PHASON STRAIN IN QUASICRYSTALS

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PHASON STRAIN IN QUASICRYSTALS

J. Socolar

To cite this version:

J. Socolar. PHASON STRAIN IN QUASICRYSTALS. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-217-C3-226. �10.1051/jphyscol:1986323�. �jpa-00225734�

JOURNAL DE PHYSIQUE

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

PHASON STRAIN IN QUASICRYSTALS

J.E.S. SOCOLAR

Department of Physics, university of Pennsylvania, Philadelphia, PA 19104, U.S.A.

RCsurnC: On decrit les phonons, les phasons et les dislocations comme des distorsions et changements de disposition des mailles. On demontre la relation entre le modkle des mailles de ces defauts et celui des ondes de densite. On discute la deformation et la relaxation des variables des phasons dans le modkle des mailles.

A b s t r a c t : Phonons, phasons and dislocations in a quasicrystal are interpreted as distor- tions and rearrangements of unit cells. The relation between the unit cell picture these defects and a simple density wave picture is demonstrated. Phason strain and relaxation is discussed in the context of the unit cell picture.

1 I n t r o d u c t i o n

The close correspondence between the observed diffraction pattern of the icosahedral phase of A1-Mn (11 and the diffraction pattern expected of an ideal icosahedral quasicrystal 123 has made the quasicrystal model a leading candidate for the description of this phase. An ideal icosahedral quasicrystal, having both long range orientational order and long range quasiperiodic translational order, would produce an icosahedraliy symmetric diffraction pattern consisting of true Bragg peaks with varying intensities. The observed icosahedral patterns exhibit fairly sharp peaks with intensities consistent with the quasicrystal model.

One does not expect, of course, to observe perfectly sharp Bragg peaks in a real experiment.

In the real samples, deviations from the perfect quasicrystal structure such as anisotropic strains or dislocations could shift and/or broaden the peaks, just as is the case for periodic crystals.

Recent measurements of the shifts of the peaks in single grain electron diffraction (31 and of the peak widths in x-ray powder diffraction [ 4 ] as well as observations of high resolution electron micrographs (HREM) ; 5 ] , indicate that anisotropic phason strain is present in these materials.'6]

This type of strain has no analogue in periodic structures. The application of standard hydrody- namic theory to quasicrystalline systems reveals that both continuuous translational symmetry and an additional continuous symmetry are broken by the quasicrystal ground state.17-91 Phason strain is associated with this extra symmetry in exactly the same way that conventional strain is associated with continuous translational symmetry.

The origin of this extra symmetry is best exhibited in terms of a simple density wave descrip- tion of a solid. Any crystal or quasicrystal can be described as the superposition of density waves with an infinite set of wavevectors. all of which can be expressed as integer linear combinations of those in some fundamental set. For d-dimensional periodic crystals this fundamental set is the set of basis vectors of the reciprocal lattice, which contains d elements. Continuous translational symmetry is broken by the choices of the phases of these d density waves and any change in these phases can be compensated by a redefinition of the origin. For quasicrystals, on the other hand, d fundamental wavevectors are not enough. An independent wavevector is needed for each incommensurate length in each dimension. Thus for an icosahedral (or pentagonal) quasicrystal, which has two incommensurate length scales in each of d = 3 (or d = 2) directions, six (or four) independent wavevectors are required. The phase of each of these independent density waves

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986323

C3-2 18 JOURNAL DE PHYSIQUE

is associated with a continuous symmetry: i e none of the physical properties of the system de- pend upon the choices of these phases. Translational symmetry accounts for d of these degrees of freedom, just as it does for periodic crystals. The other d degrees of freedom correspond to relative shifts of the phases of independent density waves and hence are termed 'phason' degrees of freedom. Phason strain arises when the variables associated with these degrees of freedom are nontrivial functions of position.

Although the density wave description of a quasicrystal is useful for the analysis of elasticity [Q]

and hydrodynamics 1101, it does not provide much insight into the microscopic structure. It is even conceivable that the microscopic structure of phason strain could be inconsistent with certain assumptions implicit in the standard hydrodynamic treatment. It is therefore important to develop a microscopic picture of phason strain and relaxation.

The quasicrystal model provides a means of achieving an understanding of microscopic struc- ture using a %nit cell" description.[ll] A quasicrystal can be regarded as consisting of two or more unit cells, or atomic motifs, that are repeated in an ordered, though nonperiodic pattern.

In this paper, unit cell configurations are analyzed so as to obtain an understanding of the micro- scopic~ of phason strain and relaxation. For each relevant result of the density wave description, a corresponding result for the unit cell description is exhibited, including the identification of the appropriate degrees of freedom, the description of phason (and phonon) strain and relaxation, and the characterization of dislocations. An attempt has been made to outline the reasoning that leads to the results discussed here, but many details have been omitted in favor of the elucidation of the basic concepts. More detailed arguments are presented in Ref. 12.

All of the concepts required for the unit cell analysis are developed in this paper for the case of 2D pentagonal quasicrystals. There are three reasons for this choice: (1) 2D quasicrystals are easier to visualize and illustrate than are 3D quasicrystals. The generalization to 3D is straightforward and requires only a few modifications, which are noted. 2) The diffraction pattern of a 2D pentagonal quasicrystal has peaks in the same positions and wit

k

(3) A special case of a 2D pentagonal quasicrystal tiling, the Penrose tiling 1131, is already familiar to many readers and is therefore convenient for heuristic purposes. For the Penrose tiling simple matching rules exist which limit the possible relations between adjacent tiles. It is therefore easy to detect defects in a Penrose pattern, a s they involve violations of these rules. In spite of the use made in this paper of certain properties of the Penrose tiles, the conceptual results can be extended to other pentagonal quasicrystals and to quasicrystals with other symmetries.

2 S y m m e t r i e s of the Free E n e r g y and Local I s o m o r p h i s m Class

The defects expected to be relevant for the elastic and hydrodynamic properties of a solid are generally those that correspond to low energy excitations of the ground state. Spatial varia- tions in the variables associated with broken continuous symmetries have the property that their energy approaches zero a t long wavelengths and are therefore of primary significance. These variables are usually identified in the context of a simple density wave description of the solid.

To construct the simplest model for a quasicrystal with a given symmetry, a minimal set of wavevectors is chosen such that the symmetry operations map the set into itself. In the 2D pentagonal case five wavevectors, G , , are chosen pointing to the vertices of a regular pentagon;

Gn = $ ( c o s y , s i n

y),

In general, a Landau free energy density for a system can be used to determine properties of the system that depend only on its symmetry. The free energy density, F, is expanded in powers of p ( x ) , the k f h order term being

where V is the volume and A:, is a constant coefficient. To find the ground state of the system one must minimize F. Eq. 2 implies that the minimum value of F must correspond t o some specific value of C 4, over every set of n's for which 2 6, = 0 .

Figure 1: Density wave images of pentagonal quasicrystals. Left:

r

r

In the minimal 2D pentagonal case there is one such set; C,!,=, ^{G , =} 0. Thus the ground state must correspond to some specific value of C:=, 4, which will be called I'. Figure 1 shows images of p(x) for two different choices of

r.

essential^^ arbitrary) cutoff. Note that these images. which correspond to systems with different free energies, differ visibly in their structure.

Any changes in the 4,'s which do not affect

r

4, = u . G,

+

r

In the unit cell description of a quasicrysta1, the parameters that correspond to u and w can be identified through an analysis of the Fourier transform of a set of atoms placed a t positions which constitute a decoration of the unit cells. Each peak in the transform represents a density wave with some amplitude and phase, and the Landau expansion of the free energy can be obtained as before. The situation is exactly the same as for the density wave analysis except that in the unit cell Fourier transforms all harmonics of the minimal set are present. When all of the constraints on the phases are enforced (one constraint for each set of G's with C G = 0), there remain four degrees of freedom. which can again be parametrized by two vectors. Since these vectors are exact analogues of the u and w of the minimal density wave picture, it is appropriate t o refer to them as u and w also.

These energetic constraints on the phases and the meaning of the broken symmetry variables u and w can also be understood from the point of view of the unit cell configurations themselves.

Cnlike crystalline structures, quasicrystalline structures are not fully determined by the shapes of their constituent unit cells. even if the frequency of occurence of each shape is specified. Several relations are possible between two quasicrystal unit cell configurations, or tilings. consisting of the same unit cells. First, the two can be identical tilings related by a pure translation. Second.

the two can be globally distinct tilings but nevertheless be indistinguishable on any finite scale:

i.e. any finite pattern contained in either one can also be found somewhere in the other. Tilings related in either of these two ways are said to belong to the same local isomorphism class (L1

class).'l4] Finally, the two tilings can contain different local configurations of unit cells and hence be distinguishable on some finite scale. An example of two such tilings, i.e. tilings belonging to different LI classes, is shown in Figure 2.

Because they are indistinguishable on any finite scale, quasicrystals based on tilings of the same LI class have identical physical properties. In particular they are all degenerate with respect to their free energy density.114~ Furthermore. barring some bizarre accidental degeneracy, tilings of different LI classes should have different free energy densities. In other words, the equivalent ground states of a real system must be members of a single LI class. For the rest of this paper.

the ground state class is taken to be that of the original Penrose tilings because it is familiar and a variety of techniques have been developed to study it. A portion of a Penrose tiling is shown in Figure 2 in the center.

For the purposes of this paper the most useful method of analyzing the Penrose tilings is to regard them as decorations of Ammann quasilatttces, which have an algebraic structure that

C3-220 JOURNAL DE PHYSIQUE

is convenient for exhibiting the significance of u and w.[11,15](') An Ammann quasilattice is composed of five grids of parallel lines, the nth grid consisting of lines that are perpendicular t o the vector en = (cos ?,sin

F )

T = (1

+

where an and 0, are real parameters and it] denotes the greatest integer less than or equal t o z.

In order for the set of five grids to form an Ammann quasilattice, certain relations must hold among the an's and 0,'s. These relations distinguish the Ammann quasilattices from a more general class of structures in exactly the same way that the Penrose tilings (a single LI class) are distinguished from the complete range of possible tilings. The Ammann quasilattices are generated by an's and P,'s of the form

where, again, u and w are arbitrary vectors.!ll] In Figure 2 a portion of an Ammann quasilattice is shown on the right. The Ammann quasilattice divides the plane into eight types of regions through which no lines pass. A Penrose tiling can be obtained by decorating each type of region the same way every time it appears in the Ammann quasilattice. The Penrose tiling and Ammann quasilattice of Figure 2 are related in this fashion. Eqs. 4 and 5 determine the way in which an .4mmann quasilattice is affected by the values of the broken symmetry variables u and w. Once u and w are understood in the context of Ammann quasilattices, their roles in the Penrose tilings follow in a straightforward manner from the decoration procedure.(2) (See Figure 5 for an illustration of the decoration.)

In summary. the energetic constraint on the phases derived from Landau theory is equivalent to the restriction to a single LI class of unit cell configurations and the choice of specific vaiues for the broken symmetry variables is equivalent to the choice of a specific member of this LI class.

The Penrose LI class is taken to be the ground state for the purposes of this paper, but the ideas expressed here can be generalized to any LI class.

Figure 2: Left and center: Pentagonal quasicrystal tilings of different local isomorphism classes.

The center figure is a portion of a Penrose tiling. The shaded tiles form a segment of a worm.

Right: The Ammann quasilattice associated with the Penrose tiling of the central panel.

3 T h e U n i t Cell Description of Phonon a n d P h a s o n S t r a i n s

From Eqs. 1 and 3 with 4, initially equal to zero, it is trivial to see that a uniform shift of u by an amount A u simply translates the entire density distribution;

(')Some readers may be more familiar with the projection technique of de Bruija, Elser, and Duneau and Katz. For a description of the results of this paper into the language of the projection technique, see Ref. 12.

I

(')The Penrose tiling can also be obtained from the Amnlann quasila~tke by application of the generalized dual met,ho#l of de Bruijn and of Socolar and Steinhardt. See Ref 11 for details.

Figure 3: Spatial variations of u in a density wave image, an Ammann quasilattice and a Penrose tiling. Note the curvature in the lines of high density points in the density wave image, the curvature in the Ammann quasilattice lines, and the distortion of the unit cells in the tiling.

Figure 4: The effect of variations in w on a density wave image, an Ammann quasilattice and a Penrose tilink. The scale of the density image has been chosen so as to make the picture easy to interpret. It annot be compared directly to the Ammann quasilattice and the tiling. Large dots in the tiling mark edges where the matching rules are violated.

Spatial variations in u represent phonon strain (conventional strain) and produce compression and,'or shear in the density wave images. In Figure 3 a density wave image with varying u (and constant k ) is shown. Note that the lines of high density points remain continuous but contain curvature.

The effects of uniform shifts and variations in u on the Ammann quasilattice and unit rell configurations are equally easy to understand. Note from Eq. 4 that a shift in an simply translates the nfh\grid in the direction of en. and for shifts of the form of Eq. 5.

i.e. a uniform A u produces an overall translation of the quasilattice. Spatial variations in LI

produce curvature in the lines of the Ammann quasilattice without affecting the sequences of spacings in the grids. which in turn causes distortions of the Penrose tiles. but no rearrangements.

(See Figure 3.) Thus variations in u, the variable associated with broken translational symmetr).

lead to distortion of the unit cells in a quasicrystal as they do in a periodic crystal.

Gniform shifts and spatial variations in w affect both the density wave images and the tilings in a rather more subtle manner. Because the changes in w enter the expression for the 4,'s of the density wave picture through the quantity w . GtznP the> cannot be absorbed in the x . G n term of the argument of the cosine of Eq. 1. The effect of such shifts is best understood and visualized in terms of the unit cell picture. as discussed belou. Nevertheless, the effect of spatial variations in w can be observed in a density wave image. as shown in Figure 4. The lines of high density points remain straight in this case, but contain regions where a given line gradually fades out and a new one emerges, displaced from the original by a specific amount (TI&). This is precisely

C3-222 JOURNAL DE PHYSIQUE

what is observed in HREM's of the icosahedral phase of A1-Mn is], an experimental indication t h a t variations in w (phason strain) are present in real materials. T h e fact t h a t these strains appear t o be quenched into the samples is consistent with the prediction of hydrodynamic theory t h a t phason relaxation is a very slow, diffusive p r o c e s s . j l ~ ]

Quasicrystal tilings (or Ammann quasilattices) that are related by a uniform shift in w are distinct members of t h e same LI class.[ll] As a n example, consider t h e Ammann quasilattice for which u and w in Eq. 5 are both zero. This quasilattice clearly has a center of pentagonal symmetry a t t h e origin. Obviously, if w is shifted by a small amount (w . e, < 1 ) the set of Pies

does not retain its five-fold symmetry. Sow the effect of a shift in P, is t o shift the position of every line in the nth grid for which the value of

+

The changes induced in a Penrose tiling by shifts in w can be obtained by consideration of the tiling as a decoration of the Ammann quasilattice. Note t h a t such shifts affect &, which occurs inside the greatest integer function in Eq. 4, and cannot be absorbed as a simple translation in t h e manner of Eq. 7. The way in which lines in the Ammann quasilattice are affected by a uniform shift in w corresponds t o a specific type of unit cell rearrangement which transforms a tiling into another of t h e same LI class. These rearrangements can be understood as taking place along 'worms' in a Penrose tiling. A worm is composed of fat and skinny rhombuses which form a connected line of hexagons. as do the tiles shaded in Figure 2. Each tiling contains a n infinite number of crisscrossing worms. most of them being interrupted a t points where they cross other worms. An entire uninterrupted worm is unique in t h a t it can b e reflect,ed about its horizontal axis, or 'flipped', without disrupting any of the Penrose matching rules.(3) A uniform shift in w causes a number of crisscrossing worms t o ffip in just such a way t h a t the matching rules a r e maintained even where the worms cross. The number density of worms t h a t flip is proportional t o t h e size of t h e change in w. (In the case of icosahedral symmetry a worm consists of a planar array of connected unit cells. These worms flip in response t o changes in w in a manner exactly analogous to the flipping of worms in the 2D case.)

Spatial variations of w produce jags in the Ammann quasilattice which result in isolated violations of the matching rules in the tiling. T h e line with index N which runs perpendicular t o the x-axis, for example, will shift by a t all points along it where

1:

⁺

-IF +

Such shifts, or jags, will occur a t various points on each of the grids in t h e Ammann quasilattice and are t h e signature of phason strain. Note t h a t phason strain introduces no curvature into the Ammann lines a n d t h a t jags are not induced by phonon strain (variations in u). Figure 4 shows an Ammann quasilattice for which u is constant but w varies about the value used in Figure 2.

Around t h e edge of t h e picture w is the same as in Figure 2 , b u t a smooth b u m p in w is centered o n t h e center of t h e picture.

From t h e point of view of the tiling picture, phason strain is evidenced by violations of the matching rules. o r local deviations from the ground state LI class, which arise when segments of worms are flipped. T h e matching rules can be realized by decorating the edges of each rhombus as shown in Figure 5 and requiring rhombuses t o join along edges with t h e same decoration color and orientation. Any edge which is shared by two segments of t h e same worm that a r e flipped with respect to each other, cannot be decorated in a manner consistent with the both of the tiles that share it. None of the other edges of tiles in the worm present any difficulty, however.

since flipping an entire worm has n o effect o n t h e consistency of t h e matching rules. Thus spatial variations in w are evidenced in the tiling by isolated, local configurations that are not consistent with the matching rules or t h e LI class. The effect of a variation in w on a portion of Penrose tiling is illustrated in Figure 4. The w field here is the same as for the Ammann quasilattice of this figure. Edges t h a t are inconsistent with the matching rules are marked with a black dot.

Kote that each of these dots corresponds a t o a jag in the Ammann quasilattice. Because the w field is constant around the edge of this figure, t h e dots come in pairs which are connected by flipped segments of worms. (Compare t o Figure 2.) This would not be the case for a constant gradient in w.

In Figure 5 a single mismatch is depicted with the Ammann quasilattice and Penrose tiling superimposed. T h e large dot indicates a n edge along which the matching rules cannot be obeyed, the vertices of t h e tiling a t the endpoints of this edge being of types t h a t are never found in

(3)When a hexagon consisting of two skinny tiles on top and one fat t.ile on the bottom, say, is rearranged within the same hexagonal boundary such that the the fat tile is now at the t o rules in the interior of the hexagon are still obeyed. (See the t,ransformation in the rightmost

diagrants on tlir right of Figure 5.) A Ripped worm is obtained by flipping all of its constituent hexagons.

Figure 5: The relation between a jag and a mismatch. The mismatch in the tiling is marked with a large dot. The worm segments depicted on the right are decorated to illustrate the Penrose matching rules, which require that two tiles can join only along edges which have the same color arrow pointing in the same direction. The worm segments on the right match the shaded segment on the left.

Figure 6: -4 mismatch in a toy atomic model of a covalently bonded structure. The decoration of the Penrose tiles employed is shown at right.

tilings of the Penrose LJ class. The right (left) half of the broken Ammann line can be shifted up (down) to restore the ideal quasilattice in the region depicted. Consequent rearrangements of the various shapes of the regions of the Ammann quasilattice cause the right (left) half of the worm to flip when the tiling decoration is applied. The segment on the lower right corresponds to the shaded segment on the left, where the rightmost hexagon (formed from one fat and two skinny rhombuses) has been flipped, causing a matching rule violation (indicated by the single arrow).

If the middle hexagon were now flipped to relieve that mismatch. a mismatch would arise along the edge indicated by the double arrow. Note, however. that the top and bottom of the hexagon are decorated in the same way, so that no other mismatches would arise. In Figure 6 the effect of a single mismatch on a toy atomic model is depicted. The two unit cells of a Penrose tiling have been decorated with atoms and bonds to produce a covalently bonded structure containing four types of atoms, distinguished here by their size and bonding configuration. The dangling bond near the middle of the picture is the indication that the pattern contains a mismatch, or a phason strain.

(In the icosahedral case, the Ammann quasilattice consists of planes rather than lines. Linear phason strains introduce steps between half planes analogous to the jags between half lines in the 2D case. Mismatches in the tiling occur all along the line that forms the boundary of the half planes. In general, phason strain produces displacements of various regions of the Ammann planes of each direction and the boundaries of these regions form closed loops of mismatches (or lines that terminate at the surface of the sample).)

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4 Energetics a n d Relaxation of P h a s o n S t r a i n

Of critical importance for the understanding of the elastic and hydrodynamic properties of quasicrystals is the understanding of the mechanism through which phason strains relax. Unfor- tunately, the process of relaxation may depend upon the detailed dynamics of a particular system and cannot be characterized completely on the basis of geometry alone. Two logical limits exist for models of this process. The first derives from the simple density wave picture and standard elasticity theory and will be called the continuum limit. The second stems naturally from tfte tiling theory and will be called the discrete limit.

In the continuum limit, the energy associated with pure phason strain is calculated to be E a S(Vw)'dx and the standard hydrodynamical treatment predicts that the phason mode is diffusive in the usual sense (w -+. q2 as q + 0). 91/10] The characteristic relaxation times are predicted to be controlled by typlcal impurity di

k

The tiling picture suggests the possibility that this assumption concerning the energy density is not appropriate. Figure 6 , for example, suggests an energy density that is strongly localized in the vicinity of a mismatch, even when V w is a constant function of position. In this picture the energy associated with phason strain is proportional to the number of mismatches induced.

From Eqs. 4 and 5 it can be seen that the number of mismatches (in a suitably large region) is proportional to I Ow / and therefore that E a $ I V w dx. Note that it remains true that E goes to zero (linearly in this case for long wavelength variations in w .

L

In the discrete picture p ason relaxation can occur through series of local rearrangements of unit cells. From Figure 5 it is clear that the location of a mismatch can be moved by the flipping of either of the hexagons that border on it. In the figure on the right containing a mismatch.

for example, the mismatch moves from the single arrow t o the double arrow when the middle hexagon is flipped. Note that the new configuration has the same energy as the old, but that the process involves the rearrangement of atoms. suggesting that the time scale for relaxation is still controlled by a diffusion rate. Kote also that this relaxation is accomplished through short range motion of atoms; no mass must be transported over distances large compared to the unit cell size. In the discrete limit, where the energy associated with a mismatch is strongly localized so that there is absolutely no interaction between mismatches, each mismatch will execute a random walk along its worm until walks off the sample or it annihilates with another mismatch on the same worm. (This annihilation corresponds to a line segment between two jags in the Ammann quasilattice shrinking to zero length.) This scenario is complicated a bit by the fact that constraints are placed on the motion of a mismatch along a worm' due to the crossings of other worms. and a fully detailed picture has yet to be worked out.

Although the unit cell description of a quasicrystal naturally suggests the discrete model of phason dynamics, it is also compatible with the continuummodel. One can adopt the point of view that the unit cells provide a template for local atomic configurations, but do not determine the dynamics. Within each unit cell in an equilibrium configuration. there is a preferred arrangement of atoms, which depends upon the unit cell shape and the matching rules (or. more generally, LI class). A matching rule violation can be interpreted as an energetically unfavorable arrangement of atoms. The presence of this disfavored arrangement will cause the positions of atoms in nearby unit cells to relax to reduce the overall energy. This relaxation provides a mechanism to spread out the energetic effect of a spatially varying w , leading again to an energy density proportional to ( V w ) ' . This relaxation is presumably a fast, non-hydrodynamic process which results in a distorted configuration with energy density proportional to ( V W ) ~ . A configuration containing such a mismatch can now relax to the state of uniform w only by rearranging atoms via a slox,, diffusive process. This picture is consistent with the elastic and hydrodynamic theories.(*) 5 Dislocations

Dislocations are variations of broken symmetry hydrodynamic variables that cannot, for t,opo- logical reasons, relax to the ground state. In a periodic crystal a dislocation is characterized by a Burgers vector which is defined as the integral of the u field taken around a closed loop containing

(*)It has been suggested by D. Frenkel. C . L . Henley and E. D. Siggia (Cornell preprint) and by Bak (private com- munica.tion) that phason relaxation must involve discrete transforn~ations for geometrical reasons. Geometrical considerations alone, however, do not determine the dynamics: the standard hydrodynamical treatment may still apply.

the dislocation core. The Burgers vector must be a lattice vector of the structure. Otherwise, there will be regions far from the core where there are large distortions the unit cells. This re- quirement can also be understood in terms of the phases of the waves in a density wave picture.

For a dislocation in a 2D periodic crystal these phases can be written as 4, = Oo,

+

.

In a quasicrystal, on the other hand, there is no exact lattice vector, so the concept of a Burgers vector must be generalized. Using the density wave picture and the requirement that the density be a smooth function of position except at the core. one arrives a t the constraint

The Burgers vector must be four dimensional, having two components that specify u(2x) - u(0) and two that specify w ( 2 s ) - w(0). For a set of G,'s corresponding t o a quasicrystal, the Burgers vector must have both nonzero u and w components.(g] Thus, any dislocation must contain some phason strain and dislocation motion in any direction is predicted to be slow.ilO] Figure 7 shows a dislocation in a density wave image. The curvature (phonon strain) and jags (phason strain) are both easiest to see when the page is viewed from a grazing angle along the direction indicated by the arrow.

As might be expected, the Burgers vector lattice can also be derived directly from the 2D unit cell picture by considering a, and ,Bn as functions of 0 and requiring that the Ammann quasilattice at ,O = 0 match smoothly onto that at 0 = 2s.116: There exists a transformation of the an's and

on

where p, and q, are integers for all n. The Burgers vector lattice is determined by the simultaneous requirements that a, and 4, be of the form of Eq. 5 for all 0 and that a,(2n) and Pn(2s) be related to a,(O) and a ( 0 ) by an umklapp. The latter requirement ensures that the quasilattice is continuous along the B = 0 ray. The Burgers vector lattice derived in this way is exactly the same as that obtained from the density wave analysis.

Figure 7 shows a dislocation in a Penrose tiling and its associated Ammann quasilattice.

Again, the presence of phason strain is visible as mismatches in the tiling and jags in the Ammann quasilattice. A straightforward analysis of the Ammann quasilattice reveals the following facts:

(1) The density of jags is inversely proprortional to the distance from the dislocation core and the total number of jags is proportional to the length of the sample. (2) The jags cannot be removed through phason relaxation, although it might appear that they could be moved off to infinity along their respective lines. The reason for this is that other lines corresponding to worms cross any given line in a way that constrains the motion of the jag. Movement of the jags beyond a certain finite distance (determined by the Burgers vector of the dislocation) necessarily involves disruptions of the local structure of the Ammann quasilattice which are energetically costly deviations from the ground state LI class.

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6 Concluding Remarks

The relevance of this discussion of phason strain to the observed icosahedral phases of AI-Mn and related alloys is suggested by a variety of scattering experiments. The resemblance of HREM's to the density wave images containing phason strain has already been noted. Further evidence for the presence of phason strain comes from the analysis of single grain electron diffraction studies. Linear phason strain produces a specific type of distortion of the peak positions which is clearly distinguishable from the usual compression or shear associated with phonon strain./l7,6]

Distortion of the type expected for pure phason strain has been observed in experiments.[3,6]

Finally, the peaks observed in x-ray powder diffraction experiments (41 have widths that depend on wavenumber in the manner expected for an anisotropic distribution of nonlinear phason strains.

(For details, see Refs. 6 and 18.)

The quasicrystal model therefore provides a reasonably complete description for the observed icosahedrai phase, including a microscopic picture of the phason strains which can explain the observed deviations from perfect quasicrystal structure.

7 Acknowledgements

I wish t o thank Paul Steinhardt and Tom Lubensky, my collaborators in this work, for their ideas and their patience. I also benefitted from conversations with R. Ammann, P. A. Bancel, P.

A. Heiney and D. Levine.

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[21 D. Let-ine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).

,3] M. Tanaka. M. Terauchi, K. Hiraga and M. Hirabayashi, Ultramicroscopy 17,279 (1985).

i4; P. A. Bancel, P. A. Heiney, P. W. Stephens, A. I. Goldman and P. M. Horn, Phys. Rev.

Lett. 5 4 , 2422 (1985).

j5] See, for example, K. Hiraga, M. Hirabayashi, A Inoue and T. Masumoto, Sci. Rep.

RITU .4-Vol. 32, Yo. 2, 309 (1985).

I6j T. C. Lubensky. J. E. S. Socolar, P. J. Steinhardt. P. A. Bancel and P. A. Heiney, Penn.

preprint (1986), submitted to Phys. Rev. Lett.

j7] P. .4. Kalugin. A. Kitaev. L. Levitov, J E T P 4 1 . 119 (1985).

18: P. Bak Phys. Rev. Lett. 5 4 , 1517, (1985); Phys. Rev. 832, 5764, (1985).

[9: D. Levine. T. C. Lubensky, S. Ostlund, S. Ramaswamy, P. J. Steinhardt and 3 . Toner, Phys. Rev. Lett. 5 4 , 1520 (1985).

:lo; T. C. Lubensky, S. Ramaswarny and J. Toner, Phys. Rev. B 32. 7444, (1985); to appear Phys. Rev. B (1986).

1111 J. E. S. Socolar and P. J. Steinhardt, Penn. preprint (1985). to appear in Phys. Rev.

B (1986).

!Ij J. E. S. Socolar, T. C. Lubensky and P. J. Steinhardt, Penn. preprint (1986), submitted to Phys. Rev. B.

113 R. Penrose, Bull. Inst. Math. and Its Appl. 10. 266 (1974).

114, D. Levine and P. J. Steinhardt. Penn. preprint (1985), to appear in Phys. Rev. B (1986).

ll5, See B. Grunbaum and G. C. Shephard Tilings and Patterns (Freeman, San Fransisco) to be published, for a discussion of the results of R. Ammann.

i16 M. KlBman, Y. Gefen, and A. Pavlovitch. Europhys. Lett. 1, 61, (1986) have discussed a different approach to dislocations in Penrose tilings.

jl7 P. A. Kalugin. A. Kitaev and S. Levitov. J. Physiques Lett. 4 6 (1985) L-601, have discussed peak shifts due to linear phason strains.

j18: D. P. DiVincenzo, elsewhere in this volume.

COMMENTS AFTER THE J . SOCOLAR TALK :

M.V. JARIC.-

Remark :

I suggest that we should make a distinction between the terms "tiling"

and "(quasi)lattice", reserving the latter for the sets of points or point scatterrers. Thus, I would call your "Amrnann lattice" either

"Ammann tiling" or "Ammann pentagrid".

In 2-D Penrose tilings phasons correspond to flipping of the interior vertex in certain "prolate" and "oblate" hexagons. Therefore, it is hard to imagine that such motions could correspond to a hydrodynamic variable. On the other hand, I suggest that phasons should be viewed as low-frequency long-wavelength excitations of a coarse grained single particle density (probability) field. In this case a phason describes a continuous deformation of the density. Then, although the maxima of the density are the most probable occupation sites, there is always a distribution of occupied sites. It is this distribution which is continuously changed. Note that although the atomic surfaces should not cross upon a phason displacement, the loci of the maxima of the distribution are free to cross and to form closed or branched surfaces. Therefore, it would not necessarily be possible to identify a particular peak in the density and to follow its evolution uniquely as one moves in the phason directions.