius ratio for MoSi2-type phases isRA/RB = 1. However, the actual values of RA/RB varyfrom <0.8 to 1.25.

The structure of MoSi2-type phases has beendescribed as a superstructure of the b.c.c. or CsClstructure with three subcells stacked along [001]. Thisis illustrated in Figure 2 for CuTi2. The c/a ratio forthe MoSi2-type tetragonal cell varies from 2.207 to4.684 corresponding to the distortion of the b.c.c. cells

Figure 1. Histogram of the number of phases with the MoSi2 structure having axial ratios lying about the indicated values. Thicklines indicate phases for which RB>RA. Above are indicated the three modes of stacking triangular layers of atoms. (Reproducedby permission of R. Oldenbourg Verlag from Hellner and Pearson, 1984)

Num

ber o

f pha

ses

Figure 2. The unit cell of CuTi2, an MoSi2-type structure.(Reproduced by permission of John Wiley & Sons from Nevitt,1967)

to c/a ratios of 0.736 to 1.561 respectively. Projectionson (110) of the MoSi2 structure for three 'ideal' axialratios are given in Figure 3. For the AB2 stoichiometryof MoSi2-type phases, all but two phases groupedabout c/a = 2.45 or c/a = 3.0 have A as the larger atom

or RA « RB. In contrast to this, all known phases withc/a ratios grouped about 3.5 have B as the larger atom.From geometry, when A is the larger atom the axial ratioc/a = 2.45 provides a structure that gives a betterseparation of the A atom and its four A neighbors along[ 100] and [010] than an arrangement with a large c/avalue where a0 is smaller. This geometrical factor isconsistent with the disposition of phases about an axialratio of 2.45 when A is the larger component.

For phases with an axial ratio close to 3.0, the subcellcorresponds to the nearly undistorted b.c.c. cell. It mightbe expected that phases with this axial ratio would haveRA~ RB. However, for AlCr2 and MoU2, Al (A) andU (B) are significantly larger than their partner species.

The phases occurring about the axial ratio c/a = 3.5presumably form because it gives good sphere packingwhen the major component B is the larger atom. Raynor(1974) had previously suggested that the phases withhigh c/a values were attributable to the directional d-electron bonding of the B components Ti, Zr, or Hf.Subsequently, however, additional phases with c/avalues near 3.5 have been discovered with other Bcomponents (e.g. Au, Ba, Pb, Bi), so geometry seemsto be the controlling factor.

The two phases with axial ratios near 4.24, i.e.TiAg2 with c/a = 4.014 and CdTi2 with c/a = 4.684,are not well understood. From the atomic sizes one

Figure 3 . Project ions of the MoSi2 s tructure, A B 2 , on (110) for the axial ratios indicated: ( Q ) A a toms and (O) B a toms inthe plane of the projection; ( v ) A a toms and ( A ) B a toms in the layers above and below. (Reproduced by permission of R.Oldenbourg Verlag from Hellner and Pearson , 1984)

Sum of volumes of atoms in cell (A3)

Figure 4. Observed unit-cell volume of phases with the MoSi2

structure plotted against the sum of the elemental volumes ofthe atoms in the unit cell. Left: phases with RB>RAi togetherwith AgTi2 and CrTi2 ( + ). Right: phases with /?A>/?B ,including AlCr2 ( x ) and Hg35Cd65 ( A ) . (Reproduced bypermission of R. Oldenbourg Verlag from Hellner and Pearson,1984)

might predict an axial ratio of —2.45 (Hellner andPearson, 1984).

Figure 4 illustrates that the volume of the unit cellof phases with the MoSi2 structure follows closely thesum of the atomic volumes of the atoms in the unit cell.This is especially true for the phases where RB>RA.

Polymorphism has been observed for MoSi2. Thehigh-temperature phase, stable above 185O°C, has theCrSi2-type structure (Svechnikov et ai, 1971). TheCrSi2 structure, hP9, space group P6222, is made upof three close-packed 36 layers of atoms (3 specifies atriangle, and 6 gives the multiplicity) one above the otheralong [OO1 ] in the unit cell (Villars and Calvert, 1991).The close-packed layers are stacked in b.c.c. [110]stacking positions. Thus, CrSi2 is part of the samefamily of structures based on b.c.c. [110] stacking ofclose-packed AB2 layers as MoSi2 (and TiSi2) (Pearson,1972).

While the atomic sizes and atomic packing of thecomponents are important for the occurrence of theMoSi2-type phases, and for the axial ratios observed,electronic factors also play a role in their occurrence.Figure 5 (Siegrist et al., 1983) gives existence regionsof transition-element MX2 phases (M = transitionelement, X = p element (metalloid)) in a diagram ofvalence-electron number ( = E group numbers) vs.

Obs

erve

d ce

ll vo

lume

(A3)

Figure 5. Existence regions of transition-element MX2 phases (X = p element) in a diagram of valence-electron number (nVE)vs electronegativity difference (xx~ XM: ( A ) t n e non-metallic exceptions among the silicides (ReSi2, CrSi2, FeSi2, and OsSi2).The formulae refer to structure types. (Reproduced by permission of Elsevier Sequoia SA from Siegrist et al.t 1983)

Figure 6. The AB2 structural map developed by Pettifor. (From Pettifor, 1988)

MN

B

electronegativity difference. The region for MoSi2-typestructure phases overlaps those for CrSi2- andTiSi2-type phases.

The most complete structural correlation using'classical' concepts has been presented by Villars andco-workers (Villars and Hulliger, 1987; Villars, 1984).Three-dimensional maps (Ax, AR, EVE) are used toseparate the structure types. Here Ax is the magnitudeof the Martynov-Batsanov electronegativities, AR themagnitude of the difference of Zunger's pseudopotentialradii sums, and EVE the sum of the valence-electronnumbers. Stability domains give good separation intoMoSi2 structure regions in this three-dimensionalspace. Of the 56 MoSi2 structure compounds, only onewas misplaced and only four non-MoSi2 structurecompounds were found in the MoSi2 domains.

While the method does provide separation, theangular dependence of the valence orbitals, i.e. whetherthe electrons have s-, p-, or d-like quantum character,is not given by these classical coordinates. Pettifor(1986a, b, 1988) has proposed a phenomenologicalstructure-map scheme based on assignment of numbers

(Mendeleev numbers, MN) to the elements in theperiodic table that does reflect the quantum characterof the electronic structure and can form a two-dimensional map. This is accomplished by running aone-dimensional string through the periodic table.Excellent structural separation of binary compounds isachieved by plotting MNA vs MN8 for a givenstoichiometry such as AB2. The Pettifor map for AB2

stoichiometry is presented in Figure 6. There are severalmajor groupings of MoSi2 phases apparent in thismap, and the two largest groupings represent thecompounds with c/a^2.45 and c/a» 3.45. However,there are many isolated MoSi2 regions as well.

Another phenomenological alloying model has beenapplied to the MoSi2 phases by Rajasekharan andGirgis (1983). These authors used Miedema's theory forthe heat of formation of intermetallic phases (Miedemaet al.9 1975). They plotted the Miedema parameter A<£*,which measures the chemical potential (related toelectronegativity), versus (A/2WS)

1/3, which representsthe electron density at the surface of the Wigner-Seitzcell for each element. A A0* vs (A/zws)

1/3 map gave

Figure 7. A plot of <£J -<£* vs. (/i* s)1 / 3-(rt^ s)

I / 3 for the binary systems in which MoSi2-type phases occur. The upper curverepresents compounds with c/a~ 3.5, the lower curve those with c/a «2.5, and labeled points in between these curves for c/a * 3.0.(Reproduced by permission of the American Physical Society from Rajasekharan and Girgis, 1983)

good resolution among the binary systems in whichdifferent structure types occur, indicating theimportance of the enthalpy of formation of anintermetallic compound in deciding the crystal structurethat it adopts. That is, the heat of formation isdetermined by the values of A$ and An in Miedema'smodel (Miedema et al., 1975). Such a plot is shown forMoSi2 phases in Figure 7. Very good separation isobserved for the MoSi2-type compounds withc/a ̂ 2.45 and c/a« 3.45, with the few compounds withc/a« 3.0 in between. Thus electronic structure as wellas atomic packing plays a role in determining thestructure of the MoSi2-type structure phases.

The MoSi2-type structure phases contain severalcompounds with interesting properties. MoSi2, theprototype of these structures, has long been a practicalmaterial used for high-temperature furnace heatingelements (see Chapter 10 by Kumar and Chapter 20 byVedernikov in Volume 2). There is much interestpresently in developing MoSi2 as a high-temperaturestructural material because of its high meltingtemperature, low density, and excellent oxidationresistance (Fleischer, 1985; Petrovic and Vasudevan,1992; Almane/tf/., 1992).

3. CuAl2-Type Phases

The CuAl2-type structure, Pearson designation tI12(Strukturbericht C16) is b.c.t. with the c/a ratio in therange 0.74 to 0.89 (Villars and Hulliger, 1987; Villarsand Calvert, 1991). It exhibits the AB2 stoichiometrywith deviations from stoichiometry of either sign up toI0Io. The space group is I4/mcm and space groupnumber 140. The atomic positions in the unit cell of theprototype (Meetsma et al., 1989) are given in Table 2.

There are 101 binary and ternary phases as tabulatedby Villars and Calvert (1991). The unit cell for theCuAl2 structure is illustrated in Figure 8. Figure 8(a)shows the unit cell and Figure 8(b) gives the atomicarrangement down [001].

Each Cu atom is surrounded by eight Al atomssituated at the corners of an Archimedean squareantiprism, and it has two Cu atoms at a distance of c/2,forming linear chains. Each Al atom has four Cu atoms

Table 2. Prototype CuAl2

Figure 8. (a) Unit cell of CuAl2. (Reproduced by permissionof Chapman & Hall from Porter and Easterling, 1981, partof Figure 5.29). (b) Atomic arrangement in the CuAl2structure projected down [001]: large open and solid circlesare Al atoms, at z = 0 and Z = ̂ . Small open circles are Cuat z = -, -. (Reproduced by permission of John Wiley &Sons from Pearson, 1972)

as nearest neighbors and is also surrounded by a totalof 11 (1+2 + 4 + 4) other Al atoms at distances dx, d2,d2, and d4, respectively, near to 2rB. The d2 and d4

distances are >2rB while dx is always, and d2 almostalways, smaller. These atoms form a convex CN 15polyhedron about the Al atom. The main feature of thisAB2 structure type appears to be an interlockinghoneycomb network of the Al atoms in hexagons in the(110) and (HO) planes, while the Cu atoms lie inchannels parallel to the c-axis.

Twenty-three quasibinary CuAl2-type alloy systemsare described by Havinga et al. (1972). In those systemswhere both end members have the CuAl2 structure, 11cases (all but three) exhibit continuous solid solubility.Of course, in systems where one end member is notisostructural, only a limited range of solubility will be

Atoms

CuAl

Wyckoffnotation

4(a)8(h)

Symmetry

422m.2m

X

00.1581

y

00.6581

Z

1/40

Occupancy

1.01.0

observed. For example, for Ni1^CuxZr2 theCuAl2-type structure is found up to Jt=0.4. For x>0.6an MoSi2-type structure phase is stable, and a two-phase field exists for 0.4<x<0.6. Interstitial elementcontamination can influence the structure in certainsystems. In Fe1^CuxZr2 an NiTi2-type phase isobserved for x< 0.8 if the samples are contaminated byC, N, or O. However, when sample preparation iscarried out such as to make interstitial elementcontamination negligible, the FeZr2 end member alsocrystallizes in the CuAl2-type structure (as does CoZr2,independent of contamination).

The atomic coordinate x and the axial ratio c/a exhibitmaxima when plotted against the concentration ofvalence electrons per atom, e/a, at e/a values near 2and 5 with a minimum in between, as illustrated inFigure 9 (Havinga et al.t 1972). This behavior suggeststhat e/a is the major parameter determining c/a.Illustrative of this suggestion is the behavior of thequasibinary system AlTh2 -AgTh2. The end membershave very similar atomic volumes and c/a ratios. Whilethe atomic volumes remain essentially constant withalloying, i.e. changing e/a, the c/a ratio has a markedminimum, as shown in Figure 10. In addition, chemically

very different compounds show similar c/a ratios forthe same values of e/a. Havinga (1972) has presenteda qualitative explanation for this oscillatory dependenceof c/a ratio with e/a for the Cl6 (til2) structurecompounds based on pseudopotential calculations ofband structure.

Havinga and Damsa (1972) analysed the CuAl2-type(C 16) structure with the aid of near-neighbor diagrams.They conclude that the C16 structure is stabilized bythe presence of A-B as well as B-B contacts. Thestructure becomes unstable when the A-A or B-B pairsare very close to making contact. These geometric ideaspredict that the C16 structure is limited to radius ratiosof the components, RA/RB, in the range 0.655 < RA /RB

<0.905+ 1.35(0.8165-c/a). It was recognized earlier(Laves, 1956) that A-B contacts were important instabilizing the structure. It was suggested (Nevitt, 1967)that its less restrictive atomic-size requirements favorits occurrence with respect to the similar NiTi2- andMoSi2-type structures. When RA/RB becomesunfavorably large for these structure types, theCuAl2-type structure was thought to be favored.However, this generalization now seems less secure, aswill be discussed in the next section.

n (e/a)

Figure 9. The atomic coordinate x and the axial ratio c/a as functions of valence-electron concentration per atom, /t, forCuAl2-type compounds and alloys. (Reproduced by permission of Elsevier Sequoia SA from Havinga et al., 1972, Figure 7)

c/a

Figure 10. Lattice constants, a and c, and volume, K, of theunit cell for the system Ag1ÂlxTh2. (Reproduced bypermission of Elsvier Sequoia SA from Havinga et al.y 1972)

Villars' three-dimensional structural stability diagram(Villars, 1984) (see discussion of MoSi2-type structurephases) for AB2 binary compounds provides excellentseparation for the 56 examples tested. Only twocompounds, AuPb2 and VSb2, with the CuAl2-typestructure were misplaced in the structural stabilitydiagram. Pettifor's structural map for AB2 compounds(Pettifor, 1986a, b, 1988) also separates the phases intoseveral domains, albeit with some admixture of otherstructure types in some cases.

4. NiTi2-Type and Fe3W3C-Type Phases

In this section the AB2 phases having the NiTi2-typeand the Fe3W3C-type structures will be discussed. TheNiTi2-type structure is cubic with 96 atoms per unitcell, Pearson symbol cF96. The space group is Fd3m,carrying space group number 227. The prototype hasthe atomic positions filled as shown in Table 3 (Muellerand Knott, 1963).

The structure is composed of interpenetrating arraysof Ti octahedra and Ni tetrahedra. The large f .c.c. cellcan be visualized as eight cubic subcells. The twoalternating patterns (a) and (b) shown in Figure 11(Mueller and Knott, 1963) are stacked as shown inFigure ll(c). In pure NiTi2, i.e. in the absence ofinterstitial H, O, N or C, the 'oxygen' positions (ii) inthe subcell of Figure 1 l(b) are unoccupied. In the binaryphases the B component is dominated by the group IVtransition metals Ti, Zr, and Hf—40 out of the list ofabout 60 reported compounds (Villars and Calvert,1991)—with minor representation by group III (Sc) andgroup V (Nb, Ta). Transition metals from the Fe, Co,and Ni groups are about equally represented as Aelements. Pettifor's AB2 structure map (Pettifor,1986a, b, 1988) places NiTi2-type phases in closeproximity to MoSi2- and CuAl2-type phases, leadingone to expect that more than one of them may coexistas polymorphic forms in some alloy systems. This pointwill be discussed further.

Ternary phases, which now outnumber the binaryphases in the published compilation, fall into twocategories. The first of these involves H as the thirdcomponent (see also Chapter 21 by Schlapbach et al.in Volume 2). Reported compounds include H3FeHf2,H3NiTi2, H3MnHf2, and H2RhHf2 (Villars andCalvert, 1991). These H concentrations are probablynominal. It has been observed (Yvon and Fischer, 1988)that NiTi2-type compounds, along with certaincompounds having the MoSi2- and CuAl2-typestructures, dissolve H interstitially. The specificinterstitial sites occupied depend on the structure of thehost lattice, but certain criteria apply in all cases: (1)octahedral and tetrahedral interstitial sites in the metalliclattice are preferred; (2) H atoms are mobile at ambienttemperature; (3) the number of interstitial sites availableis much larger than the number of H atoms absorbed.H is reported (Bonhomme et al., 1992) to occur inthe NiTi2-type compound H11RhMg2, a hithertounreported compound with this structure type that doesnot exist in the absence of interstitial H.

It is noteworthy that the two AB2 families previouslycovered (MoSi2 type and CuAl2 type), as well as theMn23Th6-type and CFe3-type compounds yet to be

Table 3. Prototype NiTi2

Atoms

Ti1NiTi2

Wyckoffnotation

16(d)32(e)48(0

Symmetry

.3m

.3m2.mm

X

5/80.7870.186

y

5/80.7870

Z

5/80.7870

Occupancy

1.01.01.0

discussed, are also reported to dissolve interstitial H(Yvon and Fischer, 1988). The same generalizationsapply as regards the occupied sites.

The second group of ternary phases are solid solutionsinvolving multiple A components. These are primarilyfrom the transition-metal groups that form the binaryphases. Compositions given in the compilation of Villarsand Calvert (1991) are nominal. More detailedinformation of the location of the phase fields in theternary systems is given in Nevitt et al. (1960).

Another group, smaller in number, appears to beuniquely ternary phases involving Si partially substitutingfor the A component (Gladyshevskii et al., 1963;

Kuz'ma et al., 1964; Bardos and Beck, 1966). Theatomic positions of Si are not certain in the absence ofdetailed crystallographic studies.

The Fe3W3C-type structure, Pearson symbol cF112(Strukturbericht E93), is cubic with 112 atoms per unitcell. The space group and_number are the same as thoseof the NiTi2-type, i.e. Fd3m and 227. In the prototypethe atomic positions indicated in Table 4 are occupied(Bojarski and Leciejewicz, 1967).

W atoms form an array of octahedra with C atomsin interspersed positions similar to those occupied byO atoms in subcell (b) of Figure 11. Fe atoms formtetrahedra arrayed between the W octahedra. Twelve

Figure 11. Schematic diagrams showing crystallographic subcells and their stacking in NiTi2 and Ni2Ti4O (Reprinted withpermission from Mueller and Knott, Transactions of the Metallurgical Society, Vol. 227 (1963) p. 674-8, a publication of TheMinerals, Metals & Materials Society, Warrendale, Pennsylvania 15086)

Figure 12. Composition profiles for (a) carbon-containing M6C, and (b) nitrogen-containing M6N Fe3W3C-type phases

Num

ber

of

com

poun

ds r

epor

ted

Num

ber

of

com

poun

ds r

epor

ted

Table 4. Prototype Fe3W3C

Atoms

C

Fe2

W

Wyckoffnotation

16(c)16(d)32(e)48(f)

Symmetry

.3m

.3m

.3m2.mm

X

1/85/80.82970.1828

y

1/85/80.82970

Z

1/85/80.82970

Occupancy

1.01.01.01.0

atoms, two C, four W and six Fe, form a distortedicosahedron around a W atom. Fe2 atoms are alsosurrounded by a distorted icosahedron of six Fe and sixW. Subcells (a) and (b) are stacked to form the arrayshown in Figure 1 l(c).

There are now approximately 70 phases reported tohave this structure (Villars and Calvert, 1991). Thisfamily of compounds is frequently called ^-carbide orM6C. However, Yakel (1985) takes into account itswider compositional range and assigns the generalformula (TxM12-x)Xy, where 6<JC<8 and 0<y<3. Tis a transition element from the Ti, V or Cr group; Mis an element from the Fe, Co or Ni group or a non-transition element; and X is a non-metal, C, N or O.The previously discussed NiTi2 type corresponds toj> = 0, and the widely occurring M6C carbidescorrespond to y = 2. There is no conclusive evidence thatcompounds with different values of x coexist in the samesystem, e.g. T2M4C and T3M3C, or that intermediatecompositions generally occur (Yakel, 1985). Thisdiscussion will focus therefore on the nominal com-positions A2B4X and A3B3X, where A and Bcorrespond, respectively, to YakePs T and M, and Xis C or N.

Known carbon-containing phases outnumber nitrogen-containing phases in a roughly 8/5 ratio (Villars andCalvert, 1991). The carbon and nitrogen phases differin their composition profiles in two respects. First,A3B3X compounds exceed A2B4X compounds in thecarbon-containing phases, while the reverse is true inthe nitrogen-containing phases. Second, the metal-atompopulations differ, as shown in Figure 12. For bothfamilies there is a minimum in the vicinity of the Mngroup, indicating a natural division between theA elements to the left and B elements to the right.The M6C (carbon-phase) population has a morepeaked distribution with rather sharp maxima at theV and Co groups. There is a blunter distributionin the M6N (nitrogen-phase) population. Figure 12does not suggest a tight atomic-size constraint. Forexample, the CN 12 metal-atom radius ratio varieswidely (1.06</?A//?B<1.29) for the M6C and M6Nphases treated as a single population. A morequantitative correlation such as that of Villars andHulliger (1987) may be more definitive, but it has notyet been generated for this multi-coordination structure.

As was mentioned earlier, the three structure types,MoSi2, CuAl2, and NiTi2, appear in overlapping orcontiguous domains in the AB2 Pettifor map (Pettifor,1986a, b, 1988) shown in Figure 6. We might expect,therefore, that the three structure types are competingfor stability in some two-component and higher-order

Figure 13. Reported coexistence of MoSi2-, CuAl2-, NiTi2-,and nitrogen-containing Fe3W3C-type phases in compoundsinvolving Zr or Hf

systems, and this competition might reveal itself incoexisting polymorphic forms, whose stability dependson composition, temperature, pressure, or on otherfactors including interstitial ('stabilizing') components.At present, only an incomplete viewpoint can bepresented. Figure 13, based on the compilation of Villarsand Calvert (1991), puts this point in perspective. Itshows that in a group of compounds that pairs Zr or Hfwith Co- and Ni-group elements sub-groups are reportedto form more than one of the three AB2 structuretypes. Most noteworthy is the apparent coexistence ofCuAl2-type structures and structures of the NiTi2 typeand/or M6N/Fe3W3C type. It is not unreasonable tosuspect that some of the entries in the NiTi2 columnmay be N-stabilized Fe3W3C-type phases, but there ispresently insufficient information to clarify this point.

5. Mn23 Th6-Type Phases

Carrying the Pearson symbol cF116 {StrukturberichtD8a), the Mn23Th6 structure type is reported to occurin 200 compounds. The space group is FmJm andnumber 225. In the prototype phase (Florio et ai, 1952),Mn and Th atoms are located as indicated in Table 5.

Table 5. Prototype Mn23Th6


Mn1 4(b) m3m 1/2 1/2 1/2 1.0Mn2 24(d) m.mm 0 1/4 1/4 1.0Th 24(e) 4m.m 0.203 0 0 1.0Mn3 32(f) .3m 0.378 0.378 0.378 1.0Mn4 32(f) .3m 0.178 0.178 0.178 1.0

Figure 14. Polyhedron of 44Mn atoms (small circles)surrounding octahedral cluster of six Th atoms (large dottedcircles). (Reproduced by permission of John Wiley & Sons fromPearson, 1972, p. 546)

The structure is cubic with 116 atoms per unit cell. Theprominent structural feature is a large polyhedron of44 Mn2, Mn3 and Mn4 atoms that encases anoctahedron of Th atoms, as shown in Figure 14. Thepolyhedra are packed together in a face-sharing patternand Mn1 atoms occupy the interstitial holes betweenthe polyhedra.

About one-quarter of the reported compounds arebinary A23B6 phases in which Mn or another first-long-period element, Fe, Co, Ni, Cu or Zn, is the A element.A lanthanide including Y, an actinide, Zr, or Hf servesas the B element. Most of the compounds are ternaryphases, and they fall into two groups. One grouprepresents miscibility between two of the binary phases.Examples are Mn23Th3Y3 and Fe25Mn21Sm12.Available information does not indicate the extent towhich complete or partial miscibility occurs. Thesecond, and larger, group consists of free-standingternary phases, which from their composition cantentatively be assumed to have the slightly differentCu16Si7Mg6-type structure. In the prototype, Si atomsreplace Mn1 and Mn2 atoms, Mg atoms occupy the Thpositions, and Cu atoms replace Mn3 and Mn4 atoms.The two structures differ in a minor way in severalatomic parameters (Pearson, 1972).

Although there are some stoichiometric variants, mostof the reported compounds in the latter group have theformula (A,C)23 B6, and most frequently A16C7B6,where A and C have a very wide periodic-table range,from Be through the transition metals of the Fe, Co andNi groups, and on through the Cu, Al and Si groups.This wide range, along with a corresponding widevariation in atomic radii, makes it difficult to discern

controlling valence and atomic-size effects, especiallyso in the absence of the detailed crystallographic studiesneeded to fix the locations and coordinations of thecomponent atoms. The B-atom component for thisgroup of ternary compounds excludes for the most partthe lanthanides and actinides, but enlarges the range ofthe transition-metal partners to include Sc, Y, Ti, Nb,and Ta, as well as Zr and Hf. A more completeunderstanding of this widely occurring compound typemust await more detailed compositional andcrystallographic studies.

6. NaZn13-Type Phases

The structure of NaZn13 is f.c.c., carrying Pearsonsymbol cF112 (Strukturbericht D23). The space groupis Fm3c, number 226. One-hundred and twelve atomsare arranged in the prototype unit cell (Shoemakeret aL, 1952) as shown in Table 6.

In the prototype, the large alkali-metal atom, Na, issurrounded by 24 Zn atoms forming a structureresembling a cube flattened at its corners. This largeatom-small atom 'fit' is the primary feature ofthe structure. Pearson (1972) has used a near-neighbordiagram to show that this structure type has a particularlyfavorable chance of occurring when the atomic diameterratio DA/DB lies between 1.6 and 1.7, providingthereby a high coordination factor in the structure.Figure 15, also from Pearson (1972), shows that inNaZn13-type phases, and in several other structuretypes containing one large component atom, thereported compounds occur for a wide range of atomic-diameter ratios, but nevertheless the ability of thisstructure type to accommodate a large A-B atomic-size disparity is clear. The small Be atom(RBt = 1.128 A) is paired with a wide range of larger Apartners in over a third of the 90 reported compounds(Villars and Calvert, 1991). More obvious yet is theinvolvement of the large lanthanide A partners(including Sc and Y) and the actinides; more thantwo-thirds of the reported compounds have atoms fromthese periods as the A component. Ternary phasesaccount for a substantial fraction of the compounds;

Table 6. Prototype NaZnn


Na 8(a) 43_2 1/4 1/4 1/4 1.0Zn1 8(b) m3. 0 0 0 1.0Zn2 96(i) m. . 0 0.1806 0.1192 1.0

multiple components representing A and B occurextensively.

7. Fe3C-Type Phases

The Fe3C-type structure phases have AB3 stoichiometryand the orthorhombic structure with Pearson symbol0PI6 {Strukturbericht DO11). This structure has thespace group Pnma with space group number 62. Theatomic positions for the prototype Fe3C (Meinhardtand Krisement, 1962) are given in Table 7, with celldimensions a = 0.5089 nm, b = 0.6743 nm, andc = 0.4524 nm.

Villars and Calvert (1991) list 137 binary or ternarycompounds that crystallize in the Fe3C-type structure.The structure of the prototype can be described asdistorted close-packed layers of iron atoms 'rumpled'into a series of troughs and crests running parallel tothe ffo-axis. The atomic arrangement in Fe3C vieweddown [001] is illustrated in Figure 16 (Pearson, 1972).The structure has been described as rt^r-close-packingof iron atoms, with carbon in the interstices. Therefore,the Fe3 C-type structure has often been classified as an

Table 7. Prototype Fe3C


C 4(c) .m. 0.881 0.25 0.432 1.0Fe1 4(c) .m. 0.044 0.25 0.837 1.0Fe2 8(d) 1 0.181 0.063 0.337 1.0

Figure 16. Atomic arrangement in Fe3C (oP16) viewed down[001] : large open and solid circles are Fe at z —- and z~~A-Small open circles are C at the indicated fractional elevationsalong z direction. (Reproduced by permission of John Wiley& Sons from Pearson, 1972)

'interstitial phase', which is mainly controlled by atomic-size factors (Hume-Rothery et al.y 1969). While manyof the Fe3 C-type phases do contain a metalloid'interstitial' atom such as carbon or boron, more thanone-half of the compounds reported are metal-metalsystems. Many of these metallic AB3 structures have alanthanide element as the B atom and a transition ornoble metal as the A atom. The radius ratios of thephases lie between 0.61 and 0.87.

Figure 15. Distribution of NaZn13-type phases and other high-coordination phases with atomic diameter ratio. The shaded bandindicates the radius ratio range satisfying a high coordination factor. (Reproduced by permission of John Wiley & Sons fromPearson, 1972, p. 75)

Several ternary silicide or germanide compoundswith an ordered version of the Fe3C-type structurehave been synthesized. Moreau et al. (1982) reportedon isotypic compounds RPd2Si with R = Ce, Pr, Nd,Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y, and RPt2Siwith R = Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y. The Rand Pd or Pt atoms occupy the Fe sites in thisordered version of Fe3C. Similarly, Jorda et al. (1983)found RPd2Ge and RPd2Ga to be of the orderedFe3C-type derivative structure. Here R stands for allthe Ianthanide elements except Pm and Yb, and alsoincludes Y.

The prototype for this structure class, Fe3C,cementite, is of course well known as an importantconstituent of ferrous alloys.

8. Th3P4-Type Phases

This highly populous family of compounds, whoseprototype is Th3 P4, has a cubic crystal structure with28 atoms per unit cell. The Pearson symbol is cI28(Strukturbericht D73) and the space group is I43d,number 220. In the prototype phase (Meisel, 1939),atoms lie as shown in Table 8.

Table 8. Prototype Th3P4


Th 12(a) 4.. 3/8 0 1/4 1.0P 16(c) .3. 0.083 0.083 0.083 1.0

Compared to the structures previously discussed, thisstructure is characterized by polyhedra with lowercoordination numbers. P atoms have six Th neighborsin a greatly distorted octahedron, while Th has eightP neighbors in a non-cubic configuration (Pearson,1972) as shown in Figure 17. There are short planarstrips of atoms lying normal to (001) planes andzigzagging alternately + and - about <100> directions.An array of interpenetrating * staircases1 results fromthis configuration (Pearson, 1972).

The reported binary phases, numbering over 100,show high selectivity as regards the component atoms.A is a member of the P or S group, while B is aIanthanide (including Y) or an actinide. A still largerfamily of isostructural compounds—over 200—areternary phases, and these fall into two categories. Onecategory involves multiple A or multiple B components,where A or B have the identities cited above. In the

Figure 17. The Th3P4 structure viewed down [001]. The coordinations of the P and Th atoms are shown. (Reproduced bypermission of John Wiley & Sons from Pearson, 1972, p. 773)

second category a divalent atom (Be, Mg, Ca, Sr, Baor Cd), or monovalent Ag, substitutes in part for thelanthanide B atom when A is S or Se. Pettifor (1986a, b,1988) is able to sequester a limited number of thesecompounds on his A3B4 structure map, noting anobservation by Wells (1975) that the 8:6 coordinationof Th3P4 arises from the fact that the stoichiometricbasis is built around eight PTh6 vertex-sharingoctahedra.

9. Summary

Over 1000 intermetallic compounds are representedby the nine structure types treated in this chapter.Given the expanded number of reported compounds andthe recent availability of comprehensive compilations,it is now possible to clarify relationships betweenthe structure types and to observe periodic-tableand atomic-size regularities that were not previouslyobvious.

With few exceptions, these families of compoundsoccur at fixed, characteristic AmBn stoichiometries; Aor B is frequently a composite of several isovalentelements. The roles of the non-metals H, O, C, and Nare clarified, particularly for the 77-carbide compounds.Phenomenologically based structural stability modelsshow promise as a means of correlating the occurrenceof these structure types. These tools have been appliedto the MoSi2-, CuAl2- and NiTi2-type compounds, butnot to the more complex, multi-coordination structuresthat are also discussed in this chapter. Further progressin the development of such models should lead tosignificant advances in the understanding of theseintermetallic compounds.

10. References

Alman, D. E., Shaw, K. G., Stoloff, N. S., and Rajan, K.(1992). Mater. ScL Eng., A155, 95.

Bardos, D. M., and Beck, P. A. (1966). Trans. AIME, 236,40-48.

Bojarski, Z., and Leciejewicz, J. (1967). Arch. Hutn., 12,255-263.

Bonhomme, F., Selvam, P., Yoshida, M., Yvon, K., andFischer, P. (1992). / . Alloys Compounds, 178, 167-172.

Fleischer, R. L. (1985). J. Met., 37, 16.Florio, J. V., Rundle, R. E., and Snow, A. I. (1952). Acta

Crystallogr., 5, 449-457.Gladyshevskii, E. I., Kuz'ma, Yu. B., and Kripyakevich, P. I.

(1963). J. Struct. Chem., 4, 343-349.

Hahn, T. (ed.) (1989). International Tables for Crys-tallography. International Union of Crystallography;KIuwer Academic, Boston.

Havinga, E. E. (1972). J. Less-Common Met., 27, 187.Havinga, E. E., and Damsa, H. (1972). J. Less-Common Met.,

27, 269.Havinga, E. E., Damsa, H., and Hokkeling, P. (1972). J. Less-

Common Met., 27, 169.Hellner, E., and Pearson, W. B. (1984). Z. Kristall., 168,

273-291.Hume-Rothery, W., Smallman, R. E., and Haworth, C. W.

(1969). The Structure of Metals and Alloys. Institute ofMetals, London, pp. 273-275.

Jorda, J. L., Ishikawa, M., and Hovestreydt, E. (1983). J. Less-Common Met., 92, 155-161.

Kuz'ma, Yu. B., Gladyshevskii, E. I., and Byk, D. S. (1964)./ . Struct. Chem., 5, 518-522.

Laves, F. (1956). Theory of Alloy Phases. American Societyfor Metals, Cleveland, OH.

Meetsma, A., de Boer, J. L., and van Smaalen, S. (1989). / .Solid State Chem., 83, 370-372.

Meinhardt, D., and Krisement, O. (1962). Arch Eisenhuettenwes.,33, 493-499.

Meisel, K. (1939). Z. Anorg. AUg. Chem., 240, 300-312.Miedema, A. R., Boom, R., and de Boer, F. R. (1975). / . Less-

Common Met., 41, 283.Moreau, J. M., Le Roy, J., and Paccard, D. (1982). Acta

Crystallogr., B38, 2446-2448.Mueller, M. H., and Knott, H. W. (1963). Trans. AIME, 227,

674-678.Nevitt, M. V. (1967). In Intermetallic Compounds (ed. J. H.

Westbrook). Wiley, New York, pp. 224-227.Nevitt, M. V., Downey, J. W., and Morris, R. A. (1960).

Trans. AIME, 218, 1019-1023.Pearson, W. B. (1972). The Crystal Chemistry and Physics of

Metals and Alloys. Wiley-Interscience, New York, pp. 75,546, 592, and 773.

Petrovic, J. J., and Vasudevan, A. K. (1992). IntermetallicMatrix Composites II (eds D. B. Miracle, D. L. Anton,and J. A. Graves). Mater. Res. Soc. Symp. Proc,273, 229.

Pettifor, D. G. (1986a). New Scientist, 110, 1510.Pettifor, D. G. (1986b). J. Phys., C19, 285.Pettifor, D. G. (1988). Mater. Sci. Technol., 4, 675-691.Porter, D. A., and Easterling, K. E. (1981). Phase

Transformations in Metals and Alloys. Van NostrandReinhoid, Wokingham, UK, p. 297.

Rajasekharan, T., and Girgis, K. (1983). Phys. Rev., B27,910-920.

Raynor, G. V. (1974). / . Less-Common Met., 37, 247-255.Shoemaker, D. P., Marsh, R. E., Ewing, F. J., and Pauling, L.

(1952). Acta Crystallogr., 5, 637-644.Siegrist, T., Hulliger, F., and Travaglini, G. (1983). J. Less-

Common Met., 92, 119-129.Svechnikov, V. N., Kocherzhinskii, Yu. A., and Yupko, L. M.

(1971). Diagrammy Sostoyaniya Metal Sistemi. Nauka,Moscow, pp. 116-119.

Thomas, O., Senateur, J. P., Madar, R., Laborde, O., andRosencher, E. (1985). Solid State Commun., 55, 629-632.

Villars, P. (1984). / . Less-Common Met., 99, 33-43.Villars, P. and Calvert, L. D. (1991). Pearson's Handbook of

Crystallographic Data for Intermetallic Phases, VoIs 1-3.ASM International, Metals Park, OH, Vol. 1, pp. 135,176-7, 257, 260, 264, 270, and Vol. 3, p. 2742.

Villars, P. and Hulliger, F. (1987). J. Less-Common Met., 132,289-315.

Wells, A. F. (1975). Structural Inorganic Chemistry, 4th edn.Clarendon, Oxford.

Westbrook, J. H. (ed.) (1967). Intermetallic Compounds.Wiley, New York.

Yakel, H. L. (1985). Int. Met. Rev., 30, 17-40.Yvon, K. and Fischer, P. (1988). In Hydrogen in

Intermetallic Compounds (ed. L. Schlapbach).Springer-Verlag, Berlin, pp. 87-138, and referencestherein.

This chapter was originally published in 1995 as Chapter 16 in Intermetallic Compounds,Vol. 1: Principles, edited by J. H. Westbrook and R. L. Fleischer.

1. Introduction

One of the most important problems in the crystalchemistry of intermetallic compounds is the deter-mination of the relations between structure types. It isknown that there are more than 2700 intermetallic-compound structure types,* that is to say varieties ofatomic arrangements in crystals that are described bya unit cell of a certain symmetry, lattice parameters,Wyckoff sites of a certain space group, and atomicarrangement on those sites with certain positional co-ordinates. The lattice parameters, angles, and positionalcoordinates of a structure type can have a range ofvalues within which the variation of the values does notlead to essential changes in the arrangement of the atomsand the coordination features (coordination numbers(CN) and coordination polyhedra (CP)).

Classifications of the structure types may be con-structed on the basis of the following different themes:relation to the structure types of the elements (simplesubstances); quantitative ratios of component species(stoichiometric type); symmetry (type of Bravais lattice);presence of closely connected atoms (island, chain, net,and frame structures); or atomic coordination. Forexample, Pearson (1972) used as the basis of hisclassification the type of Bravais lattice, but he further

*A11 known structure types of intermetallic compounds andall representatives of these types are systematized and collectedby Villars and Calvert (1991). An essential and very importantsupplement to this indispensable reference book is themonograph of Parthe et al. (1994). Questions of the chemicalbonds of intermetallic compounds are considered in a numberof K. Schubert's works (see e.g. Schubert, 1964).

classified intermetallics by employing other indicators(the presence of similar nets, atomic coordinations, andtype of chemical bonds). Another classification of allthe structure types, based on the various relationsbetween the different types of crystal structures, wassuggested by Smirnova (1975).

Considering Belov's (1947) classification of thestructure types as more preferable, since it is basedprimarily on the coordination features, we shall classifystructure types of intermetallic compounds (IMCs) bythe coordination number (CN) and coordination poly-hedron (CP) of the atoms according to Kripyakevich's(1977) systematic scheme.

In the structures of IMCs there are CN from 24 to2. The atoms that form the structure can have equalor different radius values. Atoms of smaller size havea smaller assortment of CN (from 12 to 2). Therefore,the structure-type systematization is based on CN andCP of the atoms with smaller size. Then all of thestructure types can be divided among this small numberof classes (Figure 1).

Usually, for equal atomic sizes, the structures foundare those for which CN= 12 and which have as CP acubooctahedron or its hexagonal analog (class 1 of thesystematization). Only in a few structures with atomsof equal sizes is their CN =14 and their CP a rhombo-dodecahedron, as in the structures related to a-Fe (class2), or a 14-vertices polyhedron, as in the disilicides ofMo, Cr, and Ti (class 3). Both atoms in the FeSistructure have CN= 13 and a 13-vertices polyhedron(class 4).

If atoms of smaller size are placed in the center ofa cubooctahedron and then the cubooctahedron is

Chapter 7

Topologically Close-Packed Structures

Evgen I. Gladyshevskii and Oksana BodakDepartment of Inorganic Chemistry, L'viv State University, Lomonosova str. 6, 290005, Lviv 5, Ukraine


twisted, a 12-vertices polyhedron with only triangularfaces, the icosahedron, is more favorable. The largenumber of structure types with icosahedral coordinationof the smaller atoms belong to class 5. With further

decrease in CN of the smaller atom, the number ofstructure-type classes is increased.

The structure types with CN = 8 and CP as a trigonalprism belong to class 10. Structure types with CN = 4

Figure 1 Coordination figures in the structure of IMCs. The classes are as follows: 1, cubooctahedron and its hexagonal analogs,CN= 12 (examples of the structure types—Cu, TiCd, CdMg3); 2, rhombododecahedron, cube plus octahedron, CN = 8 + 6 (cx-Fe,Cr2Al, Fe3Si); 3, 14-vertices polyhedra of two kinds, CN= 14 (MoSi2, CrSi2, TiSi2); 4, 13-vertices polyhedron for both atoms,CN= 13 (FeSi); 5, icosahedron, CN =12 (Cr3Si, MgZn2, CaCu5, ThMn12); 6, 12-vertices polyhedron, pentagonal prism pluspentagonal dipyramid, CN= 12 (Hg5Mn2); 7, defect icosahedron, CN= 10 (MnAl6, Al13Os4); 7 ' , normal 10-vertices polyhedron,CN= 10 (In4Ti3); 8, defect rhombododecahedron, cube plus defect octahedron, CN = ( 8 - m ) + 6, or defect cube plus octahedron,CN = 8 + ( 6 - m ) , or cube, CN = 8(7-brass, Cu2Sb, CaF2); 9, tetragonal antiprism with one or two additional vertices, CN = 8to 10 (Al2Cu, Mg2Cu, Ni3P, BaAl4); 9 ' , normal eight-vertices polyhedron, CN = 8 (Mn11Si19, Zr2Al3); 9", hexagonal dipyramid,CN = 8 (Hg017Sn083); 10, trigonal prism and its derivatives, CN = 6 to 11 (AlB2, a-ThSi2, PbCl2, FeB, U3Si2, Fe3C); 11,octahedron, CN = 6 (Fe2N, AsNi, NaCl); 12, trigonal dipyramid, CN = 5 (InNi2, UPt2); 13, tetragonal pyramid, CN = 5 (CaC 2 - I ,BaSi2, Si2Ni3); 14, tetrahedron, CN = 4 (SZn, Fe4C, SiFe3, La2O3); 15, square or rectangle, CN = 4 (ZrGa2, CaB6); 16, triangle,CN = 3 (C graphite, TiCu2, B4C); 17, linear or angle, CN = 2 (Ni5P4, UB12)

and CP as a tetrahedron belong to class 14; structuretypes with linear or angular coordination (CN = 2)belong to class 17.

For the other classes of structure types, CN areintermediate between those named above. Some of theclasses are divided into two or three subclasses, in whichthe small atoms have equal CNs but related or differentCPs (classes 7 and 7', 9, 9' and 9" or 11 and 11').

Kripyakevich's systematic scheme is applicable to thestructures of elements, and binary, ternary, andmulticomponent compounds. All structure types ofintermetallic compounds are divided into 17 classes.Each class is the sum of all the structure types that have,

for the smallest atoms, the same coordinationpolyhedron, including its deformed derivatives.

We use a classification of relations between the struc-ture types of intermetallic compounds in accordancewith Kripyakevich's (1977) scheme. It is based on thecommon features of the types compared and on theways of effecting the conversion by which one structuretype may be transformed into another (Table 1).

The closest relation is observed between structure typeswith equal or nearly equal positions for all atoms. Thestructures are more distantly related if one can distin-guish only the same common motifs from the separateatoms that form the structure type in different ways.

Table 1 Relationships between structure types of intermetallic compounds showing common features and ways of transformation

"Structure types 7'-Fe4N (L' 1, cP5), CdI2 (C6, hP3), FeS2 (C2, cP12), AlB2 (C32, hP3), «ThSi2 (Cc, tI12), Zr5Si4 (tP36), Sm5Ge4(oP36), a-Fe (A2, cI2), and Ce24Co1J (hP70) do not belong to the family of close packing. They are shown for illustration ofthe transitions HIc, IVb, and Vc.Êxternal deformation is connected with changing of the c/a ratio. Internal deformation is a result of a change of the atomicparameters.c'Homeotectic'—this term, which was suggested by Laves and Witte (1935), is related to structures that have similar compositionand equal CN but different ways of mutual placement of the fragments (for example, flat layers in the close-packed structures).d'Homology' suggests the possibility of the construction of structures with similar sets of fragments (details) but with differentquantitative ratios. The term was adopted from organic chemistry (Magneli, 1953).

Common features of structuretypes

Equal positions for all atoms

Approximately equal positionsfor all atoms

Equal positions for only partof the atoms

All structure details (fragments)are equal

Details (fragments) are onlypartially equal

Ways of transformation

Ia Ordered substitution (superstructureformation)

Ib Redistribution of the differentcomponent atoms

II External deformation*Internal deformation

HIa Multiple substitutionIHb Redistribution of the atoms or

substitution of an atomic groupIHc Inclusion or elimination and

redistribution of included atoms

IVa Homeotecticsc (stackings of closelypacked networks)

IVb Certain modes of stacking ofslabs (polyhedra), homogeneoushomologous^ series

V Inhomogeneous homologous series,interchanges different in two ormore details:

Va One-dimensionalVb Two-dimensional

Vc Three dimensional

Examples of structure types*

(Parent) Transformed structure

(Cu) AuCu3, (ThMn12) CeMn4Al8

(MgCu2) AuBe5 = (AuBe) Be4,(MgCu2) MgSnCu4

(Cu) a-Mn(MgCu2) TbFe2

(CaCu5) ThMn12, (CaCu5) Zr4Al3(Th2Ni17) Th2Zn17

(Cu) 7'-Fe4N0, (Mg) CdI/,

(CdI/) FeS/

(CdMg3) BaPb3, (MgZn2) MgCu2

(AlB20) a-ThSi/, (Zr5Si4

0) Sm5Ge/

(AuCu3, CaF2) HoCoGa5, Ho2CoGa8(Cr3Si, Zr4Al3) a phase,

(Cr3Si, Zr4Al3, MgZn2) P phase,(Zr4Al3, MgZn2) /i phase

(AlB/, Ct-Fe0) Ce24Co1/

2. Close Packing of Atoms with Equal andNearly Equal Sizes

2.1 Close Packings in the Structuresof Simple Substances and Solid

Solutions Based on Them

The structures that have CN = 12 for all the atoms andform the cubooctahedron or its hexagonal analog as aCP are close packings of atoms with equal or nearlyequal sizes and are grouped in class 1 of Kripyakevich'sclassification. Superstructures of the close-packedstructures and their deformed derivatives belong to thesame class. The structures of real metals and inter-metallic compounds are frequently based on stackingsof close packings with n = 2, 3, 4, 6, 8, 9, 10, 12, wheren is the number of layers in the repeat period.

There are four known undeformed close-packedstructures that are characteristic of the elements:hexagonal two-layered, h2

f (Mg type, A3, hP2); cubicthree-layered, C3 (Cu type, Al, cF4); hexagonal four-layered, (ch)2 (a-Nd type, A3 ' , hP4); andrhombohedral nine-layered, (chh)3 (a-Sm type, C19,hR3). Another five structures of the elements areconsidered as derivatives of ideally close-packedstructures produced by deformation: e.g. for the Zn typethere is an external deformation from that of the Mgtype; the a-U type (A20, oC4) is formed from the Cutype and is connected with both an internal and anexternal deformation. The In (A6, tI2), y-Mn (tI2), anda-Hg (AlO, hRl) types are also formed from the Cutype by external deformation (see Chapter 12 by Hauckand Mika in this volume).

The crystallization of binary and ternary compoundsin the above types of close-packed structures supposesa statistical substitution of atoms in the positions of thestarting structures, which imposes certain limitsconnected with the atomic sizes. An example thatillustrates this point is the interaction between the rare-earth metals (lanthanides, Ln). All the studied binarysystems Ln-Ln' may be divided into three main groupsby the type of interaction with one another at temper-atures lower than 600-8000C (Gladyshevskii and Bodak,1982).

Systems with isostructural components (La-Ce, Pr-Nd, Gd-(Tb,Dy,Ho,Er,Lu), and Yb-(Ho,Er,Lu))forming continuous solid solutions with each other

fIn this notation, h indicates a stacking in which atoms inlayers immediately above and below the reference layer havethe same positions, and c indicates that these positions aredifferent. The subscript indicates the number of such stackingsbefore the sequence repeats.

belong to the first group. The second group includesthe systems La-Nd,Ce-Nd, Nd-Sm, and Sm-Gd withnon-isostructural components forming limited solidsolutions. The third group comprises systems in whichchanging the concentration leads to the formation ofvarious sequences of binary phases with the close-packedstructure types Mg, a-Sm, a-Nd, and Cu (Figure 2). Thestudies of Gschneidner and Valetta (1968) show that theinterchange of these four structure types occurs as afunction of the ratio of the metal radii and the radii oftheir 4f shells (rLn/r4f). The hexagonality (h, %) of theclose-packed structures is increased with increase of thisratio, both in the lanthanides (Ln) (excluding Eu andYb) and in their alloys.

2.2 Superstructures of the Close-PackedStructures

The structure types now compared can have equivalentpositions of all the atoms (are isopunctal) but aredistinguished by the type and number of atoms of thecomponents that occupy these positions. Then thesymmetry and the shape (unit-cell characteristics, whichinclude form and size) of the unit cell can also bechanged. Ordered substitution of at least a part of theWyckoff sites by different atoms leads to the appearanceof a superstructure or ordered structure (in German,Ueberstrukture) (Table 2). The superstructure can beformed for a particular stoichiometric ratio of thecomponents as a consequence of the ordered re-distribution of atoms on the Wyckoff sites with aconcomitant change in symmetry, for example, AuCu-I and AuCu3 (both superstructures I). Superstructureswith the same symmetry and shape as the startingstructure type are named superstructures of the secondkind (superstructures II). Some researchers do notconsider them as a separate structure type.

The superstructures of the close packings can besystematized based on the stoichiometry and on the typeof (arrangement of) the close-packed layer, that isto say on the way the different types of atom aredistributed on it. For binary compounds in which theL and M atoms have similar effective radii, 13stoichiometric types of superstructures are known: LM,L4M5, L2M3, L3M5, LM2, L2M5, L3M8, LM3, L3M10,LM4, L3M13,LM5, and LM7. The main stoichiometriccompositions are LM and LM3; the other super-structures may be constructed by combinations oflayers that contain fragments of these compositions.Composition LM may be obtained in two ways: stackingof equal numbers of close-packed layers, one composedof L atoms and the other of M atoms (LiRh (hP2) and

Figure 2 Mutual solubility and occurrence of phases with theCu, a-Nd, a-Sm, and Mg structure types in the systems Ln-Ln' at the temperatures 600-8000C (the system La-Dy isshown at 2000C)

PtCu Ll1 (hR32) types); or stacking of layers eachcontaining equal numbers of L and M atoms. Structureswith close-packed layers of the same composition LMare numerous (types a-AuCd (B 19, oP4), TiAl (tP2),AuCu-I (Ll0, tP2), a-TaRh (oP12), UPb (tP8), AuCu-II (oI40), TiCd (tP4), etc.). LM3 structures aredistinguished both by the type of layers and by the type

Table 2 Appearance of superstructures by ordering on one orboth sublattices

of close packings (Beattie, 1967). The most essentialfactor that influences the crystallization of real binaryand ternary compounds in superstructures of close-packed structures is the size factor. This may be illus-trated by using the binary LnAl3 and ternary(Ln,Ln')Al3 compounds. The binary compoundsLnAl3 crystallize in five related structure types: AuCu3

(Ll2, cP4), HoAl3 (hR20), TiNi3 (DO24, hP16). BaPb3

(hR12), and CdMg3 (DO19, hP8). The determination ofthe LnAl3 structure type is connected with thedifference rLn - r4f (where rLn is the radius of the Lnatoms calculated from the unit-cell volume of thecompound structure, considering the number of Ln andAl atoms), just as in the structures of the elements andbinary phases in the systems Ln-Ln' (Figure 3). Withincreasing difference A*Ln-r4f, as in the Ln elementsand also in the LnAl3 compounds, the hexagonality ofthe structures increases. In the structure of the Lnelements, the hexagonality of stacking increases fromlight Ln to heavy Ln, and in the LnAl3 compounds inthe opposite direction (Pearson, 1967).

There are also size-factor influences on the compo-sition of ternary compounds Ln1^ Gdx Al3. The binarycompound TbAl3 has a structure of the BaPb3 type; toform compounds with the same structure type with Dy,Ho, Er, Tm, or Lu, it is necessary to replace part of

Symmetry Shape of the unit cell

Same Different

Same Superstructure II Superstructure ICx-Mn-Ti5Re24 Mg-CdMg3

MgZn2-Mg2Cu3SiCaCu5-CeCo3B2

Different Superstructure I Superstructure ICu-AuCu3 AuCu3-TiAl3

MgCu2-MgSnCu4 Cu-AuCu-IMgCu2-AuBe5 Cu-TiAl3

Figure 3 Hexagonality of stacking variants (h, °7o) in thestructures of Ln and LnAl3 compounds depending on rLn - r4f

Figure 4 (a) The content of Gd in the compoundsLn1^GdxAl3 with the BaPb3 structure type and (b) theaverage radius of the atoms (Ln+ Gd) as functions of Lnatomic number

the Ln atoms by the bigger Gd atom (Figure 4). TheGd quantity required increases with the progressionfrom Dy to Lu in the compounds; the average radius(r) of the statistical Ln = Ln' mixture is constant,approximately equal to 1.77, corresponding to the Tbatom radius.

Superstructures of the close-packed structures maybe the basis of the description of new structure typesusing different ways of transformation. For example

(Figure 5) (Gladyshevskii et al., 1985), the structure ofPt2NiGe (oF32) is formed as the result of an internaldeformation of the AuCu3 structure with thesimultaneous formation of a superstructure. The volumeof the unit cell is increased eight times and the symmetryis lowered from cubic to orthorhombic. Anotherstructure related to the AuCu3 type is the structureU4Re7Si6 (cI34); it is formed not only as the result ofan internal deformation, but also as the result of theinclusion of two Re atoms in positions with coordinates0, 0, 0 and \ , y, \. The volume of the unit cell isincreased by eight times in comparison with AuCu3,and the face-centered cubic cell is replaced by a body-centered cubic one.

Fragments of the superstructures of the close-packedstructures may be used as 'building blocks' to form morecomplex structures. The structure types HoCoGa5

(tP7) and Ho2CoGa8 (tPll) form a one-dimensionalhomologous series that is based on fragments of theAuCu3 and PtHg2 (tP3) types (Figure 6). One canobtain a series of hypothetical structures, which aredescribed by the formula RwR3w + 2/7Xw (m and n arethe numbers of fragments of AuCu3 and PtHg2 in theunit cell, respectively), by interchanging the packingsequence and choosing certain quantitative ratios of thefragments AuCu3 and PtHg2. Examples of suchstructures known to exist are the type HoCoGa5 (m = 1,AZ = 1) and the type Ho2CoGa8 (w = 2, n=\) (Grin etal., 1979).

3. Close Packing of Atoms with Unequal Sizes

In close packings of atoms with equal or nearly equalsizes, every atom is surrounded by 12 atoms with the

Figure 5 Structure types related to (a) AuCu3, (b) Pt2NiGe, and (c) U4Re7Si6. Coordinate z is shown in multiples of 0.01

Figure 6 Structures HoCoGa5 and Ho2CoGa8—members ofthe linear homologous series based on fragments of thestructure types AuCu3 and PtHg2. Coordinate y is shown inmultiples of 0.01

same radius, forming a cubooctahedron or its hexagonalanalog. These polyhedra have square and triangularfaces (Figure 7). If in the centre of this cubooctahedronone places an X atom with a smaller radius than theR atoms, a more compact placement of the outlyingatoms is necessary to preserve the close packing; a12-coordinated polyhedron with exclusively triangularfaces (icosahedron) then appears. The ratio rR/rx forit is equal to 1.111. Not only R atoms but also X atomsmay be at the vertices of the icosahedron. Simultaneouslythe CN of the R atoms is increased. Such structures aresaid to be topologically close packed.

There is a large group of intermetallic-compoundstructure types with close packing of atoms of unequalsizes and with icosahedral coordination of the smalleratoms. They belong to class 5 of the structure types ofKripyakevich's system and will be discussed below. Thecoordination numbers of the R atoms in these structuresare between 13 and 24. Crystal structures of this sortwith CN = 12, 14, 15, and 16 were described for the firsttime by Frank and Kasper (1958); the generally acceptedname for them is 'Frank-Kasper structures'. Theprogenitors of this group of structures are the threebinary structure types Cr3Si, MgZn2, and CaCu5, in

which the CN of R atoms is equal to 14 (for Cr), 16(for Mg), and 20 (for Ca).

3.1 Structure Types Cr3Si and Zr4Al3 and theirDerivatives

The connection of the structure type Cr3Si (Al5, cP8)(Boren, 1933) with the close packings can be illustratedby the following scheme (Figure 8). Cr and Si atomsat z = 0 and 1/2 substitute at the positions of theM atoms in the LM3 layer (symmetry pmm); thepositions of the L atoms are vacant. The other Cratoms are at the origin with z= l /4 and 3/4. So,the structure Cr3Si may be represented as that formedfrom two types of nets—a compactly occupied onethat is a derivative of the close-packed layer, andan intermediate sparsely occupied one, which alternatewith each other.

The structure type Zr4Al3 (hP7) (Wilson et al.y 1960)may be considered in the same way as Cr3Si. It may berepresented as being formed from a compactly occupiednet of Al atoms (z - 0), which is a derivative of a close-packed layer LM3 (symmetry p6m) with vacancies inposition 0, 0, and a second compactly occupied net ofZr atoms (z = 1/2), which has vacancies in positions 0,0. These nets alternate with the sparsely occupied onesof Zr atoms (z= 1/4 and 3/4).

Every unit cell for the structures Cr3Si and Zr4Al3has two general and two intermediate nets. Since thedistances between nets in both structures are equal ornearly equal, complex structures with unit cells formedby linking of the side faces of the unit cells of the startingstructures can occur. In this way are formed thetetragonal unit cell of the structure Nb2Al (tP30)(Figures 9 and 10) and numerous binary and ternaryderivatives of this type (a phases), in which Wyckoffsites are occupied by atoms of different components ortheir statistical mixtures, for example CrMn3, Cr2Fe3,Cr53Co47, Cr13Ni5Si2, Mo37Mn63, Mo60Co40, and others(see transformation type Vb of Table 1).

3.2 Structure Type MgZn2 and its Derivatives

Proceeding from the values of the interatomic distances,one can see (Friauf, 1927; Laves and Witte, 1935) thatthe ratio of the effective radii rR/rx in the MgZn2 (C 14,hP12) structure equals J572 = 1.225. A value close to thiscan be found by analyzing the connection between theMgZn2 structure and the close-packed one. For itsformation, it is necessary to substitute half of the Xatoms by half as many atoms with the larger size(X4-^RX2); then rR/rx = 3j2/l = 1.260.

The structure type MgZn2 may be represented in theway described above as the sum of nets situated alongthe directions [001] or [1 1 0 ] . Nets of two k i n d s -flat (A,B,C) and crimped (a',b',c')—alternate alonga direction perpendicular to the Z axis (Figure 11). Inthe flat net LM3 (p6m), the L positions are vacant(kagome nets), and the M positions are occupied by Znatoms. The crimped net contains Zn atoms betweentriangles of two flat nets and double the number of Mgatoms between a hexagon and a triangle. Along thedirection [110] one can distinguish general netsconsisting of pentagons and triangles with Mg and Znatoms at vertices and the intermediate ones sparselyoccupied by Zn atoms (Figure 12). The relative distancesbetween nets and the mutual displacement are the sameas in the Cr3Si structure.

Structures that are derivatives of the MgZn2 type(Laves phases) arise most often as a result of changingthe stacking of the close-packed layer (homeotectic)types (Teslyuk, 1969): MgZn2 structure type (indicatedin phase diagrams as X1, two-layer packing with n = 2,stacking symbol h2, type symbol L2; MgCu2 structuretype, C15, cF24 (X2, A* = 3, C3, L3); and MgNi2

structure type, C36, hP24 (X3, « = 4, (ch2), L4). It was

found that a series of multilayer Laves phases (L5, L6,L8, L9, L10, L14, and L16) occurs among the ternaryintermetallic compounds. Multilayer structures, derivedfrom MgZn2, are formed in the Mg-Li-Zn system(Melnyk, 1974). Their formation is connected with achange in the valence-electron concentration (VEC) andthe values of the ratio rR / rx , where Tx is the averageradius of the atoms X and X ' (Li and Zn) in themixture. The hexagonality of the packing increases withthe increase of these values (Table 3). A number ofmultilayer Laves phases (L2, L4, L6, L9, L10, and L16)have been found in the system Mg-Cu-Al by meansof electron microscopy (Kitano et al., 1977). Derivativesof the types MgZn2 and MgCu2 may also arise as aresult of ordered substitutions (R 2 X 4 -R 2 X 3 X' ,R2X4-RR 'X4 , and possibly R2X4-RR' X3X') and/orof deformation. For example, Mg2Cu3Si (L2) andMnInCu4 (L2) are superstructures of the MgZn2 type;and Mg2Cu3Al (L3) and MgSnCu4 (L3) are superstruc-tures of the MgCu2 type. The structure type AuBe5

(C15b, cF24) is formed by the redistribution of X andR atoms (R2X4-RXX4) in the MgCu2 structure type(Misch, 1935). There are also deformed derivatives ofMgZn2-orthorhombic URe2 (oC24) (Hatt, 1961)—and

Figure 7 Projections of the 12-coordination polyhedra in some intermetallic compounds: (a) hexagonal cubooctahedron (MgMg12)in the structure Mg; (b) left- and (c) right-twisted icosahedra (SiCu12) in the structure Cu15Si4 (D86, cI76); (d) icosahedron(MnMn12) in the structure a-Mn (A 12, cI58); (e) 'crystallographical' icosahedron (SiCr12) in the structure Cr3Si; and (f)icosahedron (CuMg6Cu6) in the structure Cu2Mg

Figure 8 The formation of the structures of the compounds(a) Cr3Si and (b) Zr3Al4 through occupancy of a planarnetwork LM3 with the symmetries pmm and p6m, respectively

of MgCu2-rhombohedral TbFe2 (hR6) (Dwight andKimball, 1974).

There is a series of derivative structure types (classV of Table 1) in which Laves phases, Cr3Si and Zr4Al3

types are the * building blocks' (for example, the P phase,shown in Figure 10, has been investigated by Shoemakerand Shoemaker (1967)). From, the survey of thestructure types in Table 4, in which fragments of Cr3Si,Zr4Al3, and MgZn2 structures are combined, one cansee that the weighted-average coordination number(v) depends linearly (Figure 13) on the content ofatoms with icosahedral coordination (C12) (Jarmoljukand Kripyakevich, 1974):

v= 13.6-0.4C12

The increase in the ratio of the effective radii ofcomponents (rR/rx) in the series Cr3SHZr4Al3-* MgZn2

leads to an increase in CN of the R atoms; but, at thesame time, the content of the larger R atoms isdecreased, and as a result ~v remains within the limits12.9 and 13.5. This regularity was named the 'Jarmoljuk-Kripyakevich phenomenon' and may aid in the properchoice of CN and CP (E. I. Gladyshevskii et al.y 1992).

The structures with giant cells investigated by Samson(1967, 1968) also belong to class 5 (Figure 1) of thestructure types.

3.3 Structure Type CaCu5 and its Derivatives

The structure type CaCu5 (D2d, hP6) (Haucke, 1940)may be considered to be the result of the substitutionby an R atom of three of the eight X atoms with thelarger size (X8-^RX5) in the hexagonal close packing

Figure 9 Projections of the structures (a) Cr3Si, (b) Nb2Al, and (c) Zr3Al4 on planes parallel to the close-packed layers.Coordinate z is shown in multiples of 0.01

Figure 10 Formation of (a) a phase (Nb2Al) unit cell by thecombination of the Cr3Si and Zr4Al3 unit cells and (b) P phaseby combination of Cr3Si, Zr4Al3, and MgZn2 unit cells

(Figure 11). At that degree of substitution, the ratio ofatom radii is rR/rx =

3j3/l = 1.44, which is greater thanthe ratio in the MgZn2 structure. The structure has adefect layer of symmetry p6m as in the MgZn2 andZr4Al3 structures. Along the [110] direction, CaCu5

may be represented as consisting of two types of nets:general (pentagons and triangles) and intermediate(rectangles) (Figure 12). In the type CaCu5 = CaCu2Cu3',atoms of Cu occupy two crystallographic positions: Ratoms form a simple hexagonal stacking, and X atomsform a frame from the empty tetrahedron (X4).

Recently some ternary structure types R̂ MyX2 derivedfrom CaCu5 have been described by Bodak et al.(1990). One of these is the undeformed superstructureCeCo3B2 (hP6) (Kuzma et al., 1969) (Table 5, Figure14). When the conditions for simple substitution(superstructure formation) are not suitable, deformedand/or defect derivatives arise. The structure typeHoNi2 6Ga24 (hP6) (Grin and Gladyshevskii, 1989) ispartly a statistically substituted variant of CaCu5 withthe elimination and inclusion of R atoms. The deformedderivatives retain approximately the same positions ofall the atoms in the starting type, but differ in their unitcells (symmetry, unit-cell parameters). They are realizedjust as in the preceding group of structure types at theexpense of an increase in the number of componentsin the compound.

Figure 11 Atom nets perpendicular to the six and threefoldaxes in close-packed structures (nets ABC—the upper row),in the structure MgZn2 and other Laves phases (ABC,a'b'c'—the second and fourth rows), and in CaCu5 structure(ABC, abc—second and third rows). MgZn2 structure:Ca'Ba' ( = Ac'Bc'), in the crimped a' net zMg = 2/16± 1/16.MgCu2: Ca'Bc'Ab', in the a'(c'b') nets zMg = 4/24± 1 /24(=12/24±l/24, = 20/24±l/24). MgNi2: Ca'Ba'Cb'Ab', inthe a' net zMg = 4/32 (and 12/32)± 1/32, in the b ' netzMg = 20/32 (and 28/32) ±1/32. CaCu5: both nets Cc areplanar, in C nets—Cu, in c—Ca and Cu atoms

The next group of new structure types may beconsidered as defect derivatives of the CaCu5 type.Only a part of the atom positions are the same as forthe starting structure. The inclusion-elimination of Ratoms in the majority of cases occurs with thestoichiometry preserved, corresponding to that of thetype CaCu5, and leads to a mutual shift of R atomsalong a single direction. The elimination of the Mcomponent leads to a change of stoichiometry from thestarting type.

(b) P phase

(a) a phase

Figure 12 Atomic arrangement in the (a) MgZn2 and (b)CaCu5 structures projected onto the (11 0) plane. In thegeneral nets consisting of pentagons and triangles are Mg andZn or Ca and Cu atoms. The other Zn or Cu atoms are in theintermediate sparsely occupied nets

The structure types that are formed by various R atomsubstitutions by X2 pairs preserve the same part of theatomic positions as in the CaCu5 type (Table 6). Thedifferent ways of substitution and the number of Ratoms that are substituted lead to the occurrence of newstructure types. Other structure types derived fromCaCu5 are also known. Type CeAl8Mn4 is a super-structure to the ThMn12 type (D2b, tI26) (Zarechnyukand Kripyakevich, 1962). There are two more types ofundeformed superstructures R2X17-Th2Zn17 (hR19)and U2Zn17 (hP114) (Makarov and Vinogradov, 1956),which are homeotectic. The structures BaZn5 (oC24)and SrZn5 (oP24) (Baenziger and Corant, 1956) are alsoderived from the CaCu5 type and are formed as theresult of a shift of a part of the layers; and structureCeCu6 (oP28) (Cromer et aL, 1960) is formed as a resultof the inclusion of additional X atoms in the BaZn5

structure type. The structure LaNi2Al3 is an undeformedsuperstructure of the type BaZn5 (R. Gladyshevskiiet aL, 1992).

Analyzing the composition of the ternary represen-tatives of the CaCu5 type and the new structure typesthat are derivatives of CaCu5, one notices that topreserve the necessary correlation of the atomic sizesthe majority of them occur in the ternary systemsR-X-X' or R-R'-X, where R and R' are alkali,alkaline-earth, and rare-earth metals, and X and X' ared metals and the elements B, Al, Ga, Si, and Ge. In thesystems R-M-B, ternary borides have been found thatcrystallize in the structure types CeCo4B (hP12),Ce3Co11B4 (hP18), Nd3Ni13B2 (hP18), Ce2Co7B3 (hP24),and CeCo3B2 (hP6) (Figure 15); they form withthe structures of the binary compounds, which belongto the CaCu5 type, a homologous series. Type CaCu5

and a superstructure of it, CeCo3B2, are the progenitorsof this series. The unit-cell compositions of the structures

Table 3 Multilayered Laves phases in the ternary system Mg-Li-Zn (L3' and L4' are superstructures to the types L3 and L4,respectively)

Compound

MgZn2

MgLi007Zn193

MgLi011Zn189

MgLi020Zn180

MgLi023Zn177

MgLi0-25Zn175

MgLio.5oZni.5oMgLi056Zn144MgLi077Zn123

Pearsoncode

hP12hP48hP84hR18hP60hP24hP96hR6cF24

VEC

2.0001.9771.9631.9331.9231.9171.8331.8131.743

1.1491.1441.1411.1361.1341.1321.1161.1121.098

Typecode

L2

L8L14L9

L10

L4

KL3'L3

Stackingformula

h2

(hhhc)2

(hhchhhc)2

(hhc)3

(hchhc)2

(hc)2

(hc)2

C3

Hexagonalstacking(%)

100.075.071.466.760.050.050.000

Figure 13 The weighted-average coordination number v forstructures derived from Cr3Si, Zr4Al3, and MgZn2 types ((1)Cr3Si, (2) a phase, (3) Zr4Al3, P and a phases, (4) R phase,(5) \i phase, (6) T phase, (7) X1 phase) as a function of contentof the atoms with icosahedral coordination

that belong to this series are described by the formulaRm + nM5m+3nX2n> where m is the number of CaCu5

fragments and n is the number of CeCo3B2 fragments.Representatives are known of another homologousseries of structures based on the types CeCo3B2 andMgZn2 (Figure 16). The members of this series—Dy3Ni7B2 (hP24) and Ce2Co5B2 (hP36)—are describedby the formula R2m + /iM4m + 3,,X2,,, where m and n arethe numbers of MgZn2 and CeCo3B2 fragments,respectively.

The structure type CaCu5 is the progenitor of stillother homologous series of structures, both one-dimensional and two-dimensional. Type CaCu5 withtype MgZn2 form a one-dimensional homologousseries that is described by the formula Rm+2wX5m+4n (mis the number of CaCu5 layers, n is the number ofMgZn2 layers). Known representatives of this series arethe CeNi3 (hP24) type {m = 1, n = 1), the Ce2Ni7 (hP36)type (m = 2, /2 = 1), and the Sm5Co19 (hR24) type(m = 3, n=\). The structures PuNi3 (hR12), Gd2Co7

(hR18), and Ce5Co19 (hR24) comprise a homologousseries that is described by the same formula. It isconstructed from the types CaCu5 and MgCu2. Thestructures of this homologous series have no represen-tatives among the ternary compounds. The structuretypes CeNi5Sn (hP28) and CeCu6 (oP28) belong to atwo-dimensional series of structures; the progenitors ofthis series are CaCu5 and the hypothetical RX7, whichhas no real representatives.

4. Summary

From the known 2700 structure types of all intermetalliccompounds* we have considered approximately 220structure types that are characterized by close packingsof atoms with the same, nearly the same or differentsizes and with CN= 12 for all or part of the atoms(atoms that have smaller size) and CN > 12 (from 13 to24) for atoms of bigger size, R atoms. For the latter,

*Or 1600 structure types without considering compoundscontaining elements of group VI, which in most cases differfrom the structures of compounds of the metals with each otherand with the elements of groups IH-V.

T , Mo3(Mo,Cr)5(Cr,Ni)6.5, Mo3(Mo5Ni)5Ni6.R, Mo,4(Mo,Mn,Fe)12(Mn,Fe)27.T, Mg32(Zn5Al)49.

Table 4. Structure types—hybrids of the types Cr3Si, Zr4Al3, and MgZn2

Number onFigure 10

123334567

Structuretypes0

Cr3SiNb2Al (a)Zr4Al3

P phase6 phaseR phaseW6Fe7 G*)T phaseMgZn2 (X1)

Pearsoncode

cP8tP30hP7oP56oP56hR53hR13cl 162hP12

Content of atoms(at.%) with different

coordination

CN= 16

OOO7.27.2

15.115.424.733.3

15

O13.428.614.314.311.315.47.4O

14

75.053.328.635.735.722.615.47.4O

12

25.033.342.842.842.851.053.860.566.7

AverageCN, v

13.50013.46613.42813.42813.42813.39613.38513.35813.333

Number offragments informula units

R3X

12

112

R4X3

iiii313

RX2

1182

201

C12 (%)

the following name has been suggested: structure typeswith high coordination numbers (Kripyakevich, 1960).

Investigations of the conditions for the formation ofisostructural compounds, the changes in their structureswith changing temperature (or pressure), and the law-governed transitions of one structure into others onchanging the identity of the components or their

relative proportion are problems of separate aspects ofcrystal chemistry: isomorphism (isostructure), poly-morphism (allotropy), and morphotropy, which havebeen of interest to many researchers. We haveconsidered the morphotropic series that are formed withincreasing atomic number of Ln atoms in phases of thesystems Ln-Ln' and in LnAl3 compounds. Here the

Table 5. Examples of CaCu5 superstructures, their distorted and defect derivatives

Figure 14 Projection of the structure types (a) CeCo3B2 (b) HoNi26Ga24, and (c) BaNi2B2 along the [001] direction. CoordinateZ is shown in multiples of 0.01

Way of transformation

Substitution

Deformation

Elimination or inclusion of R atoms

Elimination of M atoms

Structure typeR x M ^

CeCo3B2

ErRh3Si2

LaRu3Si2

HoNi2 6Ga2 4

Ba2Ni9B6

Er103Co3Ge2

CaRh2B2

Sr5Rh14B10

Pearsoncode

hP6

oI24hP12

hP18hR34hP16

oF400FII6

Space-groupsymbol

P6/mmm

ImmmP63/m

P6/mmmR3cP63/mcm

FdddFmmm

Unit-cell parameters {a,b,c)expressed via CaCu5

parameters (A,C)

a

A

2CA

$A

A

AIA

b

$A$A

C

C

A2C

2CAC2C

2C6C

Figure 16 The linear heterogeneous homologous series of structures containing fragments of the CeCo3B2 and MgZn2 structuretypes

Figure 15 The linear quasihomogeneous homologous series with fragments of the structures CeCo5 (CaCu5 type) and CeCo3B2

Table 6. Structure types formed by the partial substitution of R atoms by X2 pairs in the structure type CaCu5

Way ofsubstitution

Statistical

Ordered

SubstitutionschemeR^X2

R^5~* Ro. 78^5.44RX5-^R05X6

Structuretype

TbCu7

YCo6Ge6

Ce3Zn22

Th2Ni17

ThMn12

LiCo6Ge6

Pearsoncode

hP8hP8

til OOhP38tI26hP13

Space-groupsymbol

P6/mmmP6/mmm

14,/amdP63/mmc14/mmmP6/mmm

Unit-cell parameters{a,c) expressed viaCaCu5 parameters

(A9Q

a

AA

$A$A

A

b

CC

AC2C2C2C

Plate I

Figure 17 Structural relationships diagram of the structure types related to Mg and Cu (AuCu3, MgZn2, MgCu2, Cr3Si, CaCu5,Zr4Al3, and others); Roman numerals I-V are related to the ways of transformation used in Table 1

general cause of the occurrence of a new structure isthe size factor. One more example of the influence ofthe size factor on structures formed with high CN is theseries of R ^ ^ compounds. For these, the content ofthe X component and the maximum CN of the R atomsdepend typically on the relative atomic size k = rK/rx

(Kripyakevich, 1952) as follows: type W6Fe7, D85,hR13 (Ar=LIl, X content 53.9 at.%, CN= 15 for Ratoms); Laves phases RX2 (1.23, 66.7, 16); CaCu5

(1.45, 83.5, 20); and NaZn13, D23, cF112 (1.56, 92.9,24).

The influence of electron factors (e.g. VEC) on theformation of the close-packed structures is undoubted.For example, a VEC-governed shift of the homogeneityrange of the a-phases in the binary systems R-X (R = V,Cr, Mo, and X = Mn, Fe, Co) and in the ternary systemsV-Mn-X and Cr-Mn-X with the substitution of Feby Co and Ni has been observed (Wernick, 1967).Isostructural and simultaneously isoelectronic (withthe same VEC) series are known in which ternarycompounds with certain structure types are formed onlywith the substitution for Zn in the binary compoundsby statistical mixtures (Cu+ Al) or (Ni+ Si) in theternary ones (Gladyshevskii, 1971). For example, thefollowing series exist: Ce2Zn17 (structure with the typeTh2Ni17)^Ce2Cu65Al105-Ce2Ni15Si2; GdZn12 (ThMn12

type)-GdCu4Al8->GdNi10Si2; and LaZn13 (NaZn13

type)^LaCulo.o-5.5A13.o-7.5~*LaNiio.5Si2.5- So> f o r theformation of compounds with Cu and Ni that areisostructural with Zn compounds, it is necessary in orderto preserve a limiting electron concentration to substitutepart of the Cu or Ni atoms by Al and Si respectively.

Consideration of the interrelationships of a largenumber of hybrid structures that belong to differenthomologous series has turned out to be very fruitful inthe search for new intermetallic compounds and in theprediction of their possible crystal structures. Thecomposition of the unit cell of a series of structuralrepresentatives can be described by a single formula,and their crystal structures are formed by the associationof planar fragments (nets) or volume fragments(polyhedra), cut out from simpler starting structures.Some examples of such homologous series constructedfrom simpler close-packed structures have beendescribed above in Sections 2.2, 3.2, and 3.3. Thehomologous series of intermetallic compounds may behomogeneous, quasihomogeneous, or heterogeneous.The first type includes fragments of a single structuretype, but joined in different ways (for example,AlB2-QJ-ThSi2); the second includes two related frag-ments of the same structure type (for example,fragments of the structure and its superstructure or

defect derivative, CeCo5-CeCo3B2); and the thirdincludes fragments of two or more different structures(MgZn2-CeCo3B2). Combinations of the fragmentsmay be one-dimensional (linear), two-dimensional(planar), or three-dimensional (spatial). Knowing thesymmetry of the fragments and the possible ways inwhich they are joined, one can predict the symmetryof the structure as a whole, and also the quantitativecomposition of the compound. Systematic investigationsof such homologous series were started at LvivUniversity at the beginning of the 1970s (Kripyakevich,1970; Gladyshevskii and Kripyakevich, 1972) and havebeen reported in more than 50 works. Similar investi-gations have been carried out in other crystal-chemistrylaboratories (Parthe and Chabot, 1984; Fornasini et al.,1989; Parthe, 1991; Rogl, 1991) and were the mainsubject of extensive discussions at the recent Inter-national School of Crystallography (Parthe, 1992).

The genetic connections of the close-packed structures,related types, and the ways of transforming from oneto another may be illustrated by a generalized diagram(Figure 17, Plate 1). Some derivatives from the face-centered cubic Cu type, which do not belong to theclose-packed structures, but whose fragments togetherwith fragments of close-packed structures form newhybrid structure types belonging to different homologousseries, are also included in the diagram.

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Laves, F., and Witte, H. (1935). Metallwirtschaft, 14, 645.Magneli, A. (1953). Acta Chem. Scand., 6, 495.Makarov, E. S., and Vinogradov, S. I. (1956). Kristallografiya,

1, 634.Melnyk, E. W. (1974). Dop. AN URSR, A, 949.Misch, L. (1935). Metallwirtschaft, 14, 897.Parthe, E. (1991). Elements of Inorganic Structural Chemistry

(A Course on Selected Topics) K. Sutter Parthe,Switzerland, p. 67.

Parthe, E. (1992). 19th Course on Modern Perspectives inInorganic Crystal Chemistry, Lecture notes and posterabstracts. International School of Crystallography, Erice,Trapani, Sicily.

Parthe, E., and Chabot, B. (1984). In Handbook on the Physicsand Chemistry of Rare Earths (eds K. A. Gschneidner,Jr and L. R. Eyring). North-Holland, Amsterdam, ch. 48,p. 113.

Parthe, E., Gelato, L., Chabot, B., Penzo, C , Cenzual, K.,and Gladyshevskii, R. (1994). Typix. Standardized Dataand Crystal Chemical Characterization of InorganicStructure Types. In Press.

Pearson, W. B. (1967). J. Less-Common Met., 13, 626.Pearson, W. B. (1972). The Crystal Chemistry and Physics of

Metals and Alloys. Wiley, New York, p. 303.Rogl, P. (1991). Existence and Crystal Chemistry of Borides

in Inorganic Reaction and Methods (ed. A. Hagen). VCHPublishers, New York, pp. 85-167.

Samson, S. (1967). Acta Crystallogr., 23, 586.Samson, S. (1968). In Structural Chemistry and Molecular

Biology. W. H. Freeman, San Francisco, p. 687.Schubert, K. (1964). Kristallstrukturen zweikomponentiger

Phasen. Springer-Verlag, Berlin.Shoemaker, C. B., and Shoemaker, D. P. (1967). Acta

Crystallogr., 23, 231.Smirnova, N. L. (1975). Krystallografia, 20, 524.Teslyuk, M. J. (1969). Metal Compounds with the Structure

of Laves Phases (in Russian). Nauka, Moscow, p. 65.Villars, P., and Calvert, L. D. (1991). Pearson's Handbook

of Crystallographic Data for Intermetallic Phases, 2ndEdn. American Society for Metals, Metals Park, OH.

Wernick, J. H. (1967). In Intermetallic Compounds (ed. J. H.Westbrook). Wiley, New York, ch. 12, p. 197.

Wilson, C. G., Thomas, D. K., and Spooner, F. J. (1960). ActaCrystallogr., 13, 56.

Zarechnyuk, O. S., and Kripyakevich, P. I. (1962).Kristallografiya, 1, 543.


1. Introduction

Crystal-structure and materials' properties are intimatelylinked. Currently, alloy developers are searching for newcubic alloys with good mechanical properties (see, forexample, Dimiduk and Miracle, 1989), new tetragonalalloys with good permanent magnetic properties (see,for example, de Mooij and Buschow, 1987), newperovskite-type ceramics with good superconducting orferroelectric properties (see, for example, Rabe et aL,1992), and new icosahedrally based alloys as stablequasicrystals (see, for example, Rabe et aL, 1991). Atthe outset, the alloy developer is faced with the problemof deciding which particular alloy constituents willstabilize a required structure type. A useful initial guidemay be provided by structure mapping, which attemptsto order the known experimental database on thestructures of binary, ternary, and quaternarycompounds within a limited number of two- or three-dimensional structure maps.

Historically, the majority of these structure maps havebeen constructed by taking coordinates that reflect themost important physical factors determining thestructural stability of the particular class of compoundsunder consideration. For example, Mooser and Pearson(1959) in their pioneering study of valence compoundstructures chose the electronegativity difference AX asone coordinate and the average value of the principalquantum number n as the other. They argued thatdirectionally bonded phases would be expected to lie inthe region where AX and n were both small, whereasionic phases would lie in the region of large AX. Phillips

and Van Vechten (1969), on the other hand, in theirhighly successful separation of the structure types withinsp-valent AB octet semiconductors and insulators, usedthe physically motivated coordinates Eh and C, thecovalent and ionic contribution to the averagespectroscopic energy gap, respectively.

The Mooser-Pearson and Phillips-Van Vechtenmaps were restricted to valence compounds. In anattempt to handle the broad spectrum of intermetalliccompounds, Zunger (1980) followed the earlier ideasof St John and Bloch (1974) and chose Ra and Rr ashis two coordinates, the latter being defined in termsof quantum-mechanically derived s and p core radii, aswe will see later in Section 2.2. He found that he wasable to obtain a reasonable structural separation of allthe known AB compounds within three separate two-dimensional maps (Ra, Rx). In order to obtain astructural separation of all isostoichiometric binarycompounds within a single framework, Villars (1983,1984a,b) introduced his three-dimensional maps (AAÂR, N), where the three coordinates reflect theimportance of the electronegativity difference AX, theatomic-size difference AR, and the average number ofvalence electrons per atom N in the determination ofthe structural stability (see, for example, Pearson, 1972;Chapter 2 by Turchi in this volume). Recently these threecoordinates have been generalized in order to treatternaries and quaternaries (Villars and Hulliger, 1987;Rabe et al., 1992; see also Chapter 11 by Villars in thisvolume).

An alternate approach to the construction of structuremaps has been proposed by the author (Pettifor, 1988a).

Chapter 8

Structure Mapping

David G. PettiforDepartment of Materials, University of Oxford, Parks Road, Oxford OXl 3PH, UK


At the microscopic level it is well-known that structuralstability is determined by at least four factors, namelythe electronegativity difference, the atomic-sizedifference, the electron-per-atom ratio, and the angularcharacter of the s, p, and d valence orbitals (see, forexample, Pettifor and Podloucky, 1986; Cressoni andPettifor, 1991). Thus, in principle, it is not possible toorder perfectly the structures of all isostoichiometricbinary compounds within a single two- or three-dimensional map with the use of physical coordinates.Logic demands that, if binary systems are to berepresented within two-dimensional plots, then eachelement in the periodic table should have a unique singlecoordinate. Since the only single physical coordinate thatallows structural prediction from first principles is theatomic number Z, and this clearly fails to order binaryAmBw compounds within the two-dimensional map(ZA, ZB), a phenomenological coordinate was proposedinstead. The coordinate was determined by running aone-dimensional string through the two-dimensionalperiodic table, the path of the string being chosen togive the best structural separation for the binary ABcompounds. Pulling the ends of the string apart ordersall the elements along a one-dimensional axis, theirsequential order being given by the relative orderingnumber JC. Structure maps based on JC have beengiven that include most binary and ternary intermetalliccompounds (Pettifor, 1986, 1988a,b).

This chapter discusses the art of structure mappingand its relevance to the search for new intermetallicswith a required structure type. In Section 2 illustrativeexamples of Mooser-Pearson, Zunger, Villars, andPettifor maps will be presented for binary compounds.In Section 3 the latter two approaches will be generalizedto ternary systems. In Section 4 we conclude.

2. Binary Structure Maps

2 A Mooser-Pearson Maps

The first successful structure maps were constructed byMooser and Pearson (1959) for valence compounds. Aswe have seen above, they took as their two coordinatesthe difference in the electronegativity AX and theaverage value of the principal quantum number n.They believed that directionally bonded phases wouldfall in the region where AX and n were both small,whereas ionic phases would fall in the region of largeAX. These expectations were borne out among therestricted class of valence compounds that theyconsidered. For example, Figure 1 shows the Mooser-Pearson map for octet sp-valent AB compounds, which

Electronegativity difference A *

Figure 1. The Mooser-Pearson map for sp-valent octetcompounds formed from group A cations only (From Mooserand Pearson, 1959. Reproduced with permission)

are formed from group A cations only and have eightvalence electrons per AB unit. Excellent separation hasbeen achieved between the Bl (NaCl) with Pearsonsymbol cF8, B2 (CsCl) cP2, B3 (cubic ZnS, zincblende)cF8, and B4 (hexagonal ZnS, wurtzite) hP4 structuretypes. Moreover, small values of AA" and JT correspondto the directionally bonded, fourfold-coordinated ZnSlattices, whereas large values of AX and n correspondto the ionically bonded eightfold-coordinated CsCllattice.

However, intermetallic compounds that do not satisfynormal valence rules were not well-separated intodifferent structural domains by the two coordinates AXand n. Zunger (1980), for example, has shown that theMooser-Pearson map for 360 non-octet AB compoundsgives extremely poor structural separation between the14 different structure types considered.

2.2 Zunger Maps

The maps of Zunger (1980) were the first to order thestructures of a broad range of intermetallic compoundswith a single consistent choice of coordinates. Watsonand Bennett (1978a,b) in two important earlier papershad obtained structure maps for AB, AB2 and AB3

compounds but had found it necessary to use a differentchoice of coordinates for transition element-transitionelement bonding compared to sp-sp or sp-transitionelement bonding. Following St John and Bloch (1974),Zunger (1980) constructed two-dimensional AB mapsbased on the dual coordinates

Aver

age

prin

cipal

qua

ntum

num

ber

n

Figure 2. The Zunger maps for non-octet AB compounds (from Zunger, 1980. Reproduced with permission)

/tfB=l(r? + r,A)-(r» + r»)| (D

and

R™ = \(r$-r?) + (r»-r?)\ (2)

where rs and rp are s and p core radii, respectively.These core radii were calculated and tabulated for thefirst five rows of the periodic table using pseudopotentialtheory (Zunger and Cohen, 1978; Zunger, 1980). Thesum (rs + rp)/2 is a direct measure of the average radiusof the core, whereas (rp - rs)/2 is an indirect measureof the s-to-p promotion energy (compare, for example,the plots of the s and p atomic energy levels across theperiodic table with that of the inverse s and p core radiiin figures 4 and 7 of Pettifor (1983)). Thus, fromequations (1) and (2), R*B is a measure of the sizemismatch between the A and B atoms, whereas R^B isa measure of the average propensity of the two atomsto form sp hybrids.

Figure 2 shows the structural separation achieved bythe Zunger maps for 437 non-octet AB compoundsdisplaying 27 different structure types. Two maps havebeen drawn for clarity, since several different structuretypes occur in the same domain. We see that a goodstructural separation has been achieved, although

structures with similar local coordinations have not beenresolved. For example, the following structure typeshave common domains: B8} (NiAs) with Pearsonsymbol hP4, B20 (FeSi) cP8, and B31 (MnP) oP8; B27(FeB) oP8 and B33 (CrB) oC8; B2 (CsCl) cP2, BIl(CuTi) tP4, B19 (AuCd) tP4, and Ll0 (CuAu) tP4,respectively. MnP is a distorted NiAs structure type; FeBand CrB have identical nearest-neighbour coordinationpolyhedra; CsCl and CuTi are b.c.c.-derivativestructures; and hexagonal AuCd and tetragonal CuAuare 12-fold-coordinated close-packed polytypes (see, forexample, Pearson, 1972; Wells, 1975).

The left-hand panel of Figure 3 from Yeh et al. (1992)shows the structural separation achieved with the (Ra,Rw) map for octet AB compounds (with eight valenceelectrons per AB unit) using the orbital radii rs and rp

predicted by pseudopotential theory. Figure 3 excludescompounds containing transition elements (except Mn).We see that reasonable structural separation has beenachieved between the cubic zinc blende, the hexagonalwurtzite, NaCl, NiAs, graphite, and cinnabar structuretypes. The last two structure types are the ground-statestructures of BN and HgO, and of HgS, respectively.Figure 3(a) has incorrectly placed BeO in the zinc-blenderather than the wurtzite domain, and MnS, MnSe, MgS,and MgSe in the zinc-blende rather than the NaCldomain. Interestingly a near-perfect separation is found

(a) Pseudopotential radii (b) Wavefunction radii (c) Dielectric model

ZB (zinc blende) W (wurzite)

Figure 3. Structure maps for octet non-transition-element AB compounds using (a) pseudopotential orbital radii, (b) maximumradial probability density radii, and (c) Phillips-Van Vechten dielectric coordinates Eh and C (see text for details). The left-hand panel (a) also includes MnX (X = O, S, Se, and Te) compounds. The right-hand panel (c) does not include HgO (FromYeh et al., 1992. Reproduced with permission)

Figure 4. Values of the valence-electron number, the Martynov-Batsanov electronegativity X, and the Zunger s and p pseudopotential radii sum R that are used toconstruct Villars maps. Electronegativities with the superscript a are Pauling values (Pauling, 1960). Pseudopotential radii sums with the superscript b are estimatedfrom a comparison of Zunger's radii with the radii of maximum radial electron density determined by Herman-Skillmann calculations (Girifalco, 1976) (From Villarset al., 1989. Reproduced with permission)

in Figure 3(b) when rs and rp are defined by the outermaxima in the radial probability density of theappropriate atomic orbitals (Yeh et al., 1992; Chapter 14by Parthe). The resultant structural separation in themiddle panel is as good as that achieved by the Phillips-Van Vechten map (Eh, C) in the right-hand panel, whereEh and C are the covalent and ionic contributions tothe average spectroscopic energy gap, respectively. Thissystemization of the cubic versus hexagonal fourfold-coordinated octet structure types is directly relevant to thedevelopment of wide-gap III-V and H-VI semiconductors,since the minimum band gap is very structure-sensitive,changing by nearly 40% in going from cubic tohexagonal SiC, for example (Yeh et al., 1992).

2.3 Villars Maps

Villars (1983, 1984a,b) has constructed three-dimensional maps (AX, AR, N) in order to display allthe isostoichiometric binary data within a singleframework. He found that the best structural separationwas achieved by using the orbital electronegativities ofMartynov and Batsanov (1980) and the orbital radii(rs + rp) of Zunger (1980). These, together with thechoice of the number of valence electrons associatedwith each element, are given in Figure 4.

Figure 5 shows two cross-sections through the three-dimensional Villars map for AB compounds,corresponding to the octet compounds with N= 4 andthe non-octet compounds with N= 6.5, respectively.Sixteen such figures with different values of N arerequired to display all the structural information for thisparticular stoichiometry. We see that reasonablestructural separation is achieved, although again, likethe non-octet Zunger maps in Figure 2, the domainsoften contain more than one structure type. However,a major drawback is that the fifth most commonstructure type, BS1 (NiAs) hP4, was unable to beseparated and was omitted from the AB map (Villars,1983). This probably reflects the importance of theangular character of the valence orbitals, since this hasnot been included explicitly in the choice of coordinates(AX, AR, N). It is clear that the structures of theelements would not be separated within this mapbecause they lie along the N axis corresponding to(A^=O, AR = O), so that tetrahedrally coordinatedsilicon and close-packed titanium, for example, wouldboth be characterized by the same point N= 4.

2.4 Pettifor Maps

Rather than using physical coordinates, the author hassuggested using a phenomenological coordinate, which

Figure 5. The Villars maps for AB compounds corresponding to the average electron-per-atom ratio of N= 4 and 6.5 respectively(From Villars, 1983. Reproduced with permission), (a) N= 4; (b) N = 6.5

is obtained by running a string through the periodic tableas shown in Figure 6. Pulling the ends of the string apartorders all the elements along a one-dimensional axis,their sequential order defining the relative order numberJt. This simple procedure, which defines a purelyphenomenological coordinate, is found to provideexcellent structural separation of all binary compoundswith a given stoichiometry AmBw within a single two-dimensional map (JtK, Jt^) (Pettifor, 1984, 1986,1988a,b).

Figure 7 shows the AB ground-state structuremap using the experimental database of Villars andCalvert (1985). The bare patches correspond toregions where compounds do not form owing to eitherpositive heats of formation or the competing stabilityof neighboring phases with different stoichiometry. Theboundaries do not have any significance other than thatthey were drawn to separate compounds of different

structure type. In regions where there is a paucity ofdata, the boundary is usually chosen as the lineseparating adjoining groups in the periodic table. Wesee that excellent structural separation has been achievedbetween the 52 different AB structure types that havemore than one representative compound each. The twomost common structure types, namely Bl (NaCl) cF8and B2 (CsCl) cP2, are well-separated, the NaCl latticebeing found only outside the region defined by JtA,^#B<81, which encloses the main metallic CsCldomain. We see that at normal temperature and pressurethe ionic B2 (CsCl) lattice is restricted to the very smallregion of Cs-containing salts. The AB structure mapsuccessfully demarcates even closely related structuretypes such as B27 (FeB) oP8 and B33 (CrB) oC8; BS1

(NiAs) hP4 and B31 (MnP) oP8; and B3 (cubic ZnS,zinc blende) cF8 and B4 (hexagonal ZnS, wurtzite) hP4.Moreover, coherent phases with respect to the b.c.c

Figure 6. The string through this modified periodic table puts all the elements in sequential order, thereby defining the relativeordering number Jt. Note that the group HA elements Be and Mg have been grouped with group HB, the divalent rare earthshave been separated from the trivalent, and Y has been slotted between Tb and Dy (From Pettifor, 1988a)

Figure 7. The Pettifor map for AB compounds (from Pettifor, 1988a)

lattice, namely B2 (CsCl) cP2, Bl 1 (CuTi) tP4, and B32(NaTl) cF16, are also well-separated, as too are theclose-packed polytypes cubic Ll0 (CuAu) tP4 andhexagonal B19 (AuCd) oP4.

As might be expected, neighboring domains oftenhave similar local coordination (see figure 2 of Pettifor,1986). Recently, Villars et al. (1989) have assigned localcoordination polyhedra to all binary structure types withmore than five representative compounds each. Figure8 illustrates some of the more commonly occurringcoordination polyhedra, labeled according to ageneralized Jensen notation (see table 5 of Jensen, 1989,

and figures 3 and 4 of Villars et al., 1989). Structuretypes can then be characterized not simply by theirstoichiomethc formulas, such as NaCl and NiAs, butby their crystal coordination formulas, such as 3[NaCl676] and 3 [NiAs676,] (Jensen, 1989). Thisimmediately informs us that NaCl and NiAs are infinitethree-dimensional framework structures. Moreover, thesymmetry of the coordination polyhedron about the Na,Cl, or Ni sites is that of an octahedron (denoted by 6),whereas that about the As site is that of a trigonal prism(denoted by 6'), as can be seen from Figure 8. It is,therefore, not surprising that the NaCl and NiAs

Figure 8. Some commonly occurring local coordination polyhedra with generalized Jensen notation

domains adjoin each other in Figure 7 and that there is asmall domain of 3 [NbAs676/] stability in their midst.

Figure 9 shows that excellent structural separa-tion has been achieved between the 84 differentAB2 structure types that have more than onerepresentative compound each. The 8:4 coordinatedfluorite structure Cl (CaF2) cF12 with crystalcoordination formula 3 [CaF874] is observed in theionic regions at the top left and bottom right of Figure9, although it also occurs elsewhere, as for example inthe small domains centered on Mg2Si, Rh2P, and Al2Ptrespectively. The latter domains are metallic. In thesecases it is more sensible to define the local coordinationpolyhedron about the fluorine site by including the sixnext-nearest-neighbor fluorine atoms in addition to thefour tetrahedrally configured calcium first-nearestneighbors. This leads to the 10-atom configurationpolyhedron 10IV that is shown in Figure 8 (Villars etal.y 1989). A similar type of assignment must be madefor the B2 (CsCl) cP2 lattice: in the ionic region onlyfirst-nearest-neighbor unlike atoms are retained so thatthe crystal coordination formula is 3 [CsCl878],whereas in a very large metallic domain of Figure 7 thesix second-nearest-neighbor like atoms (which are only14% more distant) are also included to define the14-atom configuration polyhedron 14 that is drawn inFigure 8 (see section 2.8 of Jensen, 1989).

The 6:3 coordinated rutile structure C4 (TiO2) tP6with its crystal coordination formula 3 [TiO673] isstable only with the very electronegative constituentshydrogen, oxygen, and fluorine. The correspondingcompounds with the less electronegative halogens formthe two-dimensional layer structures C19 (CdCl2) hR3and C6 (CdI2) hP3 in which the metal atom is sixfoldoctahedrally coordinated. They thus have the crystalcoordination formula 2 [CdCl673] and 2 [CdI673]respectively. These two structures are very similar inenergy because they differ only in the stacking of thecomposite layers, which are held together by weak vander Waals interactions. CdCl2 has the halogen atomsarranged on an f.c.c. lattice, whereas CdI2 has them onan h.c.p. lattice. The metal atoms can also besandwiched so that they have sixfold trigonal symmetry;a- and /S-MoS2 with crystal coordination formula2 [MoS6Z73] correspond to the different stackingsequences 3R and 2H1 respectively (see Figure 4.11 ofWells, 1975). They are found among the early transition-metal sulfides, selenides, and halides.

The two forms of FeS2, namely C2 (pyrites) cP12and Cl8 (marcasite) oP6, are well-separated in Figure9. They have the same sixfold octahedral coordinationabout the Fe site and are characterized by the S atoms

occurring in pairs. The pyrites structure is derived fromthe NaCl structure by replacing Na with Fe and Cl withS2 dimers pointing along <111 > directions.

Cllb (MoSi2) tI6, C40 (CrSi2) hP9, and C54 (TiSi2)oF24 are polytypes that result from stacking close-packed planes of AB2 stoichiometry in a b.c.c. (110)stacking sequence, so that it is not surprising thatMoSi2 is characterized by the distorted b.c.c. localcoordination polyhedron 14, and CrSi2 and TiSi2 bythe variant 14' (see Figure 8). The C16 (CuAl2) til2structure type neighbors both these polytypes andCaF2, which is not unexpected since it can be derivedfrom the b.c.c. lattice (Burdett, 1982; see also Chapter13 by Hellner in this volume). Finally, the well-knownspace-filling AB2 Laves phases are found almost intheir entirety above the diagonal in Figure 9 and are seento be well-separated among the C14 (MgZn2) hP12, Cl5(MgCu2) cF24, and C36 (MgNi2) hP24 structure types.The maps show only the ground-state structures; polytypismis often exhibited by those compounds falling neardomain boundaries. The larger A atom is surroundedby 16 atoms within the Frank-Kasper coordinationpolyhedron 16, whereas the smaller B atom is surroundedicosahedrally by 12 atoms as illustrated by 12" in Figure 8.See also Chapter 17 by Gladyshevskii and Bodak.

Figure 10 shows that excellent structural separationhas also been achieved between the 52 differentAB3 structure types that have more than onerepresentative compound each. The simplest three-dimensional framework structure with AB3

stoichiometry that can be built from the octahedralAB6 complex is cubic ReO3 in which every octahedronis joined to six others through their vertices. Thisstructure with crystal coordination formula 3 [ReO672]is adopted by the trifluorides ZrF3, TaF3, and NbF3.Like their AB2 counterparts, the less-electronegativehalogens form two-dimensional layer structures but withthe A atoms occupying only two-thirds of the octahedralholes within the close-packed sandwich of B atoms.2 [AlCl672] has the halogen atoms arranged on anf.c.c. lattice, whereas 2 [FeCl672] has them on an h.c.p.lattice. The halogens with group IVA form the one-dimensional chain structures TiCl3 and TiI3 in whichthe AB6 octahedra share opposite faces. Distortedtricapped trigonal prismatic coordination is shown bythe halogens within the actinides and some rare-earthand group IIIA elements; YF3, LaF3, and UCl3structure types have ninefold coordination about metalsites (see Chapter 12 by Hauck and Mika in thisvolume).

It is clear from Figure 10 that many 1:3 stoichiometriccompounds take close-packed structure types with either

Figure 9. The Pettifor map for AB2 compounds (from Pettifor, 1988a)

Figure 10. The Pettifor map for AB3 compounds (from Pettifor, 1988a)

the (distorted) cubic 12 or hexagonal 12' localcoordination polyhedron (see Figure 8). Consider firstthe close-packed layer of MN3 stoichiometry with atriangular arrangement of the M atoms (see Figure 7.15of Pearson, 1972). These close-packed layers may bestacked one above the other in the usual close-packedpositions A, B or C, so that the M atoms have only Natoms as nearest neighbors. The following polytypeshave been marked explicitly on the structure map: Ll2

(Cu3Au) cP4 with cubic ABC stacking sequence; DO19

(Ni3Sn) hP8 with hexagonal close-packed AB stackingsequence; DO24 (Ni3Ti) hP16 with double hexagonalclose-packed (DHCP) sequence ABAC; and BaPb3 withthe hexagonal cell containing nine close-packed layersin the sequence ACACBCBAB. The CuTi3 structuretype is a tetragonal distortion of Cu3Au. Finally, thestructure types DO22 (Al3Ti) tI8 and Cu3Ti are polytypesthat are based on the stacking of close-packed MN3

layers with rectangular arrangement of the M atoms (seeFigure 7.21 of Pearson, 1972). They are well-separatedfrom each other and from another close-packedsuperstructure DO23 (Al3Zr) til6.

Finally, most AB3 compounds with B from groupsIVA, VA or VIA take the Al5 (Cr3Si) cP8 structuretype. The A atoms form a b.c.c. lattice through whichlines of B atoms run parallel to the edges of the cubiccell. The Al5 structure is compact with the A atomssurrounded icosahedrally by 12 B atoms withcoordination polyhedron 12". The B atoms sit at thecenter of a 14-atom polyhedron 14'", which is formedfrom four A and 10 B nearest neighbors. The DO3

(BiF3) cF16 structure type is an ordered structure basedon the b.c.c. lattice with the local coordinationpolyhedron 14.

The pure elements would lie along the diagonal line^ A = ^ B m these binary structure maps. It is,therefore, not surprising that the diagonal in Figure 7for the AB stoichiometry cuts through the CsCl, CuAu,and cubic ZnS domains where the elemental b.c.c,f.c.c, and diamond lattices are stable, whereas in Figure10 for the AB3 stoichiometry it passes through theCu3Au, Ni3Sn, and Cr3Si domains where the elementalf.c.c, h.cp., and /3-W lattices are found. Thus thephenomenological relative ordering number Jt shouldalso order the elements according to their structure type.This is broadly the case as can be seen from table 1 ofPettifor (1988a), where the string runs from the close-packed noble-gas and metallic elements through themore open metalloid elements to the halogens andhydrogen, which solidify as dimers held together on thelattice by very weak van der Waals interactions. Runningthe string from right to left through the lanthanides and

actinides preserves the continuity of elemental structureand is in accordance with quantum-mechanical theory,which predicts the structural trend from h.cp. tod.h.c.p. as the core size and corresponding number ofvalence d electrons increases through these trivalentsystems (Duthie and Pettifor, 1977).

These structure maps for the ground-state structuresof the binary compounds suggest that it might bepossible to move from one structural domain to anotherby the suitable addition of a third or more alloyingelement. If it is assumed that the ternary and quaternaryadditions C and D go preferentially to the A and B sites,then the alloy (AxC1 _x)m(ByDl_y)n may be regarded asthe pseudobinary Am En characterized by the averagerelative ordering numbers

JA=xJ?A + (\-x)J<c (3)

and

Jz=yJtB + (\-y)JtD (4)

Figures 6 and 7 of Pettifor (1988a) show that this simplescheme orders the 1:1 and 1:3 pseudobinaries in thedatabase of Villars and Calvert (1985) within the samestructural domains as for the pure AB and AB3

binaries in Figures 7 and 10 respectively.It should be noted, however, that there are important

exceptions, which may occur when a particular alloysystem with a given structure type straddles a narrowdomain with a related, but different, structure type(Pettifor, 1988a). For example, the NiAs structure-typeternary alloy system (BixPb1^)Pt straddles a narrowMnP domain in Figure 7, whereas the Cu3Austructure-type ternary alloy system (PbxTl1^)3Lastraddles a narrow Ni3Sn domain in Figure 10. Boththese cases involve linking the binary alloys of heavyatoms Hg, Tl, Pb or Bi where relativistic effects lead tothe formation of lone-pair s electrons. This dependenceof the bonding on the principal quantum number issufficient to determine the delicate balance betweenthese very similar structure types, NiAs and MnP orCu3Au and Ni3Sn, respectively, as Mooser andPearson (1959) proposed.

These maps have been used as a guide in the search fornew pseudobinaries with a required structure type, inparticular cubic transition-metal aluminides with goodmechanical properties and tetragonal rare-earth-ironintermetallics with good permanent magnetic properties.The relative ordering number allows all the knowndata on the ground-state structures of binary com-pounds A1^xBx to be presented within the single

Figure 11. The titanium-aluminum neighborhood maps (from Pettifor, 1992)

three-dimensional map (J£A, JC^, x). Figure 11shows the resultant titanium-aluminum neighborhoodmaps, which are two-dimensional cross-sections forfixed x through the three-dimensional cylinder whoseaxis is centered on (JtA =51, JC^ = 80) correspondingto the coordinates of titanium and aluminum (Pettifor,1992). The sp-bonded elements Be, B, C, N, and O havebeen excluded from such neighborhood maps becausetheir small size and high electronegativity give them uniqueproperties. The key in Figure 11 gives not only the Pearsonsymbol but also the recently proposed generalizedJensen notation for the local coordination polyhedraabout the A and B sites; see table 5 of Jensen (1989),Figures 3 and 4 of Villars et al. (1989), and Figure 8.

We see that the brittle intermetallic Ti3Al andtransition-metal trialuminides HfAl3, TiAl3, TaAl3,NbAl3, and VAl3 sit in hexagonal D019(hP8) andtetragonal DO22(tI8) domains, respectively. However,the latter domains are adjacent to cubic Ll2(cP4)domains, which suggested that it might be possible to

stabilize the cubic structure (with possibly bettermechanical properties) by_alloying so that the resultantaverage coordinates (JtA, J£B) fall in the cubicdomain. This was known to be the case for tetragonalAl3Ti, which could be transformed into the cubic Ll2

phase by replacing some of the aluminum with Cu, Nior Fe (see, for example, Villars and Calvert, 1985, andChapters 7 and 8 in Volume 2 by Yamaguchi, Inui andby Das).

This hope of stabilizing cubic phases has only beenpartially realized, however. Schneibel and Porter (1989)have indeed succeeded in stabilizing cubic ZrAl3 byalloying to take the average relative ordering numberJt^ down into the cubic domain. But attempts tostabilize cubic Ti3Al from hexagonal DO19 or cubicNbAl3 from tetragonal DO22 have failed (Liu et al.,1989; Subramanian et a/., 1989). The failure to stabilizecubic Ti3Al is probably due to the adjacent cubicdomain being taken only by 4d and 5d transitionelements, not 3d ones. This is another example of the

Figure 12. The Pettifor map for AB13, AB12, AB11, and A2B17 stoichiometries. The full and open symbols denote binaries andpseudobinaries respectively (from Pettifor, 1988b)

importance of the Mooser-Pearson (1959) principal-quantum-number factor when dealing with very narrowdomains of structural stability. The failure to stabilizecubic NbAl3 is due to the rapid oscillatory behavior ofcubic versus tetragonal stability in going from group IVto group V, which is wave-mechanical in origin (Pettiforand Aoki, 1991). Although TiAl3 and NbAl3 both liein the same DO22 domain adjacent to a cubic Ll2

domain in Figure 11, the energy required to transformNbAl3 to the cubic phase is predicted to be nearly anorder of magnitude greater than that for TiAl3, asdiscussed in Chapter 3 by Carlsson and Meschter in thisvolume. It is, therefore, extremely unlikely that cubicNbAl3 will ever be stabilized by alloying additions,supporting recent experimental evidence that a 1964report of the Ll2 phase Nb2(Al5Ni) is incorrect(Subramanian et ai, 1989).

Another example of the application of structure mapsis provided by the search for new rare-earth-iron-basedpseudobinaries with the tetragonal structure typesBaCd11, ThMn12, and NaZn13 for use as permanentmagnets (Mitchell et al.y 1989). Figure 12 shows thestructure map for binary and pseudobinary phases withthe A2B17, AB11, AB12, and AB13 stoichiometries usingVillars and Calvert's (1985) database (Pettifor, 1988b).These four stoichiometries correspond to alloysA x B 1 ^ with x = 0.105, 0.083, 0.077, and 0.071,respectively. Figure 12 is, therefore, a projection on thetwo-dimensional ( JCA, Jt B) plane of the data pointslying in the three-dimensional space (JCA, JCB, x)with 0.071 ̂ x<0.105. We see that the differentstructure types are located in well-defined domains,although there is some overlap between them. Binaryrare-earth-iron phases take the 2:17 stoichiometry, ascan be seen in Figure 12, where JCB =61 for iron.Unfortunately, they have too low a Curie temperaturefor use as permanent magnets.

Figure 12 suggests that, if we are looking for cheapiron-based pseudobinaries in which the majority of theatoms are iron, then it would be best to search in thelower ThMn12 domain by replacing iron by transitionelements to its left in the periodic table (Pettifor, 1988b).Independently, de Mooij and Buschow (1987) have doneextensive experimental work in just this area. They havestabilized R(Fe,X)12 pseudobinaries with the tetragonalThMn12 structure for the cases where X = Ti, V, Cr,Mo or W, which would all fall within the lowerThMn12 domain in Figure 12. (They also stabilized theThMn12 structure with X = Si, which would fall in the2:17 domain of Figure 12. Nevertheless, silicon appearsto be exceptional since the ThMn12 structure was notfound for the other sp-valent elements Al, Ga, Ge or

Sn.) De Mooij and Buschow (1987) have found thesetetragonal pseudobinaries to be good permanentmagnets. In particular, Sm(Fe11Ti) has a Curietemperature and magnetic anisotropy that are verysimilar to the market leader, the novel ternarycompound Nd2Fe14B. However, at room temperatureit has an average iron moment of only 1.6 t̂3 comparedto the 2.0 fiB for Nd2Fe14B, so that a pseudobinary hasnot yet displaced the ternary boride from its perch asthe most powerful known permanent magnet.

3. Ternary Structure Maps

3.1 Villars Maps

The three-dimensional Villars maps have beengeneralized to ternary systems by defining thecoordinates

AX=2x(XA-XB) + 2x(XA-Xc) + 2y(XB-Xc) (5)

AR = 2x(RA-RB) + 2x(RA-Rc) + 2y(RB-Rc) (6)

N=xNA+yNB+zNc (7)

for AxByCz with x^y^z and x+y+z=\ (Villars andHulliger, 1987). Quaternaries can be included byaveraging the atomic coordinates of the two mostelectronically similar elements and using the abovecoordinate definitions for the ternaries.

Since, to date, there are more than two thousanddifferent intermetallic ternary structure types comparedto the several hundred binary structure types, Villarsand Hulliger (1987) and Villars et al (1989) ordered thestructural database with respect to the local coordinationpolyhedra or atomic environment types taken by thelattice (see, for example, Figure 8). They found, forexample, that of the 147 binary structure types withmore than five representative compounds each, 25structure types with 1347 representatives displayed onlya single atomic environment, whereas 51 structure typeswith 1952 representatives displayed two different atomicenvironments, 34 structure types with 879representatives displayed three different atomicenvironments, and 28 structure types with 543representatives displayed four different atomicenvironments. Only nine structure types with 114representatives required more than four different atomicenvironment types for a complete description of thelocal coordination.

Villars and Hulliger (1987) have found that 98% ofthe 2532 binary, ternary, and quaternary compounds

with single-environment structure types arecharacterized by one of the following five localcoordination polyhedra: the tetrahedron, theoctahedron, the cubooctahedron, the truncatedhexagonal eclipsed bipyramid, and the rhombicdodecahedron, respectively. They are labeled by 4, 6,12, 12', and 14 respectively inJFigure 8. Within thethree-dimensional map (AX, AR, N) these single-environment compounds tended to separate into manydifferent domains dominated by one or other of the fivelocal coordination polyhedra.

These generalized Villars maps are being used in thesearch for new stable quasicrystals and high-temperatureferroelectrics and superconductors (see Rabe et al., 1992,and references therein). To date, there are nine knownstable quasicrystals: CuLi3Al6, Ca20Mg7Zn43,A15Cu20Al65 (A = Fe, Ru, Os), Cu15Co20Al65,Co15Ni15Al70, and (Mn5Re)Pd2Al7. These all fallon the Villars map within the restricted range-0.6ÂA^0.3 and -0.2Â#^0.7. Unfortunately,many thousands of potential ternaries also fall withinthis domain, so that a sensible screening procedure needsto be adopted; several such strategies are considered byRabe et al. (1992). For high-temperaturesuperconductors with 7C>1OK, the Villars mapseparates the database into three domains A, B, and Cas shown in Figure 13 from Rabe et al. (1992). (Note

that copper and zinc are assumed to have one and twovalence electrons respectively in insulating compounds.)Domains A and B are dominated by the Al5 (Cr3Si)cP8 and Bl (NaCl) cF8 binary structure-typesuperconductors respectively. Domain C contains theChevrel (hR15) and perovskite (cP5) ternary structure-type superconductors. Interestingly, high-temperatureferroelectrics occupy the space in domain C between theChevrel phases and the cuprate superconductors,indicating the close connection between ferroelectricityand superconductivity stressed much earlier by Matthias(1970). Rabe et al. (1992) have discussed variousscreening strategies for sorting between the thousandsof potential ternary compounds that fall within theseregions of high-temperature superconductivity.

3.2 Pettifor Maps

Most of the non-hydrogen-, -carbon-, -nitrogen- or-oxygen-containing phases of ternary structure typecan be separated and displayed within only seven two-dimensional structure maps (Pettifor, 1988a). This isbecause the structural information for many differentternary stoichiometries A/BmCw can be presented onthe same figure. This is illustrated by the two structuremaps for the ternary chalcogenides, namelyAB(S,Se,Te)n and A/Bm>/(S,Se,Te)/7, which include

Figure 13. The Villars map for high-rc superconductors. The left-hand panel (a) shows theômpounds projected on the (AX, AR) plane,whereas the right-hand panel (b) plots their transition temperatures as a function of N for the three domains A, B, and C (fromRabe et al., 1992. Reproduced with permission)

Figure 14. The Pettifor map for the ternary chalcogenides with A,Bm>/ (S, Se, Te)n stoichiometry. The chalcogen in bracketsafter a given structure type indicates that only it is present in the ternary compound represented by the particular symbol. Morethan one symbol may be superimposed at a given point on the map (From Pettifor, 1988a)

more than one-third of all the different ternary structuretypes listed by Villars and Calvert (1985) with more thanone representative compound each. The latter mapA7Bm>/(S,Se,Te)w is shown in Figure 14. We see, forexample, that the Chevrel phases PbMo6S8 occupy thevery narrow domain corresponding to Jt^ =56 formolybdenum. The relationship between neighboringstructural domains within the ternary intermetallicstructure maps has not yet been explored in detail. Theongoing work of Daams, Villars, and colleagues (Daamset <*/., 1992; Daams and Villars, 1993) in finding the localcoordination polyhedra or atomic environment types ofall binary, ternary, and quaternary structure types willbe crucial in revealing the underlying structural patternsbeneath the surface of ternary structure maps such asFigure 14.

4. Conclusion

Structure maps, by ordering the very large database intodomains of different structure type, have provenimportant both as a pedagogical aid in structuralchemistry and as an initial guide in the search for newintermetallics with a required structure type. However,we have seen that they must be used with caution in theirpredictive mode. The difference in energy betweencompeting structure types such as cubic and hexagonalclose-packed phases will be of the order of one-thousandth of the cohesive energy. The maps cansuggest which alloys might take a given structure type,but they cannot, of course, provide any firm guaranteeowing to the subtle competition between many differentphases and stoichiometries. Electron theory provides anessential complementary input through its new-foundability to predict the relative energies of different(simple) competing phases. This is discussed in Pettiforand Cottrell (1992) and in Chapter 2 by Turchi andChapter 3 by Carlsson and Meschter in this volume.

5. References

Burden, J. K. (1982). /. Solid State Chem., 45, 399.Cressoni, J. C, and Pettifor, D. G. (1991). /. Phys.: Condens.

Matter, 3, 495.Daams, J. L. C , and Villars, P. (1993). J. Alloys Compounds,

197, 243.Daams, J. L. C , Van Vucht, J. H. N., and Villars, P. (1992).

J. Alloys Compounds, 182, 1.de Mooij, B., and Buschow, K. H. J. (1987). Philips J. Res.,

42, 246.Dimiduk, D. M., and Miracle, D. B. (1989). Mater. Res. Soc.

Symp. Proc9 133, 349.

Duthie, J. C , and Pettifor, D. G. (1977). Phys. Rev. Lett.,38, 564.

Girifalco, L. A. (1976). Acta Metall, 24, 759.Jensen, W. B. (1989). In The Structures of Binary Compounds

(eds F. R. de Boer and D. G. Pettifor). North-Holland,Amsterdam, p. 105.

Liu, C. T., Horton, J. A., and Pettifor, D. G. (1989). Mater.Res. Soc. Symp. Proc, 133, 37.

Martynov, A. J., and Batsanov, J. S. (1980). Russ. J. Inorg.Chem., 25, 1737.

Matthias, B. T. (1970). Mater. Res. Bull., 5, 665.Mitchell, I. V., Coey, J. M. D., Givord, D., Harris, I. R., and

Hanitsch, R. (eds) (1989). Concerted European Action onMagnets. Elsevier Applied Science, London.

Mooser, E., and Pearson, W. B. (1959). Acta Crystallogr., 12,1015.

Pauling, L. (1960). The Nature of the Chemical Bond, 3rd Edn.Cornell University Press, Ithaca, NY.

Pearson, W. B. (1972). The Crystal Chemistry and Physics ofMetals and Alloys. Wiley-Interscience, New York.

Pettifor, D. G. (1983). In Physical Metallurgy (eds R. W. Cahnand P. Haasen). North-Holland, Amsterdam, p. 73.

Pettifor, D. G. (1984). Solid State Commun., 51, 31.Pettifor, D. G. (1986). /. Phys., 19, 285.Pettifor, D. G. (1988a). Mater. ScL Technol., 4, 2480.Pettifor, D. G. (1988b). Physica, B149, 3.Pettifor, D. G. (1992). Mater. ScL Technol., 8, 345.Pettifor, D. G., and Aoki, M. (1991). Phil. Trans. R. Soc,

A334, 439.Pettifor, D. G., and Cottrell, A. H. (eds) (1992). Electron

Theory in Alloy Design. Institute of Materials, London.Pettifor, D. G., and Podloucky, R. (1986). /. Phys., C19, 315.Phillips, J. C , and Van Vechten, J. A. (1969). Phys. Rev. Lett.,

22, 705.Rabe, K. M., Kortan, A. R., Phillips, J. C , and Villars, P.

(1991). Phys. Rev., B43, 6279.Rabe, K. M., Phillips, J. C , Villars, P., and Brown, I. D.

(1992). Phys. Rev., B45, 7650.Schneibel, J. H., and Porter, W. D. (1989). Mater. Res. Soc.

Symp. Proc, 133, 335.St John, J., and Bloch, A. N. (1974). Phys. Rev. Lett., 33, 1095.Subramahian, P. R., Simmons, J. P., Mendiratta, M. G., and

Dimiduk, D. M. (1989). Mater. Res. Soc Symp. Proc,133, 51.

Villars, P. (1983). J. Less-Common Met., 92, 215.Villars, P. (1984a). J. Less-Common Met., 99, 33.Villars, P. (1984b). /. Less-Common Met., 102, 199.Villars, P., and Calvert, L. D. (1985). Pearson's Handbook

of Crystallographic Data for Intermetallic Phases, VoIs1, 2, 3. American Society for Metals, Metals Park, OH.

Villars, P., and Hulliger, F. (1987). J. Less-Common Met.,132, 289.

Villars, P., Mathis, K., and Hulliger, F. (1989). In TheStructures of Binary Compounds (eds F. R. de Boer andD. G. Pettifor). North-Holland, Amsterdam, p. 1.

Watson, R. E., and Bennett, L. H. (1978a). /. Phys. Chem.Solids, 39, 1235.

Watson, R. E., and Bennett, L. H. (1978b). Phys. Rev., B18,6439.

Wells, A. F. (1975). Structural Inorganic Chemistry.Clarendon, Oxford.

Yeh, C-Y., Lu, Z. W., Froyen, S., and Zunger, A. (1992).Phys. Rev., B15, 46, 10 086.

Zunger, A. (1980). Phys. Rev., B22, 5839.Zunger, A., and Cohen, M. L. (1978). Phys. Rev. Lett.,41, 53.


1. Introduction

To understand the magnetic properties of a material,it is necessary to consider its magnetic structure. Thischapter gives an overview of the magnetic structures intransition metals, rare-earth metals, alloys, andintermetallic compounds, with emphasis on the latter.The intention is not to give a comprehensive review ofmagnetic structures in intermetallic compounds, butrather to discuss examples that illustrate why the atomicmoments on a particular group of atoms are arrangedin one way rather than another. The interrelation ofmagnetic properties and magnetic structure are treatedin Chapter 40 by Kouvel in this volume. Technologicalapplications (see Chapter 14 by Stadelmaier and Reinschand Chapter 19 by McGahan in Volume 2) and theextrinsic structural parameters that affect the design ofpractical materials—shape and size of the specimen,particle size, impurities—are not considered here.Following a brief description of types of magnetic order,models of magnetic exchange interactions, and methodsof determining magnetic structure, examples ofmagnetic structures are considered in five groups ofmaterials: (1) transition metals, (2) transition-metalintermetallic compounds, (3) rare-earth metals, (4) rare-earth intermetallic compounds, and (5) metallic systemswith small-scale order.

2. Exchange Interactions and Magnetic Structure

The concept of a magnetic structure as a spatiallyordered arrangement of atomic moments on a latticepreceded its experimental validation by neutron

diffraction. Weiss (1907) and Neel (1948) explainedmeasurements of magnetization and magneticsusceptibility by a 'molecular field* that aligned themoments of a paramagnetic substance. Magneticexchange interactions were supposed to result in threetypes of collinear arrangement of the atomic moments:(1) antiferromagnetic, an antiparallel alignment of equalmoments; (2) ferrimagnetic, an antiparallel alignmentof moments unequal in magnitude or number; and (3)ferromagnetic, a parallel alignment of the moments.

Ferromagnetism and ferrimagnetism are characterizedby a spontaneous magnetic moment below the Curietemperature, Tc. The transition from paramagnetismto antiferromagnetism is recognized by a well-definedkink in the susceptibility vs. temperature curve at theNeel temperature, TN . It is now known from neutrondiffraction that magnetic structures are more complexthan originally envisioned; the atomic spins are usuallynon-collinear, and they may be organized in spiral,helical, and cone-like arrays that are incommensuratewith the crystal lattice. The terms 'ferromagnetic' and'antiferromagnetic' are now taken to refer tointeractions that orient the moments on two atoms ina direction that has either the same or the opposite sense.

When some materials are cooled from theparamagnetic state, the spin directions become fixed or'frozen in' without long-range order. The names usedin the literature for such solids include amorphousferromagnets, spin glasses, cluster glasses, andmictomagnets. Amorphous ferromagnet (Luborsky,1980) usually implies a magnetically concentratedsubstance with a non-crystalline or amorphous atomicarrangement. There is less general agreement about spin

Chapter 9

Magnetic Structures

Walter L. RothDepartment of Physics, State University of New York at Albany, Albany,

New York 12222, USA


glass, a term that was originally applied to dilutecrystalline alloys containing about 1% of magneticimpurities. A proposed definition (Mydosh andNieuwenhuys, 1980) is a random, metallic, magneticsystem characterized by a random freezing of themoments without long-range order at a ratherwell-defined freezing temperature, Tf. The spinconfiguration in a spin glass is random but static belowTf, and the macroscopic moment in zero external fieldis zero. As the concentration of magnetic impurity ina spin glass is increased, concentration fluctuations andchemical attractive forces result in the formation ofmagnetic clusters with very large effective moments,~20 to 20 000/>tB. Materials in which the magneticbehavior is dominated by the presence of large magneticclusters have been called mictomagnets (Tustison andBeck, 1977) and cluster glasses (Murani et al, 1976).

Ferromagnetic interactions are essentially unalteredby the absence of a periodic lattice. The intrinsicmagnetic properties of amorphous ferromagnets arequalitatively the same as for crystalline alloys withappropriate modification to take into account that thelocal environment around each atom differs from siteto site. In contrast to ferromagnetic interactions,antiferromagnetic interactions can be 'frustrated' bydisorder. This is a situation in which it is impossible tosatisfy simultaneously the interactions between atomson different sites, resulting in a random, non-collinearconfiguration of the atomic moments.

The main experimental technique for determiningmagnetic structure is neutron diffraction (James, 1989).By coupling high-resolution instrumentation with theRietfeld method of analysis, structural detail rivalingthat previously obtained only on single crystals can nowbe obtained from powders (Rietfeld, 1969). Themagnetic structure of an ordered magnetic material isspecified by the position, magnitude, and orientationof the atomic moments in a magnetic unit cell. Theinterpretation of magnetic Bragg scattering gives twokinds of information about the magnetic moments insolids: (1) the magnetic structure, and (2) the spatialextension of the magnetic electrons. In an ordered alloy,specific kinds of atoms occupy crystallographicallydistinguishable lattice sites, and individual magneticmoments are obtained from the magnetic Braggscattering. In a disordered alloy, different kinds ofatoms are distributed randomly on a lattice, and theindividual moments are obtained from measurementsof neutron diffuse scattering.

The procedure for determining the arrangement ofmoments in magnetic materials in similar to crystal-structure analysis, differing mainly in the need to

determine the orientation of the atomic spins. Themagnetic structure factor, Fhkh for a reflection withMiller indices hkl is given by

^,(magnetic) = X)(X*Q/)A exp [2T1-(ZUr1- + kyt + Iz1) ]

The summation is over / atoms in the magnetic unit cell;X is a unit vector in the direction of polarization of theneutrons; q, is the magnetic interaction vector; and P1

is the magnetic scattering amplitude of the atom. Themagnetic scattering amplitude of the atom depends onthe spin quantum number S and the quantum numberL if there is an orbital contribution to the magneticmoment. The values of Xn yh Zn q, and /?, are obtainedfrom the diffraction intensity data by least-squares andFourier techniques.

Other methods that are available for investigatingmagnetic structure are nuclear magnetic resonance andMossbauer spectroscopy (Chien, 1981). These techniquesmeasure the effective magnetic field at the nucleus; theyare particularly useful as local probes for investigatingvariations in the local environment about nuclei inamorphous and chemically disordered solids.

The magnetic structure of a substance is the resultof exchange interactions. It is not possible to obtainexact solutions for exchange interactions in three-dimensional crystals, and approximate models—directexchange, spin-density waves, the Ruderman-Kittel-Kasuya-Yoshida interaction, super exchange, doubleexchange—are used to explain the magnetic arrangementsfound by neutron diffraction.

Direct exchange is due to overlap of the wavefunctionsof nearest neighbor atoms; it is the main interactionthat orders the moments in 3d transition metals.Direct exchange is short-range, and its dependenceon interatomic distance often is represented by aSlater-Neel-type diagram, which predicts a negativeexchange integral (antiferromagnetic interaction) atshort interatomic distance and a positive exchangeintegral (ferromagnetic interaction) at large interatomicdistance (Somerfeld and Bethe, 1933). The treatmentof exchange interactions between 3d transition elementsis complicated by uncertainty about whether themagnetic electrons are localized or itinerant. Amechanism proposed for propagating magneticinteractions over long distances by itinerant electronsis spin-density waves (SDW) (Overhauser and Arrott,1960). In the rare earths and dilute alloys of thetransition elements, the primary exchange mechanismis the Ruderman-Kittel-Kasuya-Yoshida (RKKY)interaction (Ruderman and Kittel, 1954). The RKKYmechanism depends on conduction electrons that carry

information on the orientation of the localized spin fromone atom to another; it is long-range and gives rise toan oscillatory spatial polarization as a function ofinteratomic distance. The main interaction mechanismsin ionic compounds are super exchange and doubleexchange. Both depend on overlap of the d orbitals ofthe magnetic atoms with the p orbitals of an intermediateanion. Superexchange usually is antiferromagnetic anda maximum when the cation-anion-cation angle is180°; double exchange is ferromagnetic and arises fromcharge transfer between multivalent ions in solidsolution. Superexchange and double exchange are notimportant exchange mechanisms in metallic systems butmay have a role in composite materials.

3. Magnetic Structures

The magnetic structures of most intermetalliccompounds are non-collinear arrangements of unequalmoments on two or more chemically different atoms.They are difficult to display in two-dimensionaldrawings and usually are presented in the form of tablesof parameters of the position, magnitude, andorientation of the magnetic moments in the unit cell(see Cox, 1973). Magnetic structures of current interestare difficult to describe in words for, as Eddington(1948) remarked when discussing an elementary particle,words are a kind of Jabberwocky applied to 'somethingunknown' that is 'doing we don't know what.' HeedingAlice's query in Wonderland when she asked 'what isthe use of a book without pictures or conversation,'schematic, simplified drawings are given of severalunusual types of magnetic structure that are importantissues of current research. Readers are referred toreviews by Bacon (1962a), Willis (1970), Nathans andPickart (1963), Goodenough (1963), and Kouvel (1967)for descriptions of magnetic structures of intermetalliccompounds that were solved in the initial periodfollowing the discovery that neutron diffractioncould be used to investigate magnetism in solids.Examples of more recent reviews are: Nakamura andFranse (1987), Moon and Nicklow (1991), Gignouxand Schmitt (1991), and Jensen and Mackintosh(1991).

3.1 Transition Metals

Many of the principles that determine the magneticarrangements of intermetallic compounds and alloyshave been obtained from studies of the magneticstructures in the transition-element metals. The

magnetism of the transition elements is a central issuein the theory of magnetism. Iron, cobalt, and nickel areferromagnets with collinear spin arrangements;manganese and chromium are antiferromagnets withcomplicated spin arrangements; vanadium, niobium,molybdenum, and tungsten do not order at all, at leastat temperatures as low as 4 K. The different magneticbehavior has its origin in the electronic structure of the3d transition metals, which is intermediate between thecases where the electrons can be regarded as localizedor itinerant.

Iron (body-centered cubic, structure type A2 (cI2)),cobalt (both hexagonal, structure type A3 (hP2), andface-centered cubic, structure type Al (cF4)), and nickel(face-centered cubic, structure type Al (cF4)) areferromagnets with collinear arrangements of the atomicmoments. The magnetic moment of iron is less than thefree-ion value, and an objective of early neutrondiffraction studies was to determine if iron was aferrimagnet, i.e. that the reduced moment resulted fromantiferromagnetic coupling between iron atoms withdifferent moments. The ferrimagnet hypothesis wasrejected after it was established that the momentmeasured by neutron diffraction was the same asdetermined by magnetic measurements, and that therewas no evidence for either an ordered or a disorderedarrangement of atoms with different magnetic moments.

Subsequent studies to understand the reducedmagnetic moment of iron focused on the distributionand symmetry of the magnetic electrons (Shull andYamada, 1962). Measurements of the magnetic formfactor show that the total magnetic moment of iron,2.177 /AB, is the result of a positive contribution of2.39 [iB from the 3d electrons and a negativecontribution of 0.21 fiB from the 4s electrons. Figure 1displays a portion of the (110) plane intersecting thetwo iron atoms in the unit cell. The magnetic electrondensity is not spherically symmetric around the nucleus.The electron density contours are not circular, as theyshould be for a spherical atom, but compressed in the[111] space diagonal, or nearest-neighbor, direction,with very small density between nearest-neighbor ironatoms. The asymmetry of the magnetic spin density isexplained by attributing 53% of the 3d quenchedelectron magnetization to electrons with Eg symmetry(the concentration along the [100] cube edges) and47% to electrons with T2g symmetry (the concentrationalong the [111] cube diagonals).

Magnetic form-factor studies on nickel give for theindividual contributions to the magnetization: 3d spin,+ 0.656/AB; 3d orbit, +0.005 /AB; conduction electrons,-0.105/iB.

In contrast to a-iron, which has the same body-centeredcrystal structure (structure type A2 (cI2)) and isferromagnetic, chromium is antiferromagnetic. Themagnetic structure of chromium has been the subjectof many investigations, originally to understand whyit shows no macroscopic magnetic properties eventhough it possesses strong atomic moments. Theproblem was resolved when it was shown by neutrondiffraction that chromium is an antiferromagnet.Subsequent studies have been concerned with theresolution of its complex spin arrangement. In additionto the fundamental Bragg scattering peaks that areexpected for antiferromagnetic coupling of twomagnetic atoms in the cubic unit cell, the neutrondiffraction pattern displays peaks of a magneticsuperlattice and a magnetic structure change at 121 K.The model that was originally advanced to explain themagnetic superstructure was antiphase domains in whichthere is a 180° spin reversal every 14 unit cells.

Alternatives to the antiphase domain model that havebeen proposed are a spiral arrangement of the spins withthe same periodicity, a sinusoidal modulation of themagnetic scattering amplitude, and spin-density waves.An investigation (Shirane and Takei, 1962) of thestructure on single crystals concluded that the sinusoidalmodel was preferred and obtained for the atomicmoment fiCr =0.59 fiB at 78 K and ^tCr0.41 /iB at 125 K(Figure 2). The chromium moments are parallel to thepropagation vector below the 121 K transition andperpendicular to the propagation vector above thetransition. The temperature dependence of the intensitiesof the satellites is consistent with spin-density waves;however, the field-cooling effect predicted by the spin-density-wave model did not give conclusive results.

Manganese is also antiferromagnetic, but in contrastto chromium the magnetic electrons appear to belocalized. The crystal structure of a-Mn (body-centeredcubic, structure type A12 (cI58) is a powerful tool for

Figure 1. Magnetic spin-density distribution in the (110) diagonal plane of the iron unit cell. Contours are shown in units oftiB A"3. Two nearest-neighbor iron atoms are shown in this section. Very small density is to be noted in the mid-region betweenthese atoms (After Shull and Yamada, 1962)

Ironnucleus

Ironnucleus

investigating the effect of the chemical environment onthe atomic moment of manganese. The unit cell contains58 Mn atoms in four non-equivalent sites, and the fourcrystallographically independent manganese atoms havedifferent numbers of neighbors at different distances.

The atomic spins are arranged in a single non-collinearconfiguration through the temperature range below theNeel temperature (Yamada et al.9 1970) (Figure 3).Single-crystal structure refinements indicate that alocalized-moment model is preferable to a spin-density-

Antiphase

Sinusoidal

Spiral

Antiferro

Figure 2. Models for Cr spin arrangement with their magnetic intensities, calculated with a periodicity of 22 unit cells. Thisspin direction corresponds to the high-temperature phase (After Shirane and Takei, 1962)

Figure 3. Magnetic structure of a-Mn. A view of the atomic configuration in a cell (left) and projections of the magneticconfiguration onto the i-j and j-k planes (After Yamada et al.y 1970)

wave model, and that the moments of manganese at4.4 K in the four different sites are pMn, = J - 9 ^ B *^Mnn = 1.7 pB , ^Mn111 = °-6 ^B > ̂ MnIV = °-2 ^B •

3.2 Transition-Metal Intermetallic Compounds

The magnetic structures in transition-metal alloys ingeneral are similar to metals after taking into accountthat the magnetic moments and exchange interactionsdepend on the chemical environment of the constituents.In only a few instances, such as Fe3Al, have thestructures of alloys been analyzed in sufficient detail todetermine the effect of alloying on the atomic momentand the electronic configuration of the constituentatoms.

Fe3Al (cubic, BiF3(DO3) structure type (cF16)) is aferromagnet with Tc ~ 750 K. There are two chemicallydifferent Fe atoms in the unit cell: Fe1 with eight Fenearest neighbors and six Al next-nearest neighbors; andFe11 with four Al and four Fe nearest neighbors. Themagnetic moment and the electronic structure of the ironatoms have been investigated by neutron diffractionwith the polarized beam technique by Pickart andNathans (1961). Their results in Figure 4 show that thelocal chemical environment has a large influence on boththe moment and the electronic configuration of iron.The iron moment in the two sites is pFe = 2.18 pB (thesame as in pure iron) and pFen = 1.50 pB. The magnetic

symmetry of the two iron atoms is: Fe1, 60% Eg, 40%T2g; and Fe11, 48% Eg, 52% T2g.

FeRh is an example of an alloy in which a change inelectronic structure with temperature changes itsmagnetic properties. At about 350 K, FeRh (cubic, CsCl(B2) structure type (cP2)) undergoes a first-ordertransformation from antiferromagnetism to ferro-magnetism. There is an isotropic change in volume atthe transition, and no change is observed in crystalstructure or symmetry. In the antiferromagneticstate, the atomic spins on nearest-neighbor atoms arecollinear and antiparallel: pFe = 3.17 pB, pRh = 0.9 pB.Approximately the same moments were found byBertaut et al. (1962) in the ferromagnetic state:pFe=3.2pB and pR h=0.9pB . The implication ofsubsequent neutron diffraction and Mossbauerexperiments by Shirane et al. (1963) is that the magneticpolarization of the conduction electrons is opposite tothat of the d electrons and sensitive to the nearest-neighbor environment of the iron atoms.

The magnetic properties of manganese alloys aredifficult to predict, in large part because the momentsmay be either localized or delocalized. Their magneticstructures typically are complicated and not completelyresolved. An example is Ni3Mn, which is non-magneticin the disordered state and magnetic in the (Cu3Au,Ll2 (cP4)) ordered state (Goldman, 1953). Amanganese alloy in the which the moments are localized

Figure 4. Projection of excess scattering density above calculated spherical density in the (110) plane of Fe3Al. This illustratesthe different magnetic electron symmetry of the two species of iron atoms. A pronounced deficiency is noted along the nearest-neighbor connecting line and an excess along the cube edge (After Pickart and Nathans, 1961)

is MnNi (tetragonal, CuAu (Ll0) structure type (tP4)),which is antiferromagnetic with Neel temperature above600 K. The spins on nearest-neighbor manganese atomsare antiparallel and perpendicular to the tetragonal axes.The Mn moments at 77 K are /*Mn = 4.0 ± 0.1 fiB andHNi<0.6fiB (Kasper and Kouvel, 1959).

In contrast to MnNi, the magnetic structures inMn-Au and Mn-Si alloys indicate that the momentsare partially delocalized. In the composition range near50at.% Mn, Mn-Au alloys have a tetragonallydistorted variant of the cubic, CsCl (B2 (cP2)) crystalstructure. Bacon (1962b) found that the magneticstructure is built up of ferromagnetic planes of Mnseparated by planes of non-magnetic Au atoms. Themanganese moments lie within the ferromagnetic sheets,A1Mn = 4.2jiB, whose orientation varies with compositionbut remains perpendicular to the shorter crystal axis.Increasing the gold content to MnAu2 (complexMoSi2, CIl5 (tI6) crystal structure) results in an atomicarrangement in which the sheets of Mn atoms are

separated by two sheets of Au atoms. The Mn momentslie within ferromagnetic sheets, and the momentdirection in adjacent sheets is rotated by an angle <£,which varies with temperature (</> = 51° at roomtemperature). The magnetic structure is a spiralor helical arrangement of spins with propagationvector parallel to the tetragonal axis (Herpin and Meriel,1960).

Mn3Si is a Heusler-type alloy with a body-centeredcubic Fe3Al (DO3) type (cF16) crystal structure atlow temperature. The Mn moments in Mn3Si aredelocalized, as evidenced by the magnetic structure.The spins on the two Mn atoms in the unit cell werefound by Tomiyoshi and Watanabe (1975) to bearranged in a sinusoidal or a screw structure withmoments /iMni = 1.72 /*B and /iMnn =0.19/iB (Figure 5).The temperature dependence of the propagation vectoris similar to that in antiferromagnetic chromium, andthe modulated structure may be due to spin-densitywaves.

Figure 5. Magnetic structure of Mn3Si along the < 111 > direction: (a) transverse sinusoidal structure; (b) proper screw structure(After Tomiyoshi and Watanabe, 1975)

3.3 Rare-Earth Metals and Alloys

Magnetic structure in the rare earths is a compromisebetween crystal-electric-field interactions and magneticexchange interactions of the RKKY type. With theexception of gadolinium, the atomic moments areordered in modulated arrangements with long repetitionperiods. The magnitude of the propagation vectors and

the spin orientations are sensitive to magnetostrictivestrain, and the structures transform into a multiplicityof spin arrangements at different temperatures.

Gadolinium (h.c.p. A3 structure type (hP2)) isferromagnetic (Tc =293.4 K), and the easy direction ofmagnetization is parallel to the crystallographic c-axisat room temperature. Gd exhibits a change inmagnetization at 140K, and neutron diffraction was

Gadolinium Terbium Dysprosium

Ferro Ferro Para Ferro Helix Para Ferro Helix Para

Holmium Erbium Thulium

Ferrocone

+ helix Helix Para

Ferrocone

+ helix

ADPcone

+ helix CAM Para Ferro CAM Para

Figure 6. Magnetic ordering in the heavy rare-earth metals (After Legvold, 1980)

used to determine if the magnetization anomaly was theresult of a change in the magnetic structure. Thestructure is ferromagnetic at all temperatures, and thechange in magnetization is the result of a change inorientation of the atomic spins (Cable and Wollan,1968). As the temperature is lowered, the spin directionrotates from the c-axis, reaches a maximum deviationof 75° at 195 K, and rotates back toward the c-axis untilthe deviation has fallen to 30° at 4.2 K.

Legvold (1980) has summarized the magnetic orderingin the heavy rare earths as found by neutron diffraction(Figure 6). The magnetic structures range from theferromagnetic arrangement found in gadolinium tocomplicated sinusoidal and helical structures in theelements beyond gadolinium. The magnetic periodicityis not necessarily commensurate with the crystalreciprocal lattice, and magnetic ordering may involveseveral sublattices.

The moments in terbium (h.c.p. A3 structure type(hP2)) form a helical arrangement in the temperatureinterval from 219 to 231.5 K, the Neel point, with aninterlayer turn angle ~ 18-20°, corresponding to aperiodicity of about 20 atomic layers. Terbium isferromagnetic below 219.5 K, and the moments lie inthe basal plane. Dysprosium has a helical spin structurefrom 179 to 89 K; it then becomes ferromagnetic withthe easy axis of magnetization parallel to thecrystallographic 0-axis. The magnetic moments inholmium are parallel to the basal plane belowTN = 132 K and ordered in a spiral arrangement. Atlower temperature the moments tilt out of the basalplane and form a conical structure. The magneticstructure of erbium is still more complicated. In additionto helical spin arrangements above 85 K, lowering thetemperature results in incommensurate c-axis-modulated(CAM) sinusoidal arrangements, which transform toconical ferromagnetism below 2OK. The magneticstructure of thulium below 32 K is built up of antiphasedomains in which four moments point up along thec-axis followed by three moments that point down.

The magnetic structure of neodymium is extremelycomplex, and, although investigated extensively, manydetails of the arrangement are not resolved (Moon andNicklow, 1991). The crystal structure of neodymium isa double-hexagonal close-packed atomic arrangement(hP4), built up of alternating cubic close-packed (C) andhexagonal close-packed (H) layers. On cooling belowTN = 19.1 K, the spins order in an antiferromagneticstructure, which on further cooling undergoes second-order transitions to different magnetic structures at 8.2,7.5, 6.2, and 5.7 K. The complex magnetic phasediagram is explained by competing antiferromagnetic

interactions in the H and C layers. The 19 K transitionis to an antiferromagnetic structure in which the spinsin the H sites are ordered in longitudinally polarizedincommensurate domains. The 8.2 K transition isexplained by ordering of the moments on the C sites.At still lower temperature, multiple propagation vectors,multiple moment components, and multiple domainsare required to represent the spin arrangements.

The magnetic structures of alloys of rare earths withother rare earths and with non-magnetic elements ingeneral are similar to the rare-earth metals. They areexplained by competition between RKKY exchange andcrystal-field interactions, and the spin arrangementsspan a broad range of helical, cone, in-plane, and out-of-plane ferromagnetic and c-axis-modulated structures.The stability ranges of the magnetic structures areconveniently displayed in magnetic phase diagrams(Moon and Nicklow, 1991).

3.4 Rare-Earth Intermetallic Compounds

The magnetic structures of intermetallic compounds ofrare earths (R) with transition elements (M) are the resultof three types of magnetic interaction: R-R, M-M, andR-M interactions. Of these, the M-M interaction is thestrongest, owing to the large radial distribution of the3d wavefunctions and the strong overlap of thewavefunctions between neighboring atoms. The rareearths interact by spin polarization of the 4s electrons(the RKKY interaction), which is long-range andoscillatory. There is little overlap of the rare-earth 4fwavefunctions, and the R-R interactions are theweakest.

Intermetallic compounds with the Th6Mn23 structure(cFl 16) have been studied extensively because the latticeis a good framework in which to substitute rare-earthand transition elements in order to determine the relativeimportance of direct exchange and RKKY exchange(Crowder and James, 1983). The crystal structure ofR6M23 (D8a (cF116) structure type) is built up of sixoctahedrally coordinated clusters of rare-earth atomssurrounded by 50 transition-metal atoms on the facesand corners of the octahedron. The transition elements,either Mn or Fe, occupy four crystallographicallyindependent sites: b, d, fl, and f2 (Wyckoff notation).Figure 7 displays the magnetic structure of Y6Mn23.Yttrium does not carry a moment, and in Y6Mn23 eachof the crystallographically independent manganeseatoms has a unique moment: b, 2.25 (2.81) fiB; d, 1.72(2.07) /iB; fl, -1.52 (-1.79)jiB; f2, -1.27(-1.77) fiB. The first moment shown is at roomtemperature and the second (in parentheses) at 4.2 K.

The moments on the b and d sites coupleantiferromagnetically with those on the f 1 and f2 sites.If R is a heavy rare-earth element and M = Fe, theR-M interactions are antiparallel. If M = Mn, themanganese lattice is ferrimagnetic and the rare-earthmoments couple parallel to the net manganese moment.

The magnetic properties of mixed compounds,Y6(Fe1^MnxJ23, are strongly dependent oncomposition. For example, there is a marked decreasein magnetization and Curie temperature as Mn issubstituted for Fe and Fe is substituted for Mn. Nomagnetic order is observed, even at 4 K, for x between0.48 and 0.69, and the structure of compounds in

the intermediate concentration range has not beenresolved.

Spectacular changes occur in the magnetic propertiesof Y6(Fe1^MoJ23 compounds that have absorbedhydrogen: Y6Mn23 is ferrimagnetic with r c=486K,but after hydrogen absorption shows no netmagnetization, even at 4.2 K. The changes have beenattributed to disorder and changes in the electronicstructure due to the introduction of hydrogen.

Magnetic ordering in RMn12 compounds (D2b

structure type (tI26)) is antiferromagnetic and occurswell below room temperature (magnetic properties ofthis family are discussed in Chapter 14 by Stadelmaier

Figure 7. The Th6Mn23 structure: large circle, Mn 4b site; diamond, Mn 24d site; triangle, Mn 36f, site; square, Mn 36f2 site;hexagon, Y 24e site (After Crowder and James, 1983)

Figure 8. Magnetic structure of YMn12. Y atoms on the sites2(a) and Mn atoms on the sites 8(i) and 8G) lie in the planesZ = O (full lines) or z = 1/2 (broken lines). Mn atoms on the sites8(f) lie in z = 1/4 and z = 3/4 (After Deportes et al, 1977)

and Reinsch in Volume 2). The manganese moments inYMn12 were found by Deportes et al. (1977) to orderin a complicated non-collinear antiferromagneticarrangement below TN = 120 K. A schematicrepresentation of the non-collinear spin arrangement is

shown in Figure 8. The mean magnetic moment ofmanganese is 0.4 fiB, and the magnetic interactionsbetween the Mn moments vary strongly with theinteratomic separation. The yttrium and manganesesublattices appear to order more or less independentlyat temperatures TR < TN .

3.5 Small-Scale Magnetic Order

Materials in which the atomic moments are not orderedon a long-range scale are of scientific and technologicalinterest for investigating magnetic materials withcompositions and structures that are not accessible intheir crystalline counterparts. Examples of thesematerials are amorphous ferromagnets, spin glasses, andmagnetic rare-earth 'superlattices.'

Amorphous ferromagnets are mainly alloys oftransition metals with metalloids and rare earths madeby very rapid quenching from the molten state(Luborsky, 1980). The atomic arrangements ofamorphous ferromagnets, inferred from radialdistribution functions computed from X-ray andneutron scattering intensity data, are approximated bymodels based on dense random packing of hard spheres.Magnetic interactions are short-range, and the localmagnetic structure in systems characterized by small-scale magnetic order is expected to be about the sameas in substances with long-range order. The intrinsicmagnetic properties of amorphous ferromagnets are

Figure 9. Magnetic moment density as a function of the distance from the solute site in dilute PdFe and PdCo alloys (AfterLow and Holden, 1966)

consistent with assuming that the local magneticstructure is not significantly affected by the absence ofa crystal lattice.

Spin-glass alloys are examples of magnetic materialsin which the small concentration of the magneticcomponent does not allow the moments to order on asignificant scale. Nuclear magnetic resonance andMossbauer studies of the effective magnetic field in thevicinity of the magnetic atoms show that the magneticclusters in spin-glass alloys are predominantlyferromagnetic and have very large effective moments,- 20 to 20 000 fiB. The giant moments are attributedto the bare moment of the magnetic atom anda surrounding cloud of partly polarized host atoms.The existence of magnetic clusters has been confirmedby Low and Holden (1966) using diffuse magneticneutron scattering. Figure 9 shows the magneticmoment density as a function of the distance from thesolute site in disordered PdFe and PdCo solid-solutionalloys: the magnetic moment density peaks at thesolute site, and the rest of the giant moment is spatially

distributed over the matrix for a distance greaterthan 6 A.

Rare-earth 'superlattices' are unusual magneticmaterials in which magnetic order is long-range in twodimensions and short-range in the third dimension.Advances in molecular-beam epitaxy techniqueshave made it possible to grow stratified magneticlayers consisting of a discrete number of atomicplanes deposited alternately with non-magnetic orother magnetic layers (Majkrzak et al.9 1988). Thesematerials are of theoretical interest for investigatingthe consequences of reduced dimensionality and oftechnological interest for fabricating devices withnovel magnetic properties. The magnetic structure inDy-Y multilayers is a helical spin arrangement withthe propagation vector along the c-axis, the sameas pure dysprosium (Figure 10). Different behavior isfound in Gd-Y multilayers. The Gd momentswithin a single layer are coupled ferromagnetically andoriented in the basal plane. In going from one Gdlayer to the next, there is a phase change of either 0 or7T, depending on the thickness of the interveningyttrium layer. The structure is attributed to coherentpropagation by the RKKY interaction of magneticcorrelations across the non-magnetic yttriumlayer.

4. Remarks

Although it is not possible to formulate a generaltheory for predicting magnetic structure fromcomposition and crystal structure, approximateexchange interaction models can be used to explain whythe magnetic moments on a particular group ofatoms interact in one way rather than another. One ofthe main issues that limits understanding of themagnetic structures and properties of transition-metalintermetallic compounds and alloys is determiningwhether in a particular collection of atoms themagnetic electrons are to be treated as localized ordelocalized. A possible appoach to better understandingof the behavior of magnetic electrons is to determinethe distribution of magnetic moments and spin densityin model systems with polarized neutron scatteringand resonance techniques with the expectation thatthe knowledge would provide a basis for theoreticalmodels relating electron delocalization to composition.The magnetic structures of materials in whichcompeting interactions lead to frustration oflong-range order could be investigated with similartechniques.

Figure 10. Schematic representation of the basal-planecomponent directions in three rare-earth 'superlattice' systems:(a) Gd-Y canted antiphase domain structure; (b) Dy-Ycoherent, incommensurate spiral; and (c) Gd-Dy 'asymmetric'state (After Majkrzak et al.y 1988)

5. References

Bacon, G. E. (1962a). Neutron Diffraction, 2nd Edn. OxfordUniversity Press, London.

Bacon, G. E. (1962b). Proc. Phys. Soc, 79, 938.Bertaut, E. F., Delapalme, A., Forrat, F., Roult, G., de

Bergevin, F., and Pauthenet, R. (1962). / . Appl. Phys.Suppl., 33, 1123.

Cable, J. W., and Wollan, E. O. (1968). Phys. Rev., 168, 733.Chien, C. L. (1981). In Nuclear and Electron Resonance

Spectroscopies Applied to Materials Science (eds E. N.Kaufmann and G. K. Shenoy). North-Holland, Amsterdam.

Cox, D. E. (1973). Magnetic Structures Data Sheets. Preparedunder the Auspices of the Neutron Diffraction Commissionof the International Union of Crystallography.

Crowder, C. E., and James, W. J. (1983). A Review of theMagnetic Structures and Properties of the R6M23

Compounds and their Hydrides. J. Less-Common Met.,95, 1.

Deportes, J., Givord, D., Lemaire, R., and Nagai, H. (1977).Physica, 86-88B, 69.

Eddington, A. S. (1948). The Nature of the Physical World.Cambridge University Press, London.

Gignoux, D., and Schmitt, D. (1991). J. Mag. Mag. Mat., 100,139.

Goldman, J. E. (1953). Rev. Mod. Phys., 25, 108.Goodenough, J. B. (1963). Magnetism and the Chemical Bond.

Wiley-Interscience, New York.Herpin, A., and Meriel, P. (1960). C. R. Acad. Sci. Paris, 250,

1450.James, W. J. (1989). Neutron Scattering Characterization of

Magnetic Materials. Mater. Sci. Eng., B3, 387.Jensen, J., and Mackintosh, A. R. (1991). Rare Earth

Magnetism: Structures and Excitations, InternationalMonographs on Physics, No. 81. Oxford University Press,London.

Kasper, J. S., and Kouvel, J. S. (1959). / . Phys. Chem. Solids,11, 231.

Kouvel, J. S. (1967). In Intermetallic Compounds (ed. J. H.Westbrook). Wiley, New York.

Legvold, S. (1980). In Ferromagnetic Materials, Vol. 1 (ed.E. P. Wohlfarth). North-Holland, Amsterdam, p. 183.

Low, G. G., and Holden, T. M. (1966). Proc. Phys. Soc., 89,119.Luborsky, F. E. (1980). In Ferromagnetic Materials, Vol. 1

(ed. E. P. Wohlfarth). North-Holland, Amsterdam, p. 451.

Majkrzak, C. F., Gibbs, D., Boni, B., Goldman, A. L,Kwo, J., Hong, M., Hsieh, T. C , Fleming, R. M.,McWhan, D. B., Yafet, Y., Cable, J. W., Bohr, J.,Grimm, H., and Chien, C. L. (1988). J. Appl. Phys., 63,3447.

Moon, R. M., and Nicklow, R. M. (1991). J. Magn. Magn.Mater., 100, 139.

Murani, A. P., Roth, S., Radhakrishna, P.,Rainford, B. D., Coles, B. R., Ibel, K., Goeltz, G., andMezei, F. (1976). J. Phys., F6, 425.

Mydosh, J. A., and Nieuwenhuys, G. J. (1980). InFerromagnetic Materials, Vol. 1 (ed. E. P. Wohlfarth).North-Holland, Amsterdam, p. 71.

Nakamura, Y. and Franse, J. J. M. (eds) (1987). Magnetismof Intermetallic Compounds. J. Magn. Magn. Mater., 70,462.

Nathans, R., and Pickart, S. J. (1963). Spin Arrangements inMetals. In Magnetism, Vol. Ill (eds G. T. Rado and H.Suhl). Academic Press, New York.

Neel, L. (1948). Ann. Phys., 3, 137.Overhauser, A. W., and Arrott, A. (1960). Phys. Rev. Lett.,

4, 226.Pickart, S. J., and Nathans, R. (1961). Phys. Rev., 123,

1163.Rietfeld, H. M. (1969). / . Appl. Crystallogr., 2, 65.Ruderman, M. A., and Kittel, C. (1954). Phys. Rev., 96, 99.Shirane, G., and Takei, W. J. (1962). / . Phys. Soc. Japan,

17, 35.Shirane, G., Chen, C. W., Flinn, P. A., and Nathans, R.

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and Shirane, G. (1970). J. Phys. Soc. Japan,28, 615.


1. Introduction

The icosahedron, a Platonic solid consisting of 20slightly distorted tetrahedra packed around a commonvertex, is a compact, stable, configuration (Frank,1952), possibly explaining why this structure dominatesthe local symmetry of many complex crystallineintermetallic phases, and is presumed to occurfrequently in undercooled liquids and glasses. It is wellknown, however, that the rotational symmetries of theicosahedron are incompatible with translationalperiodicity. It came as a tremendous surprise, then,when Shechtman et al. (1984) announced theunprecedented discovery of a new non-crystallographicphase in a rapidly quenched alloy of aluminum andmanganese that produced diffraction patternscontaining a distribution of sharp peaks showing anicosahedral rotational symmetry. These new phases,now called quasicrystals, are the subject of this chapter.

Sharp criticism of this radical interpretation of theobserved diffraction patterns was quickly raised byPauling (1985,1987, 1988) and others (Field and Fraser,1985; Carr, 1986; Anantharaman, 1988). Various, moreconventional, explanations, ranging from twinning ofordinary crystals, to previously unknown, large-unit-cell crystals containing a significant amount of localicosahedral order, were offered. Dark-field images(Shechtman and Blech, 1985), high-resolution electronmicroscopy (Shechtman et al, 1985; Hiraga et al, 1985;Bursill and Lin, 1985; Knowles et al, 1985), convergent-beam studies (Kelton and Wu, 1985; Bendersky, 1985),X-ray diffraction (Bancel et al, 1985), and field ionmicroscopy (Melmed and Klein, 1986; Elswij et al,1988) have ruled out the possibilities of twins. Large-unit-

cell crystals are more of a problem, though diffraction linewidths of largely defect-free Al-Cu-Fe, an equilibriumquasicrystal, are resolution-limited; the best data wouldimply a crystal unit-cell size greater than 185 A for theirexplanation (Bancel et al, 1989).

The evidence appears to be overwhelming:quasicrystals constitute a new phase of condensed matterthat is characterized by a non-crystallographicorientational symmetry and quasiperiodicity (cf. Section2). Their non-crystallographic orientational symmetryis the key, setting quasicrystals apart from other physicalsystems that also exhibit quasiperiodicity, such asincommensurate crystals, which also give sharpdiffraction patterns that are different from normalcrystalline patterns, requiring more than three indicesto index them (de Wolff, 1974; Janner and Janssen,1977; Janssen, 1992).

Quasicrystals are now common. Besides theicosahedral quasicrystal, phases displaying otherforbidden rotational symmetries have also beendiscovered. These include the octagonal phase (eightfoldsymmetry), the decagonal phase (a two-dimensionalquasicrystal showing 10-fold symmetry in twodimensions but is periodic in the third), and several one-dimensional quasicrystals. Although the earliestquasicrystals were metastable, transforming uponheating to more stable crystalline phases, equilibriumquasicrystal phases have now been discovered. Some ofthese are largely defect-free, demonstrating the existenceof long-range quasiperiodic translational order.

Most attention has centered on determining thequasilattice (the equivalent of the Bravais lattice forcrystalline systems) and the atomic decorations of thatlattice. We now know that quasicrystals can be related

Chapter 10

Quasicrystals and Related Structures

Kenneth F. KeltonDepartment of Physics, Washington University, St. Louis, MO 63130, USA


to complex crystalline phases with large cell sizes thatwere previously undreamed-of for metallic alloys. Whysome alloys have a strong tendency to form quasicrystalsand complex crystals is unknown, although somephenomenological approaches have led to the discoveryof new quasicrystals. Little is known about the materialsproperties of these new phases, but the recent availabilityof better samples is leading quickly to an improvedunderstanding; some applications are even beingdiscussed.

In this chapter the prominent structural features ofquasicrystals are discussed, placing this new class ofintermetallic solids firmly within the framework of moretraditional, crystalline intermetallic compounds. Generalcharacteristics, emphasizing diffraction features, of theicosahedral and decagonal quasicrystals are discussedin Section 2 and 3, related crystalline structures arediscussed in Section 4, while the similarities betweenquasicrystals and liquids and glasses are briefly reviewedin Section 5. Real-space models are discussed in Section6, listing the competing models for the quasilattice andthe methods for decorating that framework with atoms.A very brief description of their properties appears nearthe end of this chapter (Section 7).

For further information, the reader is referred to thereview articles and books enumerated under the heading'Further Reading', following the list of references. Aseries of books entitled Aperiodicity and Order containsseveral articles on the structures of quasicrystals, theirrelation to other incommensurate phases, their stabilityand metallurgical properties, and related topics.Quasicrystals is a compilation of many of the earlypapers in the field and provides a source for many ofthe ideas that now guide the field. Quasicrystals: TheState of the Art is a recent collection of review articles.

2. The Icosahedral Phase

2.1 General Metallurgy

Quasicrystals are now common; a recent compilationlists over one hundred alloys in which they have beenreported (Kelton, 1993). Most familiar are thequasicrystals with icosahedral symmetry (icosahedral ori-phases), forming readily in Al-3d transition-metal(TM), i(Al-TM), and Ti-3d transition-metal, i(Ti-TM),alloys, though there have been a few reports in othersystems such as i(PdUSi) (Poon et al., 1985), i(CdCu)(Bendersky and Biancaniello, 1987), i(PdBi) (Lilienfeld,1988), and i(GaMgZn) (Ohashi and Spaepen, 1987).Icosahedral phase formation has even been reported instainless steel (Hu et al., 1990).

Although first discovered by rapid quenching,quasicrystals have now been obtained by devitrificationof glasses (Lilienfeld et al., 1985; Holzer and Kelton,1991), vapor condensation (Csanady et al., 1987; Saitoet al., 1987), solidification under high pressure (Sekharand Rajasekharan, 1986), electrodeposition (Grushkoand Stafford, 1989a,b), solid-state precipitation(Nishitani et al., 1986; Cassada et al., 1986), inter-diffusion of multilayers (Follstaedt and Knapp, 1986),mechanical alloying (Eckert et al., 1989), and ionimplantation (Budai and Aziz, 1986). In a few cases,the quasicrystal appears to be the equilibrium phase ina limited temperature range; these include 1(AlLiCu),i(GaMgZn), i(AlCuFe), and the decagonal phases (cf.Section 3) in (Al1Si)-Co-Cu, Al-Pd-Mn, and Al-Pd-Fealloys. There, quasicrystals can form even in slowlycooled ingots.

Most structural, metallurgical, and phase-stabilitystudies have concentrated on the Al-TM alloys (forreviews see Schaefer and Bendersky, 1988; Kelton,1989); the properties of the Ti-TM quasicrystals areless well known. There are many similarities between theAl-TM and the Ti-TM icosahedral phases: (a) in mostcases, the i-phase occurs as a metastable phase in amultiphase mixture; (b) the grains are typically mottledwith a coral-like, weakly dendritic, shape, and areusually 1 to 2/xm in diameter (Shechtman et al., 1984,1985; Kelton and Wu, 1985; Kelton et al., 1988); (c) thei-phase often grows with a strong orientational relationto complex crystalline phases, suggesting a similarity intheir short-range atomic order (Guyot and Audier, 1985;Koskenmaki et al., 1986; Zhang and Kelton, 1991a); and(d) Si often plays an important, but little understood,role in i-phase formation (Schaefer et al., 1986; Chenand Chen, 1986; Bendersky and Kaufman, 1986;Dunlap and Dini, 1986a; Sabes et al., 1989; Zhang andKelton, 1991b). There are also important differences:(a) the binary phase diagrams of the Al-TM alloysfrequently contain a series of intermetallic compoundsformed by peritectic reactions on the Al-rich side(Schaefer et al., 1986), while Ti-TM phase diagramsare generally much simpler, with metastablequasicrystals often forming with a solid solution andsimple crystalline phase; (b) the concentration of the 3dtransition metal in Al-TM alloys is generally near20at.% (Inoue et al., 1986; Schaefer et al., 1986;Kimura et al., 1985; Krishnan et al., 1986), while it iscloser to 37at.% in the Ti-TM alloys (Sabes et al.,1989); and (c) although Al-TM i-phases frequently showlocalized diffuse scattering in the diffraction patterns,this diffuse scattering is much stronger and occurs inall of the Ti-TM i-phases (cf. Section 2.5).

Figure 1. (a) Single grains of i(AlCuFe) with a pentagonal dodecahedralshape, (b) Grains of i(AlLiCu) with the shape of a rhombic triacontahedron(courtesy of F. W. Gayle). (c) Decaprismatic grains of decagonal(Al,Si)-Co-Cu (by permission of L. X. He)

Figure 2. TEM selected-area diffraction (SAD) patterns of i(AlMn): (a) threefold; (b) fivefold; (c) [ T I O ] , in the notation of Chattopadhyay et al. (1985b);(d) twofold. Arrows are drawn from the patterns to the zone axes on the stereographic projection for which they are approximately aligned

In spite of their lack of translational periodicity,quasicrystals have a tendency to facet; several theoreticalexplanations for this have been offered (Jaszczak et al.,1989; Ingersent and Steinhardt, 1989; Ho et al., 1989;Stephens, 1989). While metastable Al-Mn i-phasesdevelop pentagonal dodecahedral growth morphologies(Robertson et al., 1986; Nishitani et al., 1986; Nissenet al., 1988; Yu-Zhang et al., 1988a), the most strikingexamples of faceting are found in the stable i-phases,forming pentagonal dodecahedral grains in i(GaMgZn)(Ohashi and Spaepen, 1987) and i(AlCuFe) (Tsai et al.,1987) (Figure l(a)) and rhombic triacontahedral grainsin i(AlLiCu) (Gayle, 1987) (Figure l(b)). Stabledecagonal phases frequently contain decagonalprismatic grains (Kortan et al., 1989; He et al., 1990)(Figure l(c)). For most i-phases, growth from the liquidis most rapid along the threefold direction (Fan et al.,1987; Nissen et al., 1988; Macko et al., 1988). Ini(AlLiCu) (Gayle, 1987) and KTiFeMnSi) (Zhang andKelton, 1992), however, the fivefold direction is thedirection of most rapid growth.

2.2 Diffraction

Figure 2 shows the locations and symmetries of thediffraction patterns obtained by transmission electronmicroscopy (TEM) rotation studies on the i-phase. Thezone axes are indexed following Chattopadhyay et al.(1985b). This phase has the rotational symmetry of theicosahedral point group (m35), having zones thatdisplay the expected two-, three-, and fivefold symmetricdiffraction patterns, which are separated by the correctangular distances. While the fivefold diffraction patternin Figure 2 appears to have a 10-fold symmetry, rotationstudies about axes that are 36° apart produce a differentsequence of diffraction patterns, demonstrating that theapparent 10-fold symmetry is actually the fivefoldinversion symmetry of the icosahedron. Several high-symmetry mirror patterns are also observed; one ofthese, the (rlO), is included in Figure 2.

In addition to their non-crystallographic rotationalsymmetry, quasicrystals are quasiperiodic. Anexamination along any of the axes of the fivefold patternshown in Figure 2 shows that the ratio of distances toconsecutive bright diffraction spots is an irrationalnumber, not an integer as expected for crystals. Thespots cannot, therefore, be indexed using a single lengthscale; instead, two incommensurate length scales, /and T/, are required, where r is the golden mean(r = (1 + V5)/2 = 2 cos(x/5)), an irrational numberassociated with the geometries of the pentagon andicosahedron. The distances between prominent

diffraction spots in the fivefold directions of the twofoldpatterns from most quasicrystals scale as r3 (Figure3(a)). The spots at r and r2 are forbidden by parityconstraints (Elser, 1985a,b). Those spots are present inthe diffraction patterns of a few icosahedral phases,such as i(AlCuFe), i(AlCuRu), and i(AlPdMn) (Figure3(b)), and are explained by chemical ordering leadingto centering (Ebalard and Spaepen, 1989).

The most common example of quasiperiodicity is theFibonacci series (Struik, 1967), a one-dimensionalquasiperiodic series, such as 0 ,1 ,1 ,2 ,3 ,5 ,8 ,13,21, . . . ,where each successive element of the series is thesum of the previous two elements. As will be discussedin Section 2.3, the relation between this series andquasicrystals is best understood by noting thatthe Fibonacci series can also be represented by aparticular sequence of short (S) and long (L) segments:. . .SLSLLSLSL..., where the ratio of the number oflong to short segments equals r. Other phases such asincommensurate crystals have the property of quasi-periodicity. Quasicrystals are set apart by their additionalnon-crystallographic rotational symmetry.

2.3 Projection From a Higher-DimensionalSpace

The icosahedral quasiperiodic lattice is convenientlyviewed as a projection from a six-dimensional (6D)hypercubic lattice onto a space that has an irrationalorientation with respect to the original lattice (de Bruijn,1981; Duneau and Katz, 1985; Elser, 1986; Kalugin etal., 1985; Kramer and Neri, 1984). This technique, calledthe strip-projection method, allows quasicrystals andrelated crystalline structures to be studied, enables thesystematic introduction of disorder into the quasilattice,and provides the most direct method for determiningthe atomic decoration. It is useful for calculating theFourier transform of the quasiperiodic lattice, relatedto the diffraction peaks, thus providing a method forindexing the quasicrystal diffraction patterns.

The strip-projection method is best illustrated byconsidering a projection from a two-dimensional (2D)square lattice, E2, to obtain a one-dimensional (ID)incommensurate crystal (Figure 4(a)). This also providesa useful method for describing the experimentallyobserved one-dimensional quasicrystals reported in(Al,Si)-Cu-Co, Al-Ni-Si, and Al-Cu-Mn (He et al.,1988a; Daulton and Kelton, 1991). Here, the projectionspace, JC||, is an axis that is inclined with an irrationalslope to the periodic square lattice; x± is perpendicularto JC,| . Two additional lines are constructed parallel tojc,|, forming a window or strip denoted by W(xx ). All

Figure 3. SAD patterns taken along the twofold direction of (a) i(AlMn), a simple i-phase, and (b) i(AlCuFe), a body-centered i-phase

Repeating sequence

Figure 4. Projection from two dimensions to one dimension,(a) Projection in real space from 2D periodic latticefrom a window onto a line, both inclined with slope T"1. (b)Projection in reciprocal space showing the intensity abouteach reciprocal lattice point from the Fourier transform of thewindow function, (c) Projection in real space with thewindow inclined with a slope that is a rational approximantto T"1, here 1/2; the resulting ID structure approximatesa segment of a Fibonacci sequence in each repeat distance

points of E2 that lie within this window are projectedonto Xn as illustrated. The width of the strip is chosento obtain structures with physically reasonable densities,generally containing exactly one unit cell of the originallattice. The resulting structure depends critically on theangle of inclination of both the window and x^. (Notethat the window and JC,S are not constrained to have thesame slope.) To preserve the fundamental length betweenrelative projections, only the slope of the window willbe changed in this chapter. The slope of *„ will be fixedat r"1.) If the slope of the window is irrational withrespect to the original lattice, the projected sequencenever repeats; if the slope is chosen equal to r"1,the Fibonacci sequence is obtained. If the slope of thewindow is taken as an integer approximant to T~\the projected sequence of points repeats with a longperiod; locally the spots will follow the Fibonaccisequence (Figure 4(c)). This is called a rationalapproximant (cf. Section 4).

Since the subspace chosen, Z2, is the region of E1

truncated by W(x±),

Z2 = E2(x,y)xW(x±) (1)

the reciprocal lattice is given by

Q2^F[Z2] =F[E2(Xiy)]®F[W(x±)] (2)

As illustrated in Figure 4(b), along q^, true Braggdiffraction peaks are predicted since the Fouriertransform of each lattice point is a delta function. Theintensity of the diffraction spot is determined by theamplitude of the Fourier transform along q L , whichis f(g±) = sin(g ̂ L)Zgx L, within normalizations andphase factors for this simple case. Spots near theprojection window, therefore (with low values of g±) ,are more intense than those farther away (high valuesof g±). Even though the projection space is denselyfilled, those spots with g± greater than some value willnot be observed.

The strip-projection method has limitations. It canonly produce a limited subset of the possible quasilatticesand, except for special cases, it cannot be used tocalculate the diffraction pattern from a decoratedquasilattice. A more general approach, the cut-projection method proposed by Bak (1986), obtains theprojection by the intersection of the lower-dimensionalspace with a basis embedded in the higher-dimensionalspace. More complicated structures are then possibleand chemical decorations are readily obtained bychanging the basis in the hyper cubic lattice. The stripprojection is a special case of this method.

2/1 approximant

Fibonacci sequence

2.4 Projections and Diffraction-PatternIndexing

For a crystalline diffraction pattern, the position of anydiffraction peak can be indexed by the integer multiplesh, kt / (the Miller indices) of three basis vectors (a, b, c),which are inversely related to the physical vectorsdefining the unit cell. Owing to the incommensurabilityof the length scales, six vectors are required to indexthe diffraction patterns from the icosahedral phase. Twoindexing schemes for the icosahedral phase are usedmost frequently. One, due to Elser (1985a,b), emphasizesthe structural relations to the icosahedron; the other,due to Cahn et al. (1986), emphasizes the relations tocubic symmetry.

Let a1 to a6 be the unit basis vectors of a six-dimensional cubic lattice (E6). The construction of aquasicrystal, with a fivefold rotational axis, requires thatE6 be projected onto the three-dimensional parallelspace, E\, SO that the projected basis vectors, ej to t\,point to six vertices of an icosahedron as definedin Figure 5. This is accomplished using a projectionmatrix P

giving

( e ; , . . . , e t ) = V2(a ' , . . . , a 6 )P (4)

The basis vectors in the perpendicular space, E*±, arefound using the projector P' = I - P :

(e 1 ^. . . , C6J = V ^ a 1 , . . . ,a6)P' (5)

Since a vector in the 6D space has components in theparallel and perpendicular spaces, the factor of V2 inboth equations is required if the basis vectors in the 3Dand 6D spaces have unit magnitude.

The Bragg vector for each diffraction peak can beexpressed as a linear combination of the basis vectors,multiplied by integer indices and scaled by someprefactor,

6

g i i = £ o 2 > / e i i (6)

Figure 5. A stereopair showing the icosahedral basis vectorsused in the Elser indexing scheme: (a) for the parallel spaceand (b) for the perpendicular space. To see the 3D image, stareat a point between the left and right figures and cross youreyes. This stereopair may also be viewed with stereoglasses ifthe figure is photocopied and the members of the pair areexchanged (by permission of T. L. Daulton)

Owing to their incommensurability, the choice of theprefactor g0, is not unique in quasicrystals; it can onlybe determined to within a factor of T3 (Elser, 1986).One popular choice for /(Al-Mn) (Elser, 1986) takesthe weak peak at # = 0.68 A along the fivefold axis asthe fundamental (100000), giving g0 = ic/aR, where aR

is the edge length of the rhombohedral bricks in thePenrose tiling model of a quasicrystal (cf. Section 6).Bancel et al. (1985) assigned the fundamental to thebright spot at q = 2.894 A, producing an indexingscheme that differs from Elser's by a scale factor of T3.

The basis vectors may be expressed using differentcoordinate systems. Elser arbitrarily chose the z-axis tolie along ej, giving

ejj =(0,0,1) (7)

ej = (sin 0 cos -ir(n - 2) L sin /5 sin\-Tc(n - 2) L cos 0)

where cos /3=1/5 and /2 = 2 , . . . ,6 . These vectors pointto the upper vertices of an icosahedron (Figure 5(a));

the vectors in the perpendicular space, e± , are orientedas shown in Figure 5(b).

It is often convenient to express the basis vectors inthe parallel and perpendicular spaces using a coordinatesystem aligned with three perpendicular twofold axesof the icosahedron (Cahn et al, 1986). This isparticularly convenient for indexing cubic crystallineapproximants (cf. Section 4). The unit vectors in is,,and E3

Lare related to the basis vectors in the 6D latticeby a rotation matrix, R,

( « i , a j , a t « » Z J = (a1, • •., a6)R (8)

where

1 T 0 - 1 T 0'T 0 1 T 0 - 1

ft * 0 1 T 0 - 1 T

1 0 -T 1 0 r. 0 - T 1 0 T 1.

The six columns of the top three rows of R give the 3Dcoordinates of the 6D basis vectors, projected into thephysical space, e,,. The columns of the bottom threerows give the 3D coordinates of e ± . Noticing the formof the vertex vectors expressed within this coordinatesystem, Cahn et al. (1986) introduced an indexingsystem that employs three pairs of integer indices(h/h\k/k\l/l'\whecth/h'=h + h'T,k/k'=k+k'T,and///'=/+/'T.

2.5 Structural Defects

The X-ray diffraction peaks from the i-phase aregenerally broad, indicating a considerable amount ofstructural disorder. Frequently they are shifted fromtheir expected positions, with the least intense spots(largest value of g x ) deviating most. This is due to anovel type of disorder, not found in crystalline phases,called a phason strain (Levine et al, 1985; Lubenskyet al, 1985, 1986; Socolar et al, 1986), originating fromthe three additional degrees of freedom in thequasicrystal arising from its incommensurability.Phason strain arises from non-uniform displacementsin the perpendicular space; within the quasilattice model(cf. Section 6.1), this is equivalent to modifications inthe local tiling arrangements (Socolar et al, 1986).

Phason strain is also evident in the real-spacestructures. For example, viewing the high-resolutionelectron microscopy (HREM) image shown in Figure6(a) at a grazing incidence shows rows of contrast spaced

in a Fibonacci sequence and aligned along the fivetwofold directions. Locations where the rows of contrastshift discontinuously (jog) arise from phason strain. Thehigh density of jogs in this rapidly quenched quasicrystalindicate a high density of phason strain, a conclusionsupported by the broad peaks and peak shifts observedin its diffraction patterns.

The essential features of phasons can be obtainedusing a density-functional approach (Levine et al, 1985;Lubensky et al, 1985). The mass density of atoms isgiven by

P(r) = J]pq exp(iq • r) = 2 W exp(i<^) exp(iq • r)Q Q

(10)

Since p(r) is real, pj=p_9, and </>q= -<$>-q- Minimizingthe free energy with respect to the phases of the densitywaves fixes all but six of the phases, generally expressedin terms of the six possible degrees of freedom:

^ = S?;, ",+!>>/ (IDi = l I = I

where q1^ and ^ x are the components of the real-spaceand perpendicular-space reciprocal lattice vectors,respectively. The U1 are displacements in real space,phonons in normal crystalline materials, and the w, arethe displacements in the perpendicular space, thephasons. The peak displacements and the large peakwidths are explained by linear variations in the phasonfield. Assuming that w = M -r, where M is a second-ranktensor (the phason strain tensor), peak positions at gare shifted by Ag = g±-M, proportional to g± ratherthan g,|, as would be the case for an elastic strain in thephonon field (Warren, 1990). Diffraction peaks alsofrequently show a strong shape anisotropy (Keltonet al, 1988; Zhang and Kelton, 1990, 1991a; Heineyet al., 1987; Bancel and Heiney, 1986; Urban et al1986). These can arise from a superposition of shiftedpeaks from different domains of the sample withdifferent phason strain fields (Socolar and Wright,1987), or from an extremely fine-grained, multiplytwinned rhombohedral structure (Audier and Guyot,1988). Supporting the former view, diffraction patternstaken from small domains show no spot anisotropy andhave peak shifts consistent with a uniform phasonstrain. High-g± diffraction peaks in patterns takenfrom large regions, presumably containing severaldomains, are not shifted, but have a triangular shape,owing to the superposition of diffraction peaks fromthe regions of different phason strain.

Figure 6. High-resolution electron microscopy images of (a) i(AlMn), showing many jogs and defects, and (b) i(AlCuFe), showing few defects (Courtesy of K. Hiraga)

Non-uniform spatial displacements set up elasticdeformations that carry an energy penalty. Similarly,phasons change the local atomic configuration (leadingto tiling flips within a tiling model; cf. Section 6) andare also energetically costly. While the elastic strain canrelax quickly by phonon modes, phason strain relaxationrequires atomic diffusion, giving a very slow relaxationrate. Phason strain fields resulting from samplepreparation, therefore, should remain relativelyunchanged, leading some to suggest in the early daysof the field that defect-free quasicrystals could not beobtained experimentally. With the discovery of thestable i-phases, however, this view changed. It is nowpossible to obtain essentially phason-strain-free samplesby annealing slowly cooled i(AlCuFe) and i(AlCuRu).Their narrow resolution-limited diffraction peaks,implying a coherence length for quasiperiodic order thatexceeds 8000 A (Bancel et al., 1989), demonstrate thatdisorder is not an inherent property of quasicrystals.As shown in Figure 6(b), this order is also evident inHREM images; this image is virtually free from the jogsshown in Figure 6(a).

Structural defects, including slightly misorienteddomains, twins, stacking faults, and dislocations, arealso found in quasicrystals. Like crystal dislocations,dislocations in quasicrystals (Devaud-Rzepski et al.,1989; Ebalard and Spaepen, 1990; Zhang et al., 1990;Wollgarten et al., 1992; Zhang and Zhaung, 1992; Yuet al., 1992) exhibit diffraction contrast that vanishesfor particular diffraction vectors, enabling adetermination of the direction of their Burgers' vector,b, representing the displacement field in the quasilattice.The column approximation is used to determine theconditions for extinction in crystals (Hirsch et al., 1977),by determining the scattering amplitude at some pointP on the bottom of the sample by the scattering withina column that is 10 to 20 A in diameter. Extinction isdetermined by the phase factor, T9

r = exp[2iri(g-u + ̂ ) ] (12)

where s is a parameter characterizing the deviation ofthe defect-free lattice from the Bragg orientation, z isa coordinate along the column, g is the diffractionvector, and u is the displacement-field vector in thecolumn. In crystals, the condition for extinction is thatthis phase factor be minimized, given by g-b = 0, whereb is the Burgers' vector of the dislocation. Urban et al.(1993) extended this analysis to quasicrystals, properlyassigning perpendicular and parallel components forboth g and b, and deriving the analogous condition forextinction:

g i | -b , ,+g ± -b ± =0 (13)

Two cases must be distinguished: (a) the weakextinction in which the phonon and phason phaseshiftsinside the column are nonzero, but mutually cancel, i.e.gn'bii = - g j . ' b j . ; anc* 00 t n e strong extinction inwhich the phase terms are individually zero, i.e.gn*bii = g ± b ± =°- B o t n extinction conditions havebeen observed (Urban et al., 1993; Wollgarten et al.,1992; Yu et al., 1992). Interestingly, while the strain fieldof dislocations in i-phases is preferentially localized inplanes in both parallel and reciprocal space, theamplitude in the perpendicular space is almost an orderof magnitude larger.

Strong localized diffuse scattering has been observedin diffraction patterns taken along twofold zones(Figure 7(a)); it is absent in the three- and fivefoldpatterns. Weak diffuse scattering was first observedfollowing low-temperature annealing in Al-Mn(Mukhopadhyay et al., 1987); it is particularly strongand present in all titanium-based quasicrystals (Keltonet al., 1988; Zhang and Kelton, 1990,1991a; Kelton andGibbons, 1993). In all cases, the arcs of diffusescattering are centered on odd-parity spots normallymissing in the simple icosahedral reciprocal lattice.Detailed studies of the arcs in Ti61Mn37Si2 (Gibbonset al., 1989) have shown that the arcs are formedby intersections of the Ewald sphere with approxi-mately spherical surfaces of diffuse scattering inreciprocal space that are invariant under the rotationalsymmetry of the icosahedral point group. The originof the scattering is unknown. Chemical ordering(Chattopadhyay and Mukhopadhyay, 1987; Henley,1988; Mukhopadhyay et al. 1989a), topological disorder(Goldman et al., 1988; Gibbons and Kelton, 1992;Robertson and Moss, 1993), phason strain (Levitov,1988), and the presence of extremely small grains ofanother phase (Kelton et al., 1989) have been suggested.If the diffuse scattering arises from chemical ordering,the intensity should be less in the Ti-based quasicrystalsowing to the similarities of the atomic structure factors.That the opposite is found indicates that chemicalordering models are suspect. This conclusion issupported by recent studies showing no effect of alloyingon the arc intensity or location in titanium quasicrystals(Gibbons and Kelton, 1992). The icosahedral glassmodel (cf. Section 6) provides the best agreement withthe observed diffuse scattering (Figure 7(b)), suggestingthat topological disorder is the cause (Gibbons et al.,1989). Annealing of the Al-Mn i-phase, however, leadsto the appearance of the odd-parity spots inside the ringsof diffuse scattering, supporting a chemical ordering

Figure 7. (a) SAD pattern taken along the twofold direction of i(TiMnSi) showing strong localized diffuse scattering, (b) Calculation of twofold pattern, from icosahedralglass model, demonstrating agreement with (a)

argument (Mukhopadhyay et al, 1989a), althoughannealing might also decrease the topological disorder.

2.6 Phase Transformations

Most quasicrystals are metastable; upon thermalannealing at sufficiently high temperatures, theytransform to other phases with lower free energies. Thisphase transition is typically first-order, proceeding bynucleation and growth, though apparently continuoustransformations have been reported in some cases(Ghattopadhyay and Mukhopadhyay, 1987). It is alsogenerally irreversible. Detailed crystallization studieshave been made on only a few systems, primarilyAl-Mn and Ti-Mn alloys. The transformation mode,the kinetics, and the products depend strongly on thealloy composition (Bagley and Chen, 1985; Kimura etal, 1986; McAlister et al., 1987; Yu-Zhang et a!.,1988b), as does the stability of the i-phase (Chen andChen, 1986; Schaefer et al, 1986; Dunlap and Dini,1985, 1986b; Kelton and Holzer, 1988a). Thetransformation products for Al-Mn alloys are typicallyAl6Mn, Al4Mn, and a-Al, an f.c.c. solid solution.Based on a detailed study of the transformation kinetics,Kelton and Holzer (1988a,b) placed a lower bound forthe interfacial energy between the i-phase and Al6Mnof 0.03 J m~2, much smaller than a typical orien-tationally averaged interfacial energy between unlikecrystalline phases, but similar to the interfacial energiesbetween a crystal and its melt (Turnbull, 1950; Kelton,1991). The icosahedral phase in Ti61Mn37Si2 crystallizesto a series of previously unknown, complicatedcrystalline phases, that appear to be constructed fromdifferent packings of the fundamental icosahedralclusters found in the i-phase (Holzer et al, 1989; Levineet al, 1992).

Chattopadhyay and Mukhopadhyay (1987) firstsuggested that quasicrystals might transform by passingthrough a series of rational approximants, complexcrystalline phases that approximate the local structureof the quasicrystal (cf. Section 4). Since long-wavelengthphasons locally modify the atomic structure, they might'lock' into special values to produce these approximants(Ishii, 1989). There is some evidence for such phason-based transformations. Annealing studies in rapidlyquenched (Al,Si)-Cu-Co suggest a continuousevolution from the decagonal phase (cf. Section 3) todifferent crystal approximant phases (Daulton andKelton, 1991). Further, reversible transformations havebeen observed in quasicrystals that are stable at hightemperature, to crystalline phases that are stable at

lower temperatures. X-ray and TEM investigations ofi(AlCuFe) suggest the onset of a structural transform-ation at «670°C to a lower-symmetry phase, possiblyan extremely fine-grained («100 A) rhombohedralphase (Audier and Guyot, 1988, 1989a; Zhang et al,1991a). The intensities of the high-g± peaks increasedramatically during the transformation, possiblyindicating a soft phason mechanism analogous tothe more familiar phonon softening mode for somephase transformations in crystals (Bancel, 1989).The order of this transformation is not known, butthere are theoretical reasons to suspect that it isfirst-order (Ishii, 1990). There is no evidence fornucleation, though several reports of the develop-ment of a modulated quasicrystal during the trans-formation (Audier et al, 1991) suggest that thephase transition may be spinodal (Liu and Koster,1993).

3. Decagonal Phase

3.1 General Metallurgy

Other than the i-phase, the best-studied quasicrystal isthe decagonal phase (also often called the t-phase),which has a 10-fold rotational symmetry in thequasiperiodic plane, while being periodic in the thirddimension. The point-group symmetry for mostdecagonal phases is 10/m or 10/mmm (Bendersky,1985), belonging to the cylindrical family of groupsG1. The decagonal phase is closely related to theicosahedral phase, often growing in competition with,and sometimes epitaxially on, that phase. In many waysit can be considered as an approximant structure to thei-phase. Intense activity has focused recently on thestudy of the decagonal phase, partially because it is feltthat the reduction in the dimensionality of thequasiperiodicity might make structural determinationspossible.

Chattopadhyay et al (1985a) and Bendersky (1985)first identified the decagonal phase in rapidly quenchedAl-Mn alloys. Subsequently, decagonal phase formationhas been reported in many alloys, mostly aluminum-based (see Kelton, 1993). Though icosahedral phaseformation has been reported in many titanium-transition-metal alloys, there are no substantiatedreports of decagonal phase formation there.

Like the i-phase, the decagonal phase typically ismetastable, generally formed by rapid quenching, whereit frequently occurs with the i-phase, solid solutionphases, and complicated monoclinic and orthorhombic

crystalline phases. Ledges or steps are often found inTEM images, suggesting a lateral growth mechanism(Portier et aL, 1985; Gronsky et aL, 1985). The grainsize varies from 0.1 jim to a few micrometers depending

on the sample composition and the quench conditions(Chattopadhyay et aL, 1985a; FitzGerald et aL, 1988;Daulton et aL, 1991). TEM images typically showstriations that vanish when viewed along the 10-fold

Figure 8. SAD pattern taken along the pseudo-10-fold direction of a decagonal approximant phase in (Al,Si)-Co-Cu. Partof the rhombic lattice of periodic diffraction spots has been emphasized. They correspond to an edge length of the crystal unitcell of order 100 A (Taken from Daulton et al.y 1992a)

zone axis, suggesting planar defects (Perez-Campos etal., 1986; Daulton etal., 1991). Like the i-phase, otherdefects, including twins, dislocations, dislocationloops, and planar faulting, have also been observed(Suryanarayana and Menon, 1987; Menon andSuryanarayana, 1988). A variety of microstructures havebeen reported, including rosette, nodular, cellular-dendritic, and plate-like grain morphologies(Suryanarayana and Menon, 1987; Inoue et al., 1987;Dong et al., 1987a; Li and Kuo, 1988). There are twoalloys in which the decagonal phase is believed to bestable: (Al,Si)-Co-Cu and Al-Pd-Mn. In these, thephase forms with a beautiful grain morphology,decagonal prismatic needles several millimeters in length(Menon and Suryanarayana, 1988; Hiraga et al., 1992)(see Figure l(c)).

As in many icosahedral-phase-forming alloys, Sisometimes plays a necessary, but poorly understood,role in phase selection. In Al-Mn, for example,the addition of only a few atomic percent of Sican completely suppress decagonal phase formation(Ranganathan and Chattopadhyay, 1987). In contrast,it improves decagonal phase formation in Al-Co-Cualloys, where complex crystalline phases with diffrac-tion patterns virtually identical to those from thedecagonal phase often form without its addition(Launois et al., 1991; Dong et al., 1991; Hiraga et al.,1991b; Daulton and Kelton, 1991,1992; Daulton et al.,1992a) Figure 8, for example, shows a pseudo-10-folddiffraction pattern taken from one of these crystals.The rhombic array of weak periodic spots high-lighted on the figure indicates a phase with a periodicityin one dimension of almost 100 A; a completestructural determination is lacking. Increasing theamount of Si further, beyond 4.5 at.% in this alloy,leads to a one-dimensional quasicrystal (Daulton andKelton, 1991).

If proper account is taken of the sample thicknessand TEM defocus condition, HREM images takenalong the 10-fold zone can provide an accuraterepresentation of the quasiperiodic structure. Figure9(a), for example, shows an HREM image of thedecagonal phase in (Al,Si)-Co-Cu. The light anddark regions represent the average of the electrondensity projected along the direction parallel to theincident electron beam. A characteristic feature is thepentagonal arrangement of small cluster-like features,forming wheels. Similar cluster-like objects are observedin scanning tunneling microscopy (STM) images of thequasiperiodic plane (Figure 9(b)), though the wheelsare not as evident, indicating that they likely arisefrom a projection of the atoms from several layers.

The STM image clearly shows steps (the jagged lines),supporting the notion of a ledge-growth mechanism.

3.2 Diffraction

The diffraction patterns from the decagonal phasestrongly resemble patterns obtained from theicosahedral phase, indicating a structural similaritybetween these two phases. Figure 10 shows theprominent zone axes from an irreducible section of thestereographic project for the Al-Mn decagonal phase.A 10-fold rotational symmetry with quasiperiodicdiffraction spots is obtained along direction A.A rotation to a location for an equivalent pattern in thei-phase reveals a pseudo-fivefold pattern (J). Pseudo-threefold patterns are also located at the approximatelocations for true icosahedral threefold patterns (M andI). Two different twofold patterns separated by 18° (Gand H) are both located 90° from A; G is similar tothe icosahedral twofold pattern, while H is an approxi-mation to the (T10) of the i-phase (see Figure 2). InG and H, the diffraction spots along the line pointingto A are periodic. In the perpendicular direction, thepeak locations follow a Fibonacci series, establishingthat this structure is quasiperiodic in two dimensionsand periodic in the third.

The twofold patterns of many decagonal phasesshow pronounced streaking perpendicular to theperiodic axis (Bendersky, 1985), indicating planardefects in the quasiperiodic plane, in agreement withthe TEM image studies mentioned previously. InAl-Fe, the streaking is parallel to the 10-fold zone,suggesting stacking faults along the periodic direction(Fung et al., 1986).

Decagonal phases can be classified on the basis oftheir periodicity: 0.4 nm in Al-Ni (Li and Kuo, 1988),0.8 nm in Al-Cu-TM (TM = Mn, Fe, Co or Ni) (Heet al., 1988b, 1989), 1.24 nm in Al-Mn and Al-Mn-Pd(Bendersky, 1985; Beeli et al., 1991), and 1.6nm inAl-Fe (Fung et al., 1986). All of these periodicities havebeen reported as a function of composition in Al-Co(Menon and Suryanarayana, 1988) and Al65Cu20TM15

(TM = Mn, Fe, Co or Ni) (He et al., 1988b, 1989) andin the same alloy of Al65Cu20Co15 (He et al., 1988c).That all periodicities are multiples of 0.4 nm suggeststhat decagonal phases are layered structures, built fromstackings of nearly identical atomic planes. Detailedstudies of the complete stereographic projections(Daulton and Kelton, 1992,1993a; Daulton et al., 1992b)imply two classes of decagonal phases: (1) those with0.8 nm, 1.6 nm, and presumably 0.4 nm periodicities;and (2) the 1.24 nm periodic phases.

Figure 9. (a) High-resolution electron microscopy (HREM) lattice image of decagonal (Al,Si)-Co-Cu. The pentagonal arrangement of clusters is emphasized,(b) Scanning tunneling microscopy (STM) image of layer of the same decagonal phase, showing evidence for clusters and ledges (Reproduced with permission fromKortan et aL, 1990)

Figure 10. Irreducible 18° section of the stereographic projection of the decagonal phase in Al-Mn, showing the diffractionpatterns from all pseudozones (Taken from Daulton et ai, 1991)

Figure 11. Schematic of the irreducible 18° section of thestereographic projections, showing the locations of the Kikuchibands for (a) the 12 A Al-Mn decagonal phase, (b) a perfecticosahedron with the insertion of a mirror plane, and (c) the8 A (Al,Si)-Co-Cu decagonal phase. The numbers in thefigure label the great circles of the stereograph. Great circlenumber 5 is missing in the Al-Mn phase and in the icosahedralmirror-plane construction. Great circle 2 merges with greatcircle 1 in the (Al,Si)-Co-Cu phase

Figure 11 (a) shows a diagram of the irreducible18° stereographic section for decagonal Al-Mn orAl-Mn-Pd, indicating the intersections of the greatcircles constructed from the Kikuchi bands. While thestereographic projection of the decagonal phaseresembles that predicted by the addition of a mirror

plane to the icosahedral rotational point-groupsymmetry (Figure 1 l(b)), there are systematic shifts inthe positions of the great circles between the twoprojections. These inconsistencies are removed if theicosahedron is distorted along the line connecting twoof the opposite vertices taken to point in the periodicdirection (Daulton and Kelton, 1992; Daulton et al.,1992b). Figure 1 l(c) shows a schematic diagram of thestereographic section for the 8 A (Al,Si)-Co-Cu phase.One of the great circles has vanished and the positionsof the others are shifted from their positions in Al-Mn,consistent with a different magnitude of distortion. Thedecagonal phase, then, appears itself to be anapproximant to the icosahedral phase. The Al-Mn orAl-Mn-Pd decagonal phases appear to be betterapproximants to the i-phase than is the (Al1Si)-Co-Cudecagonal phase, since the required distortion in(Al,Si)-Co-Cu removes the intersection at J, eliminatingthe pseudo-fivefold symmetry.

The availability of large single crystals of thedecagonal phase in (Al,Si)-Co-Cu and Al-Pd-Mn hasallowed precision X-ray studies of the structure to bemade (Steurer and Kuo, 1990; Steurer, 1991). Thepentagonal arrangements of clusters in this layeredstructure, indicated in the STM and HREM images(Figure 9), are evident in the projected electron densities,reconstructed using a Patterson function analysis.

3.3 Diffraction-Pattern Indexing

The restriction that one direction of the decagonal phasebe periodic reduces the required number of basis vectorsfrom six to five (one along the periodic direction andfour in the quasiperiodic plane). While one mightimagine that this would simplify the task of indexingthe diffraction patterns, no scheme currently exists thatcan unambiguously index both the reciprocal latticediffraction patterns and the real-space zone axes.

One method, similar to the four-vectors approachused to index hexagonal crystalline phases, employs sixbasis vectors (Takeuchi and Kimura, 1987; Choy et al.,1988). A set of five reciprocal lattice vectors are arrangedin the basal, quasiperiodic plane, separated by 72°; thesixth vector is perpendicular to the other five, pointingalong the decagonal periodic direction:

a* = c*z

where Rj (0) represents a rotation about the z-axisthrough the positive (counter-clockwise) angle (J- 1)0.This procedure has been used to index the diffraction

patterns of the 8, 12 and 16 A decagonal phases,though an inherent ambiguity exists, giving rise toseveral proposed methods for correctly choosing theindices (Mukhopadhyay et al.9 1989b; Ranganathanet at., 1989; Daulton et at., 1991).

The zone axes for all decagonal phases can be indexedunambiguously using 11 vectors pointing to the verticesof a properly distorted icosahedron (Daulton andKelton, 1992; Daulton etal, 1992b). This representationcan be reduced to a five-vector set using simplesymmetry arguments. The five-vector method discussedpreviously is a special case, corresponding to a particulardistortion of the icosahedral cluster that has not beenobserved. At this time, however, this method has notbeen used to index the zone axes and the diffractionpatterns consistently.

Muller (1987), considering projection methods forobtaining the decagonal phase, showed that a projection

from a five-dimensional space produces a point-groupsymmetry C5, lacking the required mirror symmetry.Two schemes have been proposed for a six-dimensionalprojection. Ho (1986) originally suggested that theprojection be made from an oblique 6D lattice, andMandal and LeIe (1991) proposed a 6D tetragonallattice. Neither method was able to describe satisfactorilyall diffraction features of the decagonal phase.

4. Crystalline Approximants

Many complex crystalline phases contain large clustersof atoms in a nearly perfect tetrahedral coordination.Similar clusters have been inferred in the structures ofliquids, glasses, and quasicrystals. These crystal phases,called crystal approximants, are closely related toquasicrystals, forming under similar conditions and

MgAl/Zn (outer)

Al/Zn (inner)

Figure 12. Fundamental clusters in 1/1 crystal approximants to the i-phase. (a) The construction of the 54-atom Mackay icosahedronfor i(AlMn). The center is vacant in a(AlMnSi); it is filled in a(TiCrSi). (Taken from Audier and Guyot, 1989b). (b) The Paulingtriacontahedron, the fundamental cluster for i(AlMgZn); the atom types are indicated (Taken from Mukhopadhyay et ai.9 1989c)

Al

Mn

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269 This page has been reformatted by Knovel to provide easier navigation.

Index

Note: compounds are listed in this index in strict alphabetic sequence, e.g. AlNi not NiAl.

Note: Figures and Tables are indicated [in this index] by italic page numbers

Index terms Links

A A2 structure 91

ABC notation 51 52 119 132

Active concentration range principle, compound-formation predicted using 31 38

Adamantane-structure compounds additional experimental rules 130 compositions 122

binary compounds 123 homogeneity range for 124 methods of calculation 122 multicomponent compounds 123 ternary compounds 123

defect compounds 118 normal compounds 118 ordered types 118 126 valence-electron rules 119 125

tolerated deviations 126

Adamantane-structure equation 120

Adamantane structures, definition and classification of 117

Adamantane-structure types defect types 134 normal types 133

Adamantine structures 117n

Ag2Nd 145

AlAu2 structure type 112

AlB2 144

270 Index terms Links


AlB2 structure type 140 164 atomic environments 24 25 143 149 150

151 atomic radii-interatomic distance relationship

binary compounds 154 ternary compounds 155

atoms mixing on one position 152 crystallographic data 140 next-neighbor histograms 144 periodic-system representation of binary compounds 152 separation in structure maps 165 205

Al4Ba structure type 140 141 150 atomic environments 149 151 crystallographic data 140

Al8CeMn4 structure type 189

Al-Cu-Mg system 186

Al2Cu structure type 161 164 165 167 172 204 205

Al-Ga-La system 154

Al3LaNi2 structure type 189

Al-Mo system 63 64

AlNb2 structure type 187 198

Alonso plots 21

Al-Ti system, neighborhood diagrams 208 209

Al3Ti structure type 206 207

Al-Ti system 63 64

Aluminum-based quasicrystals 230 238 239 241 261

Al-Zr system 63 64

Al3Zr4 structure type 185 187

Amman decoration [on Penrose tiles] 256 257 258

Amorphous ferromagnets 215 225

Angular valence-orbital factor 12 28 30 31 38 44 166 196

Antiferromagnetic model [of magnetic structures] 219



Antiferromagnetism 215

Antiphase domains (APDs), magnetic structures modeled 218 219 226

Approximants [in quasicrystals] 246

AsFeS structure type 164

As2Fe structure type 164

Aslanov’s crystal chemical model 142 157

AsNi structure type 24 25 35 198

AsNi-type compounds, in structure maps 36 197 198 200 202 207

As2Ti structure type 164

Atomic environment approach 11 140 157 combined with predictive models 157

Atomic environments (AEs) 11 47 139 in AlB2 structure 143 in CaIn2 structure 146 in close-packed structures 54 57 68 irregular 143 in space filling 25

Atomic environment types (AETs) 11 47 140 and coordination numbers 13 36 142 149 150

151 and coordination types 149 151 labeling 13 141 142 most frequently occurring AETs 12 13 33 38 number per crystal structure 29 30

Atomic number atomic properties predicted using 3 4 4 listed for various elements 7

Atomic-number factor 6 44 in AP-AN plots 4 7 atomic properties grouped under 5

Atomic properties 2 grouped under factors 5



Atomic property expressions (APEs) 2 structure maps using 16

Atomic radii close-packed crystal structures 53 56 72 and interatomic distances 27 154 and space filling 23 26

Atomic-size differences/ratio Laves phases 185 MoSi2-type compounds 162 structural stability affected by 196 198

AuBe5 structure type 186

AuCd structure type 35 198

Au-Cd system 60 61 close-packed structures 63 64

AuCu structure type 35 198 202

AuCu3 structure type 42 184 185 206 207 220

Au-Cu system 63 64

Au-Mn system 63 64

Au-Zr system 60 61

Axial ratios Al2Cu-type compounds 168 MoSi2-type compounds 162 163

B BaHg11 structure type 174 209

BaPb3 structure type 183 206 207

BaZn5 structure type 189

B2BaNi2 structure type 191

B2CeCo3 structure type 188 189 191 192

BCr structure type 35 140 148 198 atomic environments 149 151 crystallographic data 140 separation in structure maps 36 197 200 202



Belov’s classification [of structure types] 179

Bergman phase 251 259

Berthollide compounds 47 solid solubility 22 structure mapping 19

BFe structure type 35 140 148 198 atomic environments 149 151 crystallographic data 140 separation in structure maps 36 197 200 202

BiF3 structure type 206 207 220

Binary compounds stoichiometry 31 structure mapping of 15 38 196

Binary defect adamantane-structure types 127

Binary normal adamantane-structure types 126

Binary systems, compound-formation tendency 6 7 38

Bi2Tl 144

Bloch’s theory of metals 93 non-applicability 261

Body-centred cubic (b.c.c.) derivative structures 83 I framework 91 nets 110 notations for 83

Body-centred cubic (b.c.c.) lattice, packing density 55

Bond directionality 12 55 56 65

Bond valence concept 128

Borides 189

Bragg scattering, magnetic structures 216

β-Brass [CuZn] 93 94

γ-Brass [Cu5Zn8] 95 99 115

Bravais lattices, splittings of I lattices 96

Bravais-type crystal structures, number of atoms per unit cell 30

Brunner-Schwarzenbach approach (to definition of atomic environment) 11 140

B12U structure type 174



C C16 structure, stabilizing factors 168

CaCu5 structure type 187

CaF2 structure type 164 in structure maps 165 204 205

CaIn2 structure type 140 146 atomic environments 149 151 crystallographic data 140 next-neighbor histograms 147

Carbides 73 78 172

CdCl2 structure type 165 204 205

CdI2 structure type 164 165 204 205

CeCo5 structure type 192

CeCu6 structure type 189

Ce-Gd system 183

Ce-Ho system 183

Ce-Sm system 183

Ce-Tb system 183

CFe3 structure type 161 174 206

CFe3W3 structure type 161 170

Chalcopyrite [mineral name] 127 133

Charge transfer close-packed alloys 56 hydrides 72

Chelikowsky plots 20

Chemical ordering, in quasicrystals 239

χ phase 96

Cinnabar-structure compounds, separation in structure maps 198 202

Classification adamantane structures 117 structure types 179

ClCs-structure 110 198 interatomic distance relationship 27 manganese alloys 220 221



ClCs-structure (Continued) and polymorphism 35 separation in structure maps 196 197 198 200 202

ClFPb structure type 164

ClNa-structure density-melting-point plots 42 interatomic distance relationship 27 polymorphism 35 separation in structure maps 36 196 197 198 200

202

Close-packed structures 51 179 equally sized atoms 182 homologous series 67 181 homometric structures 66 nearest-neighbors 54 notation for 80 point symmetry of ordered phases 68 stacking sequences 51 unequally sized atoms 184

Cl2Pb structure type 164 165 205

Cluster formation, close-packed structures 58 61

Cluster glasses 216

Cohesive-energy factor 6 44 in AP-AN plots 4 7 atomic properties grouped under 5 relationship vs melting temperature 6

Companion crystal-structure interrelationships 35 42

Complex close-packed alloys 65 66 interstitial alloys 77

Compound-formation plots 8 9 10 11 38 39

Compound-formation tendency 6 8 44 binary systems 6 7 31 38 ternary systems 8 11 31 32 38



Concentration range rule, compound formation predicted using 31

Continuous solid solutions 182

Coordination numbers 24 36 and atomic environment types 13 36 142 149 150

151 158 close-packed structures 58 179 190 notation 13 Th3P4-type phases 175

Coordination polyhedra 12 54 83 141 illustrated 13 142 180 203 notation for 87 91 203

Coordination types 12 17 42 47 148 157

listed 149 151 in structure mapping 18 19

Co2Si structure type 140 atomic environments 149 151 atomic radii-interatomic distance relationship 155 crystallographic data 140 periodic-system representation of binary compounds 156 157 separation in structure maps 165 205

Coulombic interaction in close-packed structures 59 69 70 interstitial alloys 74 76

Covalent bonding in close-packed alloys 69 close-packed alloys 70 interstitial alloys 74 76

Covalent compounds, interstitial atoms in 75

Cowley's short-range order parameter 58

..cP2(12i) framework 109 110

CrFe structure type 23 27

Cross-substitution diagram, adamantane-structure compounds 122 123

CrSi2 structure type 164 165 166



Cr3Si structure type 185 187

CrSi2 structure type 204 205

Cr3Si structure type 206 207

Crystal approximants 247 decagonal-phase quasicrystals 242 253 259 icosahedral-phase quasicrystals 252 indexing of 237

Crystal chemical model of atomic interactions (CCMAI) 142 157

Crystallographic point groups 83 84 85 meaning of term 83n

Crystal structures of body-centred cubic structures 83 of close-packed structures 51 companion interrelationships 35 42 data availability 1 2 factors governing 1 140

strategy to find 2 glossary of terms 46 of hydrides 71 72 73 169 nomenclature xxi 47 and physical properties 38 regularities 10

active concentration range approach 31 38 44 atomic environment approach 11 38 44 chemistry principle 38 44 information-prediction systems proposed 44 simplicity approach 29 38 44 solid-solubility map approach 18 38 space-filling approach 22 stoichiometric restraint approach 31 38 44 structural relation approach 34 structure map approach 15 38 44 symmetry approach 28 30 38

solid solubility affected by 21



Cube [atomic environment] 12 13 frequency plot 14

Cubic alloys 195 207

Cubic close-packed (c.c.p.) interstitial alloys 73

Cubic close-packed (c.c.p.) structures 52 54 61

Cubic system, point groups 86

Cubooctahedron 12 12 13 55 179 180 186

frequency plot 14 structure map 19

Cu16Mg6Si7 structure type 161 173

Cu2Mg structure type 24 26 165 186 204 205

Cuprite [mineral name] 94

Cu-Sb system 60 61

Cu2Sb structure type 140 atomic environments 149 151 atomic radii-interatomic distance relationship 155 crystallographic data 140 periodic-system representation of binary compounds 156 157 separation in structure maps 165 205

Cu-Sn system 60 61

CuTi structure type 35 198 202

CuTi2 163

CuTi3 structure type 206 207

Cu3Ti structure type 206 207

Cu-Ti system 60 61

Cut-projection method, quasicrystals studied using 235

Cu5Zn8 structure type 96 99 100 101

CW structure type 140 202 atomic environments 149 151 crystallographic data 140



D D(60) framework 99 101 102 105 106

Daltonide compounds 47 solid solubility 22 stoichiometric-ratio distribution

binary compounds 32 33 ternary compounds 32 34

structure mapping 19

Darken and Gurry solubility plots 19

Decagonal-phase quasicrystals 229 241 atomic structures 259 crystal approximants 253 diffraction-pattern indexing of 246 diffraction patterns 243 245 general metallurgy 241

Defect adamantane-structure types 134

Defect icosahedron 180

Defect rhombododecahedron 180

Defect structures defect adamantane-structure compounds 118 121 123 127 129

134 tetrahedral structures 120 122

Deformation, structure transformation by 181

δ phase 190

Dense-random-packing (DRP) model 249 255

Density-functional theory (DFT), and quasicrystals 237 256

Density-melting point plots 40 42 42

Density of states (DOS), in quasicrystals 260 261

Derivative structures 181 187 defect derivatives 188 deformed derivatives 182 186 188

Deuterides 73 74

..dI2(60) framework 102 106 107

Diffraction-pattern indexing, application to quasicrystals 236 246



Directional bonding 55 56 65

Dirichlet construction 141

Disclination lines 249 250 255

Dislocations, in quasicrystals 239

Disordered close-packed alloys 78

Disorder effects, in quasicrystals 237

Dy-La system 183

Dy-Nd system 183

E Edge-centered stacking 113 115

8-N rule 120

Electrical conductivity, factors affecting 39

Electrical resistivity, of quasicrystals 260

Electrochemical factor 6 44 in AP-AN plots 4 7 atomic properties grouped under 5 and solid-solution tendency 18 22 and structure mapping 15 16

Electron density, binary alloys 56

Electronegativity as atomic property 5 6 close-packed crystal structures 53 56 72 listed for various elements 7 53 199 in solid-solubility mapping 19 21 in structure mapping 16 17 200

Electronegativity difference plot vs average of principal quantum number 131 196 plot vs electrical conductivity 39 plot vs pseudopotential radii difference 36 plot vs valence-electron number 164 structural stability affected by 196

Electroneutrality rule 75



Electronic factors close-packed structures affected by 193 MoSi2-type phases affected by 164

Electronic properties, quasicrystals 260

Electronic theories of phase stability 213

Electron-per-atom ratio, structural stability affected by 196 200

Elser indexing scheme 236

Enantiomorphous structures 66

Enargite [mineral name] 127 133

Enthalpy of formation 8 as atomic property 5 6 and phase diagrams 32 relationship with, number of equiatomic compounds per system 32

Er-Nd system 183

e-carbide compounds 172 176

External deformation 181

F Face-centred cubic (f.c.c.) lattice, and other cubic structures 55

Face symmetry 83

Famatinite [mineral name] 118 127 133 137

Ferrimagnetism 215

Ferroelectrics, quantum structure diagrams 39 40

Ferromagnetism 215

FeSi structure type 35 198 202

FeS2 structure type 164 165 204 205

Fe3W3C structure type 170

Fibonacci sequence 235 237 256

Fibonacci series 233 243 252

First-principles calculations, for crystal structures 4

Frameworks 91 of I family 99 I framework 91



Frank-Kasper phases 185 249 255 256

G Ga-Hf system 63 64

Ga24HoNi2.6 structure type 188 191

Gallides 140

Ga-Nb system 63 64

Ga-Pt system 63 64

Ga-Ti system 63 64

Ga-Zr system 63 64

Gd-La system 183

Gd-Nd system 183

Gd-Pr system 183

Generalized 8-N rule 120

GeNiPt2 structure type 184

Geometrical stability plots 23 binary compounds 24 ternary compounds 25

Ge2Os structure type 164 165 205

GeS structure type 35 202

Giant cells 187

Glasses icosahedral short-range order in 255 structural similarities with quasicrystals 255

Golden mean 233

Goodman's rule 21 23

Gray tin 51

H Hägg's rule 71

Halides 73 74 75

Heat capacity, of quasicrystals 256

Hermann-Maugin nomenclature 83



Hermann-Skillman calculations 5 15 199

Heteropolarity 93

Heusler alloys magnetic properties 221 splittings of I lattice complex 93 94

Hexagonal close-packed (h.c.p.) interstitial alloys 76

Hexagonal close-packed (h.c.p.) structures 52 61 62 65

Hexagonal cubooctahedron 186

Hexagonal dipyramid 180

Hexagonality, close-packed structures 182 183

Hexagonal layers 51 59 62 69

HgI2 layer structure 122

Hg2Pt structure type 165 185 205

HgS structure type 35

High-coordination-number structures 190

High-resolution electron microscopy (HREM) observations, quasicrystals 238 243 244 251

High-temperature ferroelectrics, structure maps 39 40 211

High-temperature superconductors, structure maps 39 40 211

Ho-La system 183

Homeotectics 181

Homologous series close-packed structures 65 67 79 181 190

192 193 heterogeneous type 192 193 homogeneous type 193 quasihomogeneous type 192 193

Homometric structures 63 66 76 examples 66

Ho-Nd system 183

Hume-Rothery mechanism, quasicrystals stabilized by 261

Hume-Rothery phases, valence-electron concentration values 93 99

Hume-Rothery rules 18 20 Goodman's extension 21 23



Hybrid close-packed structures 190 193

Hybridized bonding orbitals 120

Hydrides, crystal structures 71 72 73 169

I I framework 91

b.c.c. dervative structures described by 94

I(4t) framework 99 102 104

I(12i) framework 103 106 108

I(60) framework 99 102 104

I lattice complex 91 8th order 93 94 64th order 95 96 splittings 93 96 96 27th order 95 96 101 216th order 95 99

i phase 229 230 see also Icosahedral-phase quasicrystals

Icosahedral compounds 195 quantum structure diagram 41

Icosahedral glass model 239 258 calculated patterns 240

Icosahedral-phase quasicrystals 229 230 atomic structures 259 diffraction-pattern indexing of 236 diffraction patterns 232 233 general metallurgy 230 231 phase transformations 241 projection from higher-dimensional space 233 235 rational crystal approximants 252 structural defects 237

Icosahedron 12 13 180 185 186 229

frequency plot 14



Incommensurate crystals 229

Information-prediction systems 44 Kiselyova's system 44 45 Savitskii's system 44 45 Zhou's system 44 46

InNi2 structure type 140 150 atomic environments 149 151 crystallographic data 140 separation in structure maps 165 205

Interatomic distances, and atomic radii 27 154

Interfacial energy, crystal-liquid 256

Intergrowth concept 150

Intermetallic compounds, definitions 46

Internal deformation 181

Interstitial atoms, in close-packed alloys 70

Interstitial compounds complex close-packed alloys 77 cubic close-packed alloys 66 73 hexagonal close-packed alloys 76

Interstitial defects 126

Invariant lattice complexes 87 89 splitting of point configurations 97 symbols for 87 88

Ionic bonding in close-packed alloys 69 70 valence-electron transfer in 120

Ionic compounds, interstitial atoms in 75

Iron alloys, magnetic structures 220

Irregular atomic environment (IAE) 143

Ising model 68

Isoelectrical conductivity plot 39

Isoelectronic close-packed structures 193

Isomorphism 191

Isostructural close-packed structures 193



J Jagodzinski notation 53

Jagodzinski-Wyckoff notation 119 132 ZnS poly types 119

Jarmoljuk-Kripyakevich phenomenon 187

Jensen notation 203 208 209

K Kasper polyhedra 255

Kesterite [mineral name] 129 133

Kikuchi bands 246

Kiselyova's information-prediction system 44 45

Kripyakevich's classification (of structure types) 179 181 185

L Lattice c/a ratio 23 25 55

Al2Cu-type compounds 168 169 atomic environment affected by 146 MoSi2-type compounds 162 163

Lattice complexes examples 98 meaning of term 84 nomenclature 83 symbols for 87 88 types 87

Laves phases 164 185 193 atomic-size ratio 185 multilayer Laves phases 186 189 in structure maps 15 204 ternary Laves phases 189

Liberite [mineral name] 133

Lift-and-project technique 259

Li-Mg-Zn system, multilayer Laves phases 186 189



Line compounds 124 129 130 153 180

Liquids, ordering in 255

Lonsdaleite [mineral name] 117

M Mackay icosahedra 247 248 249 254 259

261

Madelung factors, close-packed structures 70 76

Magnetic exchange interactions 215 direct exchange 216 double exchange 217 superexchange 217

Magnetic form factor 217

Magnetic phase diagrams 223

Magnetic properties, and structure maps 207

Magnetic spin-density distribution 217 218

Magnetic structures 215 and exchange interactions 215 experimental determination 216 models 218 219 rare-earth compounds 223 rare-earth metals/alloys 222 small-scale magnetic order 225 transition-metal compounds 220 transition metals 217

Magnetic superlattices 226

Manganese alloys, magnetic structures 221

α-Manganese structure 96 99 100 101 115

β-Manganese structure 93

Marcasite [mineral name] 102 204

Martynov-Batsanov electronegativity 6 7 166 199 200

Matthias profiles 40

Maximum-convex-volume rule 141



Maximum-gap rule 11 141 examples 12 141 144

Mechanical properties, and structure maps 207

Melting temperatures as atomic property 5 6 and density 40 42 element-element contour plot 42 43 listed

various compounds 42 43 various elements 7

relationship vs cohesive energy 6

Mendeleev number in structure mapping 15 17 166 in weldability plots 40 41

MgNi2 structure type 165 186 204 205

MgZn2 structure type 165 185 188 189 204 205

Mictomagnets 216

Miedema's map 8 38

Miedema's model 6 8 32 157 166 167

MnP structure type 35 198 200 202 207

Mn12Th structure type 174 209

Mn23Th6 structure type 161 172 223

Mn23Y6 hydrides 224

Models, magnetic structures 218 219 226

Molybdenum disilicide as high-temperature material 167 see also MoSi2 structure type

Mooij correlation 260

Mooser-Pearson plots 15 16 131 131 195 196

principles behind 12 15

Morphotropy 191



MoSi2 structure type 113 114 161 162 168 172

next-neighbor histograms 162 in structure maps 165 204 205 superstructure of 112

Mössbauer spectroscopy 216 220 226 255

N NaO structure type 36 202

NaTl-type Zintl phases 93

NaZn13 structure type 161 173 209

Nearest-neighbor interactions, close-packed structures 54

Neighborhood diagrams, Al-Ti system 208 209

Nets body-centred cubic structures 110 close-packed structures 186 188

Neutron diffraction/scattering, magnetic structures determined using 215 216 218 220 223 226

Next-neighbor histograms 12 141 AlB2 structure type 144 CaIn2 structure type 147 MoSi2 structure type 162

..nI(12i) framework 103 106 108 110

Ni-Si-Ti system 32 35

Ni-Sn system 60 61

Ni3Sn structure type 206 207

NiTi2 structure type 161 168 169 172 in structure maps 165 205

Ni3Ti structure type 206 207

Nitrides 73

Non-bonding orbitals 120

Non-octet compounds crystal structure 16 structure maps 197 198 200



Non-stoichiometric compounds 73

Normal adamantane-structure types 121 123 133

Normal tetrahedral structure 120

Normal-valence compounds 120

Nuclear magnetic resonance (NMR) spectroscopy 216 255

O Octagonal-phase quasicrystals 229

Octahedral interstitial atoms 70

Octahedron 12 13 55 180 frequency plot 14 structure map 19

Octet compounds 196 crystal structure 16 structure maps 196 198 198 200

Olivine [mineral name] 102

Ordered adamantane-structure types 126 binary defect types 127 binary normal types 126 quaternary defect types 130 quaternary normal types 130 ternary defect types 124 129 ternary normal types 128 136

Ordered substitution, in close-packed structures 181 192

O3Re structure type 204 206

Orthohexagonal arrangements nets in 110 notation for 111

Orthorhombic structures 113

Orthorhombic system, point groups 85

OsSi2 structure type 164

O2Ti structure type 165 204 205

Oxides, close-packed structures 73 74 75 76



P P phase 187 188 190

Packing density close-packed structures 52 cubic structures 55

Pauling's electrovalence rule 74 75 79 128 130 136

Pauling's principles 34

Pauling's rule of parsimony 54 68 76 79

Pauling triacontahedron 247 251

Pb-U system 63 64

Pearson classification [of structure types] 179

Pearson symbols, for various crystal structure types 140 149

Penrose lattice 256 258

Penrose tiling 236 256 259

Periodic-system representation atomic properties of elements 7 199 binary AB2 compounds 152 153 Pettifor's relative order number [for structure mapping] defined 201

Permanent magnets 210

Perovskite-type ceramics 195

Pettifor [structure] maps 166 195 200 AB compounds 201 AB2 compounds 165 166 169 172 204

205 AB3 compounds 204 206 207 AB11 compounds 209 210 AB12 compounds 209 210 AB13 compounds 209 210 A2B17 compounds 209 210 A3B4 compounds 176 extension to pseudobinary phases 17 207 extension to ternary phases 17 211 relative ordering number for 201 207



Pettifor [structure] maps (Continued) ternary chalcogenides 211

Phase diagrams, magnetic structures 223

Phase transformations, quasicrystals 241

Phason-based transformations (of quasicrystals) 241

Phason strain 237 in quasicrystals 239 253 in tilings 258

Phenomenological relative ordering number 201 207

Phillips-Van Vechten plots 131 131 195 198 200

Phonon softening mode 241

Physical properties, effect of crystal structures 38

Point groups 83 84 cubic system 86 meaning of term 83n orthorhombic system 85 tetragonal system 85 trigonal system 86

Point-group symmetry 68 decagonal-phase quasicrystals 241

Point sets 29 36 149

Polyanionic-valence compounds 120

Polycationic-valence compounds 120

Polymorphic compounds 35 37

Polymorphic modifications, crystal structure affected by 34 37

Polymorphism 191

Polytope [3,3,5] 249

Polytypes stacking notations for 119 132 ZnS 118

space groups 133

Principal quantum number as atomic property 5 15 as measure of bond directionality 12



Principal quantum number (Continued) in structure mapping 16 15 131 196

Pr-Tb system 183

Pseudobinary compounds 79 structure mapping of 17 207

Pseudopotential methods 93

Pseudopotential radii 5 6 listed for various elements 7 199 plot vs atomic number for various elements 3 4 in structure mapping 16 36 198

P4Th3 structure type 161 175

Pt-Ti system 63 64

Pyrites [mineral name] 204

Q Quantum number, as measure of bond directionality 12

Quantum structure diagrams 39 40 41 158

Quasibinary/quasiternary systems 168

Quasicrystals 229 atomic structures 259 commercial applications 261 crystal approximants 242 247 decagonal phase 229 241

atomic structures 259 classification based on periodicity 243 crystal approximants 253 259 diffraction-pattern indexing 246 diffraction patterns 243 general metallurgy 241

dislocations in 239 electronic properties 260 growth morphologies 233 243 244 icosahedral phase 229 230

atomic structures 259



Quasicrystals (Continued) crystalline phases related 248 diffraction-pattern indexing of 236 diffraction patterns 232 233 general metallurgy 230 231 233 phase transformations 241 projection from higher-dimensional space 233 235 rational crystal approximants 252 structural defects 237

octagonal phase 229 prediction of formation 261 quasilattice models 256 real-space structures 256 structural similarities to liquids and glasses 255 and structure maps 39 41 211

Quasilattice 229 256

Quasilattice models, quasicrystals 256

Quasizones 261

Quaternary compounds, close-packed structures 79

Quaternary defect adamantane-structure types 130

Quaternary normal adamantane-structure types 130

R R phase 190 251

Ramsdell notation 119 132 133 134

Rare-earth compounds, magnetic structures 223

Rare-earth metals atomic properties listed 7 199 magnetic structures 222

Rational approximants [in quasicrystals] 235 241 242 252

Regularities (with crystal structures) 10

Repulsive interactions close-packed alloys 56 59 in close-packed alloys 69



ReSi2 structure type 164

Re7Si6U4 structure type 184

Rhombic dodecahedron 12 13 180 frequency plot 14 structure map 19

Rietfeld analysis 216

Ruderman-Kittel-Kasuya-Yoshida (RKKY) interactions 216 223 226

Rutile [mineral name] 102 204

S Savitskii's information-prediction system 44 45

Sb2Tl7 crystal structure 96 99 100 101 115

Scanning tunneling microscopy (STM) 243 quasicrystals 244

Segregation, close-packed structures 56 58 60 61 63

Self-coordination numbers close-packed structures 55 56 57 62

homologous structures 67 homometric structures 63 66 interstitial atoms 71 75 quaternary/ternary alloys 79

Semiconductors 200 crystal structure 200

Short-range order, in liquid alloys 255

Short-range order parameter close-packed alloys/compounds 78 79 tetragonal single-layer structures 58

σ phases 23 185 188

Silicides 164

Simplicity principle 29 38

Simulated annealing 33

Single-environment structure types 47 54 68 211

Sinusoidal model (of magnetic structures) 218 219

Site-set symbols 90



Si2Th structure type 140 148 165 205 atomic environments 149 151 crystallographic data 140

Si2Ti structure type 164 165 166 204 205

Si3W5 structure type 36

Size factor 6 44 in AP-AN plots 4 7 atomic properties grouped under 5 composition of ternary c.p. compounds affected by 183 and solid-solution tendency 18 22 in space filling 28 and structure mapping 15 16 196

Size mismatch, structural stability affected by 196 198

Si-Zr system 63 64

Si2Zr structure type 164

Slater-Néel-type diagram 216

Smirnova's classification (of structure types) 179

Solid-solubility mapping 18 38 binary systems 18 ternary systems 22 170

Space filling approach (to solubility) 22

Space-filling ratio 23

Space-filling ratio plots 23 24

Space-group numbers, distribution of compounds 28 29 33 38

Space-group operators 85

Space groups 83 85 assignment of lattice complexes 91 92 ZnS and SiC polytypes 133

Space principle 22 quantitative indicators 23

Specific heat, quasicrystals 260

Sphalerite-type compounds 94 117 cubic diamond structure compared with 118

Spin-density-wave model (of magnetic structures) 218



Spin-density waves 216 218 221

Spin-glass behavior 260

Spin glasses 215 226

Spiral model (of magnetic structures) 218 219 221 226

Splitting diagram, adamantane-structure compounds 122 123

Splitting of I-complex 93 96

SrZn5 structure type 189

Stacking faults, in quasicrystals 243

Stacking notations 53 132 182

Stacking sequences close-packed structures 51 79 hexagonal layers 51 61 homometric structures 67 NiTi2-type compounds 170

Stacking variants disordered alloys 79 interstitial alloys 77 ZnS polytypes 118

Stannite [mineral name] 118 129 133 137

Statistical substitution, in close-packed structures 182 192

Stoichiometric restraint approach, compound-formation predicted using 31 38

Stoichiometry binary compounds 31 ternary compounds 32

atomic environment approach 33 space-group approach 32

Strip-projection method quasicrystals studied using 233 235

limitations 235

Structural relation restraints 34

Structural stability, factors affecting 196 198 200

Structure maps 15 38 195 adamantane-structure compounds 130 binary compounds 15 16 38 196



Structure maps (Continued) close-packed interstitial alloys 74 77 close-packed structures 60 61 63 69 70 disordered close-packed alloys 78 hexagonal layer structures 59 65 Mooser-Pearson maps 15 16 131 195 196 ordering number defined 201 Pettifor maps 17 166 169 172 176

195 200 211 Phillips-Van Vechten maps 195 198 ternary compounds 15 17 38 210 tetragonal layer structures 56 65 Villars (three-dimensional) maps 18 19 36 131 132

157 166 169 195 200 Zunger maps 196 200

Superconducting critical/transition temperature, as function of valence electrons per atom 40

Superconductors, quantum structure diagram 39 40

Supercooling 255

Superlattices, rare-earth magnetic materials 226

Superstructures body-centred cubic structures 112 113 close-packed structures 181 182 MgCu2-type 186 MgZn2-type 186 189 MoSi2-type 162

Surface structures 59 61

Symmetry principle 28 30 38

SZn structure type 35

T T phase (in close-packed structures) 190

t phase (in decagonal-phase quasicrystals) 241

Teatum radii 142



Ternary compounds AB2 compounds with random atomic mixing 153 chalcogenides 211 close-packed structures 79 compound-formation tendency of 8 hydrides 169 NiTi2-type phases 169 solid solutions 22 170 stoichiometric ratio restraint of 32 structure maps for 15 17 38 210 structure types possible 210 Th3P4-type phases 175 total number possible 2 15

Ternary defect adamantane-structure types 124 129

Ternary normal adamantane-structure types 128

Tetragonal alloys 195 207

Tetragonal lattice 55

Tetragonal layers 62 65 69 structure mapping of 56

Tetragonal pyramid 180

Tetragonal structures 113

Tetragonal system, point groups for 85

Tetrahedral interstitial atoms 70

Tetrahedral packing, short-range 255

Tetrahedral-structure compounds, valence-electron rules 119

Tetrahedral-structure equation 120

Tetrahedral structures 12 14

Tetrahedron 12 13 180 181 frequency plot 14

Th3P4 structure type see P4Th3 structure type

Thiogallate [mineral name] 129

Three-dimensional maps compound-formation maps 10 11 38



Three-dimensional maps (Continued) structure maps 18 19 36 131 132

157 166 169 195 200

III-V compounds 117

β-Tin structure type 51

Titanium-based quasicrystals 233 239 240

Titanium-transition-metal quasicrystals 230

Topological disorder, in quasicrystals 239

Transition-metal-aluminum quasicrystals 230 238 239

Transition-metal compounds magnetic structures 220 relative solubilities 23 valence-electron number vs. electronegativity difference 164

Transition metals atomic properties listed 7 199 magnetic structures 217

Transition-metal-titanium quasicrystals 230 233 239

Transmission electron microscopy (TEM) observations, quasicrystals 232 232 233 234 240 242 242 250 254

Trialuminides, stabilization of cubic structure 209

Triangle 180

Trigonal dipyramid 180

Trigonal prism 180

Trigonal system, point groups in 86

Tungsten, crystal structure 55

Twinned cubooctahedron 13

U Unit-cell dimension ratio c/a plots 24 25

Unit-cell-parameter-radii relationships 24 26 27 28

V Vacancies 126

Vacancy-ordered phases 73 74



Valence-electron concentration (VEC) 120 adamantane compounds 121 Al2Cu-type compounds 168 close-packed structures 186 189 193 Hume-Rothery phases 93 99 tetrahedral-structure compounds 120

Valence-electron factor 6 44 in AP-AN plots 4 7 atomic properties grouped under 5 and solid-solution tendency 18 22 and structure mapping 15 16

Valence-electron number as atomic property 5 6 7 listed for various elements 7 199 plot vs electronegativity difference 164

Valence-electron rules, adamantane-structure compounds 119

Valence electrons per atom, superconducting critical/transition temperature plotted against 40

Valence orbitals, angular characteristics 166 196

Villars [compound-formation] model 8

Villars three-dimensional compound-formation map 10 11 38

Villars three-dimensional structure maps 18 19 36 131 132 157 166 169 195

application to ternary systems 39 40 210 binary compounds 200 ternary compounds 210

Voronoi polyhedron construction 141 255

V-Zn system 63 64

W Weldability

factors affecting 40 plots 40 41

Wigner-Seitz cell 141



Wirkungsbereich [of atom] 141

Wurtzite-structure compounds 117 hexagonal diamond structure compared with 118 separation in structure maps 196 198 200 202

Wurtzstannite [mineral name] 129 133

Wyckoff notation 161

Wyckoff positions 85 88 91 92 93

Wyckoff sequence 134

Wyckoff sets 85 92

Z Zeolite ZK-5 103

Zhang Bangwei plots 21 22

Zhdanov notation 119 133 134

Zhou's information-prediction system 44 46

Zinc-blende compounds see Sphalerite; Wurtzite

Zinc-blende structures separation in structure maps 36 196 198 200 space groups listed 133 134

Zintl compounds/phases 93

ZnS-type compounds, stacking variants 118

Zunger pseudopotential radius 6 7 166 198 199

Zunger structure maps 131 132 195 196 200

Documents

Crystal Structure of Intermetallic Compounds