Influence of the Chemical Composition of Alloys on the Ideal Local Order in Approximants of the Icosahedral Quasicrystals

ISSN 1063�7745, Crystallography Reports, 2010, Vol. 55, No. 6, pp. 953–963. © Pleiades Publishing, Inc., 2010.Original Russian Text © V.A. Chizhikov, V.E. Dmitrienko, 2010, published in Kristallografiya, 2010, Vol. 55, No. 6, pp. 1013–1023.

953

INTRODUCTION

One of the most intriguing philosophical problemsin the theory of cognition is the question of possiblelimits of understanding the world. There is good rea�son to believe that these limits cannot be established inprinciple; however, beginning in the 19th century, sci�entific optimists have ascertained from time to timethat the conceptual framework of science was con�structed as a whole and the only thing for the next gen�erations of researchers to do is to refine particularquestions of an applied instead of fundamental char�acter. It is most pleasing for skeptics (and maybe foroptimists as well) to make an unexpected discoveryspecifically in a field of knowledge where all possiblephenomena appear to be described and the ideality oflaws suggests no other conceivable improvements.

This situation was typical of crystallography at theend of the 19th century, when the scientific paradigmrelated the habit and optical anisotropy of crystals withtheir internal order, which in turn was characterized byso�called translational symmetry or periodicity. Allpossible types of periodic structures were mathemati�cally rigorously enumerated (230 Fedorov groups inthe 3D Euclidean space), and until the 1980s it wasbelieved that crystal structure must be described withinone of these space symmetry groups.

The discovery of icosahedral quasicrystals in 1982[1] and, a short time after, experimental findings ofcrystals with other “forbidden” point symmetries(octagonal, decagonal, and dodecagonal) made crys�tallographers reject the old intuitive concepts andimprove the age�old injustice for some point groups byextending the set of crystalline classes from 32 to atleast 36 elements. Indeed, excluding periodicity, aquasicrystal has all the characteristic properties of

conventional crystals: natural faceting; a finite pointsymmetry group; and a long�range positional order,which yields a diffraction pattern composed of sharpBragg peaks [2].

Despite their clear similarity with crystals, quasic�rystals were first considered as exotic structures (as isreflected in their name). The “isolation” of quasicrys�tals was broken when another class of periodic struc�tures, similar to quasicrystals in both chemical com�position and local atomic order (the so�called crystal�line approximants of quasicrystals), was revealed. Withan increase in the unit�cell period, approximantsapproach a quasicrystal in both structure and physicalproperties, and the quasicrystal itself can conditionallybe considered as an infinite limit of a sequence ofapproximants. It is of interest that many crystal struc�tures that became thereafter approximants were foundlong before quasicrystals (for example, in [3] in 1957,where the well�known Bergman cluster was describedfor the first time for a Mg32(A1, Zn)49 crystal). Crystalssimilar to that described in [3] were considered exoticand, paradoxical as it may seem, it was the discovery ofquasicrystals with even more unusual structures thatallowed researchers to gain insight into the nature ofthese objects and establish their position in crystallog�raphy.

Approximants for icosahedral quasicrystals are theso�called Fibonacci crystals, which can be orderedaccording to the approximations of the golden mean

by rational ratios of successive terms of

the Fibonacci series, which is defined by the recur�rence formula fn = fn – 1 + fn – 2, f1 = f2 = 1:

1 52+

τ =

τ fn/fn 1– , fn/fn 1–n ∞→

lim≈ τ.=

Influence of the Chemical Composition of Alloys on the Ideal Local Order in Approximants of the Icosahedral Quasicrystals

V. A. Chizhikov and V. E. DmitrienkoShubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiі pr. 59, Moscow, 119333 Russia

e�mail: [email protected] May 17, 2010

Abstract—Correlations between atomic displacements from ideal structural positions in the physical spaceand the atomic coordinates in the perpendicular space are revealed for approximants of icosahedral quasic�rystals. On average, the displacement magnitude increases with an increase in the distance from the center ofprojection on the perpendicular space; this indicates that disordering in quasicrystals is concentrated at theperiphery of atomic surfaces. It is found that the chemical composition of an alloy significantly affects thespecific features of distortion of the ideal structure of quasicrystals and their approximants.

DOI: 10.1134/S106377451006009X

THEORY

Dedicated to the memory of B.N. Grechushnikov

954

CRYSTALLOGRAPHY REPORTS Vol. 55 No. 6 2010

CHIZHIKOV, DMITRIENKO

Most often, the case in point is cubic Fibonacci crys�tals, whose tetrahedral point symmetry groups are sub�groups of the icosahedron group. The order of thisapproximants is the ratio of the correspondingFibonacci numbers: fn/fn – 1, for example, 13/8. Thereis some arbitrariness in indexing approximants, and itwas historically accepted to assign the order 1/1 = f2/f1

to the above�mentioned Mg32(A1, Zn)49 crystal andsimilar approximants. Then the simplest cubicapproximant (B2 structure of the CsCl type) has theorder 1/–1 = f–1/f–2, the next compound FeSi(B20 structure) has the order 0/1 =f0/f–1, etc.

Currently, Fibonacci crystals up to an order of 3/2have been found and described, although there areexperimental indications of 8/5�order approximants[4] (in that paper they were denoted as ⟨34/21⟩).Determining the atomic structure of approximants oforder 3/2 and higher is hindered by the large size oftheir unit cells: when passing to the next order, theunit�cell size increases by a factor of about τ and thenumber of atomic positions increases as τ3(~4.2). Forexample, the cubic approximants 2/1 have a period of~20 Å and approximately 500–700 atoms per unit cell[5]. This difficulty is overcome by considering thestructures of large approximants as composed of large(containing up to several hundreds of atoms) icosahe�dral clusters similar to the above�mentioned Bergmanclusters.

However, with all advantages of cluster description,the most important of which appear to be physical andchemical clarity, this approach has a number of draw�backs. With an increase in the approximant order, thecluster size should increase. Simultaneously, the num�ber of possible types of clusters, which is determinedby the possibility of attaching different outer shells,increases in a geometric progression. In addition, thecluster description is not quite appropriate for the sub�sequent idealization of the structure, because clustershells are not fixed and may change continuously (e.g.,the icosahedron radius may behave so).

The ideal structure of quasicrystals and theirapproximants is often described using the concept ofideal local configuration [6]. For icosahedral quasic�rystals and Fibonacci crystals, the best ideal local con�figuration is dodecahedral local ordering (DLO) [7,8]. In the initial DLO model, the distance betweenneighboring atoms was determined by the vectors r3,r5, and r2, directed, respectively, along the third�, fifth�,and second�order symmetry axes of the icosahedron

(the length ratio is r3 : r5 : r2 = ). There�

after, vectors (which are smaller by a factorof τ and related to phason defects) were added todescribe the structures of real crystals.

We have previously described the ideal DLO struc�tures of the cubic approximants 1/0 and 2/1 [5, 8]. Inthis paper we analyze in detail the deviation of atomsfrom ideal positions and the effect of the chemical

3 : 2 : 2+ τ

and3 5 2' ' ',r r r

composition of quasicrystalline alloys on their struc�tural features by the example of the crystals consid�ered in [5, 8] and the approximant 3/2–2/1–2/1with the orthorhombic sp. gr. Cmc21 and chemicalformula (Ga, Zn)175 – δMg97 + δ [9].

6D FORMALISM

One important advantage of the DLO model is thatit allows one to use multidimensional descriptive for�malism, passing from the geometry language to thelanguage of linear algebra (matrices and vectors). Thisdescription can uniformly be used for both quasicrys�tals and their approximants.

We will introduce the matrix approach convention�ally, using the well�known setting of the Cartesiancoordinate system, where the axes coincide with thedirections of three second�order orthogonal symmetryaxes of the icosahedron. Then the vectors setting thebonds between the nearest neighbors can be written as

bonds r3 : (±1, ±1, ±1) (8 vectors),

(±τ, 0, ±τ–1) (12 vectors);

bonds r5 : (±τ, ±1, 0) (12 vectors);

bonds r2 : (±2, 0, 0) (6 vectors),

(±τ, ±τ–1, ±1) (24 vectors); (1)

bonds : (±τ–1, ±τ–1, ±τ–1) (8 vectors),

(±1, 0, ±τ–2) (12 vectors);

bonds : (±1, ±τ–1, 0) (12 vectors);

bonds : (±2τ–1, 0, 0) (6 vectors),

(±1, ±τ–2, ±τ–1) (24 vectors).

Here, we use a conditional scale, in which a half dis�tance r2 is taken to be the length unit. The plus andminus signs are chosen independently for each coordi�nate, and the symbol means all possible cyclic per�mutations.

Thus, any distance between the nearest neighbor�ing atoms in a DLO structure is described by one of the124 vectors (1). It is easy to find that any vector (1) is alinear combination of the six vectors

a1 = (τ, 1,0),

a2 = (0,τ, 1),

a3 = (1,0, τ), (2)

a4 = (–τ, 1,0),

a5 = (0,–τ, 1),

a6 = ( 1 , 0 , –τ),

with all integer or all half�integer coefficients, forexample:

r3'

r5'

r2'

τ τ 1–1, ,( ) a2 a4,–=


INFLUENCE OF THE CHEMICAL COMPOSITION OF ALLOYS ON THE IDEAL LOCAL 955

The six vectors (2) can be chosen as a basis.By moving from an atom to its nearest neighbor,

then to the neighbor’s nearest neighbor, etc., one canpass from an arbitrary atom to any other atom of thestructure, thus linking them by a nearest neighborchain. Hence, the coordinates of each atom can bepresented in the form of a linear combination of vec�tors (2):

(3)

where

(4)

n is a six�component vector, all coordinates ni of whichare simultaneously integers or half�integers, and r0 isan arbitrary vector setting the origin of coordinates.

The matrix Eph projects the sites n of a 6D body�centered hypercubic lattice on some 3D subspace,which is nothing other than the physical space. Obvi�ously, not all sites of the 6D lattice are projected on thephysical space Eph; otherwise, the structure describedwould have an infinite density, because the projectiondirection is irrational. It is generally assumed that theprojected sites fall into a 3D “tube” in the 6D space.This tube infinitely extends along a 3D subspace Е||

and has a bounded projection on the 3D subspace Е⊥

(complementary to Е||), which is referred to as the per�pendicular space of the quasicrystal (approximant).Without any loss of generality, accurate to the insignif�icant constant factors, one can set the perpendicularspace of icosahedral quasicrystal or cubic approximantby the matrix

(5)

the 3D subspace Е||, complementary to it, can be set bythe matrix

(6)

On the assumption that α = τ, formulas (5) and (6) setthe perpendicular and parallel spaces of the icosahe�

dral quasicrystal, and , respectively. Here it can

be seen that coincides with the physical space Eph,which often leads to the incorrect conclusion that theparallel and physical spaces always coincide. However,

1 τ 1–0, ,( ) –1

2��a1

12��a2

12��a3

12��a4– 1

2��a5– 1

2��a6.+ + +=

r r0– niai Ephn,= =

Eph = a1 a2 a3 a4 a5 a6, , , , ,( ) = τ 0 1 τ– 0 1

1 τ 0 1 τ– 0

0 1 τ 0 1 τ–⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,

E⊥

1– 0 α 1 0 αα 1– 0 α 1 0

0 α 1– 0 α 1⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,=

E ||

α 0 1 α– 0 1

1 α 0 1 α– 0

0 1 α 0 1 α–⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=

iqE⊥

iqE�

iqE�

this does not hold true even for approximants. In addi�tion, according to (4), the space Eph is directly relatedto the choice of the basis of the ideal local configura�tion and, when the chosen basis vectors differ from (2)(e.g., six independent vectors r3 (DLO basis)), thematrices Eph, E⊥, and Е|| will change and Eph will signif�

icantly differ from .

The choice α = fn + 1/fn corresponds to the case of acubic approximant fn/fn – 1 in the above designations.We will denote the perpendicular and parallel spaces of

this approximant as and , respectively. If

the rows of the matrix are multiplied by fn, theywill set the integer coordinates of the cubic approxi�mant periods in the 6D space. (The projections of theperiods of approximant on its perpendicular space Е⊥

should have zero length; otherwise, in the case ofunbounded expansion, the approximant wouldemerge beyond the tube in the 6D space.) For exam�ple, the first row of matrix (6) yields the 6D period(fn + 1, 0, fn, –fn + 1, 0, fn), which, according to (4), cor�responds to the period (2τn + 1, 0, 0) in the physicalspace. Similarly, the second and third rows of matrix(6) set the periods (0, 2τn + 1, 0) and (0, 0, 2τn + 1) of thecubic approximant. Thus, the CsCl crystal (n = –1)has a period of 2 (in the conditional length units), as itmust be for the B2 structure. The next�order approxi�mant FeSi has a period of length 2τ, etc.

DODECAHEDRAL LOCAL ORDERINGAND ICOSAHEDRAL POSITIONS

An analysis of the structures of the best knownapproximants of icosahedral quasicrystals shows thatdodecahedral bonds r3 are much more widespread inthem than the icosahedral bond r5. This can be dis�proved by the example of the Al12Mn crystal structure,which is based on r5 bonds and does not contain r3

bonds. However, this crystal is an exception, because itis not an approximant for conventional quasicrystals(in particular, its diffraction pattern sharply differsfrom that of quasicrystals). With an increase in thecontent of transition metal atoms in the alloy (i.e.,with a change in its chemical composition toward thefirst known quasicrystal Al86Mn14 [1]), the situationsharply changes. The best proof of the inequality ofthe dodecahedral and icosahedral bonds in the struc�tures of icosahedral quasicrystals and their approxi�mants is the fact that structural integrity (connectiv�ity) can be obtained over r3 bonds, whereas r5 bondscannot provide it. Thus, one can speak about“dodecahedral local ordering” in icosahedral quasic�rystals in contrast to the “icosahedral local ordering”which is typical, e.g., of 3D Penrose tiling andAl12Mn crystals.

Note that the 6D formalism introduced in the pre�vious section describes both the dodecahedral and

iqE�

1/n nf fE −

⊥

1/n nf fE −

�

1/n nf fE −

�

956



icosahedral local ordering equally well. It is more nat�ural to use a basis different from (2) (constructed onthe vectors r3) to describe DLO structures; however,this change in the basis is equivalent to the choice ofanother coordinate system and does not affect the finalresult.

The few icosahedral bonds between the nearestneighbors are not distributed uniformly throughoutthe quasicrystal but involve peculiar atoms whichoccupy the so�called icosahedral positions. Thesepositions may have either a completely icosahedralenvironment (“icosahedral holes” with 12 nearestneighbors in the r5 directions) or a partially filled envi�ronment (8–11 r5 bonds). Icosahedral positions can beeither isolated or located in pairs or small groups(icosahedral holes with 12 neighbors at the distance r5

are always isolated). Thus, one can pick out a DLOmatrix with embedded atoms in icosahedral positionsin the structure of icosahedral quasicrystal or itsapproximant. Most DLO matrix atoms have at leastone icosahedral position in their environment andmany have more than one.

If the atomic coordinates (3) are known, one caneasily determine if this atom occupies an icosahedralposition or belongs to the DLO matrix. It follows from(1) that the irrational number τ always enters the coor�dinate vectors r3 an even number of times and thecoordinate vectors r5 an odd number of times (here wetake into account the relation τ–1 = τ – 1). As wasnoted above, the DLO matrix is a structure connectedover vectors r3; hence, all its atoms are characterizedby the same parity of sums of integer coefficients at τin their coordinates. (Without any loss of generality,these sums can be considered even, which can easilybe provided by an appropriate choice of the origin ofcoordinates r0.) At the same time, all icosahedral posi�tions are spaced from the DLO matrix atoms by thevector r5; therefore, the sum of the coefficients at τ intheir coordinates have another parity.

It can be seen in (3) and (4) that the parity of thesum of coefficients at τ in the atomic coordinates inthe physical space Eph coincides with the parity of thecombination (n1 + n2 + n3 – n4 – n5 – n6) of its coor�dinates in the 6D space.

DISTORTION OF IDEAL LOCAL ORDER

DLO and DLO�based 6D formalism set idealatomic coordinates in icosahedral quasicrystals andtheir approximants. Real alloys always contain variousdefects which distort the structure. However, atoms inquasicrystals and crystals with a complex structure(higher order approximants) should undergo displace�ments from the ideal positions (determined by DLO)due to their asymmetric environment even in theabsence of defects. Obviously, the structure is more

significantly distorted by short bonds because there is limiting spacing between atoms andthey tend to elongate short bonds to the conventionallength. At the same time, we can suggest that the atomsin icosahedral holes should distort the structure to theleast extent due to their symmetric nearest environ�ment (12 neighboring atoms form an icosahedron).

In [5] we described the ideal DLO structures of thecubic approximants 2/1 in the Al–Rh–Si [10, 11],Mg–Al–Zn [12], and Ca–Cd [13] systems; analyzedthe regions of these structures that are difficult to ide�alize; and showed that the presence of these regionsis generally explained by the presence of short inter�atomic bonds. The tables in [5] contain the follow�ing parameters for each atom: its ideal DLO coordi�nates and displacement, the coordinate of its pro�jection on Е⊥, the type of occupied position (DLOmatrix and icosahedral position), and coordinationnumbers.

In this paper a similar table contains the corre�sponding data for the approximant 3/2–2/1–2/1 withthe orthorhombic sp. gr. Cmc21 and chemical formula(Ga, Zn)175 – δMg97 + δ [9]. The unit cell of this approx�imant with the periods а = 36.84 Å, b = 22.78 Å, andс = 22.93 Å contains 1088 atoms (149 independentpositions). It can be seen that the period a exceeds τ bya factor of about b (c ~ b); this indicates that, alongone of the axes (x), this approximant has the next(after 2/1) order: 3/2. The projection on the perpen�dicular space is described by the matrix

with α1 = 5/3 and α2 = 3/2. Any two atoms of the base�centered unit cell of the crystal, which are bound by

the base�centering translation , will be projected

on the same site in Е⊥; therefore, the approximantprojection on the perpendicular space contains 544points. The point group of the projection on Е⊥ is C2v

(mm2) with a center at the intersection of the planes ofsymmetry that are perpendicular to the х⊥ and у⊥ axes;the coordinate of the center along the z⊥ axis is foundby averaging the z⊥ coordinates of all projection points.The atomic coordinates in the perpendicular space(see table) correspond to the position of the center of

approximant projection at the point ; the х⊥

and z⊥ coordinates are integers and y⊥ coordinates arehalf�integers (for convenience, the х⊥ coordinates ofall atoms are multiplied by f4 = 3 and the coordinatesy⊥ and z⊥ are multiplied by f3 = 2). The first column ofthe table contains the names of the atomic positionsfrom [9] and designations of the positions occupied bythese atoms in the structure: d and i denote, respec�

and3 5 2' ' ',r r r

E⊥

1– 0 α1 1 0 α1

α2 1– 0 α2 1 0

0 α2 1– 0 α2 1⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

( )1 1 02 2

( )130, 0,34

−



Real and ideal coordinates, projections on the perpendicular space, and the coordination numbers of

the atomic positions of the 3/2�2/1�2/1 approximant with the sp. gr. Cmc21 in the Mg–Ga–Zn system. The coordination

numbers in the ideal structure correspond to the distances

AtomCoordinates, 10–3

Δ, Åreal [9] ideal

Zn1, d 035,285,035 045,286,045 0.44 53 (012316)

Zn2, d 035,096,100 028,095,118 0.49 1 (011514)

Zn3, d 036,285,411 028,286,382 0.72 32 (002319)

Zn4, d 036,095,532 045,095,545 0.45 51 (011415)

Zn5, d 036,285,223 028,286,236 0.42 3 (004117)

Zn6, d 037,475,106 028,477,118 0.44 5 (011415)

Mg1, d 045,099,317 045,095,309 0.21 51 (000448)

Mg2, d 047,475,317 045,477,309 0.21 55 (000348)

Zn7, d 058,344,317 045,332,309 0.57 5 (003219)

Zn8, d 060,226,315 073,214,309 0.59 1 (010,10,01)

Mg3, d 067,171,440 073,168,427 0.37 7 (001338)

Zn9, d 071,192,063 073,214,073 0.55 3 (011516)

Mg4, d 071,402,430 073,405,427 0.11 (000349)

Mg5, d 071,167,196 073,168,191 0.13 71 (000447)

Zn10, d 072,000,065 073,023,045 0.68 (011516)

Mg6, d 072,023,430 073,023,427 0.08 (000637)

Zn11, d 072,384,073 073,359,073 0.55 93 (012316)

Mg7, d 072,026,195 073,023,191 0.12 1 (000348)

Mg8, d 073,401,202 073,405,191 0.26 1 (000637)

Zn12, i 076,280,134 073,286,118 0.41 3 (040080)

Zn13, d 109,098,103 118,095,118 0.50 21 (010812)

Mg9, d 116,285,387 118,286,382 0.14 232 (000349)

Zn14, i 117,402,319 118,405,309 0.24 2 (0200,10,0)

Mg10, d 119,089,517 118,095,500 0.41 210 (000825)

Zn15, d 119,499,254 118,477,264 0.54 254 (011613)

Zn16, i 119,036,307 118,023,309 0.32 2 (010093)

Mg11, d 120,290,245 118,286,236 0.23 23 (000447)

Mg12, d 120,482,120 118,477,118 0.13 25 (000439)

Mg13, d 120,281,015 118,286,000 0.38 230 (000,10,23)

Zn17, i 121,173,298 118,168,309 0.28 27 (110064)

Zn18, d 130,125,400 118,141,382 0.69 2 2 (101227)

Zn19, d 131,156,004 118,141,000 0.60 2 0 (011516)

Zn20, d 131,350,124 118,332,118 0.66 2 (011516)

Zn21, d 131,038,006 118,050,000 0.58 2,11,0 (012415)

Mg14, d 143,410,007 146,405,000 0.21 0 (00033,10)

Zn22, i 144,112,200 146,095,191 0.44 11 (010065)

r̃⊥

x⊥

2y⊥

z⊥

, ,( )=

c̃ c3' c5' c2' c3c5c2= r3' r5' r2' r3r5r2

r̃⊥ c̃

5

8 2

8

5

8 4

8 2

1

1

71

33

3 3

33

313

3

355

353

3

35

31

3 2

2

11

51

4

2

1

9

9

72

61

6

958



Table. (Contd.)



Mg15, d 147,214,124 146,214,118 0.13 (000447)

Zn23, d 153,401,412 163,405,427 0.50 7 (002318)

Zn24, d 154,407,220 146,405,236 0.48 (011415)

Zn25, d 159,017,414 163,023,427 0.40 7 (011415)

Zn26, i 173,012,222 191,977,236 1.08 5 (210153)

Zn27, i 182,211,239 191,214,236 0.35 (110064)

Zn28, i 186,212,376 191,214,382 0.25 2 (0100,10,2)

Zn29, d 189,304,444 191,286,427 0.57 3 (010802)

Mg16, d 189,332,312 191,332,309 0.11 (000448)

Mg17, d 189,473,318 191,477,309 0.24 5 (000637)

Zn30, d 189,495,449 191,477,455 0.42 55 (004116)

Mg18, d 190,099,078 191,095,073 0.15 13 (000249)

Zn31, d 190,310,184 191,286,191 0.56 31 (010813)

Zn32, d 191,314,065 191,332,073 0.44 3 (011514)

Zn33, i 191,415,121 191,405,118 0.25 (0200,10,0)

Mg19, d 191,096,313 191,095,309 0.09 1 (000178)

Zn34, i 192,211,016 191,214,000 0.37 0 (0200,10,0)

Zn35, i 194,015,517 191,023,500 0.45 0 (0200,10,0)

Zn36, d 194,114,453 191,095,455 0.44 15 (011415)

Zn37, d 225,400,413 219,405,427 0.40 (004116)

Mg20, i 226,106,200 236,095,191 0.49 411 (010093)

Zn38, d 226,406,221 236,405,236 0.50 4 (011525)

Mg21, d 233,413,010 236,405,000 0.33 4 0 (000538)

Mg22, d 235,219,124 236,214,118 0.19 4 (000349)

Mg23, d 239,017,392 236,023,382 0.28 4 2 (000736)

Zn39, d 242,192,290 236,214,264 0.80 4 4 (101227)

Zn40, d 250,349,125 236,359,118 0.57 49 (011514)

Zn41, d 251,149,397 264,141,382 0.61 2 (011415)

Zn42, d 252,158,007 264,141,000 0.62 0 (002318)

Zn43, d 252,042,007 264,050,000 0.51 ,11,0 (004116)

Zn44, i 260,394,319 264,405,309 0.36 (030190)

Zn45, d 260,498,262 264,477,264 0.48 54 (112314)

Mg24, d 261,283,390 264,286,382 0.22 32 (000439)

Mg25, d 263,483,127 264,477,118 0.25 5 (000349)

Mg26, d 264,292,238 264,286,236 0.13 3 (001338)

Mg27, d 265,286,008 264,286,000 0.19 30 (000538)

Zn46, d 270,100,104 264,095,118 0.41 1 (010802)

Zn47, d 274,089,297 281,095,309 0.41 91 (011415)

r̃⊥ c̃

632

13

614

53

1 4

134

13

1 3

171

1 1

1

1

1

17

112

1 1

13

15

1

913

14

1

32

5

3

2

49

49

4

411

4

4

4 2

4 4

4

4 2

1



Table. (Contd.)



Mg28, d 306,396,200 309,405,191 0.31 1 1 (010824)

Mg29, d 306,399,437 309,405,427 0.29 1 (000439)

Zn48, i 306,280,129 309,286,118 0.31 13 (030090)

Mg30, d 307,167,196 309,168,191 0.14 171 (000538)

Mg31, d 309,025,193 309,023,191 0.07 1 1 (000439)

Zn49, d 309,383,069 309,359,073 0.54 193 (011515)

Zn50, d 309,192,063 309,214,073 0.53 1 3 (010606)

Zn51, d 309,006,064 309,023,045 0.56 1 (002318)

Zn52, d 310,190,450 309,214,455 0.55 1 5 (011514)

Mg32, d 315,211,318 309,214,309 0.31 1 (000924)

Zn53, i 315,090,394 309,095,382 0.39 112 (0200,10,0)

Zn54, d 322,459,323 309,450,309 0.61 1, , (111415)

Zn55, d 323,343,319 337,332,309 0.61 (011515)

Zn56, d 345,099,097 354,095,118 0.60 61 (010607)

Zn57, d 345,093,292 354,095,264 0.72 614 (010617)

Zn58, d 346,286,223 354,286,236 0.44 63 (012414)

Zn59, d 346,285,413 337,286,427 0.45 3 (011514)

Mg33, d 354,286,007 354,286,000 0.16 630 (000637)

Mg34, d 356,479,121 354,477,118 0.12 65 (000249)

Zn60, d 367,345,128 354,332,118 0.63 6 (011515)

Zn61, d 368,156,005 382,168,000 0.60 70 (010604)

Zn62, d 372,038,004 382,023,000 0.53 0 (010802)

Zn63, i 378,097,189 382,095,191 0.14 11 (040080)

Mg35, d 380,213,119 382,214,118 0.08 (000637)

Mg36, d 381,168,384 382,168,382 0.07 72 (001338)

Zn64, d 381,192,255 382,214,264 0.54 4 (011614)

Mg37, d 382,407,246 382,405,236 0.24 (000448)

Mg38, d 384,020,384 382,023,382 0.10 2 (000349)

Mg39, d 385,403,386 382,405,382 0.16 2 (000637)

Zn65, i 387,283,321 382,286,309 0.33 3 (030090)

Zn66, d 419,094,292 427,095,309 0.48 31 (010912)

Zn67, i 425,412,134 427,405,118 0.41 3 (0200,10,0)

Mg40, d 426,097,073 427,095,073 0.05 313 (000637)

Zn68, i 427,213,004 427,214,000 0.09 3 0 (0100,11,0)

Zn69, d 427,311,069 427,332,073 0.48 3 3 (011415)

Mg41, d 428,291,200 427,286,191 0.24 331 (000538)

Mg42, d 430,285,432 427,286,427 0.15 33 (000439)

Zn70, i 431,404,498 427,405,500 0.15 3 0 (040080)

r̃⊥ c̃

1

13

2

5

3

55

3

31

11 1

771

2

4

7 3

2

72

2

25

2

232

2

23

214

25

21

2 1

1

12

3

7

3

1

960



Table. (Contd.)



Zn71, d 432,145,462 427,141,427 0.82 3 (101227)

Zn72, i 434,022,493 427,023,500 0.29 3 0 (0100,10,2)

Zn73, d 439,469,315 427,477,309 0.51 35 (010802)

Zn74, d 440,157,194 427,141,191 0.60 3 1 (011516)

Zn75, d 440,037,195 427,050,191 0.58 3,11,1 (011516)

Zn76, d 442,347,316 455,359,309 0.58 9 (002318)

Mg43, d 454,212,314 455,214,309 0.13 (000349)

Zn77, i 462,086,382 455,095,382 0.35 12 (110064)

Zn78, d 464,217,103 472,214,118 0.46 8 (011415)

Zn79, d 464,408,224 472,405,236 0.42 8 (004116)

Zn80, d 465,406,406 455,405,427 0.62 (010607)

Zn81, d 0,001,373 0,023,382 0.54 0 2 (010605)

Zn82, d 0,001,258 0,977,264 0.56 054 (002318)

Mg44, d 0,023,627 0,023,618 0.21 0 (000538)

Mg45, d 0,024,000 0,023,000 0.03 0 0 (000637)

Zn83, i 0,095,432 0,095,427 0.10 01 (0200,10,0)

Zn84, i 0,097,201 0,095,191 0.24 011 (0200,10,0)

Mg46, d 0,165,631 0,168,618 0.30 07 (000736)

Mg47, d 0,168,000 0,168,000 0.01 070 (000439)

Zn85, d 0,187,759 0,168,764 0.44 074 (004116)

Zn86, d 0,191,375 0,168,382 0.54 072 (010803)

Zn87, d 0,193,259 0,214,264 0.49 0 4 (00231,10)

Mg48, d 0,214,129 0,214,118 0.24 0 (000736)

Mg49, d 0,214,505 0,214,500 0.11 0 0 (00023,11)

Mg50, d 0,348,507 0,359,500 0.30 090 (001239)

Mg51, d 0,358,131 0,359,118 0.29 09 (000736)

Zn88, d 0,380,260 0,359,264 0.48 094 (004118)

Zn89, d 0,382,376 0,405,382 0.53 0 2 (010,11,00)

Mg52, d 0,407,002 0,405,000 0.07 0 0 (000169)

Zn90, i 0,480,210 0,477,191 0.44 051 (0200,10,0)

Zn91, i 0,487,419 0,477,427 0.29 05 (110064)

Zn92, d 0,537,027 0,550,000 0.69 0,11,0 (101227)

Mg53, d 0,598,269 0,595,264 0.13 014 (000249)

Mg54, d 0,599,131 0,595,118 0.32 01 (000825)

Zn93, i 0,698,426 0,714,427 0.36 0 (0100,10,2)

Zn94, i 0,715,208 0,714,191 0.38 0 1 (0200,10,0)

Zn95, d 0,805,375 0,786,382 0.45 032 (010802)

r̃⊥ c̃

93

5

1

9

5 1

531

5

32

14

513

5

52

5

3

2

3

32

3

2

1

1

3

2

33

3



tively, the DLO matrix atoms and icosahedral posi�tions.

A question arises: to what extent do the distortionsof ideal DLO structure at a specific atom depend onthe chemical type of this atom and the type of its posi�tion? As was noted above, the symmetry of the localatom environment and the presence or absence ofshort bonds in this environment can play an importantrole in this case. Concerning the chemical type of theatom, it can affect the distortions indirectly (throughthe atomic position), because atoms of different ele�ments occupy different sets of positions in the struc�ture. For example, it was suggested in [14] that, sincethe observed bond�length ratios lAl–TМ/lTМ–TМ/lAl–Al inquasicrystalline alloys of aluminum with transitionmetals (TM) approximately coincide with the ratio ofideal bond lengths, r3/r5/r2, the Al–TM bonds in thesestructures correspond to the vectors r3 and r5; the TM–TM bonds correspond to the vectors r3, r5, and r2; andthe Al–Al bonds correspond to the vectors r5 and r2.This correspondence, in turn, imposes limitations onthe positions that can be occupied by atoms of a par�ticular element. In addition, even when atoms of dif�ferent types occupy similar positions in a structure,they interact differently with their local environment.

To characterize the atomic positions, we can usethe distance r⊥ of the atomic projection on the perpen�dicular space Е⊥ from the center of the projection ofthe entire quasicrystal (approximant) as the first�orderapproximation. In the case of cubic approximants, theprojection has a tetrahedral point symmetry and its

center coincides with the point where the symmetryelements intersect; for a lower symmetry structure theweighted�mean point of projection can be chosen asthe center. Figures 1–3 show the distributions ofatoms of the approximants under consideration overthe distance from the center of the projection on theperpendicular space Е⊥ and over the displacementfrom the ideal DLO position. It can be seen that ineach figure the groups of atoms of different elementsand occupying positions of different types are locatedin particular (partially overlapping) areas.

The distribution of atoms of the cubic approximantAl66.6Rh26.1Si7.3 of order 2/1 [10] is shown in Fig. 1. Itcan be seen that the Rh atoms located closer to the

0.2

0.50 1.0 1.5 2.0 2.5 3.0 3.5 4.0r⊥, rel. units

0.4

0.6

0.8

1.0

1.2Δ, Å

Fig. 1. Correlation of the distributions over the distance r⊥from the center of the projection on the perpendicularspace and the displacement Δ from the ideal DLO posi�tions, which were reported in [5], for atoms of the cubicapproximant Al66.6Rh26.1Si7.3 [10] of order 2/1 (sp. gr.

): (circles) Al(Si) and (asterisks) Rh atoms; DLOmatrix atoms and icosahedral positions are shown byclosed and empty symbols, respectively.

3Pm

0.2

0.50 1.0 1.5 2.0 2.5 3.0 3.5r⊥,rel. units

0.4

0.6

0.8Δ, Å

(b)

0.2

0.50 1.0 1.5 2.0 2.5 3.0r⊥, rel. units

0.4

0.6

0.8Δ, Å

(a)

Fig. 2. Correlation of the distributions over the distance r⊥from the center of the projection on the perpendicularspace and the displacement Δ from the ideal DLO posi�tions, which were reported in [5], for (a) atoms of the cubicapproximant Mg27Al10.7Zn47.3 [12] of order 2/1 (sp. gr.

) and (b) atoms of the 3/2–2/1–2/1 approximant(Ga, Zn)175 – δMg97 + δ [9] (sp. gr.Стс21): (circles)Zn(Al, Ga) atoms and (asterisks) Mg atoms; DLO matrixatoms and icosahedral positions are shown by closed andempty symbols, respectively.

3Pa

962



center of the projection on Е⊥ in comparison with theAl(Si) atoms forming the DLO matrix are less dis�placed from their ideal positions. The icosahedralpositions located even closer to the projection centerare displaced even less. (The separate icosahedralposition with an anomalously large displacement isdifficult to determine, and the atom occupying it hasthe short bond .) This systematic feature is retainedin Figs. 2 and 3. This is apparently related to the factthat the atoms whose projections on Е⊥ are closer tothe center have better local environment than theperipheral atoms. These atoms are convenient to useas reference points when determining the DLO struc�ture. It would even appear that these atoms form a“framework” of the quasicrystal or approximantaround which the structure (distorted on the whole) isformed.

Figure 2 shows the atomic distributions for the 2/1cubic approximant Mg27Al10.7Zn47.3 [12] (Fig. 2a) and the3/2–2/1–2/1 approximant (Ga, Zn)175 – δMg97 + δ [9](Fig. 2b). A comparison of these two approximants(one of which is not cubic) shows that alloys of similarchemical compositions exhibit similar regularities inideal structure distortions. It is of interest that heavierZn(Ga) atoms are much more displaced from theirideal positions than light Mg atoms. The same holdstrue for the cubic approximant Ca13Cd76 [13] of theorder 2/1 (Fig. 3), where lighter Ca atoms are muchless displaced in comparison with Cd atoms. Anotherfeature of the Ca13Cd76 crystal is that it contains, along

5'r

with a small number of icosahedral holes (three emptysquares in the bottom left corner), many difficult�to�determine icosahedral positions with several short

bonds ( ).

CONCLUSIONS

A comparison of the approximants related to dif�ferent systems of alloys (Fig. 1–3), and the approxi�mants belonging to the same system (Fig. 2) showedthat the chemical composition significantly affects thedistortion of the ideal structure of quasicrystals andtheir approximants. The largest deviations from theideal structure occur at large r⊥, i.e., at the periphery ofthe tube (or atomic surfaces). The same is likely tooccur in quasicrystals, thus resembling models withdiffuse or fractal boundaries of atomic surfaces. Inour opinion, among the problems stated in this study,the most interesting is the question of why the dis�placement of light atoms from their ideal positions insome alloys is less than that of heavier atoms. It ispossible that, to answer this question, it is necessaryto propose a new approach to the problem of choos�ing the ideal local configuration to describe the idealstructure of quasicrystalline alloys. In this context,the study of other recently found structures of icosa�hedral quasicrystals and their approximants [15] canbe helpful.

ACKNOWLEDGMENTS

This study was performed within an agreementbetween the Centre Nationale de Recherche Scienti�fique (France) and the Russian Academy of Sciences(projects 19047 and 21252) and was supported by theRussian Foundation for Basic Research, projectno. 05�02�16763�a.

REFERENCES

1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn,Phys. Rev. Lett. 53, 1951 (1984).

2. C. Janot, Quasicrystals: A Primer (Clarendon, Oxford,1997).

3. G. Bergman, J. L. T. Waugh, and L. Pauling, Acta Crys�tallogr. 10, 254 (1957).

4. V. E. Dmitrienko, J. Phys. France 51, 2717 (1990).

5. V. E. Dmitrienko and V. A. Chizhikov, Kristallografiya52 (6), 1077 (2007) [Crystallogr. Rep. 52 (6), 1040(2007)].

6. R. V. Moody and J. Patera, Lett. Math. Phys. 36, 291(1996).

7. V. E. Dmitrienko, Acta Crystallogr. 50, 515 (1994).

and3 5 2' ' ',r r r

0.4

0.50 1.0 1.5 2.0 2.5 3.0 3.5r⊥, rel. units

0.6

0.8

1.2Δ, Å

1.0

0.2

Fig. 3. Correlation of the distributions over the distance r⊥from the center of the projection on the perpendicularspace and the displacement Δ from the ideal DLO posi�tions, which were reported in [5], for atoms of the cubicapproximant Ca13Cd76 [13] of order 2/1 (sp. gr. ):(asterisks) Ca and (squares) Cd atoms; DLO matrix atomsand icosahedral positions are shown by closed and emptysymbols, respectively.

3Pa



8. V. E. Dmitrienko and V. A. Chizhikov, Kristallografiya51 (4), 593 (2006) [Crystallogr. Rep. 51 (4), 552(2006)].

9. G. Kreiner, J. Alloys Compd. 338, 261 (2002).10. K. Sugiyama, W. Sun, and K. Hiraga, J. Non�Cryst.

Solids 334–335, 156 (2004).11. T. Takeuchi, N. Koshikawa, E. Abe, et al., J. Non�

Cryst. Solids 334–335, 161 (2004).12. Q. Lin and J. D. Corbett, Proc. Nat. Acad. Sci. USA

103, 13 589 (2006).

13. C. P. Gómez and S. Lidin, Angew. Chem. Int. Ed. 40,4037 (2001).

14. V. A. Chizhikov, Kristallografiya 49 (4), 610 (2004)[Crystallogr. Rep. 49 (4), 537 (2004)].

15. W. Steurer and S. Deloudi, Crystallography of Quasic�rystals: Concepts, Methods and Structures (Springer,Berlin, 2009).

Translated by Yu. Sin’kov

Documents

Influence of the Chemical Composition of Alloys on the Ideal Local Order in Approximants of the Icosahedral Quasicrystals