Proc. Nati. Acad. Sci. USAVol. 86, pp. 9637-9641, December 1989Chemistry

Icosahedral and decagonal quasicrystals of intermetallic compoundsare multiple twins of cubic or orthorhombic crystals composed ofvery large atomic complexes with icosahedral point-groupsymmetry in cubic close packing or body-centered packing:Structure of decagonal Al6Pd

(AlI3Cu4Fe3/AlsMn/AL6CuLi3/multiply icosahedral complexes)

LINUS PAULINGLinus Pauling Institute of Science and Medicine, 440 Page Mill Road, Palo Alto, CA 94306

Contributed by Linus Pauling, September 18, 1989

ABSTRACT A doubly icosahedral complex involvesroughly spherical clusters of atoms with icosahedral point-group symmetry, which are themselves, in parallel orientation,icosahedrally packed. These complexes may form cubic crys-tallites; three structures of this sort have been identified.Analysis of electron diffraction photographs of the decagonalquasicrystal Al6Pd has led to its description as involvingpentagonal twinning of an orthorhombic crystal with a = 51.6A,b = 37.6 A, and c = 33.24 A, with about 4202 atoms in theunit, comprising two 1980-atom doubly icosahedral complexes,each involving icosahedral packing of 45 44-atom icosahedralcomplexes (at 0 0 0 and 1/2 1/2 1/2) and 242 interstitialatoms. The complexes and clusters are oriented with one oftheir fivefold axes in the c-axis direction.

The question of the nature of icosahedral quasicrystals andother quasicrystals (decagonal, dodecagonal, octagonal) con-tinues to attract much attention. Steinhardt (1) has reportedthat 488 papers in this field were published in 1988, twice asmany as in 1987. Samples of alloys have been found to givediffraction patterns and show face development indicatingpoint-group symmetry with twelvefold, tenfold, eightfold,and fivefold axes and other features not crystallographicallyacceptable. None of the three structural models that havebeen proposed has been generally accepted.

Levine and Steinhardt (2) proposed the quasicrystal model,which states that the diffraction patterns result from a quasi-periodic arrangement of two or more distinct structural units.The twinned-crystal models (3-8) involve cubic, rhombohe-dral, or orthorhombic crystals, usually containing icosahedralcomplexes of atoms, that by twinning produce the observedpoint-group symmetry. In the icosahedral glass model (9, 10)icosahedra or pentagons are densely packed in parallel orien-tation but without translational identity operations.

Multiple Icosahedral Packing of Atoms

The most compact packings of spheres, usually of more thanone size, are those in which all of the interstices are tetra-hedra, often with some distortion from the regular tetrahe-dron (11). Such structures are said to be tetrahedrally packed.The simplest tetrahedrally packed cluster is the centeredicosahedron, which can be described as 1-12, an atom at thecenter and a shell of 12 atoms. A second shell of 32 atoms canbe added, to give the 45-atom cluster 1-12-32, and then a thirdshell of 60, 72, or 92 atoms, to give clusters with 105, 117, or137 atoms, all with icosahedral point-group symmetry. The

117-atom triple-shell cluster was reported for Mg32(Al, Zn)49in 1952 (12, 13), with the body-centered crystal having twoclusters, sharing atoms in their outer shell, in the unit cubeand with the cluster centers largely unoccupied, so that theunit cube contains 160-162 atoms. With 1.24% of boronatoms in the alloy, the centers probably would be occupiedin a well-annealed sample.

It is well known that tetrahedra cannot fill space. To be incontact with each other and the central atom, the 12 outeratoms in the 1-12 icosahedral cluster must be 10% larger thanthe central atom. Completing a second and a third shell, withgood interatomic contacts, requires some still larger atoms.There is the possibility that a tetrahedrally packed clusterwith icosahedral point-group symmetry and with a fourthshell could exist in an alloy made with a range of metal atoms,from the smallest to the largest, but at present the limit seemsto be reached at the three-shell cluster.

This is, however, not the limit for complexes of metal atomswith icosahedral point-group symmetry. The icosahedral clus-ters described above are roughly spherical and can be ex-pected to pile together in a compact way. Their icosahedralpoint-group symmetry favors icosahedral packing. Hence weanticipate that they will form larger complexes with icosahe-dral point-group symmetry, with structures 1-12, 1-12-32,1-12-32-60, etc. (It is unlikely that the central cluster would beabsent; holes larger than one atom do not occur.) Such acomplex may be called a doubly icosahedral complex.There probably is a practical limit to the size of such a

complex at the second or third shell of clusters. However,there is the possibility that a triply icosahedral complex couldbe formed, and even a quadruply icosahedral complex, aquintuply icosahedral complex, and so on, to give, finally, amacroscopic quasicrystal with no translational identity op-erations.There are, however, good reasons for translational identity

operations to institute themselves; these reasons explain whycrystals constitute so great a part of the earth's crust.Consider a liquid, a molten alloy. The individual atoms or

small clusters of atoms exist momentarily with any one of agreat number of environments. A single atom, for example,might have 4, 5, 6, . . . 10, 11, 12, . . . near neighbors. Someof these localized structures are more stable than others, butat high temperatures the effect of thermal agitation and theentropy effect conspire to make the liquid the stable phase.One local environment for an atom or cluster of atoms is,however, enough more stable than others to permit, as thetemperature is lowered, its decrease in enthalpy to overcomethe extra entropy of the liquid, and the substance crystallizes,introducing translational symmetry operations and other

Abbreviation: EDP, electron-diffraction photograph.

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Proc. Natl. Acad. Sci. USA 86 (1989)

symmetry operations of the space group of the crystal as theselecting and preserving mechanism for the favored localenvironment operates. [Some degree of disorder may remainin the crystal, even at very low temperatures, as has beenfound for H2 (14), ice (15), and a good number of othersubstances.]

Types of Multiply Icosahedral Complexes

Let N1 be the number of atoms in an icosahedral cluster andN2 be the number of clusters in the icosahedral complexformed ofthese clusters. The number ofatoms in the complexis then given by N1N2, as in Table 1. (Here the assumption ismade that interstices between clusters may be filled by atomsreleased by sharing of atoms between clusters or by somedeformation of the clusters.)

Structures of Icosahedral Quasicrystals

By analysis of the layer lines on the twofold-axis electron-diffraction photograph (EDP) of Al13Cu4Fe3 the value a =52.60 A was assigned to the cube edge of the multiply twinnedcubic crystal for this quasicrystal (Table 2) (16). Consider-ation of atomic volumes led to aboutA = 9960 for the numberof atoms per unit cube, which is indicated by the EDP to bebody centered. The obvious structure places very largedoubly icosahedral complexes, in contact with one another,in the unit cube, with centers at 0 0 0 and 1/2 1/2 1/2,probably each with 104 x 45 = 4680 atoms, leaving 600 (6.0%)interstitial atoms.A similar analysis for AI6Mn (16) gave a = 66.306 A, A =

19,400. With four 104 x 45 complexes centered at 0 0 0,0 1/2 1/2, 1/2 0 1/2, and 1/2 1/2 0 there are -=680 (3.5%)interstitial atoms.For Al6CuLi3 the values found (16) are a = 57.8 A and A

= 11,690 in a primitive cube. A reasonable kind of packing isprovided by the /8-W arrangement, two complexes at 0 0 0and 1/2 1/2 1/2 and six at 0 1/4 1/2, etc. If the complexeshave about the same size, the number of atoms in each wouldbe =1340, assuming 5% interstitial. Hence 13 104-atomclusters, 1352 atoms (Table 1), is reasonable, giving 874(7.5%) interstitial atoms.

The Number of Interstitial Atoms

For rigid spheres of the same size, the fraction of voids islarge, 26% for cubic close packing and 32% for cubic body-centered packing. In packing roughly spherical clusters ofatoms the interstitial space is smaller, because the outeratoms of one cluster may fit into the centers of trianglesformed by the outer atoms of the adjacent clusters. Forexample, the cubic crystal Al12Mo has two 1-12 icosahedralclusters in the unit, at 0 0 0 and 1/2 1/2 1/2. The clusters fittogether, with a little distortion, in such a way that there areno cavities large enough for even a single atom. Moreover,the clusters shift some of the superficial atoms by small

Table 1. The number of atoms in a doubly icosahedral complexof N2 icosahedral clusters, each with N1 atoms

N2

N1 13 45 105 137

13 169 585 1,365 1,78144 572 1980 4,620 6,02845 585 2025 4,725 6,165104 1352 4680 10,920 14,248105 1365 4725 11,025 14,385136 1768 6120 14,280 18,632137 1781 6165 14,385 18,769

Table 2. Values of the cube edge, a, number of atoms in the unitcube, A, number of multiply icosahedral complexes per unit cube,n, number of atoms per complex, A', and number of interstitialatoms per unit cube, A", for three icosahedral quasicrystals withdifferent structuresCompound a, A A n A' A"Al13Cu4Fe3 52.00 9,960 2 4680 600 (6.0%o)AI5Mn 66.306 19,400 4 4680 680 (3.5%)Al6CuLi3 57.8 11,690 8 1352 874 (7.5%)

amounts so as to tend to fill the voids. It is hence notsurprising that the fraction of interstitial atoms shown inTable 2 is small, 3.5-7.5%.

Decagonal Quasicrystals

The first decagonal quasicrystal to be reported was A17(Mn,Fe)2 (17). It shows an apparent tenfold symmetry axis and atranslational operation along this axis. An interpretation ofthis quasicrystal as a decagonal twin of an orthorhombiccrystal has been proposed (18). Several other decagonalquasicrystals have been reported, including AlCoCu (19) andAl6Pd (ref. 20; S. H. J. Idziak and P. A. Heiney, personalcommunication).A reasonable suggestion can be made about why some

alloys form icosahedral quasicrystals and others form deca-gonal quasicrystals. The three known structures of icosahe-dral quasicrystals, described above, involve large doublyicosahedral clusters in parallel orientation with three of theirtwofold axes parallel to the unit cube edges. If, instead ofhaving this orientation, the complexes were to have a fivefoldaxis parallel to an edge of the unit of an orthorhombic crystal,there would be a tendency to fivefold or tenfold twinning,with all of the twinned crystals having one translationalsymmetry operation in common.

It is not surprising that there are several different struc-tures of icosahedral quasicrystals and probably also of de-cagonal quasicrystals. The structure of a crystal of an inter-metallic compound is a sensitive function of its composition;-2500 structures have so far been determined, and manyothers are reported each year.

Decagonal AI6Pd

Idziak and Heiney (personal communication) made samplesof the decagonal quasicrystal of Al and Pd by forming ribbonson a 23-cm copper wheel from melts with compositionsbetween Al3Pd and Al8Pd, and they made EDPs with aPhillips 400T electron microscope with a two-tilt stage; andalso made x-ray and electron-diffraction powder patterns.They carried out a detailed analysis of the peak shapes,positions, and intensities seen in the electron and high-resolution x-ray powder diffraction patterns on the basis oftwo different models: that of a decagonal quasicrystal withthe diffraction patterns resulting from five incommensuratevectors in a plane and one vector normal to that plane, andthat of a decagonally twinned aggregate of microcrystals.Reasonable agreement was found with each of these twomodels.

Analysis of the EDPs of AI6Pd

I have now carried out another analysis of the EDPs andpowder patterns. Three EDPs, made with the incident elec-tron beam along three mutually orthogonal directions, areshown in Fig. 1; for Fig. 1A the beam is along the decagonalaxis and for Fig. 1 B and C it is along the directionsconventionally called D and P in the plane perpendicular tothis axis. Idziak and Heiney (personal communication) state

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Proc. Nati. Acad. Sci. USA 86 (1989) 9639

FIG. 1. EDPs of the decagonal quasicrystal Al6Pd (courtesy of S. H. J. Idziak and P. A. Heiney). (A) Electron beam along the fivefold axis(c axis of the orthorhombic crystallites). (B) An enlarged region of this EDP. (C) Beam along the D direction (a axis). (D) Beam along the Pdirection (b axis).

that the c-axis repeat distance is 16.62 A. I have used thisvalue to calibrate the EDPs. The lines marked 10 on Fig. 1 Band C contain orders of reflection 0 0 1 from the basal plane,with the strongest spots corresponding to interplanar dis-tance d = 16.62/8 = 2.0775 A (Q = 2ir/d = 3.0244 A1; thevalue 16.62 A is from Idziak and Heiney, personal commu-nication). With this calibration, measurement of the P and Dlayer lines gives the values of Q shown in Table 3.

In assigning values of h and k I have assumed that theymust be even, as in the cubic quasicrystals. The simplestallowed sequences are 6, 10, 16 and 10, 16, 26, assigned as inTable 3. The spots in Fig. ID are elongated, and two radii

Table 3. Analysis of P and D layer lines (Fig. 1 D and C,respectively) of AI6Pd

P layer lines D layer lines

d,A h a,A d,A k b,A5.09 10 (50.9)* 6.22 6 (37.3)*3.148 16 50.37 3.772 10 37.721.954 26 50.80 2.339 16 37.43

Average 50.6 Average 37.6

The scales are determined by the 0 0 1 reflections, c = 33.24 A.Even values of h and k are required by the assumed body-centered(or face-centered) structure. Average values of the lattice constantsare a = 50.6 A and b = 37.6 A.*Not included in average because the spots are caused largely bydouble diffraction.

were measured for each spot. The values in Table 3 are forthe larger estimated radii of each pair; the other values arediscussed below.

This analysis leads to the dimensions a = 50.6 A, b = 37.6A, and c = 33.24 A for the orthorhombic unit of structure.

The Mean Atomic Volume in AL6Pd

To calculate the number of atoms in the unit we may dividethe volume of the unit by the mean atomic volume, correctedfor the effects of partial ionic character, electron transfer, andthe difference between icosahedral packing and cubic closepacking. Values for seven binary AlPd alloys are given inTable 4 (22). I shall use the average, 7.9%.

The Structure of Decagonal AI6Pd

The volume of the unit is 63,240 A3. With 7.9% contraction(Table 4) and atomic volumes 16.61 A3 for Al and 14.71 A3 forPd (23), the calculated number of atoms in the unit is 4202. Ifwe assume that there are two large complexes at 0 0 0 and1/2 1/2 1/2 in the body-centered orthorhombic unit (four inapproximate cubic close packing in the corresponding 50-Acube, as was suggested for one of the icosahedral quasicrys-tals), each complex would contain somewhat fewer than 2101atoms. In the earlier study (17) the large complexes weredescribed as icosahedrally packed triple-shell 104-atomicosahedral clusters. We propose that the clusters in Al6Pd

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Proc. Natl. Acad. Sci. USA 86 (1989)

Table 4. Volume contraction in binary AlPd alloys (data fromref. 15)*Composition Structure a, A b, A c, A Change in V, %

Al82Pd18 c F 4 3.866 -11.2AlPd c P 2 3.049 -9.7AlPd c P 8 4.867 -8.0AlPd h R 26 15.659 5.251 -8.7AlPd2 o P 12 5.407 4.061 7.769 -7.4Al3Pd5 o P 16 5.373 10.471 4.035 -8.0Al4Pd h P 90 13.085 9.633 9.633 -2.2

Average -7.9

Values of atomic volume for cubic close packing: Al, 16.61 A3; Pd,14.71 A3 (16).*Four compounds are not included in the table because the numberof atoms in the unit is not known.

are the 44-atom double-shell clusters (12 atoms in the firstshell, 32 in the second shell), forming a rhombic triaconta-hedron (Fig. 2). If there is an atom in the center, the clustercontains 45 atoms. If 45 of these 44-atom clusters are ar-ranged icosahedrally in two shells around a central cluster,the complex would contain 1980 atoms, leaving 242 atoms(5.8%) for the interstices in the orthorhombic unit, about thesame fraction as for the icosahedral quasicrystals. With largedoubly icosahedral complexes at 0 0 0 and 1/2 1/2 1/2, theymay be expected to interact in accordance with their icosa-hedral point-group symmetry, which in the basal plane in-volves the angles 360 and 720. It is accordingly satisfying thatthe trace in this plane of the vector from 0 0 0 to 1/2 1/2 1/2makes the angle 36.62° with the a axis, only 0.620 from thedirection of the a axis of a tenfold twin.With this structure, each doubly icosahedral complex is

surrounded by 12 similar complexes at approximately thecuboctahedral positions of cubic close packing, 8 at 35.6 A,2 at 33.24 A, and 2 at 37.6 A (average, 35.54 A).

FIG. 2. The rhombic triacontahedron. With atoms at the 32corners, this polyhedron represents the 44-atom and 45-atom icosa-hedral clusters (12-32 and 1-12-32). With a 44-atom cluster at eachcorner (and 13 inside), it represents the doubly icosahedral com-

plex of 1980 atoms proposed as the building block of decagonalAl6Pd.

The Fine Structure of the EDP Spots

Many of the spots in Fig. 1A are not round. For example, thevery strong spotfis square, suggesting that, in addition to aspot in the center, on the symmetry plane, there are twoothers, one on each side of this plane, corresponding to asomewhat smaller value of d, and also another pair corre-sponding to a somewhat larger value. This square or crossappears also on other spots, probably in all cases as a resultof multiple diffraction involving the very strong square.The central component of the square is the spot 1690 0,

with d = 3.163 A. One of the pairs is 9 9 0, d = 3.353 A, atangle ±0.620, and the other is 16 4 0, d = 2.997 A, at angle±0.59°. These values agree reasonably well with measure-ments of the square.

Overlapping of the spots h 0 0 and h' h' 0 also accounts forthe elongation of the spots in Fig. 1D. The outer componentsfor the second and third elongated spots are 16 0 0, d = 3.148A, and 26 0 0, d = 1.954 A, and the inner components are9 9 0, d = 3.353 A, and 15 15 0, d = 2.012 A. The innercomponent of the first elongated spot is formed by doublediffraction from the other two.

Comparison of Icosahedral and Decagonal Quasicrystals

In each of the three kinds of icosahedral quasicrystals [withbody-centered, face-centered, or ,3-W packing of large icosa-hedral complexes of atoms (16)] the icosahedral complexes,in parallel orientation, have three twofold axes in the direc-tions of the edges of the cubic unit. In decagonal quasicrys-tals, however, the large icosahedral complexes have a five-fold axis in the c direction of the orthorhombic crystals, thedirection of one of the translational identity operations. Thecomposition of the alloy determines the structure of the cubicor orthorhombic crystals and also the way they twin.The four structures described in this paper and the pre-

ceding one (16) have been derived by combining the analysisof the diffraction data with arguments based on the extensiveinformation now available about the properties of the atoms,especially how they interact with one another in intermetalliccompounds. The principle that a compound will have as closepacked a structure as is permitted by the relative sizes and thevalences of the metal atoms indicates that icosahedral clus-ters will be formed in the molten alloy and retained in thequasicrystal. A new idea is that these clusters can packicosahedrally to form multiply icosahedral complexes, eachconsisting of thousands of atoms. These very large, roughlyspherical, complexes can then pack together in the usual wayfor crystals, cubic close packing or body-centered cubicpacking, or f3-W packing; probably also hexagonal closepacking (not yet observed for a quasicrystal).The very large icosahedral complexes dominate the dif-

fraction patterns, but the structure factor for the translationaloperations also shows up in the patterns. As a greater andgreater effort is made to observe the very weak diffractionmaxima it should be possible in the course of time to evaluatethe hundreds of parameters giving the locations of the atomsin the unit cells of the crystals. Crystallographers havesucceeded in making complete structure determinations ofintermetallic compounds with more than a thousand atoms inthe unit cell; with improved diffractometers and computers,they should now be able to attack twinned crystals with10,000 or 20,000 atoms in a unit cube or parallelepiped.Paul J. Steinhardt (University of Pennsylvania) has kindly

given me earlier references to multiply icosahedral com-plexes. It is mentioned in ref. 24 that J. D. Bernal, in a privatecommunication in 1960, had suggested the possible existenceof a class of hierarchic structures (multiply icosahedral), suchas 13 spheres in a centered icosahedral quasi-spherical unit,13 of which could be icosahedrally packed to make a quasi-

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Proc. Natl. Acad. Sci. USA 86 (1989) 9641

sphere of the next order. M. Hoare (25) discussed severalsuch structures, including one built of 3965 atoms; he stated,however, that "I hasten to add that the above exercise is onlyan exploration of geometrical possibilities, not in any way anattempt to postulate the occurrence of such large structuresin any particular material under realistic conditions." I, onthe other hand, believe that the extra stability associated withicosahedral packing, resulting in large part from the energy ofresonance of the metal-metal covalent bonds among theavailable bond positions (21) and the consequent trendtoward the maximum ligancy and toward tetrahedral inter-stices, leads directly to the formation of large multiplyicosahedral complexes when the composition of the alloy(atoms with different metallic radii) is favorable. With somealloys the crystal containing the icosahedral complexes is thestable form (Al13Cu4Fe3, Al6CuLi3), whereas for others theseparate large icosahedral complexes are found only whenthe molten alloy is quenched, with a rearrangement, such assharing ofthe outer shell in Mg32(Al, Zn)49, occurring on slowsolidification or annealing.

I thank S. H. J. Idziak and P. A. Heiney for giving me a copy ofFig. 1 and Dorothy Munro, Barclay Kamb, Zelek S. Herman, andGerald Latter for their help. This investigation was supported in partby a grant from the Japan Shipbuilding Industry Foundation (Tokyo).

1. Steinhardt, P. J. (1989) The Scientist, May 29, 12-18.2. Levine, I. & Steinhardt, P. J. (1984) Phys. Rev. Lett. 53,

2477-2480.3. Field, R. D. & Fraser, H. L. (1984) Mat. Sci. Eng. 68, L17-

L21.4. Pauling, L. (1985) Nature (London) 317, 512-514.

5. Pauling, L. (1987) Phys. Rev. Lett. 58, 365-368.6. Carr, M. J. (1986) J. Appl. Phys. 59, 1063-1067.7. Pauling, L. (1987) Proc. Natl. Acad. Sci. USA 84, 3951-3953.8. Pauling, L. (1988) Proc. Natl. Acad. Sci. USA 85, 8376-8380.9. Shechtman, D. & Blech, 1. (1985) Mat. Trans. 16A, 1005-1012.

10. Stephens, P. & Goldman, A. 1. (1986) Phys. Rev. Lett. 56,1168-1171.

11. Shoemaker, C. B. & Shoemaker, D. P. (1972) Acta Crystal-logr. B 28, 2957-2965.

12. Bergman, G., Waugh, J. L. T. & Pauling, L. (1952) Nature(London) 169, 1057.

13. Bergman, G., Waugh, J. L. T. & Pauling, L. (1957) ActaCrystallogr. 10, 254-259.

14. Pauling, L. (1930) Phys. Rev. 36, 430-443.15. Pauling, L. (1935) J. Am. Chem. Soc. 57, 2680-2684.16. Pauling, L. (1989) Proc. Natl. Acad. Sci. USA, 86, 8595-8599.17. Bendersky, L. (1985) Phys. Rev. Lett. 55, 1461-1463.18. Pauling, L. (1988) Proc. Natl. Acad. Sci. USA 85, 2422-2423.19. Tsai, A. P., Inoue, A. & Masumoto, T. (1989) Mat. Trans. Jpn.

Inst. Met. 30, 300-308.20. Thangaraj, N., Chattopadhyay, K., Ranganathan, S., Small, C.

& Davies, H. A. (1986) in Proceedings ofthe 11th InternationalCongress of Electron Microscopy, Kyoto, eds. Imura, T.,Maruse, S. & Susaki, T. (Japanese Society of Electron Micros-copy, Tokyo), pp. 1525-1526.

21. Pauling, L. (1984) J. Solid State Chem. 54, 297-307.22. Villars, P. & Calvert, L. D. (1985) Pearson's Handbook of

Crystallographic Data for Intermetallic Phases (Am. Soc.Metals, Metals Park, OH), 3 vols.

23. Pauling, L. & Kamb, B. (1986) Proc. Natl. Acad. Sci. USA 83,3569-3571.

24. Mackay, A. (1962) Acta Crystallogr. 15, 916-920.25. Hoare, M. (1976) Ann. NYAcad. Sci. 279, 186-207.

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