THE CRYSTALLOGRAPHY OF APERIODIC CRYSTALS

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THE CRYSTALLOGRAPHY OF APERIODIC CRYSTALS

S. Alexander

To cite this version:

S. Alexander. THE CRYSTALLOGRAPHY OF APERIODIC CRYSTALS. Journal de Physique

Colloques, 1986, 47 (C3), pp.C3-143-C3-151. �10.1051/jphyscol:1986314�. �jpa-00225725�

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THE CRYSTALLOGRAPHY OF APERIODIC CRYSTALS

S. ALEXANDER

The Racah Institute of Physics, The Hebrew University

of

Jerusalem, Jerusalem, Israel

Abstract

-

I t i s shown t h a t a complete crystallographic c l a s s i f i c a t i o n of aperiodic c r y s t a l s

i2

possible s t a r t i n g from t h e known point groups. We f i r s t construct t h e q-space Bravais s e t s consistentwith the point group.

This leads t o a hyperspace representation i n t h e space ofincomensuratehy- drodynamic t r a n s l a t i o n s . A unique h y p e r l a t t i c e dual t o t h e Bravais s e t de- s c r i b e s the d i s c r e t e hydrodynamic symmetry of t h e aperiodic c r y s t a l . From the transformation properties o f t h e phases an algorithm f o r t h e construc- t i o n of the relevant hyperspace groups i s constructed i n which the hydrodynamic representations of t h e group play a s p e c i a l r o l e .

Results a r e only discussed b r i e f l y . I t i s shown t h a t t h e r e a r e s i x cubic s e t s and four tetragonal s e t s . There a r e four icosahedral s e t s only two o f which give a s i x dimensional h y p e r l a t t i c e . There i s only one pentagonal s e t .

1

-

INTRODUCTION

Since they are not periodic,quasicrystals andincommensuratecrystals cannot be c l a s - s i f i e d i n t h e 230 Fedorov-Schonfliess crystallographicspace groups. The f u l l symmetry group of aperiodic c r y s t a l s i s c l e a r l y not a subgroup of t h e t r a n s l a t i o n r o t a - t i o n inversion group of space. Since t h i s i s t h e symmetry which i s broken one never- t h e l e s s expects t h a t the symmetry of aperiodic c r y s t a l s can be r e l a t e d t o t h a t of cartesian space i n a systematic way. We s h a l l show t h a t t h i s i s indeed possible and s h a l l develop an algorithm which, i n p r i n c i p l e , allows t h e construction o f a l 1 pos- s i b l e aperiodic c r y s t a l symmetries.

The e s s e n t i a l idea i s t o construct t h e aperiodic c r y s t a l density from t h e point group symmetries and f r o m t h e i r r e d u c i b l e representations of t h e f u l l continuous t r a n s l a t i o n group. Similar problems were e n c o ~ j e r e d long ago i n t h e c l a s s i f i c a t i o n of magnetic s t r u c t u r e s where it was a l s o found t h a t a description i n terms of t h e representa- t i o n s of the broken symmetry group was both r i c h e r and more convenient than other ap- proaches

.

Our approach i s closely related@ofhe ideas and techniques involved i n applying the Landau theory t o s o l i d i f i c a t i o n

.

The most systematic attempt t o work out t h e implications of t h i s approach is i n t h e work of8Shtrikman and ~ o r n r e i c h ( 7 ) on t h e blue phases and i n t h e ecent work of Kalugin e.a.

.

I n t h e context of icosahedral aperiodic c r y s t a sf9) s i m i l a r ideas have aitp3yeen used recently by Per 8ak(1°), by Léensky e s a . ( l l f , by Iielson(12), by Memin

,

by ~ a r i c ( l ~ ) and probably by others

.

Since t h i s work w i l l presumably be discussed elsewhere i n t h i s volume we make no a t - tempt t o describe it here. The emphasis of t h i s work i s mainly on t h e physics while we concentrate on t h e geometric crystallographicaspects.

Janner and c o w o r k e r ~ ( ~ ~ ^a16) have developed techniques f o r describing t h e synmietry of aperiodic c r y s t a l s i n terms of projections of periodic space group symmetries i n hyperspaces. Similar techniques have been used extensively r e c e n t l y i n t h e context

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986314

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C3-144 JOURNAL DE PHYSIQUE

of t h ~ l t i l i n g s of the plane(17) and t h e icosahedral t i l i n g s of 3 dimensional ~ ~ ~ ~ e space

.

Our a n a l y s i s leads t o a description i n t e r m s d a l a t t i c e i n t h e hyperspace odynamic variables. This description i s c e r t a i n l y r e l a t e d t o

Janner

4fs1"f.

It can a l s o be r e l a t e d t o other projection techniques (17

-i6yat ge

e s s e n t i a l difference i s t h a t we derive the h y p e r l a t t i c e s and symmetries from t h e point symmetries and hydrodynamic t r a n s l a t i o n a l symmetries of t h e aperiodic c r y s t a l i n physical space. The h y p e r l a t t i c e s and hyperspace groups one can obtain i n t h i s way a r e r a t h e r s p e c i a l . They provide a f u l l c l a s s i f i c a t i o n of aperiodic c r y s t a l s . The projections a r e well defined by t h e construction and q u i t e unique, and do not r e - q u i r e p r i o r knowledge of t h e l a t t i c e s and space groups i n hyperspace.

The r e l a t i o n ç h i p t o t h e Quasi c r y s t a l formalism of Levine and ~ t e i n h a r d t ( ~ ' ) i s l e s s d i r e c t .

In s e c t i o n 2 we discuss t h e construction of the Bravais s e t s i n q space on which t h e 3

Fourier expansion of the aperiodic density can be indexed. The procedure allows a complete l i s t i n g f o r each point group and i s relevant t o t h e indexation of d i f f r a c - t i o n points. We discuss t h e cubic s e t s i n d e t a i l t o i l l u s t r a t e t h e procedure and give r e s u l t s f o r the tetragonal icosahedral and pentagonal point groups. Contrary t o claims i n t h e l i t e r a t u r e ( 8 * 2 1 ) only two s i x dimensional l a t t i c e s a r e relevant t o t h e indexation of icosahedral p a t t e r n s .

Zn s e c t i o n 3 we derive the incommensuratehydrodynamic t r a n s l a t i o n s of aperiodic crystals from the s t r u c t u r e of t h e Landau expansion of t h e f r e e energy. We use t h i s t o obtain h y p e r l a t t i c e representations of t h e Bravais s e t s i n

9

space and the hyper- l a t t i c e s dual t o them. These h y p e r l a t t i c e s describe t h e d i s c r e t e hydrodynamic trans- l a t i o n a l symmetry of aperiodic c r y s t a l s and a r e the relevant h y p e r l a t t i c e s f o r projection.

In s e c t i o n 4 we analyse the symmetries of aperiodic c r y s t a l s using t h e transformation properties of t h e phases under t h e operations of t h e point group. The hydrodynamic representations r e l a t e d t o hydrodynamic phase s h i f t s play a s p e c i a l r o l e . We a l s o discuss b r i e f l y t h e construction of both symmorphic and assymorphic hyperspace groups which provide a f u l l c l a s s i f i c a t i o n of aperiodic c r y s t a l s .

Section 5 is a summaa A more complete discussion and d e t a i l e d derivations w i l l be published elsewherec

.

I I

-

BRAVAIS SETS IN RECIPROCAL SPACE 2.1

-

General Derivation

Consider a density

where P+ and 8 3 a r e r e a l . For a (periodic) c r y s t a l t h e vectors q (in 3-d) can always 3

be indeged on &e of t h e 14 Brayais l a t t i c e s i n reciprocal space. Moreover p+ de-

pends only on t h e magnitude of q(q): 9

and is therefore t h e same f o r a l 1

q

belonging t o t h e same s t a r i . e . when

+

9' = Ga9 (2.3)

whsre G i s an operqtion of the point g r o u p ( 9 . In addition, because of time inver- s i o n (TT symmetry -q belongs t o t h e s t a r of q even when t h e two a r e not r e l a t e d by an operation of G (eqn. 2 . 3 ) .

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version. The Bravais s e t i s then de?ined by a l 1 t h e vectors

where t h e n1 a r e i n t e g e r s . By construction t h e s e t has the f u l l point symmetry(G

*

T) a t a l 1 poin?s. I t i s a l s o of measure zero. The indexation by t h e n i may of course be redundant.

I n general one can choose an a r b i t r a r y number of generating vectors. We r e s t r i c t ourselves t o t h e minimum nwnber of generating vectors necessary t o obtain a Bravais s e t spanning space. The r e s u l t i n g s e t s a r e minimal Bravais s e t s . The (fourteen) crystallographicBravais l a t t i c e s a r e s p e c i a l cases obtained by s u i t a b l e choices of t h e generating vectors. Quite generally t h e minimal Bravais s e t s can be constructed and l i s t e d f o r a l 1 point groups. Except f o r the Eravais l a t t i c e s they a r e always dense.

2.2 We consider

a,

few examples: Consider t h e cubic groups. Because of t h e high symmetry one generating vector K i s always s u f f i c i e n t . The s e t s d i f f e r according t o the o r i e n t a t i o n of

2

with respect t o the axes of the point group.

Choosing

2

along a fourfold a x i s (çay (001)) generates a simple cubic l a t t i c e . The generating s t a r contains s i x vectors.

-+ K along a t h r e e f o l d axis ((111)) has e i g h t vectors i n i t s s t a r and generates a body centered l a t t i c e .

2

along a twofold axis ((011)) generates a face centered cubic l a t t i c e with twelve vectors i n t h e generating s t a r .

I t i s f a i r l y easy t o check t h a t any

ik

with commensurateprojections on t h e cubic axes a l s o generates one of these three l a t t i c e s .

There a r e t h r e e minimalincommensuratesets. Two a r i s e when 3 K i s chosen perpendicular t o a twofold a x i s and have generating s t a r s of 24 vectors:

The four centered s e t (K -+ = (klk20)) and t h e two centered s e t

(ik

⁼(klklk2)).

F i n a t J ~ ~ t h e r e i s the general case K = (k k k ) with 48 vectors i n the generating

s t a r

.

^1' ^{2' 3}

In a l 1 cases t h e ki a r e assumed mutually incommensurate.

Thus t h e r e a r e s i x minimal cubic Bravais s e t s . ^{A l 1}other cubic s e t s require more generating vectors and can be described as (vector) sums of these s e t s .

An analogous analysis f o r the a x i a l tetragonal group gives four s e t s including the two l a t t i c e s . The minimal s e t s are a l 1 periodic along the axis.

For non crystallographic point groups t h e r e are, obviously, no l a t t i c e s and t h e s e t s a r e a l 1 dense. Clearly two s e t s can d i f f e r only when t h e choice of generating vec- t o r s d i f f e r s i n i t s r e l a t i o n t o t h e symmetry operations of t h e group. Some care i s however required because d i f f e r e n t choices may i n f a c t reduce t o d i f f e r e n t indexa- t i o n s of i d e n t i c a l s e t s .

Like the cubic s e t s icosahedral s e t s require only one generating vector. There a r e f i v e d i s t i n c t choices f o r t h e o r i e n t a t i o n of t h e generating vector.

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C3-146 JOURNAL

DE

PHYSIQUE

a) along a diagonal of t h e icosahedron (ID

-

a f i v e f o l d axis) with 12 vectors i n the generating s t a r .

b) along a diagonal of the dodecahedron (DD

-

a threefold axis) with 20 vectors i n t h e s t a r .

c) along an edge ( a twofold a x i s ) . This gives t h i r t y vectors i n t h e generating s t a r .

d) A general d i r e c t i o n perpendicular t o a twofold a x i s (a symmetry plane) gives a generating s t a r of s i x t y .

Time reversa1 i s redundant i n a l 1 these cases.

e. For a general d i r e c t i o n (no symmetry) t h e r e a r e 120 vectors i n t h e generating s t a r .

These choices give r i s e t o four d i s t i n c t Bravais s e t s . Contrary t o some statements i n t h e l i t e r a t u r e the choices 3.a) and 3.b) give i d e n t i c a l s e t s which can be mapped on a simple cubic l a t t i c e i n a 6 dimensional hyperspace. The edge s e t

-

3.c i s d i f - f e r e n t and maps on an FCC h y p e r l a t t i c e f o r which the sum of the s i x indices i s a l - ways even.

The two other s e t s require twelve and eighteen dimensional representations respective- l Y .

Analysis of the pentagonal case shows t h a t t h e r e i s only one minimal Bravais s e t both f o r t h e planar (Penrose) case and i n 3 dimensional space. Like t h e tetragonal s e t s

(and a l 1 other a x i a l s e t s ) the minimal pentagonal Bravais s e t i s periodic along t h e a i s .

III. INCOMMENSURATE TRANSLATIONS AND KYDRODYNAMIC VARIABLES

The symmetry of a c r y s t a l i s not completely determined by t h e Bravais s e t i n

4

^space.

A s i n ordinarycrystallographyit i s n a t u r a l t o assume t h a t t h e amplitudes ^(p-t) i n t h e Fourier expansion of t h e density (eqn. 2.1) a r e t h e same f o r a l 1 members of

%

s t a r

(eqn. 2.2.).

For each s t a r one then has a geometrical s t r u c t u r e factor(23) A = C exp i ( q a -? r + 8 - + )

a qa

whose symmetry depends on t h e choice of t h e phases (O+). The overall symmetry of t h e c r y s t a l i s t h a t which i s common t o a l 1 t h e s t r u c t u r e q f a c t o r s (eqn. 3.1.) f o r a l 1 s t a r s . This includes both t r a n s l a t i o n a l and r o t a t i o n a l symmetries.

For periodic c r y s t a l s t h e (pure) t r a n s l a t i o n a l symmetry i s alyays described by one of t h e 14 Bravais l a t t i c e s and i s uniquely determined by t h e q space Bravais l a t t i c e . The r e a l and q-space l a t t i c e s a r e dual t o each other. This can be r e l a t e d t o t h e f a c t t h a t r i g i d t r a n s l a t i o n s a r e t h e only hydrodynamic va

Tfi3:1IS

f o r periodic c r y s t a l s

.

Aperiodic c r y s t a l s have additional hydrodynarnic variables and t h e i r symmetry can be represented by a h y p e r l a t t i c e i n t h e hyperspace ofincommensuratehydrodynamic t r a n s l a t i o n s which i s d u a l t o t h e Bravais s e t .

We first introduce t h e hydrodynamic t r a n s l a t i o n s i n a systematic way. A general term i n t h e Landau density expansion of t h e f r e e energ@-6) has the form

with a s u i t a b l e mode1 dependent, c o e f f i c i e n t s and using t h e d e f i n i t i o n s of eqn. 2.1.

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c q i = 0

s o t h a t the qi form a closed polygon. 3

A r i g i d t r a n s l a t i o n

(g)

i s represented by phase s h i f t s :

3 -+

& @ + = q a t 9

f o r a l 1 -f q.

Thus from eqns. 3 . 2 and 3.3

-f

and t h e f r e e energy i s i n v a r i a n t f o r any value of the d dimensional vector t . For a periodic c r y s t a l t h e qi a r e vectors of the Bravais l a t t i c e and the t r a n s l a t i o n s -t

f o r which

f o r a l 1

qi

form a l a t t i c e .

-

There a r e no s o l u t i o n s t o equation 3.6 f o r ageriodic c r y s t a l ~ ( ~ ~ ) ( i . e . when t h e Bravais s e t i s not a l a t t i c e ) . The vectors q. haveincommensuratecomponents along t h e axes of any coordinate system i n d dimens$03al space. Choosing a s p e c i f i c coordinate system we can decompose t h e vectors (q) i n t o t h e i r incommensuratecomponents

3

e

^+,

q = q (3.7)

v= 1

where the components of d i f f e r e n t

Q)

a r e mutually incomensurateand Q i s t h e number ofincommensuratecomponents required f o r the p a r t i c u l a r s e t . This amounts t o a d.9.

dimensional h y p e r l a t t i c e representation of t h e Bravais s e t . The decomposition depends on t h e choice of coordinates but !?. does ~ o t .

From t h e t r a n s l a t i o n a l invariance requirement (eqn. 3.3) it now follows t h a t c q i = o -ry

t ^(3.8)

f o r a l 1 terms i n t h e f r e e energy. Thus t h e phase s h i f t s

sov

=

q .

^tv

f o r a r b i t r a r y choice of t h e Q(d dimensional) vectors

2,

a r e a l 1 hydrodynamic. The aperiodic c r y s t a l has d.!?.hydrodynamic t r q l a t i o n s which do not change t h e f r e e energy. In the hyperspace defined by t h e t r i g i d t r a n s l a t i o n s define a subspace

Pz?

(3.10)

f o r a l 1 v. A general hyperspace t r a n s l a t i o n

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defines a hydrodynamic phase s h i f t through eqn. 3.9 and i s t h e r e f o r e an invariance operation of t h e f r e e energy of t h e aperiodic c r y s t a l analogous t o a r i g i d t r a n s l a t i o n . I n general two r e a l i s a t i o n s r e l a t e d by incomensuratetranslations

fi)

do of course look q u i t e d i f f e r e n t . They a r e however equivalent i n having i d e n t i c a l f r e e energies. For example a l 1 Penrose t i l i n g s with i d e n t i c a l t i l e s and construction r u l e s a r e r e l a t e d i n t h i s way.

I n t h e hyperspace

( T )

one can a l s o define a h y p e r l a t t i c e

(E)

dual t o the Bravais s e t

- -

: * r i =

~ ? . p = 2 r n (3.11)

v

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C3-148 JOURNAL DE PHYSIQUE

The h y p e r l a t t i c e describes t h e @ncommensurateJ t r a n s l a t i o n a l invariance operations of the aperiodic c r y s t a l . Since the

';f"

and therefore t h e a r e incornensurate t h e hyper- l a t t i c e (R) contains no r i g i d t r a n s l a t i o n s i . e . no vectors i n t h e subspace defined by eqn. 3.10.

A h y p e r l a t t i c e construction r e l a t e d t o t h e above has been given by Per ~ a k ( ' > l ~ ) f o r t h e simple icosahedral s e t . It i s also r e l a t e d i n a sim l e way t o t h e projection techniques which have been developed by various a u t h o r ~ ( j ~ - ~ ~ ) t o obtain aperiodic pentagonal t i l i n g s o f t h e plane o r icosahedral t i l i n g s o f t h e space. We note however t h a t our derivation defines the h y p e r l a t t i c e s uniquelyfromthe d dimensional Bravais s e t s .

Quite generally one can extend t h e s t r u c t u r e f a c t o r (Aq eqn. 3.1) t o hyperspace by writing

The density:

p(T) = L

A (F)

(3.13)

s t a r s '9 q

then describes a proper periodic c y s t a l i n hyperspace with t h e Bravais l a t t i c e

R.

In t h e space of physical vectors ( r )

À i s i d e n t i c a l t o A (eqn. 3.1). The density i n t h e u n i t ce11 of t h e periodic hyper- c&stal i s arbitra$ One obtains an aperiodic t i l i n g of hyperspace from any periodic t i l i n g of hyperspace. Our construction assures t h a t t h e t i l i n g i s described by t h e chosen Bravais s e t .

IV. SYbbETRIES RELATED TO POINT OPERATIONS AND WDRODYNAMIC REPRESENTATIONS The ( d i s c r e t e ) t r a n s l a t i o n a l symmetry of a l 1 c r y s t a l s i s described by t h e fourteen Bravais l a t t i c e s . The much r i c h e r space group synmietry r e s u l t s when one a l s o con- s i d e r s t h e e f f e c t s of t h e operations of t h e point group on t h e phases i n t h e s t r u c - t u r e f a c t o r s . The usual procedure i s t o use t h e known symmetries of t h e 230 space groups t o determine r e l a t i o n s between t h e phases i n each s t a r . The r e s u l t s a r e l i s t - ed i n r e f . 23. An e x p l i c i t procedure f o r deriving these r e s u l t s i s described e.g.

i n r e f s . 5 and 6 . Since t h e relevant hyperspace groups a r e not known we need a more general forma1 procedure. We s h a l l derive t h i s by considering t h e e f f e c t of the point group operations on t h e phases e x p l i c i t l y . The relevant hyperspace groups con- t a i n t h e h y p e r l a t t i c e t r a n s l a t i o n s a s a subgroup. Symmorphic hyperspace groups a l s o have Points with t h e f u l l point group symmetry. Non symmorphic hyperspace groups only contain combinations of some of the point operations with s u i t a b l e h y p e r l a t t i c e t r a n s l a t i o n s . They therefore have no points with the f u l l point symmetry. Only very s p e c i a l Bravais l a t t i c e s i n hyperspace a r e a c t u a l l y generated by t h e procedure o f section 3 . The point groups are a l s o very s p e c i a l and r a t h e r a r t i f i c i a l a s

groups i n hyperspace. Thw the f u l l cubic group i n 6-d has 16080 elements(ZgO?! ) of which only 60 (or 120) a r e elements of t h e relevant icosahedral group. I t i s thus c e r t a i n l y much more economical t o construct t h e relevant space groups e x p l i c i t l y . Consider the e f f e c t of these point operations on tge s t r u c t u r e f a c t o r s (eqn. 3.1).

The point group permutes the vectors of t h e s t a r (q ) . Its e f f e c t on the s t r u c t u r e f a c t o r o i i t h e r e f o r e be described by a permutation O!? t h e phases O

.

^{I t}i s convenient t o describe t h i s i n terms of the i r r e d u c i b l e representations of tfie point group

nvn mv

where 10

,

i s a vector whose dimension ( s ) i s equal t o the number of vectors i n

9 9

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ixh f o r some y and i n general w i l l depend on the type o f star(qY.

To have a r e a l density one must have O + = - & .

-9 9 (4.2)

3 3

which r e s t r i c t s t h e possible representations when q and -q a r e r e l a t e d by G. Also because of eqn. 4.2

1 ₃

The choice of o r i g i n i n t h e A ( r ) i s a r b i t r a r y . S h i f t i n g the o r i g i n r e s u l t s i n hydrodynamic phase s h i f t s corre$onding t o a r i g i d t r a n s l a t i o n (eqn. 3.4). These phase s h i f t s transform with a s p e c i a l representation of G ( t h e vector representation v) which i s allowed by the symmetry of a l 1 s t a r s .

For symmorphic space groups one c m , by s u i t a b l e choice of o r i g i n , remove t h e vector representation f o r a l 1 s t a r s . For these high symmetry choices of o r i g i n t h e i d e n t i t y representation i s t h e only one appearing, s o t h a t a l 1 s t a r s have t h e f u l l symmetry G.

We note t h a t t h e difference between r e l a t e d point groups

-

such a s the four cubic point groups, appears a t t h i s stage.

We can t r e a t symmorphic aperiodic c r y s t a l s i n an analogous way. The hydrodynamic phase s h i f t s (eqn. 3.9) transform according t o some representations which we s h a l l c a l 1 t h e hydrodynamic representationsof G.For c r y s t a l l o ~ r a p h i c p o i n t groups these a r e always r e p e t i t i o n s o Y t h e vector representation. For such point groups each incom- mensuratecomponent (q

,

tv, 80' ) transforms separately under t h e operations of G.

Equivalently one can choose a c$ordinate system which i s invariant under G and use i t t o d e f i n e t h e s e p a r a t i o n of t h e q. i n t o theirincommensuratecomponents. This i s pos- s i b l e even f o r non t r i v i a l incommlensurate s e t s l i k e those considered i n section 2.

F o r n o n c r y s t a l l o ~ g r a p h i c p o i n t groups t h e r e i s always more than one type of hydrodynam- i c representation. For example

-

t h e two t h r e e dimensional representations of t h e icosahedral group a r e both hydrodynamic. S i m i l a r i l y t h e two two dimensional represen- t a r i o n s of t h e pentagonal group are t h e hydrodynamic representations f o r the Penrose p a t t e r n s . I n both cases t h e vector representation i s one of these two. Since hydrodynamic phase s h i f t s a r e possible f o r a l 1 s t a r s each i r r e d u c i b l e hydrodynamic representation must show up a t l e a s t once f o r every d dimensional s t a r i n t h e Bravais s e t . One a l s o can not choose an invariant coordinate system f o r these point groups.

A symmorphic aperiodic c r y s t a l i s defined by t h e existence of high symmetry points, with t h e f u l l symmetry G, i n t h e hydrodynamic hyperspace. For t h i s choice of o r i g i n a l 1 phases i n a s t a r have t o be equal. When t h e i d e n t i y representation i s not an allowed representation f o r a s t a r [eqn. 4.1) t h e phases a r e determined.From t h e point group symmetry one then has only two p o s s i b i l i t i e s :

Thus t h e difference between t h e two icosahedral groups (Y;Yh ) i s t h a t eqn. 4.4 holds f o r a l 1 s t a r s i n Yh. For Y t h e r e i s a f r e e phase f o r t h e large, low symmetry, s t a r s of 120 vectors.

From the h y p e r l a t t i c e symmetry it i s c l e a r t h a t t h e high symnetry points form a l a t - t i c e i n hyperspace which has a t most one point i n the physical space (eqn. 3 .IO).

For a general point t h e r e a r e , i n addition, hydrodynamic phase s h i f t s . A l 1 equiva- l e n t r e a l i s a t i o n s of the aperiodic c r y s t a l can be r e l a t e d t o each other by t h e hyperspace t r a n s l a t i o n s i n a Wigner S e i t z ce11 of t h e h y p e r l a t t i c e .

Most of t h e 230 space groups a r e non symmorphic. The point group G i s not an e x p l i c i t subgroup of these space groups. Some of t h e point operations only appear i n combina-

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C3-150 JOURNAL DE PHYSIQUE

t i o n with s u i t a b l e t r a n s l a t i o n s (screw a x i s , g l i d e planes). While the p o s s i b i l i t y of non symmorphic aperiodic c r y s t a l s has not been c o n s i d e r e d i n t h e recent l i t e r a t u r e on icosahedral and pentagonal c r y s t a l s t h e r e i s n o a p r i o r i r e a s o n t o disregard t h i s pos- s i b i l i t y . For incommensuratecrystals with c r y s t a l l o g r a p h i c a l l y a l l o w e d points groups such synunetries have been observed and analysed. We note t h e d e t a i l e d work of Buit- ing e t . a l . (16)

.

In t h e language we have developed non symmorphic space groups appear through phase s h i f t s which transform with t h e hydrodynamic representations but cannot be removed by a s h i f t of o r i g i n . In general we c m w r i t e f o r a non symmorphic element

where G i s an element of G, 10> i s defined as i n eqn. 4.1 and

1%

i s - a hydrodynamic s h i f t rgpresenting t h e e f f e c t of a s u i t a b l e hydrodynamic t r a n s l a f i o n (hl on the phases i n t h e s t a r .

-iï

i s a vector with i n t e g e r components. Without l o s s of generality we can assume t h a t h i s contained i n t h e Wigner-Seitz ce11 of t h e h y p e r l a t t i c e . The meaning of eqn. 4.5 i s of course t h a t t h e e f f e c t of Gaon t h e phases can be r e - moved by a s u i t a b l e (hyperspace) t r a n s l a t i - (-5 )

.

^Anonsymmorphic space group a r i s e s i f one can choose a s e t of vectors

ha

f o r a i l Ga consistent with t h e point group algebra. Listing the p o s s i b i l i t i e s i s f a i r l y straightforward f o r a x i a l point groups but becomes complicated f o r t h e cubic and icosahedral groups. We only note the forma1 implications of eqn. 4.5.

Let p be t h e period of G ( ( G )' = l ) L The t r a n s l a t i o n corresponding t o ( G , ) ~ must therefore be a hyperlatt?ce Bector (fia). By repeated operation one f i n d s

= P-1

(4 6 ) ll=o

Thus one must have

where g a i s the component of which i s invariant under Ga.

In addition, since an equation analogous t o 4.5 must hold f o r a l 1 elements of the point group f o r t h e same t h e decomposition i n t o i r r e d u c i b l e representations can contain only the i d e n t i t y and the hydrodynamic representations.

Using these r e s u l t s t h e construction of t h e relevant hyperspace groups i s s t r a i g h t - forward.

V. CONCLUSIONS

The main point of our discussion was t o show t h a t t h e symmetry c l a s s i f i c a t i o n of aperiodic c r y s t a l s i s simply r e l a t e d t o t h e i r hydrodynamic t r a n s l a t i o n s and can be carried out i n a systematic way s t a r t i n g from t h e known d i s c r e t e point groups. Our procedure constructs t h e h y p e r l a t t i c e and space groups i n hyperspace e x p l i c i t l y from the point group and Bravais s e t i n physical space. The algorithm we have described allows a complete l i s t i n g of t h e Bravais s e t s , h y p e r l a t t i c e s and hyperspace groups f o r any point group.

This work was supported i n p a r t by t h e fund f o r Basic Research administered by t h e I s r a e l Academy o f Science and Humanities and by t h e National Science Foundation Grant No. DMR-84-12898. The h o s p i t a l i t y of Exxon Research Laboratories where p a r t of t h i s work was done i s g r a t e f u l l y acknowledged.

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/1/ Alexander, S., Phys. Rev. 127 420 (1962).

/2/ Landau, L.D., Phys. Z . S O W ~ .

g

26, 545 (1937).

/3/ Alexander, S. and Mctague, J . , Phys. Rev. L e t t . 41 702 (1978).

/4/ Baym, G . , Bethe, H.A. and Pathick, C.J., Nucl. ~ h y s . (A).

175

25 (1971).

/5/ Alexander, S. i n Symmetries and Broken Symmetries i n Condensed Matter Physics N. Boccara e d t . IDSET P a r i s 1981, p. 141.

/6/ Alexander, S . , J . de Physique Coll. C3 Vol. 46 33 (1985).

/7/ Hornreich, R.M. and Shtrikman, S., J. de ~ h y x q u e 335 (1980), i b i d . 367 (1981).

Grebel, H, Hornreich, R.M. and Shtrikman, S., Phys. Rev. A 28 1114 (1983).

/8/ Kalugin, P.A., Kitayev, A. Yu. and Levitov, L.S., J. de ~ h y x q u e ( L e t t r e s )

5

L601 (1985), Pisma v ZETF 41 119 (1985) (JETP L e t t . 119 (1985)).

/9/ Shechtman, D., Blech, I . , G r a t i a s , D. and Cahn, J . W . , Phys, Rev. L e t t . 1951 (1984).

/ I O / Bak, P., Phys. Rev. L e t t . 1517 (1985)and Phys. Rev. B z 5764 (1985).

/ I l / Levine, D., Lubensky, T.C., Ostlund, S . , Tamaswamy, S . , S t e i n h a r d t , P.J. and Toner, J., Phys. Rev. L e t t .

54

1520 (1985)

Lubensky, T.C., Ramaswanly and Toner, J . , Phys. Rev. B32 7444 (1985).

/12/ Sachdev, S. and Nelson, D.R., Phys. Rev. B 32 4592 (1985).

/13/ Mermin, N.D. and Troian, S.M., Phys. Rev. ~zt. 1524 (1984).

/14/ J a r i c , M.V., Phys. Rev. L e t t . 55, 607 (1985).

/15/ Janner, A. and Janssen, T., ~ h c . Rev. L e t t . $- 1700 (1980), Acta C r y s t a l .

A 3 6

408 (1980).

Janssen, T., Janner, A. and deWolf, P.M., Acta Cryst. A39 658, 667, 671 (1983).

/16/ Buiting, J . J . M , Weger, M. and Mueller, F.M., S o l i d S t a t e Corn.

46

857 (1983), J . Phys. F

2

2343 (1984) Buiting, J . J . M . , Janner, A. and Weger, M. ( p r e p r i n t ) . /17/ McKay, A.L., Physica A 114 609 (1982).

/18/ E l s e r , V . , Phys. Rev.

LX.

1730 (1985).

/19/ Duneau, M. and Katz, A., Phys. Rev. L e t t . 54 2688 (1985).

/20/ Levine, D. and S t e i n h a r d t , P.J., Phys. ~ e v T ~ e t t . 53 2477 (1984).

/21/ Cahn, J . W . , Shechtman, D. and G r a t i a s , D. ( p r e p r i n q /22/ Alexander, S. ( t o be published)

/23/ I n t e r n a t i o n a l Tables f o r X-RayCrystaLlography, v o l . 1 (1959).

/24/ Note t h a t T with 12 elements does n o t generate t h i s s t a r .

/25/ This holds f o r d i r e c t i o n s i n which t h e c r y s t a l isincommensuratetherefore

net

along t h e a x i s of a x i a l s e t s .