Comput. Math. Applic. Vol. 16, No. 5-8, pp. 407-424, 1988 0097-4943/88 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1988 Pergamon Press pie

GENERALIZED SYMMETRY IN CRYSTAL PHYSICS

V. A. KOPTSIK Department of Physics, Moscow State University, Moscow 117234, U.S.S.R.

Abstract--The external and internal symmetry groups of the matter object and physical processes with a complex structure and the internal degrees of the freedom are combined into single groups of their generalized coloured symmetry by means of the wreath product of these groups. It is shown that the colour symmetry groups of different types are widely used in the tensor and magnetic crystal physics, the crystal structure analysis and in the theory of structure phase transitions for the symmetry description of the structure and physical behaviour of non-rigid molecular crystals, incommensurate crystal phases and icosahedric quasicrystals, etc. Some difficult problems in this field are solved. It is shown that the colour symmetry approach complementary to the irreducible and induced representation of classical symmetry groups expand the whole field of the symmetry analysis of modern physical problems.

1. C O N T R I B U T I O N OF A. V. S H U B N I K O V TO T H E T H E O R Y OF S Y M M E T R Y

Academician A. V. Shubnikov was one of the most brilliant scientists of the twentieth century who worked in the field of the theory of symmetry and its applications. He was the first to comprehend that the then concepts of symmetry of natural structures and processes were, in principle, approximate and relative and he was the first to reconstruct all the science of symmetry.

In his study he has merged together the symmetry of discrete structures, which has been of interest for classical crystallography, and the symmetry of continuous and semicontinuous media and fields. He deduced the groups of antisymmetry of finite figures, of one-, two- and three- dimensional discontinua, continua and semicontinua and the groups of similarity symmetries. In a single real object he could see a whole hierarchy of symmetries, which are active at different structure levels, by considering them in their unity with the dissymmetries with respect to the senior groups of possible embracing transformations. At the present time, the pioneering works by Shubnikov on non-crystallographic and irrat ional-rotational symmetries of rods, and two- and three-dimensional semicontinua look very modern because they anticipated the present-day research on descriptions of real crystals, incommensurable crystal phases, biological macro- molecules and structures.

2. O R T H O G O N A L - P E R M U T A T I O N S Y M M E T R Y G R O U P S IN C L A S S I C A L C R Y S T A L P H Y S I C S

In fact, the ideas of the generalized symmetry have been introduced into crystal physics at the very end of the nineteenth century and in the beginning of the twentieth century when its basic equations have been formulated in the tensor form:

A,~...i, = at I . . . i s i s + I . ..inBis+l ...in" (1)

Nevertheless, this fact was understood only several decades later. Really, a matter orthogonal tensor a of the range n in the equation of the linear relationship

(1) between the tensor of action B and the tensor A of physical properties of a crystal is determined, as a geometrophysical object, by its 3" components:

C.AM.W.A, 16/5.-8--E

2 = i t . . . i ~ , # = i~+, . . . i~ ; i ~ , . . . , i ~ = 1 , 2 , 3 ;

~b = U - T S - D . E - B . H - * : o = x ' X ,

x = S , D , B , ~ , X = T , E , H , a

407

2 , # = 1 . . . . . 6,

(2)

408 V . A . KoIrrSlK

and by the law of its transformation under orthogonal transformations of the coordinate system, gX~ = Xi,:

a,~ ...on = x ( g ) cflit . . . ccninai I ...in, Cri = cos(Xr, X;), x(g) = +- 1. (3)

The sign of the correcting factor in relation (3) distinguishes polar ( x ( g ) = x(g) = + l) and axial C ( ( g ) = + l , Z ( g ) = - l ) tensors, g E m m c m ~ T = O ( 3 ) , ~ = g ] ' = ' [ g ~ O ( 3 ) , ~ o o = O * ( 3 ) c 0(3) = T x 0*(3), I being the inversion transformation, oo infinitesimal rotation. Equation (3) may be written in symbolic invariant form a' = X (g) g" a, where g ~ G ~ G ~g is the linear operator isomorphic (.-~) to the transformation g e G _ 0(3).

The invariant properties of the tensor a with respect to external (orthogonal) coordinate transformations g ~--~c~i~t... c~g~n (g)

ail...r~ = x ( g ) g ' a i ~ . . . i n =ai~...~n, g~--*g~Gc~t= 1,2,3 ,4 ,6 . . . . . m ~ m (4)

and internal (thermodynamical) permutation of indices

~e,j = ~ej~, d~jk = d, kj, s ~ = s j ~ = su~k = Skt~ . . . . . ( 5 )

were taken into account separately in every handbook on crystal physics issued in the first part of this century from Voight [1] to Nye [2]. But the conditions of the external and internal symmetry of tensors could only be strictly formulated when the concepts of the group of orthogonal symmetry of a tensor [3] and the group of its permutation symmetry in the index space [4] were determined. The group of external symmetry was defined by Shubnikov as the maximal group of trans- formations (4) which preserves the tensor's matrix

g(g)g'a~ . . . a~n = at,...~n, g e G,,~-.~G ~_ O(3), G , ~_ G , , r . r ) ~_ Gcryst (6)

The group of permutation symmetry was defined by Yahn as the maximal subgroup P, of the symmetric group S, that satisfies the invariance conditions for a fixed type of tensor

(;, p ' a i l . . .in ~ - ap~...pin, P = it Pi2 . pin ~ P I c S~. (7)

Here pi l , pi2 . . . . . pin are the same indices il,/2 . . . . . i~ ordered in a certain (the same or another) way.

The concepts of external and internal symmetry of tensors were coalesced into a unique construction [5] of direct products of groups G, of relation (6) and P, of relation (7), those products being built according to the model of senior groups of P-symmetry [6]

z (g)g tP)a t = z ( g ) ( P [ g ) al = z ( g ) ~ ( g ) a , t = at ,

a, = a~,..., n, p I - = p i ~ . . . pi~, g~')~ {(Pig>} = G <y) = P, x G,. (8)

Here {~(g)} = V ~ is the tensor representation (the nth Kronecker degree of polar-vector representation of V), ~(g) = c~i~.., c~;,. of equation (3), and the conditions of its symmetrization (or antisymmetrization) are determined by the group of permutations of indices P. of relation (7). In particular, the group G~P ) includes the subgroup 1 re,) of operators of permutational identity 1 (') ~ 1 (P'~ = {(pl l >} <--)Pi

l(P)ail . . . in = ~ ( l ) a p i I ...pi~ = a~ ...in, ~(1) = Ciii~ . . . i; , in(1) = 1, (9)

which do not change the component values of the tensor a but permutate their indices. The possibility to construct an orthogonal-permutation group P, x G, (8) as a single group of

internal-external symmetries of the tensor a is only interesting for crystal physics from the viewpoint of methodology. More essential are the inversion-permutation groups which are used in conformation analysis and in molecular spectroscopy of non-rigid crystals (see, for example, Ref. [7] and references therein for the original works by Hougen and Longuet-Higgins, 1962--63).

3. ANTISYMMETRY, COLOUR SYMMETRY AND THEIR MAGNETIC INTERPRETATIONS

During a halfcentury, crystal physics has used the supersymmetry G~Y ) = Pi x GI and thus it was like the Molirre personage who used to speak prose but never knew about it. The situation was

Generalized symmetry in crystal physics 409

realized in 1951 when Shubnikov's book "Symmetry and Antisymmetry of Finite Figures" [8] was published and Landau [9] introduced into the electrodynamics and mechanics of continuous media the operator of time inversion R = 1' which has opened a fruitful route to magnetic interpretation of antisymmetry as the subgroups of the direct product of the groups 1' x 0(3) and 1' x E(3).

Shubnikov formulated his ideas on antisymmetry in 1942-43 in connection with the problem of symmetry of directed quantities and of figures with mixed polarities, for example, of a single-axis mechanical stress tensor or of a chess-board.

On 29 April 1942, Shubnikov wrote in his diary: "It seemed to me that the positive and negative figures of a mixed polarity I have discovered are a success of the same level as the discovery of zero". On 26 December 1942 he introduced the term of "negative equality" to refer the equality of two figures one of which is positive (white outside and black inside) and the other is negative (black outside and white inside). On 17 February 1943, he discovered that a chess-board is a

symmetric figure, provided that "a new symmetry transformation--the product of mirror reflection in a plane and sign inversion--is introduced". Shubnikov considered this date as "the formal birthday of the concept of antisymmetry" (see Ref. [10]). On 20 September 1943 Shubnikov noted that antisymmetry may be considered as symmetry in four-dimensional space. Then he started to derive intensively the point groups of antisymmetry and, as the reports [11] indicate, by the 1 April 1944 all the main results in this field were obtained.

After publishing the book [8] it was discovered that the antisymmetry groups were already introduced by Heesh in 1930 [12] as the point groups of orthogonal transformations of four-dimensional space. It was revealed also that Shubnikov's groups may be obtained from Heesh groups by isomorphic projecting of the latter ones into three-dimensional matter space. Shubnikov explained the difference in the two approaches: "A mathematician constructs the science of symmetry without use of a figure. A crystallographer thinks this approach to be either useless or even senseless. I have thought that this problem must be necessarily solved because the very common three-dimensional figures were discovered whose symmetries could not be described using conventional methods." (See Ref. [10].)

The material spaces differ from purely geometrical ones because to every point r of the space R(3) we may attribute a non-geometrical property, say, spin or local magnetic moment s(r). Defining the action of the operator 1' on the magnetic moment l ' s = - s and writing the function s(r) as a set of material points s(r)= {(si, rj)[(s~, rj)e S(d) x R(3), s~ E S(d), rj~ R(3)} we establish a formal isomorphism between the transformations of vectors of a special four-dimensional space R(4) = R(3)(~R(1), whose fourth coordinate x4 is orthogonal supplement to the space R(3) and takes only two values x4 and -x4 , and a stratified geometro-physical space R(3)•S(1) in which x4 is substituted by a local axial vector s e S(1), for example,

ix IcHcc il Ix l OOlC 0 llxll X-~ C2'1 C2'2 C2"3 X2 ~ [ ¢2'1 C2'2 C2'3 0 X 2 , , x; = c3,, Ca,: c3,3 x3 /c3 ,, Ca,: c3,3 0 or = (1 s,gr) = ( - s , r ), (10) x'4 0 0 0 - x4 ~ 0 0 0 - - 1 g ' = ( l ' [ g ) e l ' x O ( 3 ) .

When generalizing this example, we arrive at the isomorphism between special orthogonal groups 0(4) and l ' x 0(3) or, in a more general way, between Euclidian groups E(4) and l' x E(3) which act in the spaces R(3)~)R(1) and R(3)OS(1), respectively:

O(4)~--~1' x O(3), E(4)*-.l ' x E(3), R(1)*-~,S(I). (11)

The crystallographical subgroups of the groups of expression (11)

G~ys t~ l ' xGc~ys t~ l ' xO(3) , ~ ' ~ l ' x ~ c l ' x E ( 3 ) (12)

are called the Shubnikov-Heesh point groups and the Shubnikov space groups, respectively. The latter ones were derived in 1953-54 by Zamorzaev [13] and by Belov et al. [14]. The groups G ~ t and ~ ' , where ¢ is a Fedorov-Sch6nflies group, describe the symmetry of homogeneous magnetic fields (magnetis tensors) of colinear magnetic structures of crystals within the framework of magnetic interpretation of colour symmetry groups. We should note that they do not include the operator l ' separately but only in the combinations ( l ' ]g) , (l '[t~) with the isometric trans- formations g ~ G~y~t, ~b ~ ¢~.

410 V.A. KOPTSIK

It is easy to see that the groups of expression (12) are constructed similarly to the groups of expression (8) which are the direct products of substitution groups Pn of n non-geometrical values (spin states, colours, etc.) with the groups of isometric transformations of crystals Gc~y,t or O. Those are a particular case of the groups of the so called P-symmetry:

G~P)~_PxG, 7r:G-~P:g~--~z(g)=p, geG, p e P ,

G(P)= {<Pig>}, <Pilg:> <Pklge> = <PiPklgjge>, (13)

which are constructed according to homomorphism n: G ~ P [6, 15]

( g . H . . , gsH "~ 7r(g)=\gg~H...gg, H / = p , H c G = g ~ H + . . . +g,H, geG, peP .

In the colour interpretation, the Shubnikov antisymmetry groups correspond to two-colour (black and white) groups. If n = 3, 4, 6, then according to the algorithm of expression (13) we obtain Belov's 3-, 4-, and 6-colour groups [16]. In the general case, the number n in the groups of P-symmetry may take any value up to infinity [17].

The magnetic interpretation of the group of coloured P-symmetry describe the symmetry of magnetic structures with predominantly exchange-type interaction (Hamiltonian)

1 ~exch = - ~ }-', J ( r - r')(s(r)'s(r')), O~P~h c (1' x 0(3)) x ¢ x G~ro~h. (14)

r , r '

Taking into account the forces of magnetic anisotropy described by the relativistic Hamiltonian ~ = ~x~. + ~m,

1 ~ = 2/~ ~ ~ ~ {(s(r)" s(r')) lr - r'l 2 - 3(s(r) . (r - r')) . (s (r) . (r - r'))} (15)

r, f '

we come to the definition of the groups of relativistic magnetic symmetry [18]

O ~ / c (1' x [G.,.y,,]) x ¢, = G.,,e,,,, c G..~o,,oh = (1' x 0 ( 3 ) ) x O, (16)

where [Geryst ] is the subgroup of rotations of Gcryst ~'~ O/T to which the space group • = TG belongs. Groups of condition (16) belong among the special groups of so-called Q-symmetry [17, 19] which are the subgroups of semidirect product

G(q) = {<Pig>} c_e2 G, <plg>(s, r) = (pgs, gr) = (s', r'),

<Pilgj><Pklgl> = <P,(g:Pkgj-t)[gjgt>, P,, gjPkg) - ~ P , gjgt~G. (17)

The action of the operator <Pig> on a point (s,r) in expression (17) differs from <plg>(s,r)=(ps, gr) in expression (13), hence the multiplications laws for the operators in expressions (13) and (17) in the groups of P- and Q-symmetry differ also. The group @(q) of Q-symmetry of rigid motions maps onto itself the vector field s(r) which describes the relativistic magnetic structure of a crystal, and this results in a rigid coupling of the field s(r) with the symmetry elements of a crystal, while the initial orientation of the s(r) field in exchange magnetic structures is arbitrary.

The common property of the P- and Q-symmetry groups is the global character of the group operation, i.e. its independence of the point (s, r).

The decisive step towards taking into account the internal symmetries P~ of geometro-physical objects is connected with the introduction of the wreath product (ordinary or twisted) of P- and G-groups into the symmetry theory [19-23]:

P, w r G = P ~ 2 G = ( P s , xP~2x . . . x P g k x . . . ) 3 , G. (18)

Here wr is the symbol of wreath product, and 2 is the symbol of semidirect product of groups, P,~ is the Cartesian product of isomorphic copies of the group P, indexed by the elements gk E G, that is, P~ is the group of all mappings f from G onto P,,

- (19) P, - { f l f :G --*P.:gk~--~f(gk)=p~*}, gkeG, pS, ~p~k ~ pG

Generalized symmetry in crystal physics

with the positional (depending on gk or rk = g~r~) law of composition

= f,(g )'fAgP

and with the natural action of G at Pa by automorphisms [24, 25] G G. g G x P , ~ P , . ( g , f ) ~ - - ~

where

o r

(*f)(gk) = gf(gk) = ~k ~ p~gk

g g k g - 1 ~ g g k g - I (*f)(gk) = *f(gk) = P ~ r, ,

411

(20)

(21)

(22)

(for the groups Wp- or Wq-symmetry) such that

g'gJf=g,(gf), J ~ f = f and g(fif j)=gf/ .~. (23)

A more clear idea about the wreath product construction P~wr G, and the action of position operators

< ~ i ~ . . . ~ i . . . I g i ) ~ P , w r G

on a matter point (s, r), and of the laws of their positional multiplication

g gygk gk ( . . . ~ . . . I g , ) ( . . . ~ . . . I g j ) - - ( . . . P , Pj . . . [gggy) ~ P~ wrpG (24a)

• . *~g, g,gff'-' ( . . ~ . .Igt)(. . .~.. .Igj)=(.. . ,- , r j . . . I g , g j ) ~ P ~ w r q a . (24b)

o r

is given in Ref. [21] for the groups Wp- and Wq-symmetry, respectively• In Figs l(a)--(c) the above definitions are exemplified and Fig. 1 (d) gives the structure schemes of the groups of P-, Q-, Wp- and Wq-symmetry which exhibit the action of their G-, P~-, and P~k-components at the correspond- ing R-, S-, and Sg~-substances of a stratified geomerophysical space.

4. MAGNETIC, COMPLEX AND PHASE SYMMETRY IN CRYSTAL PHYSICS AND IN STRUCTURE ANALYSIS OF CRYSTALS

The concepts ofcolour symmetry which takes into account the generalized (internal and external) symmetry of the subjects in question (crystals) can be transferred to their physical properties on the basis of the known principles by Neumann-Minnigerode--Curie (PNMC) and Curie-Shubnikov (PCSh) by substituting in the relationships [17, 26]

~ ) c r:.t~) c G~(Neumann), vsystom "~obm"vaetion = G pro~.y (PNMC) vcrys t - - "J a(p, 7) - -

Unbj oett'2(¢0 = ?. vpartsit'~'=(¢0 " ~srmm,~<~ = n " ~symm,(~(~) ~system(~=(~) ---- N ~partsk("~'(¢0 . ~symmi~'~) c_ G property(a) (PCSh) (25) y k

the groups of classical symmetry (in this case ~ is reduced to identification) by colour ones (a = p , q , wp, Wq) in different physical interpretations. Under the approximation of phenom- enological crystal physics, all groups in condition (25) are point ones, while in the case of microscopical crystal physics they are space groups. The symbol n denotes intersection (common subgroup) of symmetry groups of the structure parts (subsystems) of an object or a system or an action in question. In general, the symmetry of an object (system) is not reduced to the intersections of condition (25). The latter fact is reflected in the symmetrizators G<~) (7<~) i~,c~) introduced $yrnra ~ ~ symm, ~ symm into formulae in order to extend the subgroups up to the groups of generalized symmetry of objects (systems). For the phenomenological approximation, the tensors a and the tensor functions a(p, T) in relationships (25) are macroscopic, while for the microscopic approximation those tensors are

412 V.A. KoFrslK

X(o) /

Z ( c ) l

\ \ \

"\ /'" I I \-/ ",/. I ! /x

//\~ I i ~// \\ / / - \ [ i~-.. 6 \ / \I- ~I

"- . . . / \ / ' ~ \

/ ~ - ~ \

- - / _ _ ~ - . ~ 1

Figs 1. The g roups o f co lour symmet ry o f models o f real crystals and their classification.

Fig. l(a). Magnetic structure of the HoP crystal and the space group of its positional Wp-symmetry [42, 43]

{a ~'~ ":~'~, b ~'~'~'~, e ~'~''~'} ~!~"'"'~'~ ~" ~-'~ m~,..

Fig. l(b). A unit cell of a tetragonal crystal has the symmetry Gc~y,~ = 4 mm if the point defect (black triangle) is not taken into account. When it is taken into account, the unit cell point group -c~stGc~) is given by the generators [21, 42, 43] 4, m~ ~ 4 m m

1 4 4 z 43 m l m2 m3 m4

(w214)= ' ~ ' " 1 4 ) = 1 I 2 1 1 1 1 4 ,

4 42 43 i m2 m3 m4 ml

(wdmt) = (

1 4 4 2 4 3 m~ m 2 m 3 m 4

,~ . . ,m , ) = ( 2 l I l l l l 1 / • 1 1 1 1 2 1 1 1 '

/

m I m 4 m 3 m 2 1 4 ~ 42 4

Generalized symmetry in crystal physics 413

o , o o, ,22 o, ,23

t~ , 0 t 1 , t 2 tt,tz2 tt , t l / " / / /

f/// Fig. l(c). The translational group Tmod r, of a finite order corresponding to an enlarged unit cell (EUC) of a two-dimensional crystal with constant density of defects is determined by the multiplication law of positional operators

t , tk ty+ t , t ,

( . . . p~...It~)<. • 'P2'"" Itj) = ( . . . Pi P2"" It~ + tj}

and by cyclic boundary conditions [21, 42, 43] by EUC

(h,0) 3 = (0, t2) 4 = (tt, t2) u ~- (0, 0)(rood T'), T' ,-- T,

the cells with a point defect of a type (b) being black.

P

G(P)C'PX G Gcq)(~P XG

~ i g* P g~ G P f~

G(',)C--Pwrp G Gt~, ~ ~ Pwr~ G

Fig. l(d), Action of the components P, G on the sets S, R in the colour groups of P-, Q-, Wp- and Wq-symmetry.

spa t i a l a( r ) [17, 27]. A c c o r d i n g to the de f in i t i on

~(=) ~(=) N ~(") N . N ~(,o N ~ G (~) (26) -~" ~ a ( p 2, T2) • • ~'~ a ( p n , I n ) • • • a tp i . T i ) , v II(p, T) " J a ( p l , TI)

is the g r o u p o f e x t e r n a l s y m m e t r y o f the t e n s o r func t i on , w h o s e f o r m is i n v a r i a n t fo r the en t i r e d o m a i n o f s t ab i l i t y o f the c rys t a l m o d i f i c a t i o n in the p h a s e d i a g r a m (P, T). I t is d e t e r m i n e d b y

i

4 1 4 V . A . K o w r s I K

virtue of the intersection of eigen groups of the external symmetry of the tensors a (as oriented geometrical objects) related to fixed pressures and temperatures p~, T~ = const e (p, T) [17, 27].

In the generalized crystal physics, the matter macroscopic tensors a are defined as generalized geometrical objects given by the transformation laws of their components on generalized orthogonal groups 1 {po) x 0(3). In the magnetic crystal physics or in the crystal physics of complex vectors those laws coincide in their form with expression (3) but the operators g in ;((g) are taken from the groups

0(3) (~)~_ 1' x O(3), 1" x O(3), l 'w rO(3 ) , l * w r O ( 3 ) (27)

for ~t = p, q [first two groups in condition (27)] and ct = wpt Wq (two last ones), where 1' is the group of order two { 1, 1'} generated by the time inversion operator (inversion of the signs of magnetic vectors) and 1" is the group { 1, 1"} generated by the operator of complex conjugation isomorphic to the group { 1, 1'}. Formula (6) are generalized correspondingly:

z(g)g'ail.. .i, = ail...i,, g~ G(? ) ~_ 0(3) (~) c 1 (v') x 0(3). (28)

The sign of the correcting factor x ( g ) = + 1 in (3) and (28) in the case of magnetic groups G ' _ 1' x 0(3) is taken in accordance with the following table [28, 29]

T y p e o f the t e n s o r g ~ g ' ~ ' V e c t o r s y m m e t r y

E v e n 1 1 1 I E l ec t r i c a l 1 - I 1 - 1 P o l a r oo m m 1'

oo 2 ' 2 ' z ( g ) M a g n e t i c 1 1 - 1 - 1 A x i a l - - - - - - (29)

m m ' Fcl ~

o o 2 ' 2 ' M a g n e t o e l e c t r i c a l I - I - 1 1 M i x e d m ' m m

In condition (29) the type of the tensor a is given and for the first range tensors (vectors) the groups of their external magnetic symmetry are pointed out. The validity of those groups can be verified using the model interpretation of vectors [Fig. 2(a)]. The groups of external symmetry G: of tensors a = m, v, b, which are present in the equations A = all} for electric polarization ~ = ~0.Ej, magneto- electric and piezomagnetic effects ~ = vuH j, Bt= buka]k etc., can be established using PNMC G~ ___ G', N G[ by choosing the maximal group G'- which would satisfy the above relationship for the known groups G~ and G~.

Similar analysis can also be done for crystal physics equations A = aB written in terms of complex tensors A = A' + iA", a = a' + ia", B = B' + iB" [30]. Introducing the operator 1" of complex conjugation, l*(a' + ia") = a' - ia" (and similarly for the tensors A, B) and combining it with the transformations g ~ O (3), g* = g 1" = l*g e 1" x O (3) we can determine the groups of complex symmetry G*, G*, G* _ 1" x 0(3) of all tensors A, a and B and groups G*~,, c_G* describing the complex symmetry of crystals. All those groups are, evidently, isomorphic to corresponding magnetic groups

l G'A,G' , ,G[c_I 'xO(3), Gc~,tc_G',, g ' = g l ' = l ' g ~ l ' x O ( 3 ) , - - ~ l * x O ( 3 ) .

Let us carry out, for example, the analysis of complex symmetry of Fourier space using the basic relationships of the crystal structure analysis

1 +o~ +~ +~ p(x ,y ,z ) =-Vh ~ ~ ~ F(hkt)exp[-2ni(hx +ky +lz)] (30)

= - - o 0 k ~ - o o l = - o o

which express the function of crystal's density p(x ,y ,z ) through the structure amplitudes F(hkl) = JF(hkl)lexp[i(p (hkl)]. The symmetry of the physical space, Op (xyz) = p(xyz), where • is the Fedorov-Sch6nflies group of the crystal, is transferred to the reciprocal space using the formulas

F(hkl) -- F(H) = ~ fa. exp(2~iH .r), J

H . ( g r ) = H.r '* -*H' . r = (g - IH) ' r , (31)

Generalized symmetry in crystal physics 415

where~ are the atomic scattering factors and ¢ / T is the quotient-group of ¢ over the translation subgroup T c ¢ = TG isomorphic to the crystal class G=~,t. Taking into account that the operation of complex conjugation 1" inverts the sign of the phase ¢k(hki)= ~p(l-I) o f the complex structure amplitude, namely, l*~p(hkl)= -~p(hkl ) , we find the combined transformations

2n 2n g C p ) = ( p t g ) , g ~ G c ~ s t x T , p = n = - - for q ~ = - -

~p n

2n 2n or ~ p = - - for ~ p # - - , n = 1 , 2 , 3 . . . .

~p n

preserving the phase, g(P)~p(H) = p~p (gA) = ~p'(H') = ~p(H) and, hence, F(H). Such transforma- tions, which were determined in Ref. [31] for all Fedorov-Sch6nflies groups, are included in the group ¢~wp) of positional complex symmetry of the Fourier space, ¢°~P)F(H)=F(H) . In Figs 2(b)-(d) the results of this analysis are exemplified for the Fedorov-Schonflies group ¢ = P4~.

/Y

~ ¢ '-I-i0"

= ¢'._/o"irngF

Figs 2. Modelling of colour symmetry groups for different objects. Fig. 2(a). Model interpretation of electric (polar), magnetic Fig. 2(b). Colour symmetry of complex values in Gaussian (axial) and magnetoelectric (polar-axial) vectors [17, 27]. plane is described [43] by the point group G tp) = m'~, m~*2',

where I*F=F*, I 'F= -F, I '*F= -F*, 1'*= 1'1", m~ = m*, rn~*= m21'* etc. [43].

K

. . . . , ,,. •

,p=•,t" • I • • • • " ~ / L I /Y H

I

Fig. 2(c). Vector interpretation of phase space which puts into correspondence to the phase ~p(hkl) the vector p(hkl) directed at an angle ~p to the X axis in the local coordinate system (x, iv) attached to a site (hkl) of the reciprocal lattice. The vector's length ]p[ equals to the modulus of the structure amplitude

IF(hkl)l [31, 43].

G

I 2 :-

l = 4 n

G+~r

~ i I " ~.~0 ~,~

-=*"/L / / - a - ~ ~ l = 4 n + 2

11, a+-~-

~ " ~hk l ) a * ~ I . / I

/ / - a - 3~r

l =4n +1

3rr a v ~ -

) a ~ l l " . ~ ( h k , )

/ I

L-:'" L/-°-" ~ = 4 n + 3

Fig. 2(d). Phase structure of the reciprocal space corresponding to the Fedorov-Soh6ntiies group P41. To every site (hkl) of the rcciprocal lattice which belongs to the orbit of the point group 4 x T ~ 4, where 4 is the homomorphic image of the P41 group, a phase ¢(hkl) is attributed according to the extinction

laws which are active in the P4~ group [31, 43].

t t t

l ," • ' . , "- 4

¼ ~ - - I ,, ".~.. ~ . . .~ '" I ~ . ! , -~ I , "'.A.'" ] i ~ . i ,

1 ,,._._ I I . / ~ - ~ . J a I 1

\ ~ I / /

' I - '¼ ~ " - I ~.2 f h ~ c f f , -4 f~T~ T ~ ~ I 4

"P"~ ~'~ : I 4

Fig. 2(¢). The schcm© of imbedding of the sub~'oups Ol = P22~2~ *--~O0 (four unit ceUs arc drawn with solid lines) and @~ = P3z21 (. . .) into the space group ¢ = P6s22 of the prototype (---). This illustrates the genesis of the superspac¢ symmetry in the phcnole structure shown in the separate figure. At the phase

transition • ~ ¢0, the local operations of the supcrsymmetry 3z e ¢~ are preserved [34].

/

• (,

.,

J'5"~

(b)

i V

0(

""4

!(d)

A

4z(i)~

-T~

4-~(i) 4

-z ~..

-

/)

~

TII)

"(l(r,'),4

71(r~

),47z(r~

),~3(rd)

11~

G(P),

..4(4-"

I ~

(e)

~ 43

(/)\

4(i}

4(i)

7; 44

4z(/) ¢

4643 (i)

~ i4z

41i1

v 0

v

6"(

q''

4

(P

{ r/

'} I

1>

- (1

(G

')

,4z

( r2'

j,44

t

r3'1

,46

( r~

}[1~

G

tq'.

, 4

''2

'

Fig.

2(f

). D

escr

ipti

on o

f su

per-

Fed

orov

sym

met

ry o

f a m

olec

ular

bis

yste

m c

ryst

al.

Aft

er a

uni

t ce

ll {

a, b

, c }

(sol

id c

onto

ur)

is h

omog

eneo

usly

str

aine

d to

the

sta

te {

a +

Aa,

b,

c}

, ra

+A

al=

lb[

(---

),

and

the

mol

ecul

es

(r,,

r0,

(4r,

,r3)

ar

e su

bjec

ted

to

rigi

d lo

cal

shif

ts

into

th

e te

trag

onal

po

siti

ons

(r~

+u.

, rl

+u

.)=

(r;,

r;

),

(42r

i --

u,,

r 3 --

u,)

= (

r~, r

~)

such

th

at

r~ =

4r~

= r

2, r

~ =

42r~

, r~

= 43

r~ =

r 4

the

foll

owin

g lo

cal

rota

tion

s:

(...

42(r

2)..

.1 l

)(r~

, r2

) = (

4Zr~

, r2)

= (

r;',

r~);

(.

-.42

(r4)

.. .

I l)

(r~,

r4)

= (

42r~

, r4

)= (

r;',

r~)

depe

ndin

g on

the

pos

itio

n p

(rJ

=g

(ri)

,p(r

i)E

Pe'

= G

~' ~

g(rs

) tr

ansf

orm

the

ini

tial

str

uctu

re (

a) w

ith

the

spac

e sy

mm

etry

~o

= T

oGo

= {

a, b

, c}

2 in

to t

he p

roto

type

str

uctu

re (

f) w

ith

the

sym

met

ry ~

=

TG

= {

a +

Aa,

b, c

}4.

Fig

ures

(b)

, (c)

sho

w t

he s

truc

ture

orb

its

of

the

grou

ps q

~(P)

= T(

P)G

(p),

G ~p

l = 4

, 41

4 ,i

and

Fig

ures

(d)

, (e

) sh

ow t

hose

of

the

grou

ps q

B(q)

= T(

qJG

(q),

G(q

) = 4

, 4 (

42) i

som

orph

ous

to t

he g

roup

qB

= TG

. T

he s

ymm

etry

ele

men

ts 4

~ G

*--

,#/T

whi

ch

are

lost

dur

ing

the

tran

siti

on f

f~ --

-, ~0

are

tran

sfor

med

int

o th

e lo

cal

elem

ents

of

the

supe

r-F

edor

ov s

ymm

etry

4,

4 ~ m

od 4

2= (

4-1)

2= I

. T

he r

adiu

s-ve

ctor

s r k

of

the

mas

s ce

ntre

s o

f th

e m

olec

ules

(fi

gure

s) #

k ar

e in

dexe

d by

k i

nsid

e th

e tr

iang

les.

The

loc

al r

adiu

s-ve

ctor

r~

fixi

ng t

he f

igur

e's

orie

ntat

ion

is s

ubst

itut

ed b

y th

e in

dex

i. T

he t

rans

form

atio

n of

the

typ

e of

4

tr~ is

wri

tten

as

4-n(

i) e

tc.

O

o ,7

Ix

418 V.A. KoIrrSlK

R(3).I. e 4--~4.~

[

Figs 3. Symmetry description of incommensurate phases. Fig. 3(a). Isomorphic projection of the nodes of a four-dimensional lattice introduced in the reciprocal space R(4) onto the space R(3). In incommensurate crystal structures the lattice points in R (4) correspond one-to-one to the reflexes in R(3). The intensities of the reflexes modelling the colour loadings of a

diffraction lattice in the isomorphism R(3)@ S(1)~-~R(3)@ R(l) of spaces (see Ref. [41]).

0 . 5 • ¢ *

,oo..,, I : : \" ,,,, o. :t ,,,' ,,' / i / \ I / o. I / #

t I t I o . a o l / t I I \

~ / 6 o.s o ° o.~6 ~ /6o .~r o.~a o,~9 O •

Fig. 3(b). Temperature dependence of the wave vector of the modulated wave for incommensurate ~,~ (from A to C)-, 72 (from C to G)-, 7J (from G to I) and commensurate (from I to K) 6-phases of Na2CO~ crystal in the plane a*, e* of the reciprocal lattice: A--470; B---370; C--300; D--295; E--275; F--235; G--200;

H--175; 1--120; J--20; K--4.2 K [50].

Generalized symmetry in crystal physics 419

I z

Phase Xj befween 6;)0 and 300 K

/ Phase 72 between 300 and 200K

Fig. 3(c). A hypothetical scheme of remodulating phase transitions ~ ~ Y2 in the Na2CO 3 crystal. In the yrphase, the harmonic wave of atom displacements has its own (non-Fedorov) group of phase symmetry

2*(2) 2(2*) 21(2") f~p) = 1 (2~) x P(b/2)" m * ( b ) m m*(m)

which is preserved under temperature compression of the modulating wave in the direction of the local Y axis lying in the crystal's .I, Z plane. The shear deformation transforms it into a centre-affine group of symmetry

2,~'(2 A) 2(2 A') 2~(2 a°) I(2~) × P(b/2~" mAO(bA) -'~ mA.(mA),

of the ~,2-phase. The group of abstract symmetry does not change under the phase transition ~,~ ~ ~'2 but the complex group of antisymmetry transforms into centre-affine group A or A * ,-- 1" x A. At a fixed level, the group of phase symmetry D~) transforms into the subgroup t'~)= l(2")p 1 2/m 1. The elements of symmetry and complex antisymmetry are shown in the figure by black and white symbols, respectively

[42, 43].

I f necessary, the symmetry analysis o f the relationships o f tensor crystal physics can be done for some other physical interpretat ions o f colour tensors and for functions, which are properly defined on the groups o f co lour symmetry G (~) c 1 (~) x O(3), ~ = p, q, provided that the above models are

generalized.

5. S U P E R S P A C E S Y M M E T R Y OF M O L E C U L A R L Y M O D U L A T E D P O L Y S Y S T E M C R Y S T A L S

Polysystem molecular crystals are such crystals in whose unit cells there are geometrically and chemically equivalent molecules which are not symmetry equivalent in the crystal 's space g roup ¢)o = ToGa. Such crystals are a good example o f systems simulating the internal (in general, non-Fedorov-Sch6nf l ies ) symmetries P by which the g roup 40 is t ransformed into a g roup o f

420 V, A. KOPTSIK

colour symmetry

~ q ) c e w r q q b =P*~.~ , ff~gwq~4--,~ t ~ 0 . (32)

At the phase transitions which occur due to molecular motion in such crystals (under assumption of rigid or non-rigid molecules) there appears a special type of space molecular modulation accompanied (or not) with a commensurable (multiple) or non-commensurable increase of the unit cell's parameters. Here we have no opportunity to go into details and refer to a series of works [32-34], where this topic was discussed. In Figs 2(e) and (f) we give some examples which demonstrate the essence of the approach developed in those works. We should only mention that the main assumption of this approach is that the structure of a polysystem crystal with the symmetry ~c0w*~ ~ ~t c ~ is obtained from the structure of a certain hypothetical (or real) prototype (for example, molecular lattice gas or liquid) with the symmetry ~ * ~ under the phase transition ff~q~ ~ ~0wq ~ induced by the properly chosen order parameter (OP) ~/(r).

6. THE PROBLEM OF INCOMMENSURABILITY IN CRYSTAL PHYSICS

Let us now discuss the problem which appeared about three decades ago as a catastrophe in the theory of crystal state but is not completely resolved till now. What we speak about is the violation of translational invariance in incommensurably modulated (INC) phases which rather often appear under phase transitions of the first and second order (PhT- 1 and PhT-2). The concept of long-range order is intrinsic of the definition of a crystal so that some experts consider the loss of this quality as the complete loss of the crystal state. They declare that the INC-phase is a giant molecule whose symmetry cannot be described using the formalism of space (Fedorov-Schrnflies) groups. This problem can be compared with the violation of CP-invariance both in the character of the difficulties and the methods used to overcome them. We should explain these difficulties using a simple model.

Let an INC-crystal consists of a basic atomic structure p0(r) with wave vectors bi(i = 1, 2, 3) and a modulating structure 6p (r) which is realized by the waves of charge, spin, occupation, distortion, or molecular density with the wave vectors k = #~bi = #t hI + #2112 +/t3b3, where one, two, or three of the numbers #i are irrational. We denote the space groups of the structures p0(r) and 6p(r) as Opo = Tpo Gpo and Orp = T~p Grp where T are translational and G are point subgroups of those groups or the modulo T groups for symmorphic or non-symmorphic space groups ¢~ correspondingly (see the definition of modulo groups in Ref. [17, p. 246]. The symmetry group of the composed structure p(r) = p0(r) + 6p(r) defined using the Curie rule of superposition degenerates into a point one

~oon ~,o = (Teon T~e)(GeonG~.) = GoonG,e =/~e. roon r,p = 0. (33)

or in a once- or twice periodic space-group • if Too f~ T~p # O. In the limiting case, the subgroup H e reduces to identity.

Such dissymmetrization ff~p0 ~ lip is in contradiction with the physical reality. The X-ray diffraction pattern consisting of the superposition of basic and satellite reflexes is three- dimensionally periodic in the reciprocal space, this making us suspect a lattent periodicity of the direct space. The set of possible macroscopic properties of INC-structures is governed by prohibitions which are, in general, imposed by a group higher than H e. Three approaches were suggested in order to resolve this contradiction:

(a ) Representation approach Dzyaloshinskii was the first to resolve partially the puzzle of INC-crystals using the example of

helicoidal magnetic structures [18]. Using the Landau theory of PhT-2 he introduced an anisotropic invariant of the type of cosmp into the expansion of the thermodynamic potential F(r/)=F0+½~xr/2+... and obtained the soliton solutions for the minimization equations OF/dtl = 0 of the type of mathematical pendulum.

Later on, Levanyuk and Sannikov applied a similar approach to analyse ferroelectric PhT. They and other authors have thoroughly studied the structure of invariants which depend on spatial derivatives of the order parameter. As a result of this study the view-point [35, p. 214] become

Generalized symmetry in crystal physics 421

commonly shared that "the formalism of the theory of representations of space groups is well sufficient to describe the incommensurable phases".

But this solution to the problem does not answer the question what is the symmetry of the INC-phase. We can only say that the initial O~0-invariance is intrinsic for the expression for the potential F(~/), while to characterize the symmetry of INC-phase one must use the transformation properties of the OP which is transformed by irreducible representation (IR) of the group Opo whose wave vector k is incommensurable with the parameters bj~ TTo~= {b~, b2, b3}.

(b ) The approach of the "superspace" 0(3 + d)-symmetry

In 1974 de Wolff and later Yanner and Yanssen have proposed the approach [36--38] of "superspace" symmetry to the analysis of INC-structures. Let, for example, a crystal is modulated along the direction b 3 and the vector h = hjb~ + h262 + h3b3 + (m3/n3) b3 describes the position of the basic (h~, h2, h3) and satellite (q = (m3/n3)b3) reflexes. If we introduce formally a unity vector e4 which is orthogonal to the basic vectors b ~ R(3) and determine the vector b4 = q + e4, then in the basis bj~ R(4) h = hzb~ + h2b2 + h3b3 + h464. The existence of the four-dimensional lattice T -~ = {b~, b2, b3, b4} in the reciprocal space allows us to index by integers all the reflexes (hi, h2, h3, h4) and to restore from the Fourier series

+ ~

p(x~,x2,x3,x4)= ~ ~ F(h~,h2,h3,h4)exp[2ni(h,x, +h2x2+h3x3+h4x4)], (34) h l , h 2 , h 3 , h 4 = - - ~.~

the density function p (x~, x2, x3, x4) for a fictituous four-dimensional object called a supercrystal. The equality of the symmetry groups of the r.h.s, and l.h.s, of equation (34) permits us to find the four-dimensional space group 0(4) of the object p(x~, x2, x3, x4). If the crystal is modulated along d directions, the superspace group 0(3 + d), d = 1, 2, 3 . . . . . appears [36-38].

In Ref. [39] it was suggested to reformulate the approach of Refs [36-38] by studying an additional "phase" symmetry of the expression for free energy F(Q) (as a function of OP Q = Q' + iQ") instead of the non-physical model of "supercrystal". It was shown that F(QQ*) remains invariant with respect to the combined transformations

Q~ = exp(iqf~) ~ ~(k~, (glt})ltqQtj = Qt~, (35) k , j

the set of which comprise the superspace symmetry group of the expression F(Q), where the order parameter r /= Q consists of the normal coordinates of the distortion (soft) mode {Qki}, and denotes the phase shift vector accompanying the translations t, afaR(d), t eR(3), R(n) = R(3)@R(d). Having established IR by which the OP is transformed, we can determine also a superspace subgroup 0(3 +d)p = 0(3 + d )~ ) of the group of superspace symmetry of the expression F(Q). As we shall see this subgroup is isomorphic to the generalized symmetry group of the INC-phase for the OP-model involved, 0(3 + d)p.-,O(3)~, ) or ~O(3)~* ).

(c) Colour symmetry approach

The n-dimensional Fedorov groups O ( n ) = O ( 3 + d ) act in the Euclidian space R(n) = R (3)~)R (d) in which all the coordinate axes are geometrically equivalent. If we change the internal geometrical space R(d) by a non-geometrical space S(d) using the method of isomorphic projection (see Section 3) we define the groups of positional colour symmetry O(3)~w,),~O(3 + d) which are isomorphic to the groups 0(3 + d) and which are the groups that describe the generalized symmetry of structure models related to this isomorphism. In particular, these groups may be the groups O(3) ~p) ~ 0(3) with a continuous spectrum of the values of p ~ P [17]. The continuous spectrum of values of non-geometrical coordinates can be realized on certain substructures P~kcP~a)cPnwrpO(3 ), where PncS~ for n = oo. The colour groups with such a structure describe the symmetry of incommensurably modulated phases.

We should note that the method of colour positional O(3)twp)-symmetry was suggested to describe INC-structures in 1974 [20] and in the American edition of the book [17] independently of Ref. [36], while in Ref. [40] it was clearly mentioned that the symmetry of incommensurable magnetic helicoids is described just by such groups. To illustrate this idea we give here Figs 3(a)-(c) and for

422 v, A. KOFrSXK

detai ls refer to Refs [41-43, 23], to some o f our works in [22, 19, 44, 34] and to the works on PhT

[45-47]. In this app roach , af ter the phase t rans i t ion into a space -modu la t ed s t ructure the init ial symmet ry

o f a crys ta l O(3)p0 does no t b reak bu t remains in a modif ied form at the level o f the groups O ( 3 ) ~ p ) ~ 0 ( 3 + d) . As the first a p p r o x i m a t i o n to the symmet ry o f the i ncommensu rab l e phase we

can take the j u n i o r g roup O(3)~p)*-~O(3)p0 which is a subg roup o f the g roup O(3)~op), O ( 3 ) ~ ) , - O(3)~p ). U n d e r this a p p r o x i m a t i o n (see Refs [17,26]) the subsys tem proper t i e s o f INC-c rys ta l s may be classified accord ing to the k n o w n I R o f the init ial F e d o r o v g roups O(3)p0, since the I R massives o f i somorph ic g roups O(3)~'p)*-,O(3)p 0 coincide. The tensor p roper t i e s o f I N C - p h a s e are par t ly de te rmined by the tensor represen ta t ions o f the groups O(3)p0,--, O(3)~'~). In o rde r to descr ibe more comple te ly the physical p roper t i e s o f IN C-pha se s it is necessary to use the fo rmal i sm o f I R for the subgroups O(3)(p~ ) or O(3)~q ) o f wrea th p roduc t s o f the P , and 4~(3)p0 g roups [48, 49], i.e. to use the pos i t iona l dependence o f I R on the subs t ruc ture involved.

(3

73

73

C3

73

C3

C3 (3

Fig. 4(a). Though the groups G~i,t = 5 mm of the local symmetry of pentagons do not belong to the group Aut T = 4 ram of automorphisms of a translational square lattice T, the symmetry of the entire structure can be described by the direct product of two groups 5 mm x P4 mm which belong to the type of senior groups

of P-symmetry [43].

J

Figs 4. Two-dimensional quasicrystais with pentagonal local symmetry.

Fig. 4(b). A fragment of the two-dimensional Penrose mosaic. A cell of the two-dimensional net is shown in whose nodes decagonal figures are centred, their contour being drawn with solid fines. Every such figure has internal mosaic structure consisting of rhombs of two kinds. Under trans- lations by the vectors a, b, a + b the contour of Fig. 1 transforms into the contour of Figs 2-4, etc., but their internal structures do not coincide. The internal structures of those figures may be put into coincidence using positional local compensating transformations of restructuring 11~')), 1 ~)), 11~°+b~) etc. Combining those local trans- formations with translations we find generalized colour translations a ~ ' ) ) = a l (e~')) = l (~ ' ) )a , b c~)) = b l t~b)) = l ~ ) ) b , (a + by ~'+.)) = (a + b)1 <e~'+b)) = l¢~'+b)(a + b) . . . . . which belong to the group of positional colour symmetry of the structure fragment in question. The structure fragment can be put into coincidence to adjacent ones in a domain of certain enlarged unit cell by means of local positional operators of the second type which change the choice of the contour of local figures centering the nodes of the plane net. Continuing this process we arrive to space groups of Wp-symmetry of pentagonal (in the case of three dimen- sions they are icosahedric) quasicrystals according to the model of constructing of such groups for incommensurate modulated structures. This process may be done finite under the approximation of the enlarged unit cells with the

constant density of rhombs [compare Fig. l(c)[.

Generalized symmetry in crystal physics 423

7. THE PROBLEM OF ICOSAHEDRIC QUASICRYSTALS [51-55]

This up-to-date problem consists in the loss of translational invariance T: the point group of a unit cell (UC) Guc ~ Aut T. By choosing the proper compensating transformations the violated T-invariance may be restored just as in the case of INC-phases. Figure 4(a) presents the groups • (P) for a plane net in whose unit cells the pentagons are placed. Figure 4(b) shows a possible choice of compensating transformations which restore the generalized T (w.)-invariance of two-dimensional Penrose mosaics.

It is known [56-58] that the method of enlarged unit cells (EUC) allows us to describe the average tensor properties of INC-crystals under the approximation of long-periodical commensurable phases. And the same is true for quasicrystals. In the EUC-method we neglect the differences between EUC's in the fine internal structure and we are only interested in those properties of quasicrystals which depend on continuous repetition of EUC's in three-dimensional space.

R E F E R E N C E S

1. W. Voigt, Lehrbuch der Kristalphysik. Teubner, Leipzig (1910). 2. J. F. Nye, PhysicalProperties of Crystals. Their Representation by Tensor and Matrices. Clarendon Press, Oxford (1964). 3. A. V. Shubnikov, On the vector and tensor symmetry, lzv. Akad. Nauk. SSSR, ser.fiz. 13, 347-374 (1949). See also:

Selected Works on Crystallography pp. 97-132 (in Russian). Nauka, Moscow (1975). 4. H. A. Yahn, Note on the Bhagavantam-Suryarayana method of enumerating the physical constants of crystals. Acta

Crystallogr. 2, 30--33 (1949). 5. V. A. Koptsik, Shubnikov Groups. Handbook on the Symmetry and Physical Properties of Crystals. (in Russian). Izd.

Mosk. Univ., Moscow (1966). 6. A. M. Zamorzaev, On the groups of quasisymmetry (P-symmetry) (in Russian). Kristallografiya 12, 819--825 (1967). 7. Ph. R. Bunker, Molecular Symmetry and Spectroscopy. Academic Press, New York (1979). 8. A. V. Shubnikov, Symmetry and antisymmetry of finite figures. (in Russian). Izd. Akad. Nauk SSSR, Moscow (1951).

See also: A. V. Shubnikov, N. V. Belov, Colored Symmetry pp. 1-172. Pergamon Press, New York (1964). 9. L. D. Landau and E. M. Lifshits, Statistical Physics (in Russian). Fizmatgiz, Moscow (1951). See also: L. D. Landau,

E. M. Lifshits, Course of Theoretical Physics, Vol. 5. Addison-Wesley, Reading, Mass. (1959-1971). 10. V. A. Koptsik, General sketch of the development of the theory of symmetry and its applications in physical

crystallography over the last 50 years. Kristallografiya 12, 755-774 (1967). See also: Soviet Phys. Crystallogr. 12, 667-683 (1968).

11. A. V. Shubnikov, New ideas in the theory of symmetry and its applications, In Report of the General Meeting of the Academy of Sciences of the USSR (October 14--17, 1944) pp. 212-227. Izd. Akad. Nauk SSSR, Moscow (1945).

12. H. Heesh, Uber die vierdimensionalen Gruppen der dreidimensionalen Raumen. Z. Kristallogr. 73, 325-345 (1930). 13. A. M. Zamorzaev, Generalization of Fedorov groups (in Russian). Kristallografiya 2, 15-20 (1957). See also: Soviet

Phys. Crystallogr. 2, 10-15 (1957). 14. N. V. Belov, N. N. Neronova and T. S. Smirnova, 1651 Shuhnikov groups. Trudy Inst. Kristall. 11, 33-67 (1955). See

also: A. V. Shubnikov and N. V. Belov, Colored Symmetry pp. 175-210. Pergamon Press, New York (1964). 15. B. L. van der Waerden and J. J. Burkhardt. Farbengruppen. Z. Kristallogr. 115, 231-234 (1961). 16. N. V. Belov and T. N. Tarkhova, Colour symmetry groups. Kristallografiya 1, 4-13, 619-620 (1956). See also: Soviet

Phys. Crystallogr. 2, 311-322 0957). 17. A. V. Shubnikov and A. V. Koptsik. Symmetry in Science and Art (in Russian). Nauka, Moscow (1972). See also the

American edition of this book (Ed. David Harker). Plenum Press, New York (1974). 18. I. E. Dzyaloshinskii, Thermodynamic theory of weak ferromagnetism in antiferromagnetics. Zh. eksp. teor. Fiz. 32,

1547-1562 (1957); 46, 1352-1362, 1420-1430; 47, 336-348, 992-1001 (1964). See also: Soviet Phys. JETP 5, 1259-1267 (1957), etc.

19. V. A. Koptsik, Colour symmetry and scaling in the theory of phase transitions and crystal phenomena (in Russian), In Group Theoretical Methods in Physics, vol. I, pp. 320-331. Nauka, Moscow (1983). See also the American edition of this book. Harwood Academic, London (to be published).

20. V. A. Koptsik, I. N. Kotzev, On the theory and classification of colour symmetry groups. Part 2. Wp-symmetry (in Russian), p. 4-8068. Commun. of the Joint Institute for Nuclear Research, Dubna (1974).

21. V. A. Koptsik, Advances in the theoretical crystallography. Colour symmetry of defect crystals. Krist. Tech. 10, 231-245 (1975).

22. V. A. Koptsik, Group theoretical methods in physics of imperfect crystals and the theory of structure phase transitions (in Russian), In Group Theoretical Methods in Physics vol. 1, pp. 368-381. Nauka, Moscow (1980).

23. V. A. Koptsik, New group theoretical methods in physics of imperfect crystals and the theory of structure phase transitions. J. Phys. C16, 1-22 (1983).

24. B. H. Neumann, Twisted wreath products of groups. Arch. Math. 14, 1-6 (1963). Composito Math. 13, Fasc. 1, 47-64 (1957).

25. A. W. M. Dress, A mathematical note on Koptsik's definition of imperfect crystals. MATCH (commun. math. chem.) 9, 15-20 (1980).

26. V. A. Koptsik, Symmetry principles in physics. J. Phys. C16, 23-35 (1983). 27. V. A. Koptsik, Aleksei Vasilievich Shubnikov (on the 100th Birthday) (in Russian). Kristalografiya 32, 535-539 (1987). 28. Yu. I. Sirotin and M. P. Shaskovskaya, Foundation of Crystal Physics (in Russian). Nauka, Moscow (1975). 29. R. Birss, Symmetry and Magnetism. Holland. Amsterdam (1964). 30. F. I. Fedorov, Theory of Gyrotropy (in Russian). Izd. Nauka i. Teknika, Minsk (1976).

C.A.M.W.A. 16/5-8--F

424 V . A . KOFI~IK

31. V. A. Koptsik, E. N. Ovchinnikova and A. Yu. Papaev, Symmetry of the phase space and the direct methods of X-ray analysis of crystals. Kristailografiya 28, 32-41 (1983). See also: Soviet Phys. Crystallogr. 25, (1983).

32. V. A. Koptsik and A, L. Talis, Super-Fedorov symmetry of the molecularly modulated crystals (in Russian). In Group Theoretical Methods in Physics vol. I, pp. 527-547, Nauka, Moscow (1986). See also the English edition of this book (Ed. M. A. Markov), Vol. 2, VNU Science Press, Utrecht (1986).

33. V. A. Koptsik and A. L. Tails, Super-Fedorov Symmetry of the Molecularly Modulated Crystals (in Russian). No. 437-13-86 Dep.; (1987), No. 3051-B-87 Dep. Izd. VINITI, Moscow (1986).

34. V. A. Koptsik, Super-Fedorov symmetry of the molecularly modulated crystals, (in Russian) In Problems of Crystallography (on the 100th Birthday of A. V. Shubnikov). pp. 69-89. Nauka, Moscow (1987).

35. Yu. A. Izyumov and V. N. Syromiatnikov, Phase Transitions and Symmetry of Crystals (in Russian). Nauka, Moscow (1984).

36. P. M. de Wolff, The pseudosymmetry of molecular crystal structures. Acta Crystallogr. A30, 777-785 (1974). 37. T. Janssen and A. Janner, Superspace groups. Alternative approaches to the symmetry of incommensurate crystal

phases. Physica A99, 47-76 (1979); A126, 163-176 (1984). 38. A. Janner and T. Janssen, Symmetry of incommensurate crystal phases. Acta Crystallogr. A36, 399-415 (1980). 39. J. M. Pertz-Mato, G. Madariaga and M. J. Tello, Superspace groups and Landau theory. A physical approach to

superspace symmetry in incommensurate structures. Phys. Roy. ID0, 1534-1543 (1984). 40. V. A. Koptsik, The theory of symmetry of space modulated crystal structures. Ferroelectrics 21, 499-501 (1978). 41. A. Jamamoto. Structure factor of modulated crystal structures. Acta Crystallogr. A38, 87-92 (1982). 42. V. A. Koptsik, The symmetry of imperfect crystals. On the theory of structure phase transitions in crystals with internal

degrees of freedom. MATCH (commun. math. chem.) 8, 3-20; 21-35 (1980). 43. V. A. Koptsik, On the theory of space symmetry of imperfect crystals. Experimental and theoretical study of the space

modulated crystals (in Russian). In Laws of Evolution of Complex Systems (Ed. K. O. Kratz and E. N. Eliseev) pp. 152-212. Nauka, Leningrad (1980).

44. V. A. Koptsik, Phase transitions and the symmetry of space modulated crystal media, In Group Theoretical Methods in Physics. (Ed. M. A. Markov), Vol. 1, pp. 710-714. Nauka, Moscow (1986). See also the English edition of this book, Vol. 2. VNU Science Press, Utrecht (1986).

45. D. B. Litvin, I. N. Kotzev and I. L. Birman, Physical applications of crystallographic colour groups. Landau theory of phase transitions. Phys. Roy. B7,6, 6947-6970 (1982).

46. G. M. Chechin and V. A. Koptsik, Relation between multidimensional representations of fedorov groups and groups of colour symmetry. Comput. Math. Applic. 16, 521-536 (1988).

47. V. P. Sakhnenko and G. M. Chechin, Symmetry methods and space group representations in the theory of phase transitions. Comput. Math. Applic. 16, 453-464 (1988).

48. A. Kerber, Representations of Permutation Groups. Springer, Berlin, Part 1 (1971), Part 2 (1975). 49. B. A. Men and V. L. Cherepanov, The use of plethym for classification of the permitted terms of impurity complexes

in crystals. Int. J. Quant. Chem. 7, 739-743 (1973). 50. C. J. de Pater, An experimental study of the incommensurate phase transformations in Rb2ZnBr4 and Na2CO3. Delft

Univ. of Technology (1978). 51. D. Levine and P. J. Steinhardt, Quasicrystals. A new class of ordered structures. Phys. Rev. Lett. 53, 2477-2480 (1984). 52. D. Levine and P. J. Steinhardt, Quasicrystals. 1. Definition and structure. Phys. Roy. B34, 596-617 (1986). 53. J. E. S. Socolar and P. J. Steinhardt, Quasicrystals. 2. Unit cell configuration. Phys. Roy. B34, 617-647 (1986). 54. D. R. Nelson, Quasicrystals. Scient. Am. 10, 19-28 (1986). See also: M. Gardner, Mathematical games. Scient. Am.

236, 110-121 (1977). 55. P. A. Kalugin, A. Yu. Kitaev and L. S. Levitov. Six-dimensional crystal of Alo.s6Mn0.14 (in Russian). Pis'ma Zh. Exp.

Teor. Fiz. 41, 119-121 (1985). 56. V. A. Koptsik, S. A. Ryabchikov and Yu. I. Sirotin, Method of enlarged unit cells in the theory of vibrational spectra

of crystals (quasicontinuous approach), lO'istailografiya 22, 229-241 (1977). See also: Soviet Phys. Crystallogr. 22, (1977).

57. V. A. Koptsik, R. A. Evarestov, Group theoretical analysis of the model of enlarged unit cells. Kristallografiya 25, 5-13 (1980). See also: Soviet Phys. Crystallogr. 25, 5-13 (1980).

58. V. A. Koptsik and S. A. Ryabchikov, Model of enlarged unit cell-reduced brillonin zone in the vibrational problem of crystal physics in the approach of the continuous media mechanics (in Russian), In Group Theoretical Methods in Physics, VoL I, pp. 630-642. Nauka, Moscow (1986). See also the English edition of this book (Ed. M. A. Markov), Vol. 2, pp. 441-448. VNU Science Press. Utrecht (1986).