DIFFRACTION FROM QUASICRYSTALS : ORIENTATIONAL ORDER, ATOMIC STRUCTURE, AND ELASTICITY

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DIFFRACTION FROM QUASICRYSTALS :

ORIENTATIONAL ORDER, ATOMIC STRUCTURE, AND ELASTICITY

M. Jarić

To cite this version:

M. Jarić. DIFFRACTION FROM QUASICRYSTALS : ORIENTATIONAL ORDER, ATOMIC STRUCTURE, AND ELASTICITY. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-259-C3-270.

�10.1051/jphyscol:1986328�. �jpa-00225740�

JOURNAL DE PHYSIQUE

Colloque C3, supplkment au n o 7, Tome 47, juillet 1986

DIFFRACTION FROM QUASICRYSTALS : ORIENTATIONAL ORDER, ATOMIC STRUCTURE, AND ELASTICITY

D e p a r t m e n t of P h y s i c s , Harvard U n i v e r s i t y , C a m b r i d g e , MA 0 2 1 3 8 , U.S.A.

Abstract. Work related t o several properties of the structure factor of quasicrystals is summarized. It is shown that bond-orientational order induces icosahedral quasicry- stalline order. A phase with pure icosahedral orientational order is also possible and it may lead to diffraction patterns similar t o the ones found experimentally. The geometric structure factors for the two commonly used methods of modeling quasicry- stal structures are also derived. It is shown that decoration of the quasicrystalline tiles generally lifts the nodes of the intrinsic structure factor. The effect of thermal fluctuations on the structure factor of quasicrystals is examined. Phonon and phason elastic constants are calculated t o obtain the Debye-Waller factor and the diffuse ther- mal broadening. It is suggested that the diffuse broadening (thermal, or due t o quenched phasons) might explain the observed widths of the diffraction peaks of icosahedral quasicrystals as well a s the appearance of the sheets of diffuse intensity observed in the decagonal quasicrystals.

1. Introduction

In this paper I shall summarize my recent work relevant t o the understanding of icosahedral diffraction patterns 11. There are three parts t o this work. In the first part [2,3 I shall simply

6 I

try t o understand t e origin of icosahedral diffraction patterns. That is, I shal try t o under- stand the stability of structures which may produce such patterns. The main conclusion will be that bond-orientational order plays an essential role in the process of solidification. In particu- lar, we shall see that bond-orientational order might induce positional ordering. Moreover, it follows from our approach that icosahedral and octahedral bond-orientational ordering will be universally preferred, and that they can induce icosahedral vertex or FCC structures, in agree- ment with experimental observations (41. Additionally, the purely bond-orientationally ordered phases will generally exhibit structure factors with icosahedral symmetry and Lorentzian peaks a t finite wavevectors in the directions s f would-be reciprocal lattice vectors. If these peaks were sufficiently sharp they could, through higher order interaction terms, generate peaks approxi- mately a t the harmonics of the fundamental peaks.

One can argue that the importance of bond-orientational ordering in solidification of icosahedral quasicrystals indicates the importance of packing considerations and that it reflects probable icosahedral clustering of the constituent atoms in the icosahedral structure. This view of the structure is currently pursued in several attempts a t the structure determination (5-71.

In the second part of this work [8] I shall examine the connection between the atomic structure of a quasicrystal and its diffraction pattern. The motivation behind this investigation is to try t o determine an analog of the geometric structure factor, usually defined for ordinary crystals, so that the structure of quasicrystals may be viewed in terms of a "quasilattice" and its decora- tion.

Although a quasicrystal can be viewed as a cut through a higher dimensional crystal [9,10], determination of this higher dimensional crystal structure does not seem feasible at present par-

*on leave from Department of Physics, North Dakota State University. Fargo. ND 58105. U.S.A.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986328

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tially because the higher dimensional atoms have t o be represented by surfaces. Therefore, we shall focus on quasicrystalline structures which can be obtained by two currently used and gen- erally inequivalent methods: the projection method and the tiling method [5-71. While in the first case the structure factor naturally splits into an intrinsic factor and a geometric structure factor this is not true for the tiling method where a node in the intrinsic structure factor will generally be lifted by an atomic decoration. Moreover, in both cases the geometric structure factor does not drop out of the total structure factor in the zerc-wavelength limit. The reason is that no two infinite environments are identical in a quasicrystal. Consequently, there cannot be a firm separation between the quasilattice and the decoration contributions to the scattering.

In the third part of this work [ll] I shall examine the effect of thermal fluctuations on the diffraction patterns from icosahedral quasicrystals. Therefore, it will be necessary t o calculate the elastic constants associated with long-wavelength low-frequency modes. Besides the usual phonon modes we shall find three additional modes which are called phasons.

The elastic constants will be calculated in a formalism based on the Ramakrishnan-Yussouff [12]

theory of solidification. The two constants will be related t o the ~isual, phonon strains, another two constants will be related t o the phason strains, while one coilstant will describe the cou- pling of the two strains. Starting from this elastic energy, we shall derive the Debye-Waller factor and the diffuse thermal broadening due t o both phonons and phasons. The Debye-Waller factor is similar t o the case of crystals except that it also contains a Gaussian in the phason directions. The diffuse thermal broadening is manifested in l / q 2 singularities a t the reciprocal lattice points. However, the broadening is very anisotropic and its scale is proportional t o the square root of a quadratic form in the reciprocal hyperlattice vectors. Therefore, large phason components of the vectors will dominate the peak widths. The phason dominance would be particularly strong if the phasons are quenched. It is possible that some of the observed peak broadening [4] is partially thermal in origin. One could also speculate that the sheets of diffuse intensity observed in the decagonal quasicrystals [13] are also due t o thermal fluctuations. T o verify this one would have t o calculate the in-plane elastic constants of these quasicrystals.

2. Orientational Order and Quasicrystals [2,3]

The traditional Landau-Ginzburg approach t o solidification 114,151 focuses on the positional ordering manifested through the emergence of a nonuniform equilibrium coarse grained density p(x). At equilibrium, p(x) minimizes the Landau free-energy functional F p [p] which is assumed t o be a low order expansion in the Fourier components p(q):

The expansion is also of low degree in the derivatives of the density. For example,

A ( ~ ) = A ~ + A ~ ~ ~ + . . . . (2)

This free energy was analyzed by Baym et al. [16 and by Alexander and McTague [15]. Under the assumption that the fourth-order term Fp4[d in the free energy expansion (1) is isotropic in the order parameter space, they were able to show that the universal third-order term Fp3[p]

forces the BCC structure t o be the most stable crystalline structure. Indeed, this seems to correlate very well with the fact that many monoatomic crystals share this structure.

Alexander and McTague 151 also argued that other cubic and hexagonal structures are not universal in the sense in w

I

The most natural candidate is the bond-orientational order parameter n(x,ii) which represents the density a t x of the- (geometric) bonds in the direction of the unit vector fi (171. At the coarse grained level, this order parameter is an independent order parameter although a t the microscopic level it can be completely expressed in terms of the microscopic density.

Clearly, the orientational ordering should play an even more important role in solids containing obvious atomic clusters. For example, icosahedral clustering has been observed in metallic glasses, supercooled liquids, and many crystalline alloys 118). It has also been proposed that certain icosahedral clusters are the main building blocks of icosahedral quasicrystals [5,6].

Whenever such clustering occurs, it is possible that the solid is more robust t o fluctuations in the orientational degrees of freedom than in the positional ones [7].

The importance of the orientational ordering is also evidenced in the difficulties of the purely positional Landau theories which attempt t o describe icosahedral phases. These theories are reviewed elsewhere in this volume [19/.

An additional motivation for the introduction of orientational ordering into the study of solidification is offered by recent experiments by Bancel et al. (41 who performed high resolution X-ray powder diffraction measurements of icosahedral AlMn. They found that the diffraction peaks had widths corresponding t o the positional correlation length of the order of several hun- dred A. Hence, the icosahedral AlMn could be orientationally ordered but positionally disor- dered [7].

All of the above motivation strongly suggests that solidification should be viewed as an inter- play between positional and orientational ordering 21. Hence, a new physical mechanism of solidification emerges, namely bond-orientationa 1 ly induced solidification. For such solidification we find that icosahedral and FCC positionally ordered structures are "universally"

preferred. Our analysis [2] is summarized in the rest of this section.

Studies of three-dimensional bond-orientational ordering were initiated by Nelson and Toner [17]. They proposed a scenario in which solidification proceeds through an intermediate, octahedratic phase characterized by long-range octahedral bond-orientational order without positional order. Crystal-to-octahedratic transition is driven by unbound dislocation loops while liquid-to-octahedratic transition is driven by an orientational order parameter. Consequently, two independent descriptions were necessary. This is unlike our approach in which both transi- tions are described by a single free energy functional [2].

In order t o describe orientational ordering, Nelson and Toner used the L = 4, q = 0, com- ponents of the orientational order parameter,

They constructed corresponding Landau free-energy up t o third-order terms. T o this order, besides the trivial quadratic invariant

I,(n) = C s z , m Q ~ - m (4)

there is only one thrid degree invariant:

In the absence of anisotropic quartic terms, the equilibrium density maximizes lf3(a)1.

Within the special subspace given by

n ⁼ (n4-, ⁰⁰⁰ n,, ⁰⁰⁰ a,,) ₍₆₎

it was found in [17 that I I ~ ( S ~ ) ~ is maximized by octahedral symmetry when

J-

I = I , , = 5/14 Ifl4,l.

Several years later, bond-orientational order was also investigated in molecular dynamic simula- tions of supercooled liquids revealing an extended icosahedral orientational order 1201. In search of a theoretical exy:anation of this ordering, the Landau theory approach was again used and attention was focused on the third order term in the expansion. In particular, this term was evaluated for L ⁼ 4, 6, 8, and 10 and for aLm given by FCC, BCC, SC, HCP and icosahedral clusters. It was found that L = 4 and L = 6 are the most important harmonics.

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For L = 6 the following subspaces were examined:

It was f o u e t h a t the icosahedral symmetry maximizes l13(fi)l when InF5! = m6,1 ^{= 7/11 IflG0l.} Thus, the icosahedral (L = 6) or the octahedral (t = 4) bond- orlentational order seemed t o be "universally" preferred.

However, these conclusions were based only on a partial analysis and in order t o confirm them it was necessary t o maximize II3(0)I over the full 13-dimensional (L = 6) and q-dimensional (L = 4) spaces. The same problem arose earlier in studies on the buckling of spherical shells and on convection in spherical symmetry. In this context Busse 1211 conjectured in 1975 that the absolute maximum of I13(fi)l does indeed correspond to icosahedral (L = 6) or octahedral (L = 4) symmetries. A few years later Sattinger [22] proved that these symmetries correspond to at least local maxima of l13(fi)1.

The discovery of icosahedral quasicrystals stimulated renewed interest in the icosahedral bond- orientational order and it appeared necessary t o answer the following three questions:

- What is the absolute maximum of 113(a)1 and what is its symmetry?

- How universal is this symmetry, that is, what are the anisotropic quartic invariants and what is t h e full phase diagram for bond orientational ordering?

- What is the effect of bond-orientational ordering on the positional ordering?

By analyzing the complete orientational Landau free energy'

where I,,(n) = [1~(s2)]~, 14,(S1) = d13/dn . d13/df2, and 142(n) = d2141/df22 : d2~41/dS12, it was recently [3] possible t o answer the first two questions:

- Busse's conjecture is valid.

- For L = 2, only the uniaxial phase is stable; for L = 4, the octahedral phase dominates the phase diagram but the uniaxial phase is also accessible; for L = 6, the icosahedral phase dominates but the uniaxial, octahedral, and hexagonal phases are also accessible.

The above answers were made possible largely due t o recent group theoretical developments in approaches t o the minimization of group invariant Landau potentials in phase transitions and Higgs potentials in gauge theories [24]. The solution involves two steps: First the problem is reduced t o a linear programming problem [by making a nonlinear change of variables,

fl ^4. ( I ~ ~ , I ~ ~ , I ~ ) ] . Then, the convex hull of the domain of the new variables is determined by explo~ting the knowledge of the symmetry planes in the aspace. In fact, there is a one-to-one correspondence between the phase diagram and the convex hull.

The a d w e r t o the third question can be found [2] after realizing that in the presence of both the orientational and the positional order parameters the total free energy must include an interaction term, i.e.,

Translational and rotational invariance imply that t o lowest order and for given L

where a is a coupling constant.

Here we are describing a uniform bond-orientational order, i.e., we assume the gradient term is anan.

This would not be a correct model for liquid crystals with frustration [23].

A similar interaction was introduced earlier [20], and it was observed that it implies that posi- tional ordering always generates orientational ordering while orientational ordering does not automatically generate positional ordering. However, the significance of this interaction does not seem t o have been fully appreciated.

The most interesting situation occurs when the orientational t r a ition temperature is above the positional transition temperature. As the orientational order

TR}

The first obvious consequence is that A(q) acquires a directional, 9 dependence. Therefore, the minimum of A(q) not only determines the magnitude of the wavevectors q of the density waves, but it also selects their directions consistent with the symmetry of the orientational ord- ering. In particular, icosahedral symmetry picks the icosahedral vertex or face directions while octahedral symmetry picks the corresponding octahedral directions.

The second consequence is that, since the orientational orderingtransition is discontinuous, the orientational orderin may sufficiently suppress the minima of A(q) so that p(q) orders at the minimal qils, i.e., fp(qi)) + ^0. In such a case one would have solidification induced by orien- tational ordering. Such solidification seems to be a good candidate in the case of icosahedral quasicrystals (L = 6) and it also could explain frequent solidification into the FCC crystalline structure ( L = 4 ) . An attractive feature of this view is that both of these types of solidification are universal in the sense that they are driven by the universal terms in the free energy.

It is also interesting t o analyze the structure factor of a purely orient_ationally ordered phase.

To lowest order in fluctuations the structure factor is proportional t o A(~)-', and it will conse- quently have anisotropic peaks at the qils. Depending on the interaction constant aL(q), these peaks may be fairly sharp. In addition, higher order coupling terms of the type

generally generate peaks in the structure factor near the n-th harmonics of the fundamental wavevectors qi. This can be easily verified in the perturbation expansion of the structure f a o tor which gives rise t o terms of the form

In general, a s the order is increased the peaks are shifted, broadened, and reduced.

3. Atomic Structure and Diffraction Patterns [8]

One of the most successful tools for the determination of crystal structures is the analysis of diffraction patterns. From the positions of the peaks one first determines crystal's Bravais lat- tice. Then, modulation of the peaks is related t o the atomic content (decoration) of the unit cell associated with the lattice. The modulation can be extracted from the structure factor S(q) which splits into two factors

I ( q ) is the intrinsic structure factor given by

and is equal t o the scattering from the Bravais lattice of point scatterers, while G Q) is the geometric structure factor equal t o the scattering from the atoms located a t points of a sin-

gle unit cell: t

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where f is the scattering amplitude of these atoms, and Q's are the reciprocal lattice vectors.

A natural question arises as to whether there is such a natural decomposition of the structure factor of quasicrystals and, in particular, whether there is a natural definition and classification of Bravais quasilattices.

An answer to this question can be found in earlier works on incommensurate crystals 191.

In order t o be specific we shall concentrate on the use of diffraction patterns which can be indexed by six vertex vectors of an icosahedron. As emphasized recently by Bak [lo], whatever the density pN(x") which produces such a diffraction pattern is, it can be represented as a cut through a periodic (crystalline) six-dimensional density p(x",xl),

where It refers t o the physical three-dimensional space and 1 refers to the remaining three- dimensional space. The structure factor of the quasicrystal can now be written in a simple form analogous to (14),

where

and

is the geometric structure factor of the hypercrystal. It is assumed that there is one-tc-one correspondence between the reciprocal hyper1atti:e vectors Q and their projections Q" which can be verified for the icosahedral case we study.

Therefore, a t least in principle, the analysis of a quasicrystalline structure can proceed in a similar way as in the case of crystals. One first interprets the three-dimensional diffraction pat- tern as the six-dimensional diffraction pattern of a hypercrystal. Then one uses the geometric structure factor G(Q) t o determine the decoration of the hyperunit cells. However, an immediate difficulty emerges since the fact that real atoms are point-like implies that each atom must be represented by a three-dimensional surface in the six-dimensional hypercrystal. Since nothing is known about the chemistry of such surfaces it seems unlikely that their direct deter- mination can be achieved a t present.

Consequently, there are two, more physical, special cases of the above method which are currently being used in the modeling of the icosahedral quasicrystalline structures. The first method is the projection method (25,261 in which one decorates the hypercrystalline Bravais lat- tice by pointlike atoms, cuts a slab of the hypercrystal parallel t o the physical space, and finally projects the content of the slab into the physical space. In the second method, a Bravais quasilattice is first constructed, a tiling associated with the quasilattice is identified and the fun- damental tiles are decorated [5,6].

In the projection method the quasicrystal density can be written as

p: (2) =

x x

* Reference [8] focuses on a twc-dimensional case where there is no such one-tc-one correspondence.

where R is a hyperlattice vector, the atoms with scattering amplitude f are located a t points F in the hyperunit cell, ~ ( d ) equals unity when d is inside the sla6 while it is aem otherwise, and y determines position of the slab. The structure factor associated with this density can be written as a product of an intrinsic structure factor and a geometric structure factor,

s;(~")

-

where

and

This identification of the intrinsic structure factor is indeed reasonable since S; reduces to I"

in the case of one atom per unit cell.

In the quasilattice decoration method it is first necessary t o specify a primitive, "Bravais"

quasilattice. At present it is not known what a good definition of Bravais quasilattices would be. However, a necessary condition is that a tiling associated with the quasilattice should have a Jinite number of distinct tiles. For example, the Voronoi polyhedrons associated with the quasilattice may be considered as fundamental tiles in which case the above condition reduces t o the requirement that there should be a finite nunmber of distinct local environments of the quasilattice vertices.

A Bravais quasilattice may be constructed by the projection from a Bravais hyperlattice.* Two- and three-dimensional Penrose quasilattices can be constructed in this way. For the three- dimensional Penrose quasilattice there are 24 different types of environments (without distin- guishing those related by symmetry) [27] and there is the corresponding number of different types of Voronoi polygons.

Let us distinguish quasilattice points of different types by an index p. Then, all points p are projected from a slab specified by the window function wP(xL) such that

and the atomic density can be written as

where is the position of the atom with scattering amplitude fpCrll in a tile of type p.

The structure factor for density (26) can be formally written in a way similar t o (22),

s,"(qi')

-

* S. Alexander (these proceedings) gives a complete classification of relevant Bravais hyperlattices.

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Again, as one places a unit scatterer a t each quasilattice vertex, the structure factor S: reduces to the intrinsic structure factor. However, an important distinction between (28) and the usual case of crystals is that insistence on the form (27) generally leads t o singular geometric struc- ture factor. Therefore, while in ordinary crystals the atomic structure might cause extinctions without being able t o reproduce new spots, quasicrystalline diffraction spots which are not present& I " ( ~ " ) due t o a node in w(Q') generally appear due t o atomic decoration.

An interesting property of the geometric structure factors (24) and (28) is that they do not gen- erally become independent of Q" in the limit Q" -+ 0 (Q' -+ oo) contrary t o what is the case for all crystals. A physical cause of this lies in the fact that a separation between short distance (high Q") and the long distance (low Q") behaviors is not possible because each vertex has a distinct infinite environment. A further consequence of the last fact is that atoms in real quasicrystals are expected to slightly relax away from the ideal positions. Such relaxations will generally modify the structure factor in the form of a Debye-Waller-like modulation, discussed in the following section.

4. Elastic Constants and Thermal Fluctuations of Quasicrystals [ll]

In order t o calculate the effect of thermal fluctuations on the diffraction from quasicrystals we must first, like in the case of ordinary crystals, find the long wavelength, low frequency excita- tions. For crystals such excitations are phonons and they can be associated with the invariance of the Helmholtz free energy with respect t o uniform translations in the following way.

Consider a general affine transformation

where UA(X) is the displacement field associated with the transformation A . The equilibrium density changes under such a transformation as

~ ( x ) ^--+ PA+) = P(A-' . x ) I J(A ) , (30) where J(A ) is the Jacobian of the transformation. If the number of particles is constant, then the volume of the system must also change as

V -+ VA = J ( A ) V . ₍₃₁₎

The Helmholtz free energy changes under this transformation as H [ P ( ~ ) , T ] H[PA (x), V~ TI .

Generally, A can be decomposed into a product of a translation and a linear transformation.

The linear transformation can then be written as the product of an orthogonal transformation (rotation) and a symmetric transformation. The Helmholtz free energy must be invariant under uniform translations and rotations. Therefore, the elastic energy will be created by the sym- metric transformation

where 6 is the usual strain tensor. The change in the Helmholtz free energy caused by (33) consists of the elastic energy and the work against the external equilibrium pressure P. That is, to order 0 ( c 2 ) the lastic energy E is given by

Therefore, in order t o calculate the static elastic energy we need t o know the Helmholtz free energy functional and the equilibrium solid density. Then, (34) will give us the values of the elastic constants C. Once the elastic energy is calculated the long wavelength harmonic energy density can be obtained a s

where

and the effect of the long wavelength fluctuations on the diffraction pattern can be evaluated.

Hence, one can calculate the Debye-Waller factor and, t o the next order, the diffuse thermal broadening.

The Ramakrishnan-Yussouff theory of solidification [12], which is a molecular-field theory, has been recently used with great success t o evaluate the free energies of crystals and quasicrystals and their equilibrium densities near the solid to liquid transition [28,29]. The theory gives the difference between the solid grand potential and the grand potential of a reference liquid,

A G E G(ps,V,T) - G(pe,V,T) = min A W ,

P(X)

(37) where A W is the variational potential

A W = W[P(X),P,,V,TI - G(pe, V,T)

= -NSAp - kBTAN + ^kgT p(x)tn fi

v ^Pe

~ B T

-- J J [ ~ f x ) - ~ e l c e ( ~ - ~ l ) [ ~ ( ~ ) - ~ e l

v v

p is the chemical potential, N, is the number of particles, and ce is the liquid's direct pair correlation function which depends on T and the equilibrium density pe. The direct pair correl'ation function contains the information about the interactions and its Fourier transform is simply related t o the liquid structure factor,

which is accessible either from experiments or from some independent calculation. The density p(x) which minimizes A W is the equilibrium density of the solid.

At the coexistence point between the liquid and the solid, the grand potential AG, which is identical to - V A P , must be zero as must be the difference Ap in the chemical potentials.

By making the Legendre transformation we can evaluate the Helmholtz potential and write the elastic free energy (34) in terms of A W:

where AWA is given by (38) with p(x) and V replaced by pA(x) and VA, and where p(x) is the equilibrium density. The actual expressions for the elastic constants, which can be obtained by expanding (40) t o second order in 6 , are too cumbersome t o be reproduced here.

We only emphasize that the elastic constants can be expressed in terms of the liquid compressi- bility and the second derivatives of ~ ~ ( 4 ) .

The actual minimization is typically performed by choosing a crystal lattice and placing Gaus- sians a t its vertices. The variational parameters are then the lattice constant(s), the Gaussian width(s) and the average solid density p,. In the case of quasicrystals we use (17) which can then be substituted into (38). After some manipulation, this yields the identical expression as for a crystal except that the density is now the hypercrystalline density and the appropriate hyperliquid structure factor is

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c e(q) C: (q'') . More explicitly,

where p, = p:, pi = p i , I' A p = Ap", v is the hyperunit cell volume, and Q are the recipre cal hyperlattice vectors.

Starting from the structure factor of vapor deposited amorphous cobalt, Sachdev and Nelson [29] examined icosahedral state using (42) as well as FCC and BCC states using a similar expression with all variables replaced by their physical, three-dimensional counterparts. They found that for cobalt the FCC structure was the most stable, although the icosahedral structure was more stable than the liquid.

It is natural t o ask what are the elastic constants of quasicrystals and whether there are any new long-wavelength, low-frequency excitations. The answer can be found by observing that the potential (42) is invariant under a uniform six-dimensional displacement u. The three com- ponents u" of this displacement correspond to the usual translation invariance while the three ui components correspond t o some new zero frequency degrees of freedom. Clearly then, the new modes are found by letting u" and uL become slowly varying functions of xi'. This corresponds t o the six-dimensional strain which has only nonvanishing components t"," (the usual strain), and tip", the new, u p ~ o n " field strain. The corresponding static elastic free energy can now be obtained in a similar way as for the usual crystals.

The results are the two usual elastic constants, the two constants associated with phason strains, and one constant connecting the phason and the phonon strains. The tensorial struc- ture of the new elastic constants is somewhat complicated [30, 311 and it will be presented else- where [ll]. We quote here only our preliminary results for the usual elastic constants. They are given in Table I together with the experimental values for hexagonal cobalt and cubic nickel. Our numbers were obtained by assuming T = 300 " K and pe = 9.1 X loz2 ~ m - ~ . These numbers are presented only t o demonstrate that the calculation is correct t o an order of magnitude. More precise calculations based on the Lennard-Jones interactions will be presented elsewhere [Ill. However, the present result suggests that a quasicrystal should be significantly less compressible than a related crystal. A similar conclusion was also reached recently by Widom (321 who used a polytope model of icosahedral ordering.

Table I

The new, phason degrees of freedom contribute t o the thermal fluctuations in a way similar t o phonons. The effect of those fluctuations on the structure factor may be evaluated in the usual way except that one is dealing with a six-dimensional rather than three-dimensional crystal.

B [loll erg/cm3]

23.1 22.0 39.5 21.2 FCC

BCC ICO Co

cI2

14.0 19.7 24.2 16.5

cI1

41.3 26.6 70.1 30.7

c~~

13.0 18.7 22.9 7.55

Namely, one first adds six-dimensional displacements u(R) t o a hyperlattice vector R. This changes the structure factor ~ " ( q " ) into S: ($') which is then thermally averaged in the long- wavelength limit, using the elastic energy density *

where r(ql') is related t o the six-dimensional elastic energy E = 1/2 E+:C:E via the substitu- t ions

In the harmonic approximation, the average (s: ^(q")) can be evaluated t o give

where

displays the Debye-Waller factor, and

-1/2 Q-IS QSp"r-'fp 131-Q

S! (qfl) =Y

x

Q

displays the diffuse broadening caused by the r - l ( q l ' - ~ " )

-

Few remarks about these formulas are in order. The Debye-Waller factor clearly contains a phason contribution which dominates in the limit IQ'I >> IQ1II. Secondly, there is a phason contribution also in the diffuse broadening whose width is set by

where k" = qll - Q". Therefore, as Q' -+ oo the phason contribution dominates and the peak widths are proportional t o IQ'I. However, the prefactor - ~Q'.[~-'(Z;")]~.Q: is obviously anisotropic and it depends both on k " and Q'. The IQ'l contribution is particu- larly enhanced if the phasons are quenched. In this case the r U contribution dominates since it corresponds t o an effective phason temperature which is much lower than the physical tem- perature. S ecifically, r U should be replaced by + $ll.[rll"]-l.rll', so that (r-l)u = (T&ncherl 1-1-

It is interesting t o note that some recent experiments [33] show peak widths approximately pro- portional t o IQJI, thus we suggest they might be a reflection of the thermal diffuse broadening or of the quenched phasons. We also suggest that the sheets of diffuse intensity observed in the decagonal quasicrystalline phase might have similar origin and might, in fact, reflect weak elas- tic constants in the pentagonal planes.

5. Acknowledgements

I am grateful t o R. D'Arcangelo for excellent typing, and efficient typesetting, t o L. Zeise for assistance, and t o R.D. Nelson, U. Mohanty, L. Bendersky and B.I. Halperin for valuable discus- sion. This work was supported in part by Petroleum Research Fund administered by the Amer- ican Chemical Society and the National Science Foundation through grants CHE-85-12728 and DMR-85-14638 and the Harvard Materials Research Laboratory.

We shall assume the energies are in units kgT.

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References

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