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HAL Id: jpa-00225735

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Submitted on 1 Jan 1986

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DEFECTS IN APERIODIC CRYSTALS

M. Kleman, A. Pavlovitch

To cite this version:

M. Kleman, A. Pavlovitch. DEFECTS IN APERIODIC CRYSTALS. Journal de Physique Colloques,

1986, 47 (C3), pp.C3-229-C3-236. �10.1051/jphyscol:1986324�. �jpa-00225735�

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DEFECTS IN APERIODIC CRYSTALS

M. KLEMAN* and A. PAVLOVITCH**

' ~ a b o r a t o i r e de Physique des Solides, Universit6 Paris-Sud, Bat. 510, F-91405 Orsay Cedex, France

* * D . Tech./SRMP-CEN Saclay, B.P. 2, F-91191 Gif-Sur-Yvette Cedex, France

Rdsumd - On incorpore les excitations de ~ h g s e et les excitations structu- rales des pavages de Penrose usuels 2 2 dimensions dans la thdorie des oavages

~dndralisds, que l'on dGfinit, et l'on introduit un espace de paramstre d'ordre inhonogsne pour ces derniers. On utilise cet espace pour classerlesdisloca- tions des pava2es de Penrose, et l'on ddmontre l'importance des pavaees gBnB- ralisds dans l'dtude de leurs propridtCs physiques.

Abstract - After havinz incorporated phason and structural excitations of usual 2-d Penrose tilings in the framework of generalized tilings, we definean inhomogreneous parameter space and use it to classify dislocations, for which ve demonstrate the importance of considerinz generalized tilings.

The subject of aperiodic tilinys has been first developed by inspired mathenati- cians

/ I / ,

/2/ and crystallographers /3/

;

the experimental discovery- of aperiodic crystals /4/ has since been the starting point of an astonishing outburst of 1<7orks, which attest for the fundamental interest of the subject. Bowever, nost of the research still bears on structural properties, which are very exceptional. This paper is a contribution to this topic, and will develop some results specific to aperiodic tilings in comparison to and contrast with the well-known structural description of (usual periodic crystals. r7e shall limit ourselves to the 2-d case

/5/,

which is much easier to visualize

;

the extension to 3-d crystals does not brinz essential new concepts. Ue shall show that the high-dimensional space (5-d for 2-d crystals, 6-d for 3-d crystals) introduced by de Sruijn /2/ and largely used to-day to describe their structural properties /6/, /7/ is worth interpreting as the order parameter space of the aperiodic crystals, how its introduction leads quite straightforwardly to generalized Penrose tilings whose local arrangements describe physical excitations of usual tilings, and how finally this order parameter space allows for a classification of defects in aperiodic crystals, in a manner which enlar- ges the theory of defects of usual crystals /8/. In fact, aperiodic crystals must be considered as physical media with very usual symmetries (rotations, translations) but these sy!nmetries rre hidden in some sense irr the ground state, while they reveal when dislocations, which are nothing else than local symmetry breakings, show up. Usual Penrose patterns are self-similar

:

since

t h i s

symmetry property does not extend in an obvious manner to generalized patterns, it does not appear in our approach. Some con- sequences of self-similarity on the energy of defects are discussed elsewhere 19.1

Defects introduce in the description of aperiodic crystals at several levels. Apart those which break the s p e t r y as above, a perfect aperiodic crystal can be conside- red itself as a phase of disclinations

;

such a model has been introduced geometri- cally

/ l o /

as a decurving model for a (3351 polytope on the 3-d sphere. This node1 has however more than a geometrical merit

:

an aperiodic crystal is inhomogeneous, bond an~les and lengths do not keep constant

;

there are therefore internal stresses

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986324

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C3-230 JOURNAL DE PHYSIQUE

on some scale which is lareer than the atomic scale, and it is of interest to consi- der the disclinations which relieve these stresses on that scale. The sane arpnent goes for anorphous materials, for which the same structural nodel was first develo- yed

I l l / ,

/12/. This remark hints also to the fact that these must be deep relation- ships between disordered materials and aperiodic crystals. Such relationships might be enlightened by the study of generalized tilings.

OPDER PARAMETER SPACZ AND DEFECTS IN PERIODIC CRYSTALS.

Dislocations, disclinations and other 'topological' defects in usual crystals a) break locally the symmetries, b) create internal stresses.

The first property is illustrated by considering the mapping of a closed loop, which surrounds the dislocation, the so-called Burgers' circuit, on the order para- meter space /8/. In the simple case illustrated fig. la (a 2-d crystal of rectangu-

lar symmetry), the order parameter space is a 2-d torus T2 and the image of the Burgers' circuit is a closed loop which is non-homotopic to zero on T2. The classes of homotopy of closed loops on T2, alT2, conversely, classify the topologically stable dislocations in this 2-d crystal. Similar considerations extend to disclina- tions, singular points, ... 181.

The second property is illustrated by the Volterra construction of a dislocation.

For 'example, the dislocation of fis. Ib obtains by removing a slice of matter of unit thickness in the perfect crystal, and afterwards glueinz back the two lips of the cut. The zlueing process is at the origin of the internal stresses, which are conversely relaxed when a cut is performed along a surface bound by the dislocation.

ORDER PARAMETER SPACE FOR GENERALIZED PENROSE TILING'S.

An usual Penrose tiling can be constructed with two tiles T and t and associated colouring rules (fig. 2a) and b)), and is the dual figure of the de Bruijn's penta- grid, which is defined by a set a five phases Yj (j

=

0, ..., 4), obeying the condi- tion zyj

=

integer, and such that a point M of the pentagrid obeys the equation

:

0M.V.

+

Y

= K .

J ~ J

( 1 )

-?

where K. is any integer, and the vjls are unit vectors alonp, the sides of a regular pentagon. Each mesh of the pentagrrd carries five integers

3 K j ,

which are, say, the largest intezers for each j, which belong to the edges of the given mesh

;

the dual vertex R of the mesh is

:

An evident generalization of the above is to consider pentagrids with zyj # inte- ger. One can show that the dual, defined as in eq. 2, can be constructed wlth four tiles, two of them being the 'old' ones T and t

;

the new ones, T' and

t'

have the sizes of the old ones, but with different colouring rules /13/. The ratio of the

T + t

concentrations - depends only on the quantity y

=

zy and not on the individual

T'+tf j

Generalized Penrose tilings are given in fig. 3, for y

=

0.1 and Y

=

0.5. They consist in islands of old tiles separated by walls of new tiles

;

these walls grow in size with Y. For Y

=

0.5, there is an equal number of old and new tiles. The ratio -

=

-

= 5

does not depend on y. Tilincs containing only old tiles display 8

& - 1

L L

types of vertices, described by de Bruijn

;

there are 28 new vertices when new tiles

are used, 5 of them havine a mirror i m a ~ e 1131. Dhysically, arrangements with old and

new tiles should not have however extremely different energies, so that the consi-

deration of generalized tilings is worthwhile in the context of low energy excita-

tions of y

=

0 tilings, which are assumed by some people 1141 to achieve minimal

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tiles begin to aggrezate and form finally ten tiles clusters. In terms of interac- tion energy between small (t and t') and large (T and T') tiles, the Y

= 0

tiling indicates a strong actraction between tiles of different size and therefore genera- lized patterns would have a hipher energy.

The y; 's can clearly play the role of the order parameter. Their sum Y

=

Zyi (nod. l j classifies the classes of isonorphisn of various tilings

;

in a class, the various possible sets of yj's represent tilings which differ by phasons excita- tions. Consider for example the local transition between ttro hexagons of 3 tiles pictured in fig. 4a)

;

the colouring rules at the borders being not the same, this local excitation must propapate and the tiling will change step by step star- ting in the directions indicated fig. 4a). One observes indeed that in this aanner the excited state propagates to infinity and to the whole tiling, which fact contri- butes to chanee the individual yjls at constant Y. Finally a change from Y to Y + dy can be induced by a local structural transition (fig. 4b) between, say,

two hexagons

;

apain, this excitation propagates to infinity and to the whole tiling.

In the lan,oua&e of the pentagrid, an infinitesimal chano,e of yj is represented by a small translation of the bundle Fj of parallel lines perpendicular to 3 j.

The 5-d order parameter space Eg with coordinates Yj is the product of three mani- folds

:

which ars chosen as follows. D is a line in the five fold direction (11111) passins through y

=

(y o , . ~ l , y2, y3, Y

)

P is a 2-plane perpendicular to D and the two direc-

-+ 4 +

fions u l

I

(Re L')~

=

O,...l and u 2 =I m (~'1, where h

=

exp 9

;

P' is a 2-plane per-

-: *:

pendicular to

9

and P, and also to u,

=

(Re fJ) and u,

=

lm(fJ). Both P and P' intersect D at Y. 3e Bruijn has s h o d that the interseztion of + ~(7) with the reticu- lar planes of a simple hypercubic lattice in E5, with integral coordinates for the vertices, is the pentagrid described above.

-+ +

Th$ variations of y take a simple representation in Eg. If y is moved by a quanti- ty 6y which is in P, tie pentagrid is not modified and, accordizg to en+ 2, the tiliqg is moved by - 6y. Phonons are also represented by such 6~'s. If Y is moved by 6y in P', the tiling is modified by phasons excitations, but stays in the same class of isomorphism. Finally, if $ is moved by a quantity 67 along D, the class of isomorphism is changed.

It is convenient, in the spirit of the topology of the order paraneter space developed in ref. 8, to consider in E5 only an unit cell of the simple hypercubic lattice, since Yj is defined modulo 1. Topologically, thin unit cell is a five dimensional torus Tg. Now the decomposition of E5 eiven by eq. 3 is not trivial when it is mapped on Tg

:

it consists of the fold in^ of P, P' and D in an unit cell.

The image of

r)

is a closed path which is the product of the five elementary closed paths belonging to TITS

= Z 8 Z @ Z 8

Z

@

Z. The image of P

@

P S is a 4-d torus Tq imbedded in T5. And, because of the incommensurability of five-fold symmetry, the imazes of P and of P' in T4 are 2-d manifolds which span densely Tq, in the same way a line mizht span densely a 2-d torus, as an effect of the inconmensurability of two periodicities. T5 is an order parameter space which plays the same role for the generalized Penrose tilings than Tg plays for ordinary periodic crystals, but while Tg is an hono~eneous order parameter space, this is no lon~er the case for T This extension of the concept of order parameter space can find other uses

:

for 5' exanple for usual incommensurate media /15/ (with incomensurate translation symme- tries).

TOPOLOGICALLY STABLE DEFECTS IN PENROSE TILINGS /16/.

Consider the 4-d torus Tq embedded in T5 for y

=

0, integer. It is the full order

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C3-232 JOURNAL

DE PHYSIQUE

parameter space for the displacements in the usual Penrose tilings. Therefore r1T4, which is the group of the oriented loops (up to homotopy), classifies the dislo- cations of these tilings. It is easy to convince oneself that, apart the loops which can shrink to zero and are represented by the zero class in this group, the smallest loops on Tq correspond to the following displacements in E

4 '

The network generated by these vectors is made of four 3-d f.c.c. lattices in subspa- ces orthogonal to each bi. The elementary cell of this network naps on T . Any linear combination of the zi's with integral coefficientsis a possible ~ u r g e r s ~ ~ v e c t o r % (in

E5)

;

elementary Burgers' vectors ~roject, in the arrowed rhombuses scheme, either along the short diaqonals of T or along the long diagonals of t

;

the projections on^' are completely determined, when the projections on P are known. In other words, as noted first by Levine et al. 1171, there are two parts in a dislocation

:

. an 'elastic' part, represented by the projection on P, which obeys the Volterra process in some sense. This means that by making a cut terminating on the defect, renovine alonz the cut a slice of matter of thickness %, and then zlueing back the two lips, one should obtain a dislocation. Eowever the two lips never match perfect- ly, since the tiling is not periodic with period %, and some further operation must be done, belonging to the second part, as described now.

. a 'phase distortion' part, which is represented by the projection on P'. This phase distortion visualizes in Tq, since the oriented loop corresponding to the dislocation crosses infinitely many tines the manifolds P and P' with different values of the yjls. The faulted matching along the cut surface is also a represen- tation of these phase distortions, but the existence of a continuous napping in Tq insures us that they can be dispersed away, either in localized 'phasons' (i.e.

there is around the dislocation a density of wrong vertices, as in the Socolar and Steinhardt model /18/), or on the boundaries of the specimen. This is this latter case which is dratm fig. 5, where now the two lips of the cut surface are correctly matching. This result has been obtained by a step by step process, with the wrong natchin~s along the cut removed one after the other by exciting phasons. The boun- daries of the specimen are still wrong, in the sense that the forcing rules for extending the tiling lead to contradictions, which have to be relieved the same way.

The same considerations apply to dislocations of the Penrose patterns belonging to any class of isomorphismY

:

they are classified by ITlT5(Y), i.e. the sane group, and obviously the same Burgers' vectors. But now more tiles and more vertices are allowed. These new possibilities should make the construction of dislocations, with still the two parts, elastic distortions and phase distortions) a lot easier.

Finally, if one considers all the Penrose tilings for different y's as belonging to the same set, the dislocations are classified by TlT5. The oriented loops of T4(y) are on T5, and can be deformed smoothly to allow for various values of Y along their path. But there is also a new set of dislocations with Burgers' vector in Eg, of the type

:

and projection in P along an edge of the rhonbuses. These dislocations have the

smallest Burgers' vector allowed, and are therefore of smaller elastic energy, but

they necessitate (with respect to usual Penrose patterns) many localized 'phasons',

which participate to the total energy, The Volterra construction is easily achieved

by removing a half-row of rhombuses with commun edge direction (i.e. belonging to

a given half-line of the pentazrid)

;

if the lips are glued immediately after such

an operation, we are left with a typical 'stackinz fault' 1191. But this wrong

matchinz can be dispersed away, and the whole tiling then obtained with the four

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Acknowledgements

- We t h a n k Prof.

Y. G e f e n and D r . J. S o c o l a r f o r d i s c u s s i o n s .

BIBLIOGRAPHY

/1/ Penrose R. (1974), Bull. Inst. Math. Appl., 10, 266.

(1979), Math. ~ntelligencer, 2, 32.

/2/ de Bruijn N.G. (1981), Kon. Nederl. Akad. Wetensch. Proc.

(=

Indig. Math.) A E , 51.

/3/ Mackay A.L. (1982) Physica A, 114, 609.

/4/ Shechtman D., Blech I., Gratias D. and Cahn J.W. (1984), Phys. Rev. Lett. 53,

1951.

/5/ Experimental evidence has recently been gi'ven for the 2-d case (Bendersky L., these proceedings).

/6/ Kramer P. and Neri R. (1984), Acta Cryst. A%, 580.

/7/ Duneau M. and Katz A. (1985), Pphys. Rev. Lett. 56, 2688.

/8/ Toulouse G. and Klgman

t!.

(1976), J. Phys. (Paris) Lett., 37, L-149.

Kldman M., Michel L. and Toulouse G. (1977),

J.

Phys. (paris) Lett. 38, L-195.

191 Gefen Y., Kldman M., Pavlovitch A. and PeyriOre

J.

(1986) subm. to Phys. Rev.

Lett.

/lo/ Mosseri R. and Sadoc J.F. (1985)

J.

Phys. (paris) 6, 1809.

/11/ Kldman M. and Sadoc J.F. (19)9)

J.

Phys. (Paris) Lett- 40, L-569.

/12/ Kldman M. (1983),

J.

Phys. (Paris) Lett. 66, L-295.

1131 Pavlovitch A. and Klgman M. (1986) subm. to J. physics A.

/14/ Bak P. (1985) Phys. Rev. Lett. 54, 1517.

Biham 0. and Mukame1 U. (1986),ijeizmann Inst. preprint.

/I5/ Janner A., These proceedings.

1161 Kldman M., Gefen Y. and Pavlovitch A. (1986), Europhysics Lett. 1, 61.

/ 1 7 /

Levine D., Lubensky T.C., Oslund S., Ramaswamy S., Steinhardt P.J. and Toner

J.

(1985) Phys. Rev. Lett. 6, 1520.

1181 Socolar

J.

and Steinhardt P.J., personal communication and these proceedings.

/I91 J. Friedel, Dislocations, Pergamon Press (1964).

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JOURNAL DE PHYSIQUE

Fig. la - Dislocation in a rec- Fig.

I b

- Dislocation in a rec- tangular 2-d crystal. Mapping of tangular 2-d crystal. Volterra a closed loop r surroundinn it on process for the dislocation

:

the order parameter space, a 2-d the elueing of the two lips of tours along, which the crystal is the cut surface will introduce folded when periodic boundary con- internal stresses.

ditions are taken into account.

Fir. 2a - Tiles and their laws of association in usual Penrose tilings (old tiles).

Fig. 2b - Usual

(Cy =

integer) Penrose

tiling.

j

Fig. 3a - New tiles.

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Fie. 4a - A local phason excita- tion of an hexagon.

Fie. 4 b - A local structural

excitation qf an hexagon.

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JOURNAL

DE PHYSIQUE

Fig. 5 - A dislocation of Eurgers vector + b alone the long diagonal of a thin rhombus

;

matter has been removed in a strip of width %, and the two lips not glued back. Localized phase distortions subsist on the boundaries of the specimen

:

the tiling cannot be extended without further phason excitations.

Fig. 6 - Stacking fault in a

Penrose tiling

(y =

0).

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