Irreducible representations of magnetic groups of quantum-mechanical operators

J. Phys. Chem.

Solids

Pergamon Press 1970. Vol. 3 1,pp. 149- 161.

Printed in Great Britain.

IRREDUCIBLE REPRESENTATIONS OF MAGNETIC GROUPS OF QUANTUM-MECHANICAL OPERATORS Physico-Technical

0. V. KOVALEV and A. G. GORBANYUK Institute, Academy of Sciences, Yumovskii Tupik, Khar’Kov, U.S.S.R. (Received

25 April 1969)

Abstract-The

methods of constructions of Wigner corepresentations in all those cases which can be met in considering all 674 magnetic space groups, are discussed. In order to use known tables of the irreducible representations of usual space groups, it is necessary to change the description of many magnetic groups. All cases of such changes are considered in detail. The examples of direct calculations of corepresentations are given.

1. INTRODUCTION

tions of the geometrical and time symmetry and are defined over functions I,!Jwhich are the solutions of Schrodinger or Pauly equations[6]. These equations must be invariant with respect to those operators. If operation R is not taken into account, the group of operators for Schrodinger equation is isomorphic to that of the geometrical symmetry of physical system, and the group of operators for Pauly equation is isomorphic to that of two-dimensional unimodular unitary matrices [ SU(2)]. Odd representations of unitary group is called “double-valued” representations of geometrical operation group. In accordance with Wigner theory[7], when the time symmetry is taken into account, it is necessary to confront the operation R with time reversal operator 0. According to this, the magnetic group reflect oneself over the group G+GfI of operators, half of them containing 8. Since the operator 8 is antiunitary, the group G + Go contains both unitary g and antiunitary 0s operators. These last contain complex conjugate operation. Let us note that for Schriidinger equation e2 = 1 and for Pauly equation e2 = - 1. Operators g form a unitary subgroup G of the group G + Go. Matrix forms of operators defined over the wave functions $, which are the solutions of correspondent equation, are called corepresentations. Under the time

IRREDUCIBLE

representations of the magnetic space groups are used to obtain physical consequences from the magnetic symmetry of crystals with non-vanishing current density. Representations consist of all (or nearly all) information about the magnetic symmetry and, besides, in a form which is the most suitable for physical applications. Magnetic space groups (1191 Shubnicov groups) have been found[l, 21. In these groups the antisymmetry operation is not an element of the group itself, but inserts as product by any geometrical symmetry operation (rotation, rotation with inversion, translation). In physics, the antisymmetry operation corresponds to the time reversal operation R : t + -t. In particular, the operation R changes a sign of the magnetic moment. The representations of such group considering as a set of operations of the geometrical and time symmetry, obviously coincide with those of the space group which is obtained if the operation R is replaced by the identical operation in the magnetic group (‘isomorphism’ noted by Indenbom[3]). The representations of all space groups are presented in [4,5]. However, for quantum-mechanical problems these representations are not to be used, but the representations of groups of operators g which are corresponded to opera149

150

0. V. KOVALEV and A. G.

reversaling and the rotating and reflecting allowed by crystal symmetry the wave functions are transformed by corepresentations. The corepresentations of group G + G0 are constructed with usual irreducible representations of the unitary subgroup G. In the present work we shall show how one can (1) use known representations of the space groups [5] for obtaining those of unitary subgroups G and (2) carry out real construction of the corepresentations. We will limit ourselves to those 674 magnetic groups, in which the operator 8 is contained as a product by the rotation operator or rotation with inversion operator. For such groups the chemical and magnetic unit cells coincide, and operators g and g are different. Further, the operators g or g and antiunitary operators 8g will be called simply “elements” and “antiunitary elements”, respectively. Any point operator hi will be called “rotation”. The method of constructing the corepresentations of such groups is based on following simple speculation. The unitary subgroup G of a magnetic group is any usual space group G’ which, of course, is in the list of all space groups. Since the representations of all space groups are known[5], we shall use them for constructing the corepresentations. In accordance with it, we shall denote sometimes the magnetic group by two symbols G’(Ifj) [lo], the first being Shoenflies symbol of a space group C + G and the second (near the brackets) being that of a unitary subgroup. So, for instance, Ohl(O1) is the magnetic group which reduces to Oh1 in replacing R by unity; 0’ is the unitary subgroup of O,l(Ol). Analogical symbols are used for denoting of magnetic classes.

2. UNITARY SUBGROUPS AND THEIR REPRESENTATIONS

Here we shall show how it should describe the group G + &II and the unitary subgroup G of this group to coincide G with a space group G’ from the group list[51 and to

GORBANYUK

use then the representations of the group G’ for constructing those of G. We suppose that the description of a magnetic group is known in some kind. It means we know all symmetry elements g and g. The set of these elements forms a usual group G + G which is in the space group list[5]. The elements of every space group in[5] have been described as ((YJhJ where hi and cyi is the rotation and the subsequent translation, respectively. Under such description all axes and planes intersect in one point (point group center). This point is chosen as an origin of Cartesian coordinate system. Every rotation hi inserts into the group description the only one time. The translations (Yi associated to rotations are defined by the position of the coordinate system origin in the crystal cell. If the coordinate system origin is displaced be a vector R, a space group takes up other description, that is, an element (ailhi) must be replaced by an element (pi/h: ) where pi=ai+hiR-R==i+yi.

(1)

Here hiR the vector to which vector R is changed under the operation hi- The rotation hi is made around a new coordinate system origin. For the consecutive application of an element g, and then g their product corresponds gg, = (a/h) (al/h)

= (a+ hdhh,

1.

The product table of rotations hi is in [5] Further we shall deal with “small” representations of the wave vector Kj group GKj. Let us remind, that group Gxj consist of elements of G, which being applied to Kj produce hiKj that differs from Kj not more than by a translation of the reciprocal lattice. The representations t(gi) and n(gi) of the group GKj is called “small”. Side by side with them loaded representations 7( h,) = exp [iKj, Iyi]T(gi) and m(hi) = exp [iKj, cw&r(gi) are introduced[8,5]. It is very important that matrices of loaded representations of elements

IRREDUCIBLE

REPRESENTATIONS

g, and (a,/gJ - where a, is lattice translation are identical. Our aim is to obtain loaded representations of G with the help of loaded representations of G’ for all vectors Kj which form Brillouin zone. Let us deduce a formula, needed below, showing how the loaded representation is changed under displacement R of the coordinate system origin. It is clear that matrix T(gi) assigned to element gi does not depend on the choice of the coordinate system origin. Therefore, if (ailhi) and (&/hi’) is the same element described in the different coordinate systems, exp [- iKj , CXi]T(hi) = exp [where r( h,) and r’ (hi ) are matrices before and after displacement, respectively. Using (I), we have iKj

pi]+

(hi

1

)

&(I$)

(2)

= hils(hi)

where Ai = exp [X,, n]. The factor Ai is the same for R and R-t an, since (Kj,

hfan-&g)

=

(h$-lKj-Kjj,

U,)

=

bnan

=

2?T6,,.

Later on we shall suppose that G + c has description as it is in [S]. Therefore, for the use of rules given here the reader must describe the considered group with regard to [5]. As it turned out for aims of the present work, it is suitable to change some of the data[5] and add something. This is done in the Appendix. So $1 contains the description and representations of groups D&‘O; $2 contains the description of the lattices I‘,n and I‘,,!. Some corrections of [5] (Russian edition, 1961) are in $7. We shall show further how it should identify the groups G and G’ and determine representations of G. We shah pick out three essential different cases which, in his turn, are divided into subcases. Case A Magnetic classes of cubic, hexagonal, trigonal system and magnetic classes D&C&,

OF MAGNETIC

GROUPS

151

Group G + G and subgroup G considered as a some space group belong to the same system and have the same type of Bravais lattice. For all systems except orthorhombic. which we consider apart, a set of rotations h, of G coincides with one of some group G’. The group G’ should be sick among groups of the same lattice lY and the same class G which the group G belong to. However, the description of G can appear different than that of G’ in [5]. There are the following possible cases. A-l. There is a group G’ among groups of corresponding lattice T and class G, the description of which differs from that of G no more than by the lattice translations for vectors q. it is clear that G = G’. Loaded representations t( h,) and n(hi) of the group GKj of vector Kj, adduced In [5] for the group G’, are those of t(hJ and p(hi) of the group GKj of the same vector Kj for the unitary subgroup G . A-2. For the group of lattice r and class G in the list[5], there is no description that coincides with that of G in sense discussed above. In this case, in order to identify G with some group G’ of the class G in the list[5], it should change the description of G’ in just degree by displacing of the coordinate system origin by a vector R in latter. Let us show how one can determine the vector R and the group G’. Let us assume we have p groups G;, G.i,. . ., CL of class G and lattice r in the list [5]. Choosingqgeneratingelements in a class G (that is, such ones which generate the group G by multiplying), let us then displace the coordinate origin in the groups GA by a vector R. The description of these groups must be changed and new translations associated to rotations hi in elements Sim

giL1l* pi,=cui,+hR_R(m=1,2,...,p; i= 1,2,...,q)

152

0. V. KOVALEV

and A. G. GORBANYUK

where (yi, is the translation associated to hi before displacement. Demand now Pim = (Yi-t II, where (Y~is the translation associated to hi in G and a, is a lattice translation which can be different for different i. We shah obtain p systems of equations, the total number of equations in everyone being q. Only one of them is compatible. The to a compatible group G&, corresponding system, must be identified with G. In order to make the description of Gk coincide with that of G with accuracy to the lattice translations, we must displace the coordinate origin in GA, by a vector R determined from this compatible system. The following rule is useful. helping to determine R and G,: the displacement of the coordinate origin along a symmetry axis or reflection plane do not change the translation along these elements. Formula (1) allows verification of this rule and deduction of other ones, but we do not consider it here. Example 1. Magnetic group D,3(D3”). rh. With 151 De3 = G(h,, (2ff‘~~~), (4(~‘/&). (a’lh,), (3a’/h,o), (5(11’/h12)}-tG{((y’/hZ). (3cr’/h,), (Sa’lhs). h,, (2a’lh,), (‘Ia’/&)) where ix1 = 2 = (O,O, 7,/3. It is easy true, if we displace the coordinate origin in the group D,5[5] by a vector R -3 -cx’/~, its description will coincide with that of G . hence DSn = D35. Loaded representations of G are obtained with those of D3’ in ]5] by multiplying them by factors Al. For instance, for K = bb,, A1= X3= A5= 1 and AR= Alo = A,., = ein@= u. Using. with Ref. [S], T 18 we find the representations t(hi) and t(sl) of G

t'

h

h3

hs

1

v*

v4

hs

ho

h,,

A-3. Magnetic class DPh(C&,), lattices I,,> TOv, IJ In 151, the only one variant of the space group description of class C,, has been given. Nevertheless, the groups of this class allow six variants of the description differing from each other by the orientation of the axis C2 and the reflection planes respect to coordinate axes (variants, corresponding to the same position of axis CB, can coincide for some groups). In real magnetic crystal, axis C, of subgroup G has been oriented definitely along x, y or z axis. It should be compared to the description of G with those two descriptions of the group C,“, in which axis C, has been oriented along the same coordinate axis. Therefore, it is convenient to have five more variants of the group C,, description, besides that given in [5]. One can obtain them from the description cited in [5] by formal replacing rotations and translations for every element g apart with agree Table 1. Table 1. Six variants of the class C,, group description, the taftices IO, IO”, Iof. u,, z+, v:, are 0 or 1 in the description of groups CZvn in [53 rx, q., Q are hair-periods along X, Y, Z axes; hi is rotation

Example 2. Magnetic group D& (C,“,,), C, [IX, To. It follows from the description of D& in [5] that G = C&,(h,, h,, (O,O.T~/h*~,h~~)~. With agree to the third and fourth rows of g1

g3

6s

g8

gl0

&?!2

IRREDUCIBLE

REPRESENTATIONS

Table 1, we replace h4, hzs. h,, by h2, h2,, h2, (a) and by hz, hz8, h,, (b) in all ten groups of the class C,,. Accordingly, we replace translations. In particular, we obtain two new descriptions for C$, : {h,, (0, TV, O/h,), h2,, (0, rU, O/h,,)} (a) and {h,, (O,O,~zlhz). h,,, (O,Orz/ h2,)} (b). If we now carry out the displacement by R = (0,0,7,/2), the latter is described as {h,, h2, (O,O,T~~h~~), (0,0.7~/h~,)}. With the accuracy to the lattice translations, it coincides with the description of the subgroup G = G?jU,that is, n = 4. If in G CZ]]y, it should be realized the replacements with agree to the fifth and sixth rows of Table 1. If in G C,]fz, one must use the description given in [S] and that, which we will obtain from it with the help of the second row of Table 1. In order to obtain representations of groups G, one should have representations of C,, ctass groups for all six variants of the description. For the description given in I.51 representations are in [5]. For other variants of description representations are in Appendix $3. Obviously, the loaded representations t and v of G’ are bound up with t and p of G on formula (2). Let us obtain t of D$(C&,) for the vector K,,=p&,+it_(b,+b,). In this case R= (O,O;r,/2). It follows from appendix (C, 11~. b, C&J that the set T12 is bound up with Kll

OF MAGNETIC

GROUPS

153

r,,*,CZ ]IZ. The group G is identified with groups C#jJ2J3 in the way of A-l or A-2, the group C&$ allowing two descriptions which are different from one another by the orientation of the glide plane respect to the coordinate system. One of them is given in [5] (a), another is following: C:i{h,, hz7, (0, 0,rJh4_ h&} (b). The description of G should be compared with both, since, if G is C.j& its description differs from (a) or (b) only by shift of the coordinate origin. The loaded representations, corresponding to these descriptions, are identical. C, II x. In the way A-l or A-2, G is identified with one of groups C.J$15*16,17 described in [5]. C, /Iy. In the way A-l or A-2, G is identified with one of groups C&$,15*‘6,‘7 described in Table 2. The representations of groups C$15*16*17 (Table 2) are given in Appendix $4.

Type

rob r*l, r-0” r**

Case B Systems S and S, which the magnetic group belongs to, are different. For instance, T12 h, the group C42(C2n) belongs to tetragonal h, hm h,, system and the group CZn belongs to monoclinic one. In any list of the space groups, as in [5], groups G + G, which we must identify with the subgroup G, are found in sections Since Xi = exp [K, hiR --RI, we have X1=I: with different lattices I and I” or, as we shall &,, = 1 and hz = xZs = -i. Hence, the loaded speak, G + G refers to lattice I and G refers representations of G are equal to to lattice I”‘. It connects with the rule (not proved, but also, apparently, not be broken), according to which, any space group is realized in nature with the least symmetrical lattice, allowing its ~onst~ctions. It is the rule which the distribution of the space groups over lattices conform to. In our case, the elements of G form a subgroup of G f G. Here, they

154

0. V. KOVALEV

and A. G. GORBANYUK

are realized with lattice I, since G + G refers to I. However, if these elements do not form a subgroup of other group, but are considered as some space group G’, then G’ must refer to lattice I’. In above example, Cd2 refers to lattice I,, however, CZn should be referred to lattice I,,, or Imb. Thus, two problems arise. One of them is to determine the lattice I’ to which the group G’ refers; the other is to choose the basic periods in such a way that their arrangement with respect to the coordinate system is the same, as in I’. After solving these problems, it is possible to compare the subgroup G with groups, referring to the lattice I’. The system S’ of the group G’ is determined in the simple way, since G and G’ refer to the same class, that is, G = G’. To determine I’ one can use the following consideration: I” is obtained from I as a result of such infinitesimal deformation which transfers the crystal from the system S to system S’. In given example, the rotation of the vector a1 of the lattice Ia by an infinitesimal angle in the plane z = 0 results the lattice I,,, (latter Imb, thus, fall away). Comparing now elements of G = C,“{h,, (O,O,r+/ h4)} with elements of class C, of the lattice Im, we see that C,” = CZ2. Below we shall show the accordance I + I’ and those combinations of vectors ai which must be taken as basic periods ai of the lattice r’. If periods ai of I” do not coincide with periods ai of I, then reciprocal lattice vectors bt and bi will not coincide also. In connection with this, the same Brillouin zone point is expressed by vectors bi and bi in a different way. From relations between a[ and ai these follow for b; and bi. These last are adduced also. B-l. Letters of rotations hi of the subgroup G coincide with those of class G groups in [5]. Magnetic classes C,,(C,,), C,(C,), S4(cZ), c2U(C2) 9 C2h(Ci), c8(CO) 5D4h(1D2h) 7 C,“(1C!m), D,(la), n&l&). h(lCm)r C, (C,) , where the figure 1 means that sym-

metry

axes and planes of the subgroup

G

coincide with coordinate axes and planes, the I, and Iqu orienting with agree [5]. The accordance I + I” is following. For Cdh(C,,) , c4(c2)* C*(G)

s4(c2)v

c2u(C2)v

C*h(Ci)*

C.s(Cd7

ra. r. -+ rm rm,rmb* rtz yai=al,bi=bl rob--, rm1

rp~,r~~~rm~,u2+u3=u;,-u,=~~,u,=u~, b, = b;, b, = b; - b;, b, = 6:. rof +

r,b, us= u;,u,-a, = &a, = a;, bl=b;,bp=-b;,b3=b;+b;.

If the group G now are considered as referring to the lattice with periods ai, its description may differ from that of some group in [5], referring to the lattice I’, only by a coordinate origin displacement. The displacement vector R is determined in the way pointed out in A-2. Example 3. Group Czh (C$) , rqv. In accordance with [5] Cl,, = G{h,, h,, (0, r, 7J2 I&E,, hzs)} + G{&, hggr (0, 7, 762 I h,,, Ml. Introduce new periods ai = u2 + u3 = (r,?, TV)+ (7,7,7J = (27,0,0)) a; = i&l= (?,7,T2), a'= aI = (T,T,TJ, as it pointed out in the above list. Hence, it appears (0,7,7,/2) = &‘+ )(a;, - a;). In accordance with A-2, for groups C$, and C$, referring to the lattice Imb, we obtain two systems of equations

(1)

(h,R-R h,,R-R

= a;, = u~~+&z:+f(a6-u~)

One makes sure that system (II) has solution R = &(a; -a;). Therefore, n = 6.

IRREDUCIBLE

Example

accordance

4.

Group with [5]

REPRESENTATIONS

D410(lDzn),

OF MAGNETIC

hl ha

TpU. In t’

Dal0 = G{ h,, h,. (O,T, 7,/2/h,, h,) 1

pare the group G with groups of the lattice IOU. Periods of IqU are considered to coincide with those of IOU (for this, it is necessary to put T, = 7y = 7 in I,“). If we displace the coordinate origin by the vector R = (0,7/2, 7,/4) in Dzs, we shall obtain the description of Dps as following D2s{ h,, (0,27,0/h,). (0, T, T2/2 1 h2), (T. 0,72/2 ) hJ} which differs only by the lattice translations from that of G. Therefore, II = 9. It is possible to determine easily the small representations of the subgroup G with the help of above adduced lists. Let us interest the representation bound up with the vector K = plbl +p2b2 + p3b3. With the help of obtained formulae in the list, we express. Kj by vectors bi. We have K = pibi + p2be + &bj. In [5]. we find the loaded representations of the group G’ bound up with this vector. Multiplying them by factors hi = exp [iK, yi], where yi = hiR -R, we obtain the loaded representations t of the group G. The same is true for p. As an example, we will construct some representations of groups considered above. GroupCah(Cg),r,u,R=~(a~-a;). Letus take, for example, K = p(b, - bp) +fb3(K = K, for I,“). In accordance with the list, we replace bi by bl and obtain K = p( b; + b: - b;) ++b;. Loaded representations of this vector for IYmb[5] coincide with those r and 7c of vector plbl + p(b2 + b3) (this is K1 for rmb) T2 h, 7l 72

h,,

P2

h,

h,,

1 1 m1 1 -1 7rz

1 1

-i

1 1

t2

It follows from the list that one should com-

i

Since y1 = 0, yz8 = *(ai -a;), then AZ8= ei(r/4) = 6. Hence, it appears

Al = 1,

GROUPS

From them, obtained

hl &a 1 1

p’ p*

63 S’

the small representations

g1

t’ t2

.6 65

155

g28

g1

V/L! p’ @/.L’ p*

1 1

are

g28

1 1

6/.LI 65j.L’

where CL’= exp [- inp]. Group D410(1 D2s), r,“, R = (0, p/2, tJ4). Here bi = bi; therefore, the loaded representations of the unitary subgroup bound up with a vector K, are obtained from those of group Dzs bound up with the same vector. For In let K = p(bl + b2 - b3)++b3. example, accordance with [5], the loaded representations T24 and P24 have bound up with this vector T24 T1

72

h,

h,

P24

1 i d l-i,rr*

Here y1 = 0, y4 = (0,7,0) Therefore, h, t’ t*

h,

h,

1 l-l

1

and A1= 1, A4= i.

h,

h,

h,

l-lp’ 1 1 p*

1 1

-:

.

Since for D410(1D2s) g, = h4 then small representations are equal to ones loaded. B-2. Rotations h of the subgroup have been denoted not the same way as it has been done in groups of class G in [5]. Magnetic classes D.&2D& C,,(2C,,), D,(2D,), D,,W,,), D2,(2D2). The figure 2 means that symmetry axes and planes oriented at an angle of 45” with respect to axis x. Tables 3, 4, 5 contain the description of all subgroups allowed in this case. Appendix $5 contains their repre-

156

0.

V. KOVALEV

and A. G. GORBANYUK

Table 3. The description of groups C~$13J8Jg Group

Type

h37

h,

h40

Table 4. The description of groups Dg,6,7 Group

h 13

Type

h 16

unitary subgroup representations, the more so, as it can be applied to case both magnetic groups and usual space groups (in [5] the method is not given). In a group G, let us select elements gi = (q/hi) which do not change the vector K or add to it the reciprocal lattice vector. The set of these elements forms the group G, of the vector K. If gi = (q/hi) and g, = (qJhK), then matrices of loaded representations satisfy the relation

h,

OoT,

where ‘l’[K= exp [i(K - hi-‘K, (Ye)] and K is the reciprocal lattice vector. Calculating qi, for all g from G, and using the

hi-‘K

Table 5. The description of groups Dirz4 Group

Type

h,,

h,,

h

h 25

h,,

h 40

h 211

ra rq

r,

rq I’,”

sentations. Subgroups G were denoted in the same way, as groups G’ of orthorombic system which the subgroups may be identified with. In real magnetic crystal, the description of the subgroup G differs from that of corresponding subgroup in Table 3 no more than by a coordinate origin displacement. The comparison of subgroups and obtaining of the loaded representations is carried out by analogy with A-l or A-2. Case C

Magnetic classes &(CZh), &(Cz), C,,.( C,). It turned out that the description of ways of obtaining the representations of subgroups G from those of groups G’ are bulky. In this case, it is more expedient to give the method of direct constructing of the

product table of hi [5], we compose the product table of matrices t(hi). With the help of it, the matrices t(h,) are determined. Selecting the simplest sets of figures or matrices, satisfying the product table, we obtain irreducible representations. Irreducible representations p ( hi) are determined in the same way with the difference that some qfK must be multiplied by - 1, as it pointed out in the product table in [5]. Example. Group @&(C&), ITO, C,ll y. In accordance with the description of D!j,, [5], G = C&{h,, (O,OvTzlh3), (7z,i,,Olhzs), (7s,Ty,TZ/h27)}. Let us consider K = Jb,. The group GK consists of all elements of the group G. It is true that \Ir3,z, = q\v,,z7= q\Ir,,,z,= ? 27,27= - 1; the rest of qi,,. is equal to 1. Taking into account product rules of rota-

IRREDUCIBLE

REPRESENTATIONS

tions h (see [5]), we will obtain the product tablesfort andp(hi).

f(h3)

f(hm) p(h)

t(b)

t(h)

-f(hm)

-f(M

p(hJ -p(h)

Wd

t(b)

-t(h)

- f(hd

p(h,s)

f(b)

t(h,,)

f(b)

t(h)

of order two can satisfy

p(h)

dh,)

i(g) and A(g) are is equivalent to some

p(b)

p(M -p(h,,)

p(b)

p(h,s)

-p(h)

-p(h,s)

1.57

GROUPS

(c) Representations inequivalent, but E(g)

t(b)

The only matrices such tables.

f(hi)

OF MAGNETIC

-p(h)

p(h,)

-_p(h,)

other representation of the group GK, for example A1(g) = p-‘A”( g)p. Two representations Al(g) and AZ(g) form one corepresentation D(g)

= (

A’(g) 0 j3-lAY(g)p

> ’

A’(&,@) D(@) p

(E)

(oy

(Oil)

(3

Corepresentations of group GK,_-K are constructed with the irreducible representations of the group GK. There are three types of corepresentations (g,K = - K + 6). (a) Representations A(g) and i(g) = A*(go-‘gg,) are equivalent, that is, A(g) = P-IA (g)p the matrix /3 satisfies the equation p/3 * = A (Pgo2). Corepresentation D is construtted by formulae [7] =

At&o-')P

where g and g6J is unitary and antiunitary element of GK, +, respectively. (b) Representations A(g) and x(g) are equivalent, but matrix p satisfies the equation pp* = -A(e2g02). Then D(g) = D(ge)

=

(A(g) 0

0 A(g) > ’ 0

-Atego-')P

Ah%,-')P 0

-‘A2(;g 6

3. COREPRESENTATIONS OF GROUPS Gx,_x

D(g) = A(g),D(@‘)

=

>

*

-1)~ 0

0

. >

In order to determine the type of the corepresentation, it is convenient to use the criterion ,q]

S=ixexp[-iK,(cu+ha)]x[h2]=

where summation is over all different rotations hi associated to antiunitary elements @ = (a/he) of the group GK,_-K;h and x is number of rotations and matrix character of a loaded representation, respectively; K = 1 fOrtandK=-1fOrp. The matrix /3 is determined by foregoing relations, being enough to consider only those which generate the group GF Example. D,3(D35), K = +b,. The description of the group is in example 1. In the group GK,_-K,rotations h2, h4, h,, h9, h,, associate to antiunitary elements. One can calculate the summation S, as is shown in Table 6. It follows from the table that St1 = St = 0 and St = 1, that is, t1 and r2 form the type (c)

0. V. KOVALEV and A. G. GORBANYUK

158

Tab/e 6. The calculation of sum S. (Y’ = (00(7,/3)) ; w = ei(x’3); cl!+ hcu is translation associated with h2; xti is matrix loaded character of representation t

hi2 a+ha exp [-ik.a+ha] XI’

=

xt*

xt3

hz

h,

h,

h,

h,

h,,

h, 2a’ ~5

h, 6a’ t-9

h, 10a’

h, 0

h, 0

h, 0

W4

1

1 1

1 1

1

w

2

2

2

2

corepresentation and t3 forms the type (a) Let g, = g, = (3a?/h,). corepresentation. Then g,-l = (3a’lh.J. Productions gg4 and &+-‘9 where g is element, belonging to set of G, are given in Table 7. For corepresentation

;2

W5

1

Corepresentations with p are constructed the analogical way.

in

4. CONCLUSION If element 13is added to the magnetic group,

Table 7. The productions gg, and gg,-’ g, Bg4 &?-’

(a&A g3

D1q2formed by one-dimentional

representation

t’ and P, p = 1. For D3, matrix p is deter-

mined by equations /3/3t = 1,/3p* = t3(gd2) = t3(a3) = - 1, Pt3*k-‘g3K4)

=

Pt3*k-1g8g4)

= t”ks)P.

f3(K3)PY

Since g4-‘g3g4 = g3 and g4-‘g8g4 = (-as/&, then we have

Finally, matrices (for some elements) g3

g7

of corepresentations

gn

88

are

(--akJ RIO

g9 (- aA g12

it results in a neutral group in which elements g and g0 are contained together. In many cases, the neutral group is a group of the chemical symmetry of the crystal and coincides with the whole group of a magnetic crystal in the nonmagnetic phase. Corepresentations of some neutral groups were considered in [ 11, 12, 131. Thus, the transition from the nonmagnetic phase to the magnetic one are accompanied b y 1owering of the symmetry and, possibly, partial or whole splitting of the energy-levels. To determine the splitting, it should be compared with the corepresentations of the neutral group with those of the magnetic group. Later on, authors intend to publish corepresentations of all neutral and magnetic groups.

R-i0 authors wish to thank Professor Akhiezer A.I. and Professor Lifshitz 1.M. for the interestshowninthework.

Acknowledgements-The

Dl.2

(i-0)

(Oyl)

(_“:o)

(yi)

REFERENCES 1. BELOV N. V., NERONOVA

N. N. and SMIR-

IRREDUCIBLE

REPRESENTATIONS

NOVA T. S., Trudy-Inst. Krist. Akad. Nauk SSSR, 11,33 (1955). ZAMORZAEV A. M., Kristallografiya 2, 15 (1957). INDEMBOM V. L.. Kristallografiya 4, 619 (I 959):

GROUPS

159

7. WIGNER E. P., Group Theory (Russian translation). Moscow (1961). 8. LYUBARSKY G. J., Group Theory and its Application in Physics. Moscow (1957). 9. CHALDiSHEV V. A., KUDRYAVTZEVA N. V. and KARAVAYEV G. F.. Izv. vuzov SSSR. Fiz. 2. 46 (1963). 10. KOPTZIK V. A., Shubnikov Groups. Moscow University edition (1966). 11. HERING C., Phys. Rev. 52,365 (1937). 12. ELLIOTT R. J., Phys. Rev. %, 266,280, (1954). 13. PARMENTER R. H., Phys. Rev. 100,573, (1955).

5,513(1960).

FADDEYEV

D. K., Tables of Basic Unitary Reof Fedorou Groups. Moscow ( 196 1). KOVALEV 0. V., Irreducible Representations of the Soace Grottos. Kiev (1961): English edition:

presentations

Gordon & Breach’, New York ( 1965). 6. DIMMOCK J. 0. and WHEELER Rev. 127.391 (1962).

OF MAGNETIC

R. G.. Phvs.

APPENDIX of Dfd”rqV §l. Description and representations Both groups contain rotations h,, h,,, h,, h.w hm hm hm h27; for DZ& = 0. for D.$(Y~~.~~.,:~..‘, = (0.0. TV).).Representations: K,, K,, K,,- 10. K,, K,, K4, KS-O. K,- 130. K,-171. K,-3. K,,- 11. K,,- 117. K,,- 172. K,,, K,,- 131. §2. r,-a, and a2 are in plane z = 0, a3 = (0.0.2~~). respect to plane Z = 0.

$3. Representations

of groups of class CZV, lattices

b). K,, K,, &r Kur K,,, K,,-9.

a, is in plane z = 0; a2 and a3 symmetrically

arranged with

To, Tot, To”

K,, K,, K,, K,,, K,,, K,,-

C~v*(Czl/z, b) KU K,, K,, K,,, KM, K,,-9.

C%‘(C,k

I’,*-

K,, K,, K,, Km K,,, K,,-

10. K,, Ks-0.

10. K,, KB-0.

Ky, K16-Kz6-

11.

Kg, K,,, K,,, K,,, K,,, K,,-

11.

K,,, K,,, K,,, K,,, K,,, K,,C~~,(C~llz~b).K,,K,,K,,K,3rK~4.K~~-9.K3,K~,K,,K~~,K,,,K,2-l0.

345.

K,,Kc-O. Kg,K,,,K,,-11. Km Km K,,-346.K,,,Hm KM- 13. K,,, K,,, K,,-345.

‘%(czII C5-‘(C&a).

XTa). KI, Kz-0. K,, K,-Q.

KS, Kdr K,, Km Km KmK,, K,, K,, K,,-K,8-

10. K,, KS, K,, K,,, K,,, K,,-2.

K,, K,,, K,,, K,,, K,,-K,,-30.

10. Kg,K6,K~,K,,-K,,-2. Kn Km Km, Km Km K,,- 30. K,,, K,,,KZl,K23,K25,

W(Cz

IIx94. K,, K-0. 4, Kq,Kg,K,,-K,,-

10. Kg, KBr K,, K,,-K,,-

2. K,, K,,, I&,-

Ku,, K,,, &s-63. ciG’(cz 11X. b).K,.

K, -0.

K,. K,. Kg. K,,-K,,-

10. Kg,KG. Kg. K,3-K,5-2.

K,,- 63.

30.

K,,, Km K,,-340.

K,. K,,. K,g-K,,.

K,,, Kz2, K,,-341.

K,,-30. K,x.K,,.K,,-K,,.Kz,-341.

C~?(CZ 11X7 b). KI, K,-0.

K,, K,,

Kg,

Km-K,e-

10. K,, Ks, K,, K,,-K,,-2.

K,, K,g, K,,-30.

KU,, Kz,. K,,-63. CZ(CZ C#CZ

IIY* b),

CL(C,

11y, b). K,, K,,

Kg,

IIY,a).K,, KM-K,,--9.

K,, Kg, KIG-K,,-9. K,, K,-0.

K,-K,,

K,, K, -0.

K,-K,,

K,,, K,,, K26-340.

K,~-K~~ -2.

K,, K,,-K,,.

K12. K,,, K,,-341. K,,-K,,

_ 3 1.

K,,-K,,-2. Kar KI,,

K,97

&,-Km-

3 1. K,,, K,,, K,i,, K,,-K2,-

342.

c80(C~ 11Y 9b). K,, K,, K,, KM, K,,, K,, - 9. K3, K, - 0. K,, KB, K,, K,,, K,,, K,, - 2. KS* K19r KU - 3 1. KU, Kpo, K,,C’S(CZ 11X, a). KI, Kw. K,,, K,,, K,,-0. C~~CZ 11X9 a). KI, K,,, K,,, K,z, K,a-0.

K,, Kg, Kg, K,,Kz,

Kg,

Km-

64. K,dr KZ4, KZ6- 343. K,,, K,,, K,,-

10. K1, K,,, K,-2.

10. K,, K,, K,-2.

Kq, K,, K,,, K,,, K,,, K,,-30.

K,, K,,, K,,-30.

K,, K,,_63.

K,,-341,

344.

160

0. V. KOVALEV

C;i’22(C, I(x, a)> C;?TC,

and A. G. GORBANYUK

K,, Kg, K,,, K,,-

(1x, b). K,, Kg>b-0.

10. K,, K,. K,,, K,,-- 2. K,.K,,.K,,.

$4. Representations

K,,-

15. K,, K,,,K,,-30.

of C&?J5J6,,7 (Table 2)

C~,4~‘5,K,.K,.K,-9grK~rKg-O.K~rK~rK~,K~,K,~,K,~-2.K

I,,, K,,,K,,,K,,,K,GrK,,-31.

C:~~‘7~K,,K~,K,-9.K~,K~-0.K3.Ks,K,~-2.K4.Kg.K,3-3S.K,o,K,4.K,5-3l.K,,,K,6,K,~-43. $5. Representations C:?.

of groups given in Tables 3.4.5

K,, K,. K,, K1, Kc,, K,, Kg, KS-O. K,, Km K,, - 118. K,,. K,,, K,,-

3. K,,, K,,, K,,, K,,, K,,, K,,-

126.

C~,9.K,.K,.K,.K~~K,,-0.KS,K7-1l8.K4,KLI-166.K9.K,O,K,2,K,3,K,4.K,5-l26. C:,9.K,.K,.K,,K~,K,,-O.K~.K7-ll8.K4rK~-l67.Ky,K,2.K,a-l68.K,o.K,4.K,5-l26. Oz.‘,. K,-K,O.

K,o, K,,-

130. K,,-K,G-

3. K,,,K,,+-172.

338 (for Dz5), 172 (for D,?

K,,, K,,-

Dz7. K,-Kc,

K,,-0.

K,-

130. K,-

171. Kg, K,o, K,,-3.

K,,, K,,, K,5-

172.

D:%,2,.K,,K2,K~,K,rKR,K~-2.KS.K~-O.K~-ll8.K,0.KI,-l46.K,S-3.K,3,K,4-126. K,s, K,,-4(for D;p=. K,, Kg, K,-2.

K,, K,, Kg-35. K,,-4

D;j$ K,, KS-O. K,, Ka-2.

K,, K,-0.

K,-

118. Klo- 146. K,,-

(for D.$), 99 (for D$,

K,-

118. K,-

166. K,-

K,6-

0:X,, 99 (for D;jJ. K,,-K,,,-

148. K,,-3.

K,,, K,,-

176.

126.

101 (for D$), 102 (for Dz2). K,,, Kls- 176. K,,,, K,,-

146. KS- 175. K,, K,,, K,,-

126. K,,-

1. K,,, K,,, K,,-

300.

176.

D~~.K,,K5-O.K2,Ks-2.K3-l18.K~-l67.K~-l46.K1-l72.Ks,Klz-l68. K,,-

126. K,,-

104. K,,-

181. K,,-

176. K,,-300.

$6. Additional representations P340

P343

, (;;,

($

t,

(A,

(Jy(J

h,

h ZB

h,

h 28

(3 (-03 (LYO)(fi)

(;;)

T338

T340 T343

t’ t2 t3 t4 T341 T342 T344 T345 T346

h, h,

;;J h, h,

1 1 1 1

1 1 -1 -1

h, h, h, h,

h, h, h, h,

P338 P344

(1,) h 21 h 26

1

h 28 h 211

p’

-1

P3 P4

-1

P’

h, h,

h IB h,

1 1 1 1

1 1 -1 -1

-1

-1

h 27 h 2~ h h :I

h h :,” h h 1:

P341 P342 P345 P346

h,

h,

h, h, h,

h, h, h,

P’ P2 P3 p4

1 1 1 1

i

-i

-i

i

i

-i

-i

h 21 h 26 h 26 h 27

h, h ZR h 21 h 26

1

1 -1 -1

h, h 2R

i

1 1

1

h ,6 h 26

-i

-i

i i

-i -i

i

i

IRREDUCIBLE h

REPRESENTATIONS h 13

h 16

ha

(A(i)

T300,t’

(A:)

(Or-o,)

(0101)

p300,p1

(A;)

(O?i)

(“;o)

OF MAGNETIC h 25

h 37

(-0:o)

(!!;;)

(3

161

GROUPS h 28

h 40

(Yi)

(z)

(-“:o)

(-“io)

(-$)

(3

t* = t’ X t5 (T176, P2 = P’ x t5 (T176). p7. Corrections of[5] h,

P12. Cl:

hz6

h 27

oar,

oar,

f’42. Add&s = pL(h+bz] +t(bz+b,)Ihl, hq, h,,, h,,. P43.C::. K,-9. K,-l&K,-11. K,-14.K,,-12.C;~,D2,~.K,s-ll.Dp’. KL8-3.D;f. K,,-38. P45. Df:. KS-31. P48. D:d. Kg- 16. D$. K,,-3. P51. D:,,. K,,- 152. P61,62. Td2,5. K,,- 127. 03. K4- 130. P63,64.AddK,,=~(l-~~)(b~+b~)+~(3~-lI)baJh,,hs,h~,,h~7,h~2.h4,. T3,Th5,05. K,,-268. Tj,T,,‘,08. K1, = 269. Td3. On’. K1, -270. Tds, Onlo. K,, - 271. P66. Hexagonal system. b, I ev. b, I e,. Add (P74) T29

h, (:;)

d:J

(Jfo) Pl60

Correct: P77. Pj(P77) = tJ(T31).

P119

(;;o)

p120

(ii)

(“;o)

(0;;

(K)

(%, h 13

h,

h 25

;;

(Z)

P*=Plxt*(T147) P3 = P’ x t”( T147) p4 = P’ x tG(T147)

(Z)

P147, P149, P152, P153. P3= P’x t5(T147),Y4 = P’x V(Tl47)

1,2,4, 5.6,7, 10 P187 P190 P200 P204 P231

P’ P’ P’ P’ P’

1 1 1 1 1

3,& 9, 11,12

-1 -1 -1 -1 -1

13,16,17, lk21.23, 24

14,15.19. 20,22

25,26,28, 29,30,31, 34

1 6 63

27,32,33, 35.36

37,40,41, 42,45,47, 48

38,39,33, 44,46

-1

65 67 i

-i