Atomic structure of quasicrystals

Materials Science and Engineering, 99 (1988) 323-329

323

Atomic Structure of Quasicrystals* T. EGAMI Department of Materials Science and Engineering and Laboratoryfor Research on the Structure of Matter, Universityof Pennsylvania, Philadelphia, PA 19104-6272 (U.S.A.) S. J. POON Department of Physics, University of Virginia, Charlottesville, VA 22901 (U.S.A.)

Abstract

Progress in the determination of the atomic structure of quasicrystalline solids via pair distribution function analysis obtained by X-ray and neutron diffraction is reviewed. Common features of the local structure in these solids are described, followed by the discussion on the mechanism of stabilization of quasi-crystals by long-range internal stress fields.

we discuss later, these models compare less favorably than the quasicrystalline model in explaining the experimental observations, and furthermore they do not really improve our understanding of the mechanism of formation of these complex structures. In this paper, we summarize our recent progress in the experimental study on the icosahedral solids by X-ray and neutron scattering, and disucss the possible mechanism with which the atoms conspire to form the quasicrystalline structure.

1. Introduction

The observation of icosahedral symmetry in A186Mn14 by Shechtman et al. [1], which apparently contradicted an axiom in crystallography, and its explanation in terms of quasicrystalline structure by Levine and Steinhardt [2] generated strong interest which resulted in a large number of publications [3]. The lattice structure of quasicrystals has two unit cells which are quasi-periodically placed, and is most conveniently described by the projection from the sixdimensional space [3, 4]. Many of the solids now regarded as quasi-crystals have icosahedral symmetry, but other symmetries are also possible. Although the concept of quasi-crystal is mathematically appealing, it is not intuitively obvious what physical mechanism would provide a driving force to form such a seemingly complex structure. In fact, it is quite perplexing to find atoms arranging themselves in a quasi-periodic manner, as if they know how to project from the sixdimensional! Perhaps partly for that reason, various other possibilities to explain the icosahedral symmetry were suggested. The idea of twinned crystals was proposed by many authors [5]. The other ideas are based on packing of icosahedral clusters, either periodically arranged and twinned as proposed by Pauling [6], or randomly arranged retaining the long-range icosahedral orientational order [7, 8]. As

2. Diffraction studies of icosahedral solids

*Paper presented at the Sixth International Conference on Rapidly Quenched Metals, Montreal, August 3 7, 1987.

S(Q_) =

0025-5416/88/$3.50

Earlier structural studies concentrated on the identification of the symmetry and the range of positional order. The symmetry of the structure can be determined by the diffraction from a single grain, most often by electron diffraction [1]. The range of positional order is best assessed by high resolution X-ray diffraction using synchrotron radiation [9]. In these studies, only the position of the reciprocal lattice points and their peak shape are measured, and the question of the actual location of atoms in the solid is not addressed. The first step in determining the atomic positions is a careful measurement of diffraction intensities. The result then has to be compared with the models. It is often useful, however, to go through an intermediate step of calculating the atomic positional correlations. For single crystals, this step is the Patterson analysis, and for amorphous or polycrystalline solids it is the radial distribution function (RDF) analysis [10]. 2.1. Radial distribution function analysis The diffraction intensity of X-rays or neutrons include contributions from various scattering mechanisms, but through standard procedures we can obtain the normalized atomic structure factor

I ~

f~(Q)f.j(Q) exp(iO " ~.)/:.f(Q)>2

(l)

© Elsevier Sequoia/Printed in The Netherlands

324 25[--

-- I -phase

o

~_ 20 [ D EL

g

Pdss.sU20.6Si2o.6 [I1], and for crystalline (FrankKasper phase) and icosahedral A156Li34Culo [12]. These data were obtained at the National Synchrotron Light Source (NSLS) of Brookhaven National Laboratory. It is clearly seen in Fig. 1 that the local structure of icosahedral solids up to 6/~ is very similar to that of amorphous phase, even though the longrange order, and thus the diffraction pattern, is very different. The icosahedral and Frank-Kasper phases have even more similar local structure up to about 20/~, as shown in Fig. 2.

|

--- A -phase

! r5~ I

~

Z

v

N 05~ '/

:l

oo; d

2

4

6

I0

8

[2

14

r(~,)

Fig. I. Total atomic pair distribution functions of Pdss.sU20.6Si20.6: , in the icosahedral phase; - - - , in the amorphous phase [ 11].

where () is the scattering vector, f ( Q ) is the atomic scattering factor of the ith atom,/~u is the separation between the ith and jth atom, N is the number of atoms, and (. • .) denotes the compositional average. By the Fourier transformation of the structure factor, two-atom correlation functions can be obtained. If S(Q) is spherically averaged before the transformation, for instance by using powder samples, one obtains the RDF, 4~r2p(r), and the transformation without spherical averaging gives the Patterson function. Although ideally one needs a single crystal to carry out the Patterson analysis, it still is possible to do so on powder samples by indexing the Bragg peaks. But in this case overlapping peaks cannot be separated and the diffuse scattering cannot be properly taken into account. However, the RDF can be determined much more accurately directly from the powder diffraction data. Examples are shown in Figs. 1 and 2. Here we compare the pair distribution function (PDF), p(r), for amorphous and icosahedral

2.2. Differential distribution function analysis The values of PDF shown in Figs. 1 and 2 are the X-ray weighted total pair correlation functions for all the atoms in the solids. However, since all the icosahedral solids found so far are compounds or alloys, it is important to isolate the local environment of one element" from the others. This can be done by differential anomalous scattering (DAS) of X-rays, or neutron scattering using isomorphous or isotopic substitution. In the DAS study, one makes use of the energy dependence of the atomic scattering factor which is strong only very near the absorption edge [ 11, 13]. Since a tunable high intensity X-ray source is required for this experiment, synchrotron radiation is almost always used. By this technique, one can determine the differential structure factor of element A SA(Q) = ~

c,f,(Q)SAJQ)/(,f(Q))

(2)

where c~ is the atomic concentration of element ~ and S,p(Q) is the compositionally resolved partial structure factor

1 ~ 6i~6j#exp(iQ

S~( Q ) = N7~,c5 ..

•/~)

(3)

and TOP-FRANK-KASPER PHASE BOTTOM-ICOSAHEDRAL PHASE

A,ts. 5 Li 3 . 3 C u

6~ = 1 =0

'7 LL ,,--, CL

0

I

I

I0

20

I 30

40

r (~,) Fig.

2. Total

atomic

pair

distribution

functions

of

A156Li34Culo:(above), in the crystalline(Frank-Kasper) phase; (below), in the icosahedral phase [12].

when the ith atom is element otherwise

(4)

The differential distribution function (DDF) is obtained as a Fourier transform of SA(Q), and describes the atomic distribution around the element A. Figure 3 shows examples of a DDF, for icosahedral Pdss.sU2o.6Si2o.6 , determined using X-ray energies just below the Lm edge of uranium [11]. The uranium D D F describes U - U and U-Pd correlations, whereas the palladium D D F describes Pd-U and Pd-Pd correlations. Silicon atoms are practically invisible by Xrays in this case. A DDF can also be obtained by isomorphous substitution using neutron scattering, as shown in Fig. 4 for AlsoMn2o [14]. An equivalent of DDF can be determined by the extended X-ray

325

o

25[

F~) z

P

,' ----

20

U DDF Pd D D F

016

012

,1

00~

C-

15

lco

ok

'~

35t

'

'

A

~

i, I,'1:l~

/~

z

ooo "b', I!,,"....

i 'I

1

,~,,', <~,~-<

;,

.

tu

,

-004~

' I'

" ~" ~ h

"

,t

~

,, ;

~....1

s 0

%<,,

~_ oo

i)

m

O

2

4

6

8

I0

12

14

r(~)

Fig. 3. Differential distribution functions of icosahedral Pd58.8U20.6Si20.6:- - , for uranium atoms; , for palladium atoms [11]. Arrows indicate the distances between the vertices of the quasicrystalline lattice.

I2

04 os

g m m

0

• t," ~

:~

tt

+t

,'~

F-

T u_

o

-o~ o

I

2

3

4

5

6

7

t

il

8

9

I0

'!,

t

II

12

t

t 13

t 14

15

r (~.)

Fig. 4. Differential distribution function of icosahedral AlsoMn2ofrom manganese atoms determined by pulsed neutron scattering using isomorphous substitution of chromium for manganese [14]. Mn-Mn peaks appear as negative peaks, because of the negative scattering length of manganese. Arrows indicate the vertex-vertex distances in the quasicrystalline lattice.

absorption fine structure (EXAFS) measurement [15, 16]. But the E X A F S - D D F is not reliable beyond the nearest neighbours, and furthermore the nearest neighbour distance and the co-ordination number determined by EXAFS are often inaccurate unless a very careful calibration with a known substance is made.

3. Models of atomic structure

The correlation functions thus obtained, however, do not yield a three-dimensional structure directly, and the structure can be determined only through a comparison with structural models. Our results were compared with the quasicrystalline, icosahedral glass and crystalline models.

3.1. Quasicrystalline model By decorating the quasicrystalline lattice, i.e. by placing atoms at specific sites in the quasicrystalline

Fig. 5. Uranium DDF for icosahedral Pd58.sU2o.6Si2o.6 (A), compared with U-U PDF for the QC model (B), RQC model (D) and RQC model with additional uranium atoms to complete the QC rhombodedra (C) [17].

unit cells, models of atomic structure can be produced. The simplest model is to place atoms at the vertices, or the corners, of the lattice. The closest atomic distance in this case is about a half (0.564) of the unit cell edge length, or the quasi-lattice constant a. Therefore a has to be at least of the order of second neighbour distance. For Pd-U-Si, this model was in fact found to account for the position of uranium atoms [11, 17]. As shown in Fig. 5, the experimental uranium D D F compares well, beyond 12/~, with the quasicrystailine (QC) model with a uranium atom at each vertex. At shorter distances, the experimental D D F is dominated by U - P d correlation which is not included in the model, since the positions of palladium and silicon atoms have not yet been determined. Also the shortest U - U distance in the model (2.9 ,~) appears to be replaced by a larger one (about 4 ,~), and its effect on the structure has not yet been evaluated. The QC lattice constant was 5.14/~, and indeed corresponds to the second neighbour distance. The density of the vertices with this QC lattice constant, 0.0113 atom/~ -3, was found to be almost exactly equal to the physical density of uranium atoms in this compound, supporting the QC model. Thus, aside from details, the QC model seems to account for the structure of P d - U - S i quite well. The peak intensities of the structure factor of A156Li34Culo were well explained assuming that aluminium and copper atoms occupy both the vertices and the edge centres of the unit cells [18], as proposed by Guyot and Audier [19] and by Henley and Elser [20]. The compositional ordering between aluminium and copper atoms appears to be either very weak or absent. Lithium atoms are placed along the long diagonal of the prolate cell. However, it is not entirely clear if the structure of AlaoMn2o can be explained by the QC model as proposed by Elser and Henley [21]. Further examination is under way.

326

3.2. Icosahedral glass model The icosahedral symmetry of the diffraction pattern can be obtained by the random packing of icosahedral clusters retaining the orientational order. For this icosahedral glass (IG) model, the width of the diffraction peaks is finite, but it can be quite small, almost equivalent to the experimental peak widths [8]. Thus it is difficult to determine from the peak width data alone which of the two models, QC or IG, describes the real solids better [22]. In addition, when the icosahedral clusters are placed in such a way to share a prolate rhombohedron which is one of the quasi-crystalline unit cells between them, the structure is in fact very similar to the QC model. This model, the random QC (RQC) model, has a PDF which is very similar, except for amplitude, to that of the QC model, as shown in Fig. 5. Within a radius of 10a, 50% of the lattice points of the RQC model were found to coincide with the lattice points of the QC model [ 17]. The difference between the QC and RQC models is discussed in more detail in the next section. 3.3. Twinned crystal model However, it appears to be possible to eliminate the twinned crystal model from consideration, since the multiple twins necessary to explain the symmetry by the crystalline model have never been observed by electron microscopy, and the models have not convincingly explained both the X-ray and electron diffraction patterns simultaneously. The powder diffraction pattern from the 820-atom unit cell recently proposed by Pauling, for instance, has many peaks which are closely positioned to the peaks observed for the icosahedral solids [23]. But at the same time it has an even larger number of other peaks which do not show up in the diffraction pattern for the icosahedral (i) phase. It seems almost impossible to introduce so many extinction rules to eliminate these unwanted peaks from the diffraction pattern.

4. Comparison between QC and RQC models We will now compare with more care the two models, the quasicrystalline model and the random quasicrystalline model, both of which account for the experimental results to a certain degree.

4. I. Packing consideration Let us first consider the QC model in terms of the IG picture. The analysis of the QC model shows that 5.5% of the vertex points have an icosahedral environment, i.e. the nearest vertex points form an icosahedron [24]. Since the points on the surface of an icosahedron cannot be at the same time a centre of

another icosahedron, this means that in the QC model about 72% of the lattice points belong to the icosahedral clusters, while the remainder, 28%, are filling in between the clusters. In comparison, in the RQC model which has the same physical density as the QC model, these fractions are 50% each, since the random packing of icosahedral clusters tends to leave small gaps in which other icosahedral clusters cannot fit [ 17]. Thus the QC model represents a considerably better packing method of icosahedral clusters than the RQC model. However, the body-centred cubic (b.c.c.) packing of icosahedral clusters gives the density as 80% of the QC density. Therefore, an effort to increase the packing density of the RQC model tends to produce the b.c.c, local order, rather than the QC structure.

4.2. Size of the unit cell As we have shown, the size of the QC lattice is about 5/~, or the second neighbour distance. Consequently, the size of the isocahedral cluster for the RQC model is about 12 ,~ in diameter, and the cluster contains 50-100 atoms. Thus the unit structure of the RQC model is considerably larger than that of the QC model. Although clusters of such a size are found in the Frank-Kasper (FK) phases, it is difficult to believe that such a cluster is a molecule stabilized by chemical bondings. 4.3. Number of atomic sites As a result of a larger unit size of the RQC model, the RQC model is more complex than the QC model in terms of the atomic structure. This is in contrast with the naive notion that the QC model is more complex since it has no translational symmetry. Consequently, for instance in AI-Li-Cu, lithium has two kinds of lattice sites in the RQC model, one in the icosahedral cluster (A site) and the other between the two (B site), whereas in the QC model there is only one type of site, within the prolate rhombohedra. The first peak in the PDF of the FK phase (Fig. 2) has a small shoulder at about 3.4 A, which is a signature of the B site. In the i phase, the first peak of the PDF has no shoulder, suggesting that the B site does not exist in the i phase, supporting the QC model. 4.4. Comparison with experiments The direct comparison of the observed structure factor of Als6Li34Cu~o to the corresponding structure factor calculated for the QC model and RQC model suggests that the QC model compares with the experiment better than the RQC model does [ 18]. In addition, the absence of the B site mentioned above and the agreement of the physical density of the uranium

327

sites in the Pd-U-Si alloy with the QC model [17] indicate that the QC model provides a better description of the atomic structure of existing icosahedral solids.

the strong atomic level stresses found in metallic glasses [26] should also exist in the icosahedral solids. For an assembly of atoms interacting via a central force potential ~b(rij), the atomic level stresses are defined by [26, 27]

5. Local structure of icosahedral solids

tr/"(i) = ~ / / ~ r,j~b'(r/) Y/"(r0)

1

We will now summarize common features of the local structure of icosahedral solids as represented by the PDF to gain insights regarding the mechanism of structural stability of these solids. They are as follows. (i) The nearest neighbour (1NN) peak and the second nearest neighbour (2NN) peak are well separated. This means that the predominant local structural unit is a tetrahedron, as in the metallic glasses

[25]. (ii) The vertices of the QC lattice are almost always filled by an atom. (iii) The INN peak of the PDF and D D F is quite wide, suggesting a large distribution in the atomic distances. This is not a consequence of termination of the structure factor during the process of obtaining PDF from S(Q). We found that S(Q) of i-phase solids attenuates with Q faster than both crystalline solids and many transition metal-based amorphous solids. In particular, the last point is important for the following discussion. In Table 1, the normalized INN peak width is shown, and is compared with the thermal width, a,h.

(5)

./

where f~i is the atomic volume of the ith atom, and Y/"(~) are the spherical harmonics. The definition can easily be extended to non-spherical potentials, and the stresses can also be calculated quantum-mechanically [28]. As shown in eqn. (5), the deviation of the near neighbour distance r o. from the equilbrium distance to, defined by the minimum of the potential $, produces the stress. Thus the width of the 1NN peak of PDF is a good measure of the magnitude of the atomic level stresses. The root mean square (r.m.s.) value of the hydrostatic pressure in amorphous iron was calculated to be about 160 kbar [29], and stresses of similar magnitude are expected to be found in quasi-crystals. The origin of the atomic level stresses is the size mismatch between the atom and the hole in which the atom is placed [29]. If the strain which has to be applied to an atom i so that it fits the atomic site without disturbing the neighbours is given by [E/"(~3]T, then using the results in continuum mechanics by Eshelby [30], the local stress at the ith atom owing to this mismatch is given by l+v

B[Eo°(ri)] T

(6)

4(4 - - 5 V ) G[(.2m(~i)]T 15( 1 -- v)

(7)

[O'o°(ri)] 1 = ~

6. Atomic level stresses in quasicrystais

As shown in Table 1, the atomic distances in the icosahedral solids have a substantial spread, indicated as the width of the INN peak, and the spread is similar to that in the metallic glasses. This suggests that

TABLE 1

[a2,.(f)]l

where B and G are the bulk and shear modulus and v is the Poisson ratio. The local structural mismatch

Width of the first peak of the atomic pair distribution function (PDF)

Composition

PDF

Measurement temperature (K)

Position o f the peak a

Gaussian width a (A)

a /a

O'th/fl

(A) Icosahedral AlsoMn 15Sis a Icosahedral Pdsa.sU2o.6Si2o.6c Amorphous

Mn-DDF b

10

2.63

0.171

0.065

0.019

U-DDF d

~- 300

3.06

0.245

0.080

0.023

Pd58.8U2o.65i20.6 c

U DDF d

- 300

3.05

0.275

0.090

0.023

Face-centred cubic AIa

PDF e

10

2.80

0.088

0.032

0.019

a Ref. 14. b D D F with manganese atom at the origin, determined by pulsed neutron scattering. CRef. II. d D D F with uranium atom at the origin, determined by differential anomalous X-ray scattering using synchrotron radiation. e PDF determined by pulsed neutron scattering.

328 produces, in addition, a long-range stress field around it, which can also be well described by the theory of inclusion by Eshelby. By rewriting the results by Eshelby in the spherical representation, one obtains the long-range field at the j t h atom owing to the mismatch at the ith atomic site

[0.00(5)] LR

_ _

3BCo.2 ru3 ~ [e.2"(5)]Ty2-m(Pi)

(8)

[a=m(6)] LR = 2GC2,o [EoO(5)]T r2m(~) + ( - 1) m2GC2'2

x

~, m [E2m'(ri)]T

3 - m m' m

(

51/2\ i -- 5

m'

For quasicrystals it is convenient to use the density wave approximation in calculating the local stresses and the interaction energy. The atomic density p(~) is given by

(13)

-

where 2n ~/2 f ¢'(z)z 3 dz Go = ---if-

C°'2 - 5~ 1/2(1 - v)'

. [ 5 ] '/2 c~,~ = ysS_~ L1-T~nj

(12)

_ Pa exp(iQ " r~Ft(O)Y/"(O) ] = p(O[2. p*oGoa,,o+ ~o

where D

(11)

[E,m(r')]T (ru)

m

q- ~ ~55 ~'~[E2m(ri)]T[a2-m(ri)]LR}

P(r) = Po + ~ P0 exp(iQ • ~) o The local mismatches are then given by

r# 3

rq 3

Eint= ~/ {47Z['][EoO(ri)]T[0-oO(I~i)]LR

(14)

~ ( 1 + v)

C2.o - 2n 1/2(1 - v)'

F~(Q) = 4~ h i - ; f Jt(Qz)ck'(z)z 3 dz (Co = B, C2 = 2G)

(15)

(1o)

and (: : :) is the Wigner's 3-j symbol. The atomic level stresses can exist not only in glasses but in all kinds of crystals as long as the unit cell has more than one inequivalent atomic site. Since the presence of such atomic level stresses costs energy owing to their elastic energies, atoms attempt to arrange themselves to avoid the energy penalty arising from the stresses. The best way to do so is to eliminate the stresses altogether, and for that reason elements usually crystallize into very simple structure with only one crystallographic site (Bravais lattice). However, in multicomponent systems with significantly differing atomic sizes, forcing all atoms into one kind of site by forming a solid solution of a simple structure results in strong local stresses [31]. The condition of instability of solid solution owing to such local stresses in the spirit of H u m e Rothery rules was in fact found to be the glassforming condition for amorphous alloys [31], and is certainly the necessary condition for the formation of icosahedral solids. Therefore, usually in multicomponent systems the atomic level stresses can never be made equal to zero everywhere. To reduce the energy penalty arising from the local stresses, the system would then attempt to minimize the interaction energy among the local stresses. The elastic interaction energy between the local mismatches is given by [30]

and Jr(x) is a spherical Bessel function. The interaction energy evaluated by eqns. (6)-(15) consists of three- and four-wave terms which stabilize the structure in the Landau free energy expansion [32-34]. It can be shown that the quasicrystalline structure very effectively reduces this interaction energy, since the most important Q vectors, i.e. (100000) and (110000), are very close to each other in magnitude (different by 5%), and the high symmetry makes the structural averages of the spherical harmonics zero except when l 1> 6 [35]. Thus it is most likely that the interaction among the local mismatches via long-range stress fields is the mechanism to produce the QC structure. Acknowledgments

The authors are grateful to their collaborators for valuable contributions. This work was supported by the National Science Foundation through grants DMR85-19059, DMR86-17950 and DMR85-12869. References

1 D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett., 53(1984) 1951. 2 D. Levine and P. J. Steinhardt, Phys. Rev. Lett., 53 (1984) 2477. 3 P. J. Steinhardt and S. Ostlund (eds.), Quasicrystals, World Scientific, Singapore, 1986; C. L. Henley, preprint.

329 4 V. Elser, Acta Crystallogr. A, 42 (1986) 36. 5 R. D. Field and H. L. Fraser, Mater. Sci. Eng., 68 (1984) LI7. 6 L. Pauling, Nature, 317 (1985) 512. 7 D. Shechtman and I. Blech, Metall. Trans. A., 16 (1985) 1005. 8 P. W. Stephens and A. I. Goldman, Phys. Rev. Lett., 56 (1986) 1168. 9 P. A. Bancel, P. A. Heiney, P. W. Stephens, A. I. Goldman and P. M. Horn, Phys. Rev. Lett., 54 (1985) 2422. 10 B. E. Warren X-Ray Diffraction, Addison-Wesley, Reading, 1969. 11 D. D. Kofalt, S. Nanao, T. Egami, K. M. Wong and S. J. Poon, Phys. Rev. Lett., 57(1986) 114. 12 W. Dmowski, T. Egami, Y. Shen and S. J. Poon, Philos. Mag., Lett, 56(1987) 63. 13 P. H. Fuoss, P. Eisenberger, W. K. Warburton and A. I. Bienenstock, Phys. Rev. Lett., 46(1981) 1537. 14 S. Nanao, W. Dmowski, T. Egami, J. W. Richardson, Jr., and J. D. Jorgensen, Phys. Rev. B, 35(1987)435. 15 E. A. Stern, Y. Ma and C. E. Bouldin, Phys. Rev. Lett., 55 (1985) 2172. 16 Y. Ma, E. A. Stern and C. E. Bouldin, Phys. Rev. Lett., 57 (1986) 1611. 17 D. D. Kofalt, 1. A. Morrison, T. Egami, S. Preische, S. J. Poon and P. J. Steinhardt, Phys. Rev. B, 35(1987) 4489. 18 Y. Shen, S. J. Poon, W. Dmowski, T. Egami and G. J. Shifter, Phys. Rev. Lett., 58 (1987) 1440.

19 P. Guyot and M. Audier, Philos. Mag. B, 53 (1986) LA3. 20 C. L. Henley and V. Elser, Philos. Mag. B, 53 (1986) L59. 21 V. Elser and C. L. Henley, Phys. Rev. Lett., 55 (1985) 2883. 22 P. M. Horn, W. Malzfedt, D. P. DiVincenzo, J. Toner and R. Gambino, Phys. Rev. Lett., 57 (1986) 1444. 23 L. Pauling, Phys. Rev. Lett., 58(1987) 365. 24 C. L. Henley, Phys. Rev. B, 34 (1986) 797. 25 J. L. Finney and J. Wallace, J. Non-Cryst. Solids, 43(1981) 165. 26 T. Egami, K. Maeda and V. Vitek, Philos. Mag. A, 41 (1980) 883. 27 M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1954. 28 V. Vitek and T. Egami, Phys. Status Solidi, in the press. 29 T. Egami and V. Vitek, in V. Vitek (ed.), Amorphous Materials: Modeling of Structure and Properties, TMS-AIME, Warrendale, PA, 1983, p. 127. 30 J. D. Eshelby, Proc. Roy. Soc. A, 153 (1957) 376. 31 T. Egami and Y. Waseda, J. Non-Cryst. Solids, 64 (1984) 113. 32 P. Bak, Phys. Rev. Lett., 54(1985) 1517. 33 D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy, P. J. Steinhardt and J. Toner, Phys. Rev. Lett., 54 (1985) 1520. 34 N. D. Mermin and S. M. Troian, Phys. Rev. Left., 54(1985) 1524. 35 T. Egami, unpublished.