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The irreducible representations of the space group D 16 2h Yakel, Kent Alexander 1968

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THE IRREDUCIBLE REPRESENTATIONS OF THE SPACE GROUP D^6 2h by KENT ALEXANDER YAKEL B.Sc. (Hons.), University of Manitoba, 1 9 6 7 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1 9 6 S In presenting th is thesis in pa r t ia l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i sh Columbia, I agree that the Library sha l l make i t f ree ly ava i lab le for reference and Study. I further agree that permission for extensive copying of th is thesis for scholar ly purposes may be granted by the Head of my Department or by h lis representat ives. It is understood that copying or publ icat ion of th is thesis for f inanc ia l gain shal l not be allowed without my wr i t ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date CLvutJI I <{ L $ ABSTRACT The matrices of the inequivalent i r r e d u c i b l e repre-sentations of the space group Dl6 (pnma i n international sym-2h bols) are derived. The allowable representations of the groups of the k-vectors are obtained from certain ray representations of the corresponding point groups. From these allowable repre-sentations the ir r e d u c i b l e representations of the entire space group are induced. The r e s u l t s are presented i n a systematically arranged set of tables. - i i -, TABLE OF CONTENTS Abstract i i Table of Contents i i i L i s t of Tables i v Acknowledgement v Section 1 Introduction 1 Section 2 D e f i n i t i o n and Properties of Space Groups ... 3 Section 3 Inducing Representations of Space Groups .... 7 (a) General Theory 7 (b) Applications to Space Groups 10 (c) Applications to the Space Group 19 Section 4 Allowable Representations of the L i t t l e Groups L( k ) 37 Section 5 Irreducible Representations of 53 Bibliography 65 Appendix Ray Representations of Certain Point Groups . 36 - i i i -LIST OF TABLES (excluding those of Sections 4 and 5) Table ( 3 - 1 ) (displaying the special k-vectors and a gen-e r a l k-vector of the B r i l l o u i n zone, and t h e i r respective l i t t l e groups L( Xk ) ) 22 Table (3-2) (c) k = £b L(~X ) = Dj£ (displaying the j k <ch allowable representations of ^ o r IS ~ ) . . . . 31 Table ( 3 - 3 ) (j) k = f ^ ib_ 3 P(kT = [ l , 2 x , m , (displaying c e r t a i n i r r e d u c i b l e ray representations of the point group £(k) ) 33 Table ( 3 - 4 ) (j) k = f ^ + Jb_ 3 LTYk) = { (l)t), 2x|s), ( m !§.), (m \s_) f (displaying the allowable representa-tions of L( tfk) f o r k = . f ^ \- §b_3 ) 33 f a b l e A l (displaying c e r t a i n ray representations of the point group C2^) #5 Table A2 (displaying c e r t a i n ray representations of the point group D ? b) 86 — i v — ACKNOWLEDGEMENT I would l i k e to thank Dr. W. Opechowski for suggesting t h i s topic and f o r h i s advice during the writing of t h i s t h e s i s . I also wish to thank the National Research Council of Canada f o r f i n a n c i a l assistance during the past year, i n the form of a Bursary. -v-Section 1 I PRODUCTION The purpose of this thesis i s to derive a l l the i n -equivalent irreducible representations of the non-symmorphic space group (Pnma in international symbols). The method used i s to construct the allowable representations of the l i t t l e groups of the ^various representations of the translation sub-group, i.e. the allowable representations of the groups of the k-vectors. These allowable representations are obtained from certain of the ray representations of the point groups of the l i t t l e groups. From these allowable representations, the matrices of the in equivalent irreducible representations of are induced. In Section 2, the definition and certain properties of space groups are discussed, and the notation introduced that w i l l be used throughout. The elements of are l i s t e d . In Section 2h 3, the theory of inducing representations by the method of l i t t l e groups i s outlined. Section 3 i s divided into three parts. Part (a) deals with the general theory for arbitrary groups, part (b) deals with the theory for space groups, and explains how certain ray representations of the point groups of the l i t t l e groups are used, and part (c) shows the actual calculations for two k-vectors of the Brillouin zone. Section 4 consists entirely of tables showing the irreducible representations of the various l i t t l e groups, or groups of the k-vectors. Section 5 consists, of (1) (2) tables showing the i r r e d u c i b l e representations of induced from the representations of the l i t t l e groups. A number of paramagnetic c r y s t a l s (for example TbCrOy DyCrO-j, V^CaO^) which become magnetically ordered at s u f f i c i e n t l y low temperatures have a symmetry described by the space group For the discussion of these c r y s t a l s i t i s useful to have a re-l i a b l e survey of the i r r e d u c i b l e representations of t h i s group. There are unpublished tables of characters of the i r r e d u c i b l e representations of the space groups, by A. Casher and M. Gluck (1963). There i s a book by Kovalev (1961) which does give the actual matrices of the representations of space groups but i n the English t r a n s l a t i o n the p r i n t i n g and the display of tables i s such that the tables are useless without extensive checking. For these reasons i t i s hoped that t h i s work w i l l be of use to anyone wishing to study these paramagnetic c r y s t a l s . Section 2 D e f i n i t i o n and Properties of Space Groups We consider the group of r e a l coordinate transforma-tions which preserve lengths, of the form x» - Rx + s (2-1) where x f,x, a*1** §. are r e a l three dimensional vectors, and R i s either a r e a l orthogonal matrix or equivalent to one by a simi-l a r i t y transformation. I f we have two such transformations, x ' ^ x + S ! ( 2 _ 2 ) x t T = RgX1 +- s 2 , we may write t h e i r product as x , f - R2R1- + R 2 - l "** -2 * (2-3) I f we denote the inhomogeneous l i n e a r transformation (2-1) by the symbol ( a j s ) , where a corresponds to the r o t a t i o n a l part R of the transformation and s. i s the t r a n s l a t i o n a l part, then the m u l t i p l i c a t i o n rule between two such transformations i s , from (2-3), ( a 2 | s 2) (a^J s±) ~ ( a 2 a 1 | a2s1 +• s 2 ) . (2-4) Also, the inverse of a transformation i s ( a l s ) " 1 = ( a ^ l - a ^ s ) . The c o l l e c t i o n of a l l such transformations, with the m u l t i p l i c a t i o n r u l e defined above, forms a group, c a l l e d the general inhomogeneous group. Space groups are subgroups of the general inhomogeneous group such that a l l of the elements which have as the r o t a t i o n a l part the i d e n t i t y r o t a t i o n "1" are of the form ( l | t ) , where (3) U ) t — n±&± + n 2 ^ 2 +" n 3 ^ 3 (2-5) i n which a^, &2, §LJ are three l i n e a r l y independent vectors, c a l l e d basic primitive t r a n s l a t i o n s , and n^, are any three integers. In any space group F, the c o l l e c t i o n of elements ( l | t ) forms an invariant subgroup TCF. This follows from the r e l a t i o n ( a l s r ^ - d l t M a l s ) = ( l l a ^ t ) . ( 2 - 6 ) The subgroup T i s c a l l e d the l a t t i c e group of the space group, and the sectors t are c a l l e d primitive t r a n s l a t i o n s . The c o l l e c t i o n of r o t a t i o n a l parts a of the elements (a|s_) of F, with m u l t i p l i c a t i o n defined between the d i f f e r e n t a's, forms a group c a l l e d the point group P of F. The condition ( 2 - 6 ) , along with the r e s t r i c t i o n s on the matrices R of the rotations a of P, i s s u f f i c i e n t to re-s t r i c t the possible angles o f rotation to multiples df sixt y degrees or ninety degrees or else to products of the inversion 1 with these rotations. The l a t t e r are c a l l e d improper rota -tions and the former proper rotations. From the t o t a l number of proper and improper rotations which are permitted, i t i s possible to form thirty-two point groups. Every space group has f o r i t s point group one of these thirty-two point groups. Since the l a t t i c e of a space group i s invariant under the rotations of the point group, there are c e r t a i n r e s t r i c t i o n s on the basic primitive translations that define the l a t t i c e . ( 5 ) From these r e s t r i c t i o n s i t follows that there are exactly four-teen d i f f e r e n t classes of l a t t i c e s , c a l l e d the fourteen Bravais classes., Although the t r a n s l a t i o n a l part s of an element (als) of F, where a i s not the i d e n t i t y rotation 1 , need not be a primitive t r a n s l a t i o n of the form ( 2 - 5 ) , i t may r e a d i l y be shown that s must be of the form s - v(a) + t ( 2 - 7 ) where t i s a primitive t r a n s l a t i o n and v(a) i s either zero or a non-primitive t r a n s l a t i o n . Thus with each element a of the point group P' i s associated a vector v(a). Space groups i n which a l l the v(a).'s are zero are c a l l e d symraorphic, and those i n which some of the y j a j ' s are not zero are c a l l e d non-symmor-phic. There are 2 3 0 space groups, of which 7 3 are symmorphic and 1 5 7 are non-symmorphic. The space group under consideration, D^, has for i t s l a t t i c e the simple orthorhombic l a t t i c e , i n which the three basic primitive translations are mutually perpendicular but of a r b i t r a r y length... The point group of the space group Dl£ i s designated 2 h D 2 h ( S c h o e n * * l i e s notation), and i s made up of the i d e n t i t y 1 , three two-fold rotations 2 , 2 , 2 about the axes specified by by the three primitive t r a n s l a t i o n s , the inversion 1 , and three mirror r e f l e c t i o n s mx, niy, mz i n the planes through the o r i g i n (6) perpendicular to the three basic primitive t r a n s l a t i o n s , —1* ^2* —3* The elements of are written (l i l t ) , ( 2 x l s ) , (2 y|s) , (2 z|s) ( l i t ) , (mx|s) , (m yls) , (m zls), where s i s of the form (2-7), and v(m x) - v ( 2 x ) = + h§,2 + ^ 3 Z(« y) = v ( 2 y ) = | a 2 (2-8) v(m z) = v(2 z) =r J a x + J a ^ . Section 3 Inducing Representations of Space Groups (a) General Theory; We f i r s t review cer t a i n d e f i n i t i o n s and concepts used i n the general theory of deriving i r r e d u c i b l e representations o f a group from those of i t s normal subgroups. Let cf be an a r b i t r a r y group and Hc"G any subgroup, with =jd(h):h« H | a representation of H of dimension n y . We decompose G into a f i n i t e number of l e f t cosets o f H, G = s,H + s 0H -t-sJf-t- ... -»- s H, where s. £ G, i = ±j 1 2 3 r i 1,2, ... , r and s^ = e, the unit element. Note that H i s a subgroup of index r o f G, where r i s a f i n i t e p o s i t i v e integer. Now f o r any element g « G we define a super-matrix D(g) of dimension rn , which i s divided into r^ blocks each of dimension n y , and where the p q ^ block (p t l 1 row, q t h column) i s defined as D (g) = d ( s " 1 g S r t ) A p q pq P q _ where A * 1 i f s ^ g s t H , (3-2) pq P q = 0 otherwise. It may be shown that the set of matrices D(g) so defined for-msa representation r0t sr. r i s c a l l e d the representation of G induced by the representation 2f of the subgroup HcG. Two representations * = jd(h)j and r = |d»(h)jof ( 7 ) a ndrmal subgroup H C G are said to be "conjugate representa-tions" with respect to G by s i f there exists an element s €Gf such that for a l l he H, d(h) = d'U^hs). (>3) An "orbit" of a normal subgroup He:G" ±3 a maximal set of inequivalent irreducible representations of H which are mutually conjugate with respect to G. It may be shown that members of the same orbit of a subgroup induce equivalent rep-resentations. Let G be a group, and HcG a normal subgroup with tf an irreducible representation of H. Then the " l i t t l e group" L( tf ) associated with tf i s the subgroup of G consisting of a l l elements that generate mutually conjugate irreducible representations of H which are equivalent to tf . Let /—1= ^D(g) : g $ G | be a representation of a group G, and let HCG be a subgroup of G. Then the set of matrices P ( s ) = J D ( g ) : gel] i s a representation of H, called the representation of H subduced by the representation Pot G. An "allowable representation" of the l i t t l e group L(tf) associated with the irreducible representation tf of the normal subgroup HcG i s any irreducible representation of L{i) which subduces on H a representation which i s a multiple of 5 • We are now ready to state the steps by which one pro-(9) ceeds from the i r r e d u c i b l e representations of a normal subgroup HcG, v i a the representations of the l i t t l e groups of these representations, to induce a l l the i r r e d u c i b l e representations of a group G. The proof that t h i s procedure i s v a l i d , and that each i r r e d u c i b l e representation of G i s obtained exactly once, may be found i n the l i t e r a t u r e (see Lomont (1959) or Jansen and Boon ( 1967) )» The general procedure i s outlined below. 1) A l l the inequivalent i r r e d u c i b l e representations of H are found. 2) The inequivalent i r r e d u c i b l e representations X of H are grouped into o r b i t s of H, and one member i s selected from each o r b i t . 3) The l i t t l e groups L(tf) associated with the selected members of each o r b i t are obtained. 4) The allowable representations of these l i t t l e groups are constructed. 5) From each allowable representation of each l i t t l e group, an i r r e d u c i b l e representation of the entire group G i s induced. No two of these representations of the entire group are equivalent to each other, and a l l the i r r e d u c i b l e represen-tations of the group G are obtained by t h i s procedure. (10) (b) Applications to Space Groups We now apply the general procedure f o r obtaining the representations of a group from those of an invariant subgroup to a space group Y. We choose f o r the invariant subgroup the tr a n s l a t i o n subgroup T, 1) T i s the d i r e c t product of three abelian sub-groups, T = [ ( l U ^ ) " ! ® [(Hn 2a 2)| ( g ^ a l n ^ j , (3-5) so the matrices of i t s representations are the products of three one dimensional matrices of the representations o f i t s subgroups. Consider one of these subgroups, say ^ d l n ^ a - ^ , and l e t exp(-i2Ti f^n-^), f-^ a fixed r e a l number, correspond to the element ( l l n ^ a ^ ) . The convenience of the factor 2TT w i l l become apparent i n what follows. Clearly, the set of numbers, or one dimensional matrices, exp(-i2 n f-jn-^), f o r f ^ fixed and for a l l integers n-^ , forms a representation of the subgroup ^(lln^a^)^ , and t h i s rep-resentation i s i r r e d u c i b l e . Hence, i f we associate with each element ( l l n ^ a ^ + n 2 a 2 +• n^a-j) of T the number exp(-i2H ( f 1 n 1 + f 2 n 2 •+• f o ^ ) ) , f ^ , f 2 , f 3 f i x e d , then the set of these numbers f o r a l l integers n^,n2,n2 forms an i r r e d u c i b l e representation of T. To simplify the notation we now introduce three vectors b-j^bgjby c a l l e d the "basic r e c i p r o c a l l a t t i c e vectors", and de-fined by (11) Then i f we define a vector k by IS. = * x - l +• f2^2 f3^3 * fl* f 2 ' f 3 r e a l> {1-7) the set of (one dimensional) matrices ^ = |d f c(l|t) : ( l | t ) t ^ where d k ( l | t ) = exp(-ik.t) = exp(-i2ii ( f ^ + f 2 n 2 + f3 n3 ^ constitutes an i r r e d u c i b l e representation of T, completely speci-f i e d by the so-called "k-vector" k. The vectors b ^ b ^ j b ^ generate a l a t t i c e , c a l l e d the r e c i p r o c a l l a t t i c e . Any vector K of the form K = rn-^b^ + m2^-2 + m3b_3 m2, m^  integers, (3-9) i s c a l l e d a " r e c i p r o c a l l a t t i c e vector", and each such K cor-responds to a point on the r e c i p r o c a l l a t t i c e . I f two k-vectors k and k f d i f f e r by a r e c i p r o c a l l a t t i c e vector K, k» = k 4- K (3-10) then the representations and V^, of T corresponding to k and k* are the same. This follows from d (1 | t) - exp(-ik».t) = exp(-i(k+ K)-t) (3-11) = exp(-ik.t) = d k ( l l t ) where the t h i r d equality holds because K*t = 271(01^2^-+- m2n2 + m3 n3) - 2 TT (integer). (3-12) From t h i s we see that only a limited volume of r e c i p r o c a l space gives r i s e to inequivalent representations of T. It i s conven-ient f o r us to take as t h i s volume the so-called B r i l l o u i n zone of the r e c i p r o c a l l a t t i c e . The B r i l l o u i n zone may be defined as (12) the set of points which are closer to the origin (K - 0) of the reciprocal lattice than to any other la t t i c e site, with the con-dition that no two points k of the Brillouin zone may d i f f e r by a reciprocal lattice vector. This last condition imposes a re-striction only on points on the surface of the zone,, where they are equidistant between K = 0 and other reciprocal la t t i c e sites. By letting k vary over the entire Brillouin zone we obtain a l l the irreducible representations of the translation subgroup T C F. 2) We now show how to group the irreducible represen-tations ^ of T into orbits. If ^ and ^ t are in the same orbit, then according to the definition of "orbit" there exists an element (aIs.) of F^  such that d v , ( l | t ) = d. ((als)" 1(l\t)(a\s)) = d.dla" 1^) - expt-ik.a^t) k T k k (3-13) = exp(-iak't) for a l l t. But d k t ( l \ t ) - exp(-ik».t) for a l l t. Hence k* = ak. Furthermore, these two representations are not equivalent, which for one dimension means they are not equal. Therefore k 1 — ak zL k \ K for any reciprocal la t t i c e vector K. " ~ T3-14) We apply these conditions to a l l the k-vectors in the Brillouin zone, using a l l the rotations a in the point group P of F. If the order of the point group i s p, then for a general k-vector k there w i l l be (p-1) other k-vectors k« satisfying (3-14). The (13) set of k-vectors s a t i s f y i n g ( 3 - 1 4 ) , with each k* i n general r e-quiring a d i f f e r e n t rotation a of P, i s c a l l e d the "s t a r " of the k-vector k. A l l the representations o k of T whose k-vectors are i n the same star are, by d e f i n i t i o n , i n the same o r b i t . We must select one representative k-vector from each star i n the B r i l l o u i n zone. This w i l l complete step #2 of the general pro-cedure as applied to space groups. 3) We now show how to obtain the l i t t l e groups L(b^) for each representation of T which has been selected i n step #2). From the d e f i n i t i o n of l i t t l e groups, and equation ( 3 - 1 3 ) , the condition f o r an element (ais^of F to belong to L ( ^ ) i s that exp(-iak.t) •=- exp(-ik.t) f o r a l l t . Therefore there must exist some r e c i p r o c a l l a t t i c e vector K such that ak = k -v- K. Thus to construct the l i t t l e group L ( t f k ) , we select from the point group P those operators a^= e, a 2 , a^, a r such that a ^ = k -v K± , i = 1, 2 , ..., r k . ( 3-15) Then i t w i l l be seen that U*k) + ( a 2 l v(a 2))T / + (a3|v(a3))T + ... + ( a r | v ( a r ) )T . — • k k *- • I t s w i l l be shown i n section (c) that i n fac t i t i s not necessary to follow t h i s t h e o r e t i c a l p r e s c r i p t i o n to obtain a l l (14) the l i t t l e groups L( Xy.), because the work has already been carried out and appears i n the l i t e r a t u r e . , 4) We now show how one constructs a l l the allowable representations of the l i t t l e groups L( X k) obtained i n the preceding step. The condition f o r a representation ^ L of a l i t t l e group L( >jk) to be allowable implies i n t h i s context that those matrices,d^(l|t) representing pure translations are of the form d L ( l | t ) = exp(-ik-t)I, ( 3 - 1 6 ) where I i s the i d e n t i t y matrix of whatever dimension happens to be. We s h a l l impose condition ( 3 - 1 6 ) i n deriving the a l -lowable representations of L( X k ) . We follow the method of ray representations which i s discussed, f o r example, by Hurley ( 1 9 6 6 ) , and which we summarize below. Let P('k) be the point group of the l i t t l e group L( Y k ) . Suppose o = i d (ajv(a ) + t)] i s an allowable representation of £TVk)» Consider the set R(k) = |(a\v(a)) : a£P(k)j, and for con-venience we now write (a) = (a|v(a)) f o r a l l (a|v[a)) i n R(k). I f [aL±), (aj) are i n R(k), then ( a ± ) ( a j ) = (a ia J.|v(a i) 4 - a ^ a j ) ) = ( l \ v ( a ± ) +a.v(a.) -v( a i a . ) ) ( a . a . ) = (1 | t ( i , j)) (a-a.j, 1 J 1 ^ 1 3 ( 3 - 1 7 ) where t ( i , j ) = v U i ) -K a i v U j ) - v ( a j a j ) . Henc e d L(a i)d L(a J.) - d L ( l | t ( i , j ) ) d L ( a i a J ) = e x p ( - i k r t ( i , j ) ) d L ( a i a J . ) , (15) by equation ( 3 - 1 6 ) . Thus the representation X L of L f X ^ ) * when r e s t r i c t e d to the set R(k), may be regarded as a ray representa-t i o n of the point group P(lc), with the factor system A ( a i , a j ) = e x p ( - i k _ . t ( i , j ) ) . ( 3 - 1 9 ) Conversely, i t may be shown that any i r r e d u c i b l e ray representation = [ d < r ) ( a ) ] of P(k) with the factor system of equation ( 3 - 1 9 ) may be regarded as part of an allowable representation of the l i t t l e group L( X^) which may e a s i l y be extended to a l l of L ( ^ . ) . Also, i t i s shown that a l l the allowable representations of L( y k ) may be found i n t h i s way. To effect the extension of o^ r^ to a representation of L( , we l e t d L ( a l v ( a ) ) = d ( r ) ( a ) L ( 3 - 2 0 ) and d L(a|v(a) 4 - t ) = d L ( l | t ) d L ( a l v ( a ) ) = exp(-ik.t)d( r)(a) for a l l a i n PQT). This i s the basic idea behind the method of ray repre-sentations. As a check that a l l the i r r e d u c i b l e ray representa-tions of a point group P(k) with the given f a c t o r system have been found, one uses the f a c t that the sum of squares of the dimensions of a l l the i r r e d u c i b l e ray representations of a group, with a p a r t i c u l a r factor system, equals the order of the . group. For symmorphic space groups, i n which a l l the v ( a ) f s (16) are zero, the f a c t o r systems are a l l t r i v i a l , /l(a.,a.) — exp(-ik.O) —1 f o r a l l a±, a. i n P(k), f o r a l l k, and we can obtain the allowable representations X L o f L( X k ) d i r e c t l y from the ordinary i r r e d u c i b l e vector representations of P(k), by l e t t i n g d L(a|t) = exp(-ik-t)d(a) f o r a l l (alt) i n L( y k ) . (3-21) * = {d(a)] For representations of T i n which k i s i n the i n -t e r i o r of the B r i l l o u i n zone, the rotation operators a of the point group P(k) leave k invariant, due to t h e i r length preserv-ing property. Hence the f a c t o r system of equation (3-19) becomes /A( a i,aj) = exp(-ik» ( v f a ^ + a ^ U j ) - v(a ia J.))) exp(-ik»v(a.))exp(-iaTlk»v(a )) - : * - ± • (3-22) exp(-ik»v(a^aj)) From t h i s i t follows, by ray representation theory, that the so-c a l l e d gauge transformation d(a ±) = exp(ik_.v(a ) ) d ( r ) ( a i ) f o r a l l a± i n P ( k ) (3-23) associates the t r i v i a l factor system with the factor system of equation ( 3 - 1 9 ) , so that the matrices d{a^) form an i r r e d u c i b l e vector representation of P(k). Therefore, for those k-vectors i n the i n t e r i o r o f the B r i l l o u i n zone, we have dL(a±| v(a.) +• t) =• exp(ik.t)d^ r)(a.) = exp(-ik.t)exp(-ik.v(a i))d(a ±) (3-24) = exp(-ik* (v(a i) -r t ) ) d ( a i ) (17) which i s si m i l a r to equation (3-21) for symmorphic space groups, except that the non-primitive translations v(a^) also appear i n the exponent. I t i s only f o r non-symmorphic space groups and k-vectors on the surface of the B r i l l o u i n zone of the re c i p r o c a l l a t t i c e that one actually has to know the ray representations o f P(k) with the fac t o r system of ( 3 - 1 9 ) . These are available i n the l i t e r a t u r e , and i t w i l l be shown i n section (c) that even more si m p l i f i c a t i o n s occur f o r t h i s case. For a l l other situations, one can r e a d i l y obtain a l l the i r r e d u c i b l e vector representations of the various P(k)»s from the l i t e r a t u r e . 5) The f i n a l step of inducing representations of the space group f from the allowable representations of the various l i t t l e groups L( 2T^ ) i s straightforward. Assume the space group F'has been s p l i t into a number of cosets of a certa i n l i t t l e group F = (b^b^LT^) + (b2lv(b2))£TN?k) +- ... + (bn[v(b^ ))LT*k), where i d ^ l v d ^ ) ) = (UO0». Consider the element (als) of F. To determine the j column o f submatrices i n the induced supermatrix D(als), we consider the product ( a l s ) ( b i | v ( b j ) ) , and resolve i t into another product of the form ( b i l v{b^) ) (a'| s. f), where ( a M s . * ) belongs to £71^ )• This second product i s uniquely determined. Then i n the i ^ h r o w 0 f submatrices i n the j t n columnj we inse r t (id) the submatrix d L ( a , ) s t ) from the given allowable representation tfL = {d L(aMs')j of LTV^). A l l other submatrices i n the j t h column are zero. From each allowable representation o f each l i t t l e group L( Xk) we induce an i r r e d u c i b l e representation of'F'. (19) (c) Applications to the Space Group We s h a l l now apply the r e s u l t s of the l a s t section to the space group D^. We s h a l l show steps #4 and #5 i n d e t a i l only f o r two l i t t l e groups L( ^^)» 1) In section (b) the form of the i r r e d u c i b l e repre-sentations X of the t r a n s l a t i o n subgroup T C D ^ was determined, so i t i s only necessary to f i n d the B r i l l o u i n zone B f o r the r e c i p r o c a l l a t t i c e of D^. For the simple orthorhombic l a t t i c e D 2 h ' t h e ^ a s^- c l a t t i c e vectors a^, a 2 , a^ are mutually perpen-d i c u l a r , and so likewise are the basic r e c i p r o c a l l a t t i c e vec-tors b-^ , bg, by From t h i s i t follows that the B r i l l o u i n zone B of t h i s r e c i p r o c a l l a t t i c e i s , (3-26) B = | k = f - J ^ +- f 2 b 2 +• f ^  : - J < f ^ i , i = 1, 2, 3 j . Thus we obtain each inequivalent i r r e d u c i b l e representation Y^— • d ^ t l l t ) ^ . of T, where d k ( l | t ) = exp(-ik»t) , exactly once, by l e t t i n g >k vary over a l l the points of the B r i l l o u i n zone B. 2) For the space group Dl£, i t i s r e a d i l y seen that 2h the set of k-vectors C = [ k = f ^ 4 - f 2 b 2 4- f 3b_ 3 : O i f ^ J , i = 1, 2, 3 ] (3-27) i s a maximal set of k-vectors no two of which correspond to rep-resentations and of T'belonging to the same o r b i t . That t h i s i s so may be seen by inspection, but i t may also be seen from the International Tables for X-Ray Crystallography (1953), as the discussion i n part (3) of t h i s section w i l l i ndicate. (20) . —' We therefore select only those representations of T f o r which k belongs to C. 3) In order to f i n d the l i t t l e group L( X^ ) correspond ing to a p a r t i c u l a r k-vector k i n C, i n p r i n c i p l e one must operate on k with a l l the rotations a i n the point group P , and select each a^ that s a t i s f i e s a i k = k + K ± , i = 1, 2, 3, . . . , r k (3-15) for some r e c i p r o c a l l a t t i c e vector K^. However, i t i s not neces-sary to go through t h i s lengthy procedure i f one makes use of cer t a i n information contained i n the International Tables for>X-Ray Crystallography, i n which are l i s t e d for each gpace group F the so-called "coordinates of equivalent p o s i t i o n s " which we now define. Consider the Wigner-Seitz unit c e l l U of the l a t t i c e , which i s defined just as the B r i l l o u i n zone: i s defined, except with d i r e c t l a t t i c e t ranslations t replacing r e c i p r o c a l l a t t i c e t r a n s lations K. For a general point r = (x, y, z) i n U with no spec i a l symmetry the coordinates of the equivalent positions are the coordinates of a l l those points r ' obtained by operating on r with a l l the elements of the form (alv(a)) i n the space group. However, i f for some r i n U i t i s found that some (a|v(a)) of F produces a vector ar + v(a) d i f f e r i n g from r by a primitive t r a n s l a t i o n t , ar + v(a) = r +• t , ( 3-28) (21) t h e n r i s a s o - c a l l e d " s p e c i a l p o i n t " . T h e n , i n t h e c o o r d i n a t e s o f e q u i v a l e n t p o s i t i o n s f o r t h i s r a r e l i s t e d o n l y t h o s e p o i n t s a r + v ( a ) w h i c h do n o t d i f f e r f r o m r o r f r o m each o t h e r b y a p r i m i t i v e l a t t i c e t r a n s l a t i o n t . F o r each s p a c e g r o u p , t h e c o o r d i n a t e s o f e q u i v a l e n t p o s i t i o n s o f a g e n e r a l p o i n t and o f e a c h s p e c i a l p o i n t a r e p r e -s e n t e d . The l i s t s a r e s u c h t h a t e v e r y p o i n t i n t h e u n i t c e l l U a p p e a r s , e i t h e r as a p o i n t r on w h i c h t h e e l e m e n t s ( a | v ( a ) ) o p e r a t e , o r as a p o i n t r * = a r -+-v(a) g o t f r o m r by t h e o p e r a -t i o n o f ( a | v ( a ) ) . F o r e x a m p l e , i n t h e c a s e o f t h e s p a c e g r o u p ^ i , , t h e p o i n t s r a r e r e s t r i c t e d t o h a v i n g n o n - n e g a t i v e c o o r d i n a t e s , 2h ~~ b e c a u s e any o t h e r p o i n t i n U may be g o t f r o m any o f t h e s e p o i n t s u s i n g a s u i t a b l e r o t a t i o n (a|0) i n D 2 h . A l s o , f o r a g i v e n r and l i s t o f e q u i v a l e n t p o s i t i o n s , t h e s e t o f e l e m e n t s (a ls . ) o f F s u c h t h a t a r -y v ( a ) = r T t f o r some t f o r m a g r o u p . B e s i d e e a c h l i s t o f t h e c o o r d i n a t e s o f e q u i v a l e n t p o s i t i o n s i n t h e I n t e r n a -t i o n a l T a b l e s i s t h e i n t e r n a t i o n a l s y m b o l f o r t h i s g r o u p . I f one i g n o r e s t h e t e r m v ( a ) t h e n e q u a t i o n (3-2S) becomes v e r y s i m i l a r t o e q u a t i o n ( $ w h i c h i s t h e e q u a t i o n u s e d i n d e t e r m i n i n g t h e r o t a t i o n o p e r a t o r s i n t h e e l e m e n t s o f t h e l i t t l e g r o u p s L ( "tf^). L e t us now c o n s i d e r t h e symmorphic space g r o u p h a v i n g t h e same p o i n t g r o u p as DJIjf?' and t h e same l a t t i c e as t h e r e c i p r o c a l l a t t i c e o f D ^ . T h a t i s t h e s p a c e 1 g r o u p D 2 ^ . From t h e p r e v i o u s d i s c u s s i o n i t f o l l o w s t h a t t h e (22) l i s t of the coordinates of equivalent positions and the corres-ponding groups f o r each set o f positions f o r may be regarded as a complete l i s t of k-vectors i n the B r i l l o u i n zone, c l a s s i f i e d into stars, and with the l i t t l e group L( b^) stated for each star. Using t h i s information on D^, and regarding the groups i n the l i s t as l i t t l e groups and the points as k-vectors, we are able now to l i s t i n Table ( 3 - D a l l the k-vectors i n the volume C, along with t h e i r l i t t l e groups L( Y k ) . We also include the Wyckoff symbol for each k-vector. This completes step #3 of the general procedure as applied to D^h* Table ( 3 - D ( 0<f f<£ , i = 1 , 2 , 3) Case # Wyckoff k-vector L( X ^ . ) symbol • - ~ 1 2 3 4 5 6 7 a a b c d e f g h 0 ibx + ib_3 ( 2 3 ) Table ( 3 - D (continued) ( 0 < f i < i , i = l , 2 , 3 ) Case # Wyckoff k-vector ^ ^ k ^ symbol -*  9 i 1 0 5 1 1 k flt>l + ib2 1 2 1 1 3 m f 2 * 2 1 4 n f 2 b 2 ^ 3 1 5 0 J t ^ fbg 1 6 P Jb^ 4 - f 2bg 4 - ib_ 3 1 7 q f 3 * 3 Id r 1 9 s 2 0 t 2 1 u f 2 * 2 + f 3 ^ 3 2 2 V % + f 2 ^ 2 + * £ 3 2 3 w 2 4 X W i b - 2 + fjbj 2 5 y f 1*1 + f 2 * 2 2 6 z ' A * '2*2**3 2 7 c< f i ^ i + f 2 * 2 v f 3 * 3 l ' i ( H t ) , ( 2 x | s ) , (m \s), (mzl.s)] i [ ( H t ) , ( 2 y | s ) , (m x)s), (mz\s)] { ( H t ) , ( 2 z \ s ) , (m x\s), (m )s)]. • [ ( l i t ) , (m x|s)^ { ( H t ) , (m y|s)^ • { ( l i t ) , (m z\s)| [ ( U t ) ] (24) 4) We now show the details of the derivation of the allowable representations of the l i t t l e groups L( K^ ), for each k listed in Table (3-1). We show the calculations only for two k-vectors, k = £b. and k •= f.b, + kb~. (i) k = %by We see from Table (3-D that U*k) = D^. Thus we w i l l not have to carry out step # 5 of the general pro-cedure. According to section (b), step #4, we should now compute the factor system /\ ( i , j ) for a l l i , j , which entails knowledge of a l l the vectors t ( i , j ) . However, i f we compute the multipli-cation rules between the elements (a|v(a)) of which correspond to the generating elements a of the point group P(k) — these w i l l be sufficient to specify the entire factor system completely. The generating elements of D 2 h we shall take to be mx, niy, mz, and we now write down the necessary relations for the correspond-ing elements of D^, using the notation (a) = (a|v(a)), and equa-tion (2-6). (m x) 2 = U l a g + a 3 ) (m y) 2 = (1 \ 0 ) (m z) 2 = (mx)(my) = (l\a 2)(m y)(m x) = (l\a 2)(2 a) (3-29) (my)(mz) =: (n^Jdry) = (2 X) (•ma)(mx) =1 ( 1 ^ - a 3)(m x)^m z) z= ( l ^ a 1 ) ( 2 y ) (mx)(my)(mz) = [1\a2 +• a 3) (I) Now assume Y - ] d (a|s)( i s an allowable representa-(25) tion of L( X k) for k = |b_3, and note that restricted to the set R(k) forms an irreducible ray representation o^1^ = ^d^r^(a)| of the point group P ( k ) = D 2 h » From equations (3-16) and (3-29) we get d(r)2(mx) - exp(-ik. (ag + a_3))I = -I d( r ) 2 (my) ^ I d ( r ) 2 (mz) = exp(-ik-3^)1 = I d(r)(m )d ( r ) (m y ) = exp(- ik .a^)Id ( r }(ny )d ( r )(n^) = d(r)(m )d(r)(m ) = exp(-ik.a 0 )Id( r )(2>) y x "2 z - d(**)(2 ) z d(r^(m )d(r>(m ) = d(r)(m )d(r)(m ) - d ( r ) ( 2 ) (3-30) y z z y x d(r)(m )d(r)'cfcm ) = exp(-ik-(a- - a j ) l d ( r ) ( m )d<r)(m ) ~ ~3 x z - -d( r )(m )d( r )(m ) - exp(-ik.a.,)ld( r )(2 ) x z 1 y = d<r>(2 ) y _ d<r)(m )d(r^(m )d(r>(m ) - exp(-ik.(a 0 + aj) ld< r ) ( l ) x y z ~ ~2 —3 - -d(r ) ( l ) . Equations (3-30) specify the factor system of X^ r^ completely, so that i t is not necessary to compile a table of a l l the factors A ( i>j) . It remains now to find the irreducible ray representations which satisfy (3-30). These are available in the literature. The tables used for this thesis are those at the end of Hurley*s article. Those parts of the tables of ray representa-tions in Hurley*s article that were used are reproduced in the appendix. (26) It w i l l be seen that equations ( 3 - 3 0 ) correspond to Table A - 4 o f Hurley , with the f actors <X = 1 , 6 = 1 , where <*,/3 , ^are defined i n Table A - 4 . In the f i r s t of equations ( 3 - 3 0 ) , we have a minus sign before the i d e n t i t y matrix I , which means we must in ser t a factor o f t i i n the matrix A which we s h a l l use as d ( r ) ( m x ) . Then, us ing ; ( 3 - 3 0 ) , we have > f l 0 d ( r ) ( i ) = i -\ 0 1 t1 °\ d ( r ) ( m x ) = - i A -\ 0 i / d w ( m y , , B = (; i (0 l' d(r)(m ) - c - \ z \1 0 / d ( r > ( 2 j r - i A B = f"1 ° \ 0 1 z d(**)(2j = BC = \ l 0 / x / \ , 0 i . (2 ) = - i C A = y d ( r )(T) - iABC -\ - i 0 0 i \ i - i 0 / which i s an i r r e d u c i b l e ray representat ion of D 2 h having the de-s i r e d fac tor system. Note t h a t , as mentioned i n Hurley 's t a b l e , (27) the signs of the matrices A, B, C may be changed a r b i t r a r i l y , to get a l l other i r r e d u c i b l e ray representations o f D,,^  with the same factor system. By changing the sign of d^ r^(m y) = B, we get a d i f f e r e n t set of characters, and hence an i r r e d u c i b l e ray representation inequivalent to the f i r s t one. Since the order of T^2h i s ^ d since the sum of squares of dimensions of these two representations i s # ( = 2 2 4- 2 2 ) , we have found a l l the i r r e -ducible ray representations of D 2 h with the factor system of equation (3-19). We now extend these representations to two representa-t i o n s , tfjlUJd^Ula)] a n d * ] ! 2 ) - | d ( 2 ) ( a ) s ) ] , of U\) « Dl£, according to equations (3-20). We then obtain, with k s- ^ b^ and s — t +- v(a), d L ^ U I t ) = exp(-ik.t) ^ d L 1 ) ( 2 x | s ) - exp(-ik't) ^ (1) 1° d L ( 2 y l s ) - exp(-ik.t)( d ( D ( 2 I a) = exp(-ik.t) ( L z \ d ^ ^ l l t ) = exp(-ik.t) di 1^(m \g) exp(-ik.t) L x1;— -1 0 and ( 2 6 ) d ^ d n |s) = e x p ( - i k . t ) 3^  d j D f m i s ) - e x p ( - i k » t ) L Z d T ( 2 ) ( l | t ) = e x p ( - i k » t ) d | ) 2 , ( 2 x | s ) = e x p ( - i k . t ) d L2 ) ( 2 z | s ) -= e x p ( - i k . t ) d T ( 2 ) ( l | t ) - e x p ( - i k . t ) d j 2 ^ ( m | s ) - e x p ( - i k . t ) l i x d T ( 2 ) ( m I s ) - e x p ( - i k . t ) IJ y <*l2Umz\s) - e x p ( - i k ' t ) a 0 lO 1, '0 1\ ,1 0/ '1 0^ ,0 ll /O -1^ \ - l Oj ( - i 0, (i T \0 -It C 1) \o J r-1 0' \0 -1, '0 1 > a o, T h u s , f o r k = £by we h a v e o b t a i n e d a l l t h e a l l o w a b l e r e p r e s e n t a t i o n s o f L( X ) - D ^ . k 2 h (29) We now express these representations more concisely, by means of Table (3-2). This type of table w i l l be used through-out the remainder of t h i s thesis to show representations, so we s h a l l explain i t here i n some d e t a i l . At the top of the table the k-vector associated with these representations i s s p e c i f i e d , i n Wyckoff and vector notation, and the group being represented i s stated. The f i r s t column of the table l i s t s the elements (al,s) of t h i s group. The second column ( i j ) gives the row and column num-bers of the non-zero matrix entries of the matrix d ( 1 ) ( a | s ) representing ;each adjacent element (a|s.) in a p a r t i c u l a r repre-sentation tf^. Beside each i j pair i s the corresponding matrix entry d j ^ ( a | s _ ) , i n the t h i r d column of the table. Each additional column, la b e l l e d tf(2), V ^ ) , etc., indicates that there i s a second, or t h i r d , etc., inequivalent representation, and each entry i n these columns i s a m u l t i p l i c a t i v e f a c t o r . I f a fac t o r from the column i s multiplied with the matrix entries d ( l ) ( a l s ) to the l e f t , the products are the matrix entries d(^)(a|sj i j "~ i j for the representation Y ^ r ^ . Note that each factor i s associated with a group element (als.), rather than with the adjacent matrix entry, and i s to multiply the entire matrix d ^ ^ ( a | s ) . It turns out that the only factors needed a r e i l . I t i s never necessary to specify new i j numbers for d i f f e r e n t representations associated with the same k-vector. (30) The above explanation has the following exception. In tables (5-13) to (5-18), each m u l t i p l i c a t i v e factor 11 i n the columns l a b e l l e d p(r) i s intended to multiply not the entire matrix D ^ f a l . s ) , but merely the non-zero matrix entry D [ V(ais) ^ 3 to the l e f t , i n the same row as the fac t o r . In some tables, expressions of the form expf-iak/s.) w i l l occur, where a i s a rotation i n the point group of D2h* ^° e-^ m'" inate any possible confusion, we now indicate e x p l i c i t l y the effe c t of each such rotation on a general k-vector k = f ^ + f 2b_ 2 4- f^y l k — k 2x* — - f £ 3 2 k y- - f l * l + f 2 b 2 - f 3^3 2z*- - f l * l " f 2 b - 2 + f lk = -k m x ^ — + f 2 b 2 + f — f l * l - f 2 b 2 4 • f3^3 m k z— f l * l + f 2^2 " f 3 b 3 ( 3 D Table (c) k iH3 L( X,) = D * (a|s) d T ( 1 ) ( a l s ) Li.1 (1 t) 11 22 exp(-ik»t) exp(-ik» t) 1 ( 2 X s) 12 21 exp(-ik. t) exp(-ik> t) -1 ( 2 y s) 12 21 iexp(-ik»t) -iexp(-ik> t) 1 (2 Z s) 11 22 -iexp(-ik»t) iexp(-ik.t) -1 (1 t) 12 21 iexp(-ik»t) -iexp(-ik»t) -1 (mx s) 11 22 -iexp(-ik»t) iexp(-ik»t) 1 11 22 exp(-Ik»t) exp(-ik»t) -1 (raz s) 12 21 exp(-ik«t) exp(-ik«t) 1 (32) ( i i ) k = fjbi + |b_3. We see from Table ( 3 - D that L( X k ) = { ( l i t ) , ( 2 x \ s ) , (m y|s), ( m z l s ^ . The generators o f P(k) = j l , 2 X , my, m^ are 2 X and my, and the m u l t i p l i c a t i o n r e l a t i o n s that determine the factor system of the type i n equation (3-19) f o r the ray representations of P(k) are ( 2 x ) 2 = ( i l ^ ) (my) 2^ ( 1 1 0 ) ( 3 - 3 D (2 )(m ) = (m )(2 ) = (m ). x y y x z Now i f ^ ^ ^ ( a l s . ) ! i s an allowable representation of L( X k ) which, when r e s t r i c t e d to the set R ( k ) , forms a ray repre-sentation = | d^ r^ (a)| of P ( k ) , then just as i n case (i) we have (r)2, d ( r ) 2 ( m y ) = I (3-32) d ^ ) ( 2 x ) d ( r ) ( m y ) = d ^ ) ( m y ) d ( r ) ( 2 x ) = d ( r > ( m z ) . d [ r ) d { 2 ) = exp(-ik«a )I = e x p ( - i 2 T i f x ) I Since the generators commute, we use the ordinary vec-to r representations of L("x^), which are one-dimensional. Because of the f i r s t o f equations (3-32) we must in s e r t a factor of expt-iTTfx) i n the numbers representing 2 x and mz. The four i r r e d u c i b l e ray representations of P(k) with the desired factor system are l i s t e d i n Table ( 3 - 3 ) . (33) Table (3-3) (j) k - f ^ +• i b 3 P (k) = j l , ' 2 X , n y mz] a i j d^) ((a) X ( r ) ( 2 ) * ( r ) ( 3 ) ^ ( r ) ( 4 ) 1 n l 1 1 1 2 x n exp(-in f ) -1 1 - 1 " y n 1 1 - 1 -1 i i e x p ( - i T i f 1 ) -1 - 1 1 We now extend these representations to obtain four allowable representations of L( X k)« We rewrite the exponent - i r ? f ^ i n terms of k.v ( 2 x ) and k.v(m2) where appropriate. The four allowable representations of L( tfk) are shown i n Table ( 3 - 4 ) . Table ( 3 -4 ) (j) k = f ^  -V- ib_ 3 LT? k) = | ( l \ t ) , ( 2 x \ s ) , (m \ a ) , K\s)^ (a|s) i j d[!)(als) y [ 2 ) * [ 3 ) y ( 4 ) ( l i t ) 11 exp(-ik»t) 1 1 1 ( 2 x|s) 11 iexp(-ik«s) - 1 1 -1 (my\s) 11 exp(-ik«s.) 1 - 1 -1 (mz\s) 11 iexp(-ik«s_) - 1 - 1 1 From these representations w i l l be induced four i r r e -ducible representations of D^, i n step #5 of the general pro-cedure. (34) 5) We now show how four i r r e d u c i b l e representations of are induced from the four allowable representations u^ 1) of L( y ) f o r k = f b + £b_. F i r s t we s p l i t D££ into — 3 ^n l e f t cosets o f L( Ky.) as follows: D 2 h = f ^ k ) + (iifi)iT^) . Then the defining equations f o r the matrices D(a)s) of the i n -duced representations are ( l l t ) ( l l O ) - ( l | 0 ) ( l l t ) ( l | t ) ( l | 0 ) = ( l \ 0 ) ( l | l t ) ( 2 x | s ) ( l | 0 . J = ( l | 0 ) ( 2 x l s ) ( 2 x l s ) ( l | 0 ) = (I|0)(2 x|lsi (fflyls)(llO) = (l)p_)(m y|s) (mylsidiO) ='(l|0)(my|l£) (m z ls)(l|0) = (l)0)(m z l s ) (m zIs)(llO) = (l)0)(m z|ls) ( l i t ) ( 1 | 0 ) = (1|0)(l|It) ( l | t ) ( l | 0 ) - (110)(l|t) (mxIsJdlO) = ( l | 0 ) ( 2 x | l s ) (mx| s)(l\0) = ( l ) 0 ) ( 2 x \ s ) ( 2 y | s ) ( l l 0 ) = (TiOMm^Ls) ( 2 y l s ) ( l l 0 ) =• (110)(m | s ) (35) ( 2 z | s ) ( l | & ) = ( l | 0 ) ( m z \ l s ) ( 2 z|s)(110) = (HO)(m z|s) . The r e s u l t i n g representations of Dl6 are shown i n Table 2h ( 3 - 5 ) « In t h i s table and i n several others i n l a t e r parts of t h i s t h e s i s , we have taken advantage of the fact that for a l l rotations a i n the point group of D^, a =• a""'", k*at — a~^k»t =• ak*t, and k»as = a~^k»a = akvs. ( 3 6 ) Table ( 3 - 5 ) (j) k = f ^ +• Jb D 16 2 h (a Is) D j l (a|s) P(2) p ( 3 ) p(4) ( l i t ) 1 1 2 2 e x p ( - i k . t ) e x p ( - i l k . t ) 1 1 1 (2 xls) 1 1 2 2 i e x p ( - i k . s.) i e x p ( - i l k . s ) - 1 1 - 1 (2yU) 12 2 1 e x p ( - i k » s j exp( -ilk»s_) 1 - 1 - 1 (2 zl3) 12 2 1 i e x p ( - i k » s ) i exp ( - i l k s_) - 1 - 1 1 ( l i t ) 12 2 1 e x p ( - i k » t ) e x p ( - i l k * t ) 1 1 1 (mx\s) 12 2 1 iexp(- ik/s_) i e x p ( - i l k » s.) - 1 1 - 1 (my|s) 11 2 2 exp(-ik»js) e x p ( - i l k » s ) 1 - 1 - 1 (mz|s) 11 22 iexp(-ik«s_) iexp(-i lk»s_) - 1 - 1 1 Section 4 Allowable Representations of the L i t t l e Groups L( Table (4-D (a) k = 0 L( 7 j k ) = D: 16 2h (a|s) i j ( D d L (ajs) x , 2 ) x<3) (5) Y (7) Y (lit) 11 1 1 1 i 1 i I i (2 x |s) 11 1 -1 1 - i 1 - i I - i 11 1 1 -1 - i 1 i - l - i (2 z |s) 11 1 -1 -1 i 1 - i - l i (lit) 11 1 1 1 i -1 - i - l - i K l s ) 11 1 -1 1 - i -1 i - l i (niyls) 11 1 1 -1 - i -1 - i I i (mz\s) 11 1 -1 -1 i -1 i I - i (37) (38) Table (4-2) (b) k = ^ L ( t f k ) = - D ; (1) v ( 2 ) (als) i j d (a|s) X L i j ( l i t ) 11 exp(-•ik.t) 1 22 exp(--ik«t) (2 Is) X 12 -iexpi -ik«t) -1 21 -iexp -ik«t) 12 -iexpi -ik/t) 1 21 iexpi -ik/t) 11 expi -ik/t) -1 22 -expi -ik/t) ( l i t ) 12 -iexpi -ik/t) -1 21 iexpi -ik/t) (mx\s) 11 expi '-ik/t) 1 22 -exp [-ik-tl. (my\s) 11 exp [-ik/t) -1 22 exp [*ik/t) (mz\s) 12 -iexp ,-ik/t) 1 21 -iexp [-ik/t) (39) Table (4-3) (c) k = Jb L( (als) i j (1) dT (als) * ( 2 ) (l|t) 11 22 exp (-ik/t) exp (-ik/t) i: (2 x ls) 12 21 exp (-ik/t) exp (-ik/t) - l (2 y \s) 12 21 iexp (-ik/t) -iexp(-ik/t) l 11 22 -iexp( -ik/t) iexp (-ik/t) - l (lit) 12 21 iexp (-ik/t) -iexp( -ik/t) - l (nixls) 11 22 -iexp(-ik/t) iexp (-ik/t) l (my|s) 11 22 exp (-ik/t) exp (-ik/t) - l <»,U> 12 21 exp (-ik/t) exp (-ik/t) l (40) Table ( 4 r 4 ) (d) k - i b x + £b_3 L( tfk) = (1) % v ( 2 ) (3) ~ (4) v ( 5 ) (6) ^ 7 ) ( (a|s) i j d T (als) * X ^ S 0 I ( l i t ) 11 exp(-ik»t) 1 1 1 1 1 1 1 ( 2 x l s ) 11 iexp(-ik»t) - 1 1 - 1 1 - 1 1 - 1 ( 2 y)s) 11 exp(-ik»t) 1 - 1 - 1 1 1 - 1 - 1 ( 2 z|s) 11 iexp(-ik»t) - 1 - 1 1 1 - 1 - 1 1 ( l i t ) 11 exp(-ik»t) 1 1 1 - 1 - 1 - 1 -1 (m xls) 11 iexp(-ik»t) - 1 1 - 1 - 1 1 - 1 1 (•"ylsi) 11 exp(-ik»t) 1 - 1 - 1 - 1 - 1 1 1 (m z)s) 11 iexp(-ik-t) - 1 -1 1 - 1 1 1 - 1 (41) Table (4-5) (e) k - |b_2 L(Yk) - D; (als) i j (1) d_ (als) Li.1 (2) X ( l i t ) 11 22 exp(-ik.t) exp(-ik.t) 1 (2 xJs) 12 21 exp(-ik.t) exp(-ik.t) -1 (2 y|s) 11 22 -iexp(-ik«t) iexp(-ik.t) -1 12 21 iexp(-ik.t) -iexp(-ik»t) 1 ( l i t ) 12 21 iexp(-ik»t) -iexp(-ik.t) -1 (m xJs) 11 22 -iexp(-ik.t) iexp(-ik«t) 1 (myls) 12 21 exp(-ik»t) exp(-ik.t) 1 (n>2U) 11 22 exp(-ik.t) exp(-ik.t) -1 ( 4 2 ) Table ( 4 - 6 ) (f) - k = i b ^ + £b 2 L ( Y k ) - Dj (a|s) ( 1 ) J2) d T (a|s) L i . 1 ( l i t ) 1 1 exp(-ik»t) 1 2 2 exp(-ik»t) 1 1 -iexp(-ik»t) - 1 2 2 -iexp(-ik»t) ( 2 y | s ) 1 2 exp(-ik»t) 1 2 1 -exp(-ik«t) ( 2 z | s ) 1 2 -iexp(-Ik?£) - 1 2 1 iexp^-ik»t) U | t ) 1 2 exp(-ik»t) - 1 2 1 exp(-ik«t) (mx|s) 1 2 -iexp(-ik»t) 1 2 1 -iexp(-ik»t) 1 1 exp(-ik»t) - 1 2 2 -exp(-ik»t) (mz|s) 1 1 -iexp(-ik*t) 1 2 2 iexp(-ik«t) (43) Table (4-7) (g) k = £b_2 + Jb. L( = D; (als) (1) L i . i als) (2) (l|t) 11 expi - i k / t ) 1 22 expi - i k / t ) (2 x ls) 11 expi - i k / t ) -1 22 expi - i k . t ) (2 y |s) 12 expi - i k / t ) 1 21 -expi - i k . t ) (2 Is) z 12 expi - i k / t ) -1 21 -expi - i k / t ) (lit) 12 expi - i k - t ) -1 21 expi - i k . t ) (nys) 12 expi - i k . t ) 1 21 expi - i k . t ) (myU) 11 expi - i k / t ) -1 22 -exp1 - i k . t ) (m \s) z 11 exp' - i k / t ) 1 22 -expi -ik«t) (44) Table (4-3) (h) k = -h £b_2 + i b L( - D _ nl6 2h (als) (1; i j d L L i . .(a|s) ( l i t ) 11 exp - i k - t ) 1 22 exp1 - i k - t ) (2 |s) X 12 -iexpi -ik»t) -1 21 -iexpi -ik«t) ( 2 y l s ) 11 iexpi -ik»t) -1 22 -iexpi - i k ' t ) 12 -expl -ik«t) 1 21 expl - i k ' t ) ( l i t ) 12 -iexpi -ik«t) -1 21 iexpi - i k / t ) (mx|s) 11 expi -ik»t) 1 22 -expl -ik»t) (myls) 12 expl -ik«t) 1 21 expi - i k / t ) (mz\s) 11 -iexpi - i k ' t ) -1 22 -iexpi - i k ' t ) (45) Table (4-9) (I) k - f ^ L( k) = {(1/t), (2 xls), (myls), (razl (a»s) (1) dLi.(a|s) (2) V 3 ) (lit) 11 exp(-ik»t) 1 1 i (2 x|s) 11 exp(-ik»s.) -1 1 - i (myls) 11 exp(-ik«s_) 1 -1 - i 11 exp(-ik »jj) -1 =1 i 4-10) (j) k -: + £b_3 L( * k ) = | (Ht), (2x\s) (als) i j d L i.(ais) (lit) 11 exp(-ik*t) 1 i 1 (2x\s) 11 iexp( -ik» s.) -1 _ i -1 (my|s) 11 exp(-ik>s) 1 - i -1 ( m a l s) 11 iexp(-ik_.s_) -1 - i 1 Table (4-11) (k) k = f b + £b u Y ) = ) ( l | t ) , (2 \ s ) , ( m j s ) , (1) v (als) i j dL. .(a|s) 6 (2) ,(3) (4) (lit) 11 exp (-ik/t) 1 1 1 (2 Is) X 11 iexp( -ik» s_) -1 1 -1 (m Is) y 11 iexp( -ik»s.) 1 -1 -1 (mzjs) 11 exp(-ik»s) -1 -1 1 (46) Table (4-12) ( 1 ) k = 4- £ b _ 2 + - £ b 3 L ( V k ) = ( ( l i t ) , ( 2 X ) s ) , m \ s ) , ( r a z \ (1) " (2) (3) U ) (a|s) d L i 1 ( a i s ) ( l i t ) 11 exp (-ik/t) 1 1 1 (2 x|s) 11 -exp(-ik/s_) 1 -1 1 (myU) 11 iexp( -ik/^) 1 -1 -1 (mzU) 11 iexp( -ik/s.) -1 -1 1 Table (4-13) ^ (m) k = f 2 b 2 L( \ ) = [ ( l i t ) , ( 2 y | s ) , (mx\ s ) , (raz\s)} (a|s) i j (1) dT (a|s) V (2) (3) .(4) ( l i t ) 11 exp(-ik^t) 1 1 1 (2y|s) 11 exp(-ik.s.) -1 1 -1 (m^ l s) 11 exp(-ik«s.) 1 -1 -1 (raz| s) 11 exp(-ik»s) -1 -1 1 (47) Table (4-14) (n) k = + J b 3 L(Yk)= { ( H t ) , ( 2 | 4 ) , ( m x | s ) , (ngs.)} ( a la.) (1) d L i i ( a | s ) ( l i t ) 11 e x p ( - i k * t ) 22 e x p ( - i k » t ) 12 exp( - i k « s j 21 exp(- ik»s.) (mxU) 11 e x p ( - i k « s ) 22 - e x p ( - i k » s ) 12 e x p ( - i k * s ) 21 - e x p ( - i k •_§.) Table (4-15) (o) * = + - f 2 b 2 ; ; L ( Y k ) = [ ( H t ) , ( 2 Is .) , ( m x \ s ) , ( m ^ s ) • (1) ( a l s ) i j d L . . ( a l s ) ( l i t ) 11 exp(-ik«»t) 22 e x p ( - i k » t ) ( 2 y l s ) 12 exp(-ik»s_) 21 e x p ( - i k » s ) ( m x | s ) 11 i e x p ( - i k . s ) 22 - i e x p ( - i k » s _ ) ( m z | s ) 12 iexp(-ik»is) 21 - i e x p ( - i k » s _ ) (4B) Table (4-16) (p) k - ^ + f 2 b 2 +. Jb 3 LTT^ ) = j(l\t), ( 2 y \ s ) , (m x\s), (m z\s)] (1) 12) (3) (4) (als) i j dT (als) Li.i Y ( H t ) 11 exp (-ik/t) 1 I I (2 |s) 11 exp(-ik/s_) -1 I - l (raxis) 11 iexp(-ik/s_) 1 - l - l (mz\s) 11 iexp( -ik#s_) -1 - l i Table (4-17) ^ (q) k r f ^ L( X k ) ^ d \ t ) , ( 2 z \ s ) , (m x\s), {m\s)\ (1) (2) (3) (4) (als) i j c l L i . ( a s ) ( l \ t ) 11 exp(-ik/t) 1 1 1 11 exp(-ik*sj -1 1 -1 (mxU) 11 exp(-ik/s) 1 -1 -1 11 exp(-ik/sj -1 -1 1 (49) Table (4-13) ( r ) k = i b 2 + f 3 b 3 LU, k) = { U l t ) , ( 2 B \ s . ) , i f o n j s . ) , (my| (1) (als) i j d L (als) , , ( l i t ) 11 exp(-ik»t) 22 exp(-ik«t) (2.1ft) 12 exp(-ik»s) 21 exp(-ik»s) (mx|s) 11 -exp(-ik*s_) 22 exp(-ik*s.) (m yls) 12 -exp(-ik*s_) 21 exp(-ik»s) Table (4-19) (s) & ^^b-L LT?^) = { ( H t ) , ( 2 z l s ) , (mx\s)=, ( .(2) 3) (4) (a Js) i j dL^.Uls) 8 (lit) 11 exp(-ik«t) 1 1 1 <22 Is) 11 iexp(-ik»s.) -1 1 -1 (mx|s) 11 iexp(-ik»s[) 1 -1 -1 (my|s) 11 exp(-ik»s_) -1 -1 1 (50) Table (4-20) r It) k =r i b x -h ^b 2 +• f j o 3 L ( X ^ ) = { ( l l t ) , (22\s.), (m xls), (1) (a|s) i j d L i : i ( a | s ) (l|t) 11 exp(-ik.t) 22 exp(-ik.t) (2 z|s) 12 i e x p ( - i k . s ) 21 iexp(-ik.s) (%|s) 11 i e x p ( - i k . s ) 22 -ie x p ( - i k . j s ) (my |s) 12 exp(-ik.s_) 21 -exp(-ik . s i ) Table (4-21) -2 + f 3^3 L( K k) = {(Ht), (n^s)} (als) (1) i j d T (a|s) L i . i (Ht) 11 exp(-ik»t) 1 (mx|s) 11 exp( - i k . s.) -1 (my\s Table (4-22) . - f (v) k = 4 - f 2 b 2 + f j o 3 L( tfk)= | ( l i t ) , (m xIs)| (a |s) i j (1) d L.,(a\s) (2) (lit) 11 exp(-ik»t) 1 (mx\s) 11 iexp(-ik»sj -1 (51) Table (4-23) ^ r lw ) -k = f ^ +• L ( ^ k) - |(Ht), (my\s) (als) i j d L. .(als) V (lit) 11 exp (-ik/t) 1 (my\s) 11 exp(-ik/s) -1 Table (4-24) ~ f (x) k = + |b 2 + fjbj L( V k) = { ( l i t ) , (m.| (1)" % J 2 ) (als) i j d L . .(als) X (lit) 11 exp (-ik/t) 1 (my|s) 11 iexp( -ik/sj -1 Table (4-25) — - C ") (y) k =, +- f 2 b 2 L(X k)= | (Ht), (mz\s)i X2) (a\s) i j d L. .(als) ( l i t ) 11 exp (-ik/t) l (mz\s) 11 exp(-ik/s.) - l (52) Table (4-26) r -\ (z) k - +- f 2 b 2 + ^ b 3 L( tfk) = \ ( l | t ) , (mz\ s ) | ( I T (2) (a\s) i j ( l | t ) 11 exp(-ik>t) 1 (mz\s) 11 iexp(-ik»s) -1 Table (4-27) ~ \ ^ (<*) k = f ^ + f 2 b 2 + L( X k ) = | ( l ) t ) V ar J Section 5 Irreducible Representations of DJ-6 - 2h For the f i r s t 3 k-vectors l i s t e d i n Table (3-D, we have L( = D^, so the f i r s t 3 tables i n Section 4 give i r r e -ducible representations of D^. We do not rep r i n t those tables here, but instead we star t with case #9, k — f-jk^. Table (5-1) ( i ) k = f_b_ DJ£ l 1 <cn (a |s) i j D ^ U l s ) 1.1 p(2) p(3) p(4) (lit) 11 exp(-ik t) 1 1 1 22 exp(-ilk»t) (2 x ls) 11 exp(-ik»s) -1 1 -1 22 exp (-ilk's.) (2 y ls) 12 exp(-ik»js) 1 -1 -1 21 exp(-ilk»s_) 12 exp(-ik»s.) -1 -1 1 21 exp (-ilk. s.) ( l | t ) 12 exp(-ik.t) 1 1 1 21 exp(-ilk.t) (mx\s) 12 exp(-ik.s_) -1 1 -1 21 exp (-ilk. s.) (^U) 11 exp(-ik.s_) 1 -1 -1 22 exp(-ilk.s_) (*BU> 11 exp(-ik.js) -1 -1 1 22 exp (-ilk«s.) (53) (54) T a b l e (5-2) ( j ) k = f b 4- £b, Dj£ (als) i j DJV(als) i.1 p(2) p(3) p(4) (lit) 11 22 exp(-ik*t) exp(-ilk*t) 1 1 1 (2 x \s) 11 22 iexp(-ik* si) iexp(-ilk*'s_) -1 1 -1 (2 y ls) 12 21 exp(-ik*s_) exp(-ilk*s_) 1 -1 -1 (2 ZU) 12 21 iexp(-ik*s.) iexp (-ilk* s_) -1 -1 1 (Ht) 12 21 exp(-ik*t) exp(-ilk*t) 1 1 1 (mx\s) 12 21 iexp(-ik*s) iexp (-ilk* s.) -1 1 -1 U y|s) 11 22 exp(-ik*s.) exp( -ilk*s_) 1 -1 -1 (mz|s) 11 22 iexp(-ik*s) iexp(-ilk*s) -1 -1 1 (55) Table (5-3) (k) k = + £b_2 D 16 2h ( a l s ) i j D ^ } ( a \ s ) p ( 2 > p ( 3) p(4) (ljt) 11 exp(-ik«t) 1 1 1 22 exp(-ilk*t) (2x\s) 11 iexp(-ik».s) -1 1 -1 22 iexp(-ilk*s.) <Vft> 12 iexp(-ik*s_) 1 -1 -1 21 iexp(-ilk*si) (2 ZU) 12 exp( -ik* s.) -1 -1 1 21 exp(-ilk«s.) (l|t) 12 exp(-ik*t) 1 1 1 21 exp(-ilk*t) (mx|s) 12 iexp(-ik*s>) -1 1 -1 21 iexp(-il^ *,s) (Ms) 11 iexp(-ik*s) 1 -1 -1 22 i exp (-ilk* s.) KU> 11 exp(-ik*s.) -1 _ -1 1 22 exp (-ilk* sj (56) Table (5-4) (1) k - f ^  +• |b_2 +• ^  D 16 2h (als) i j D ^ U l s ) p t 2 ) p<3) p(4) (lit) 11 22 exp(-ik't) exp(-ilk«t) 1 1 1 (2 x |s) 11 22 -exp(-ik«s_) -exp(-ilk•§) -1 1 -1 (2 y |s) 12 21 iexp(-ik* s.) iexp(-ilk's_) 1 -1 -1 (2 z |s) 12 21 iexp(-ik»s_) iexp (-ilk »s.) -1 -1 1 (lit) 12 21 exp(-ik»t) exp(-ilk>t) 1 1 1 (mx|s) 12 21 -exp(-ik» sj -exp (-ilk» s.) -1 1 -1 (myls) 11 22 iexp(-ik« s.) iexp(-ilk*s) 1 -1 -1 (mzIs) 11 22 iexp(-ik»s_) iexp (-ilk» s.) -1 -1 1 (57) Table ( 5 -5 ) (m) k r f ^ (ajs) i j P e t a l s ) p 2 ) p(3) p(4) (lit) 11 22 exp(-ik*t) exp(-ilk«t) 1 1 1 ( 2 J s ) 12 21 exp(-ik»^) exp( -ilk»j3) 1 -1 -1 (2yU) 11 22 exp( -ik« s_) exp (-ilk* s.) -1 1 -1 12 21 exp(-ik»s) exp(-ilk»s.) -1 -1 1 (T|t) 12 21 exp (-ik/t) exp(-ilk*t) 1 1 1 (mx|s) 11 22 exp(-ik/sj exp(-ilk/s) 1 -1 -1 12 21 exp(-ik»s) exp (-ilk* s_) -1 1 -1 ( m z l s ) 11 22 exp(-ik/s_) exp( - i l k * js) -1 -1 1 (5*) Table (5-6) (n) k = + £b_3 (a|s) i j D d i . i Ua|s) (lit) 11 exp [-ik«t) 22 exp1 -ik-t) 33 exp1 ,-ilk-t) 44 exp1 -ilk*t) (2 X U) 13 expi -ik.s) 24 -expi -ik«s_) 31 expl -ilk* s.) 42 -expl -ilk*s_) (2 y ls) 12 expl -ik.s) 21 expl -ik.s.) 34 expl -ilk* s.) 43 expl -ilk*s) (2 z\s) 14 expl -ik*s) 23 -expl -ik* s.) 32 exp( -ilk* ss) 41 -expl -ilk*sj (59) Table (5-6) (continued) (n) k = f 2p_ 2 + §b D: (a|s) i.i (a\s) ( l i t ) 1 3 expi - i k . t ) 2 4 expi - i k . t ) 31 expi - i l k - t ) 4 2 expi - i l k - t ) (mx|s) 11 exp' -ik_» s_) 22 -expi - i k * s.) 3 3 expl - i l k - s ) 4 4 -expl - i l k - s ) (my|s) 14 expl -ik»s) 2 3 expl -ik-s.) 32 expl - i l k . a ) 41 expi - i l k - s ) (m zls) 12 expl -ik.s) 21 -expl -ik.s) 34 expl - i l k - s ) 4 3 -expl - i l k - s ) (60) Table (5-7) (o) k = Jb^  + f ^ D (a|s) I.I '(als) (l|t) 1 1 exp [-ik-t) 22 exp [-ik.t) 3 3 exp ,-ilk-t) 4 4 exp ,-ilk-t) (2 x)s) 1 3 iexp1 -ik«s_) 2 4 -iexp -ik»s_) 31 iexpi -ilk-s.) 4 2 -iexpi -ilk-sj ( 2 y U ) 1 2 expl -ik-s.) 2 1 expl -ik.s.) 34 expl -ilk-s.) 4 3 expl -ilk-s_) (2 z | s ) 1 4 iexp( -ik-s) 2 3 -iexpl -ik-s.) 32 iexpi -ilk. sj 4 1 -iexpl -ilk* s.) ( 6 1 ) Table (5-7) (cont inued) (o) k = Jb^ + f 2 b_ 2 D; ( a l s ) i . i >(a\s) ( l i t ) 1 3 exp ( - i k . t ) 2 4 exp [ - i k . t ) 31 exp ( - i l k - t ) 4 2 exp - i l k - t ) (m x )s) 1 1 i e x p 1 - i k - s . ) 2 2 - i e x p - i k - s . ) 3 3 i e x p i - i l k - s ) 4 4 - i e x p l - i l k - s ) (my U ) 1 4 expl - i k - s ) 2 3 expl - i k - s . ) 32 expl - i l k - s ) 4 1 expl - i l k - s ) ( m z | s ) 1 2 i e x p l - i k - s . ) 2 1 - i e x p l - i k . s . ) 34 i exp( - i l k - s ) 4 3 - i e x p ( - i l k - s ) (62) Table (5-8) (p) k = Ip^ 4- + £b D 16 2h (a|s) D!V(a|s) l . l p(2) p(3) p(4) ( l ) t ) 11 22 exp(-ik»t) exp(-ilk«t) 1 1 1 (2 x|s) 12 21 iexp(-ik'js) i e x p ( - i l k * s ) 1 -1 -1 (2y|s) 11 22 exp(-ik's) exp(-ilk»s.) -1 1 -1 (2 2|s) 12 21 iexp(-ik»s) iexp (-ilk» s.) -1 -1 1 ( l i t ) 12 21 exp(-ik«t) exp(-ilk»t) 1 1 1 (mx| s) 11 22 iexp( -ik's.) iexp (-ilk «s_) 1 -1 -1 (»yla) 12 21 exp(-ik« s.) exp (-ilk* sj -1 1 -1 (mz|s) 11 22 iexp(-ik«s) iexp(-ilk.s_) -1 -1 1 (63) Table (5-9) (q) k = fj>_ D £ £ (a|s) i j DJ^(als) P<2) p( 3) p(4) (lit) n 22 exp(-ik»t) exp(-ilk»t) 1 1 1 (2x)s) 12 21 exp(-ik»s_) exp(-ilk*s) 1 -1 -1 ( 2y |s) 12 21 exp(-ik* s.) exp(Lilk»s_) -1 -1 1 (2 zls) 11 22 exp(-ik*s) exp(-ilk»s_) -1 1 -1 (lit) 12 21 exp(-ik«t) exp(-ilk«t) 1 1 1 [ m x \ s ) 11 22 exp(-ik«s_) exp (-ilk* s.) 1 -1 -1 (myjs) 11 22 expt-ik'S.) exp(-ilk.s_) -1 -1 1 12 21 exp( -ik* s.) exp (-ilk ' S i ) -1 1 -1 ( 6 4 ) Table ( 5 - 1 0 ) (r) k = £b_2 +- f (a|s) i j D(l i.i >(a|s) (Ht) 1 1 exp L-ik«t) 2 2 exp -ik«t) 3 3 expi -ilk*t) 4 4 expi -ilk-t) (2 x | s ) 1 3 -expi -ik-s.) 2 4 expl -ik-s.) 31 -expl -ilk-s) 4 2 expl -ilk-s.) (2 y | s ) 14 -expl -ik-s.) 2 3 expl -ik/s<); 32 -exp( -ilk-sj 4 1 exp( -ilk-s) (2 z | s ) 1 2 exp( -ik-s) 2 1 exp( -ik-s) 34 exp( -ilk-s.) 4 3 exp( -ilk-s.) (65) T a b l e (5-10) ( c o n t i n u e d ) ( r ) k = £b + fjo D; ( a l s ) D d i . i »(a|s) ( l i t ) 1 3 exp [ - i k * t ) 2 4 exp - i k ^ t ) 31 expi - i l k - t ) 4 2 expi - i l k * t ) ( m x l s ) 11 -expi -ik*s_) 22 expi - i k * _§_) 3 3 -expi - i l k ' s ) 4 4 expi - i l k ' s ) (myU) 12 -expi -ik*s) 21 exp( -ik's.) 34 -expi - i l k * s ) 4 3 expi - i l k ' s ) (m 2\s) 1 4 expi -ik«s) 2 3 expi -ik._s) 32 expi - i l k - s ) 4 1 expi - i l k . s ) (66) Table (5-11) (s) k = Jb^ + f £ D 16 2h (a |s) i j (1 (a|s) P ( 2 ) p ( 3 ) p ,(4) ( l i t ) 11 22 exp(-ik/t) exp(-ilk/;t) 1 1 1 (2 x|s) 12 21 iexp(-ik/s_) iexp (-ilk* s_) 1 -1 -1 (2yU) 12 21 exp(-ik/s) exp (-ilk ' S . ) -1 -1 1 (2 z|s) 11 22 iexp(-ik/s_) iexp(-ilk»s_) -1 1 -1 ( l i t ) 12 21 exp (-ik/t) exp(-ilk/t) 1 1 1 ("xla) 11 22 iexp(-ik/s_) iexp(-ilk*s ) 1 -1 -1 (m yls) 11 22 exp(-ik/s_) exp( -ilk/s_) -1 -1 1 (m zls) 12 21 iexp(-ik« s_) iexp(-ilk«s) -1 1 -1 (67) Table (5-12) (t) k = Jb^ + Jbg + f _b D; (als) Uals) ( H t ) 11 exp i - i k ' t ) 22 exp) - i k . t ) 33 exp - i l k - t ) 44 expi - i l k - t ) (2x]s) 13 iexpi -ik-ss) 24 -iexpl -ik-s) 31 iexpl -ilk - s ) 42 -iexpl -ilk - s ) (2 y\s) 14 expl -ik-s) 23 -expl -ik-s) 32 exp( -ilk - s ) 41 -expl -ilk-s.) (2 zls) 12 iexpl -ik-s) 21 iexpl -ik-s_) 34 iexp( -ilk-s.) 43 iexpl -ilk - s ) (63) Table ( 5 - 1 2 ) (continued) (t) k = + i b g + f_b D (als) D d i. i U a l s ) ( l i t ) 1 3 exp - i k . t ) 2 4 exp1 - i k . t ) 31 exp1 - i l k - t ) 4 2 expi - i l k ' t ) (m xls) 1 1 iexpl -ik.s) 2 2 -iexpi -ik« s.) 3 3 iexpl - i l k ' s ) 4 4 -iexpl - i l k . s ) (my\i). 1 2 expl -ik« s.) 2 1 -expl -ik« s.) 34 expi - i l k . s ) 4 3 -expl - i l k - s ) (mz|s) 1 4 iexpl -ik's.) 2 3 iexpl - i k * s_) 32 iexpl - i l k . s ) 4 1 iexpl - i l k . s ) (69) Table {5-1}) (u) k = f ^ + f ^ 1)16 (als) ^ a l s ) p ( 2 ) ( l i t ) 1 1 exp (-ik.t) 1 2 2 exp (-ilk-t) 1 3 3 exp l-imyk't.) 1 4 4 exp ^ -imjjk.t) 1 ( 2 x | s ) 1 2 expi - i k * s_) - 1 2 1 expi - i l k - s ) - 1 34 expi -iniyk. (t -f a^)) 1 4 3 expi -inigk't) 1 ( 2 y l s ) 1 4 exp( - i k . (t + a 2 - v(m x))) - 1 2 3 exp( - i l k - ( t + a 2)) 1 32 exp( -im yk.t) 1 4 1 exp( -im k«(s - v(m ))) z— — — z - 1 ( 2 z l s ) 1 3 exp( - i k . ( t - a 2 + v(m x))) - 1 2 4 exp( - i l k . ( t + a 3)) 1 31 exp( -iniyk. (s - l ( r o y ) ) ) - 1 4 2 exp( -im zk.t) 1 (70) Table (5-13) (continued) (u) k - f b -h f b DJ-£ 2~2 3~3 2h (a|s) }(a|s) p (2) (1 )t) 12 exp (-ik-t) 1 21 exp (-ilk-t) 1 34 expi -im yk-(t - v(m x))) -1 43 expl -im zk.(t - v(mx))) -1 (mx|s) 11 expl -ik -s_) -1 22 expl -ilk-s) -1 33 exp( -irriyk-s.) -1 44 exp( -imzk- (s - a^)) -1 13 exp( -ik-t) 1 24 exp( - i l k - ( t + v(mx))) -1 31 exp( -iniyk-t) 1 42 exp( -ira zk-(g - v(m z))) -1 (mz|s) 14 exp( - i k - ( t -1- a x)) 1 23 exp( - i l k - ( t + v(ra x))) -1 32 exp( -imyk-(s - v(my))) -1 41 exp( -im zk-t) 1 (71) Table (5-14) (•) .k = Jb^ + + f £ D** (als) i j D!V(a|s) i.i p ( 2 ) (lit) 1 1 exp(-ik*t) 1 2 2 exp(-ilk*t) 1 3 3 exp (-iniyk* t) 1 4 4 exp(-imzk*t) 1 (2 X U) 1 2 iexp( -ik* s,) - 1 2 1 iexp(-ilk*s.) - 1 34 exp(-iniyk* (t +• a^ )) 1 4 3 exp(-imzk*t) 1 . (2 y | s ) 14 iexp(-ik*(t + a2 - v(mx))) - 1 2 3 exp(-ilk*(t + a2)) 1 32 exp (-iniyk *t) 1 41 iexp(-imzk*(s - v(mz))) - 1 (2 z l s ) 1 3 iexp(-ik* (t - a2 4- v(mx))) - 1 2 4 exp(-ilk*(t + a-j)) 1 31 iexp(-iniyk* (s - v(my))) - 1 4 2 exp(-imzk*t) 1 ( 7 2 ) Table (5-14) (continued) (v) k = Jb •¥ f b + f h D 1 6 2 h (als) D<1 i j ) (ajs) p ( 2 ) ( l \ t ) 1 2 exp (-ik-t) 1 2 1 exp (-ilk-t) 1 34 iexp t-iniyk. (t - v(m x))) - 1 4 3 iexpi -im zk.(t - v(m x))) - 1 (m xls) 1 1 iexpl -ik-s) - 1 2 2 iexpl - i l k - s ) - 1 3 3 iexpl -irriyk-s.) - 1 4 4 iexp( -im zk»(s - a^)) - 1 (ni y|s) 1 3 2 4 exp( iexp( -ik«t) - i l k - ( t + v(m x))) 1 - 1 31 exp( -irriyk-t) 1 4 2 iexp( -im zk«(s - v(m z))) - 1 (m z)s) 1 4 exp( - i k . (t + a^)) 1 2 3 iexp( - i l k - ( t +• v(m x))) - 1 32 iexp( -imyk-(s - v(my))) - 1 4 1 exp( -im zk«t) 1 ( 7 3 ) Table (5-15) (w) k - + f D 1 6 2 h (a|s) i j D ( l ) ( a l s ) p ( 2 ) ( l i t ) 1 1 exp(-ik»t) 1 2 2 exp(-ilk't) 1 33 exp(-im xk»t) 1 4 4 exp(-im zk»t) 1 ( 2 x | s ) 1 4 exp(-ik/(t + a^ + v(m y))) - 1 2 3 exp(-ilk»(t +• a^) ) 1 32 exp(-im xk«t) 1 4 1 exp(-im zk»(s - v(m z))) - 1 1 2 exp(-ik»s.) - 1 2 1 exp(-ilk»s_;) - 1 34 exp(-im xk.(s - aj_ - £13 - v(m y))) 1 43 exp(-im zk.(s - - a 1 - a^ + v(my))) 1 ( 2 z l s ) 1 3 exp(-ik«(t + a-j - yjniy))) - 1 2 4 e x p ( - i l k . ( t + a^)) 1 31 exp(-im xk.(s - y_(mx))) - 1 42 exp(-im zk»t) 1 (74) Table (5-15) (continued) (w) k = f h + f b 1 1 3~3 2h (als) D(V(a|s) i . i p(2) ( l i t ) 12 exp(-ik»t) 1 21 exp(-ilk*t) 1 34 exp(-im xk« (t - § _ ] _ - - v(m y))) -1 43 exp(-im zk'(t - a^ - a^ - yjriiy))) -1 (mx|s) 13 exp(-ik• (t +• a 2 a^)). 1 24 exp(-ilk» (t •+ a^ +- yjniy.))) -1 31 exp(-im xk.t) 1 42 exp(-im zk. (s_ - v(m z))) -1 (*y|s) 11 exp(-ik» s_) -1 22 exp(-ilk*s_) -1 33 exp(-iraxk» (s_ - a^)) -1 44 exp(-irazk»s_) -1 (mz|s) 14 exp(-ik.(t + a )) 1 23 e x p ( - i l k * ( t +• a x +• v(m y))) -1 32 exp(-im xk.(s - v(m x))) -1 41 exp(-im zk.t) 1 (75) Table (5-16) (x) k = f b + £b + f b Dl£ (a|s) D(l)(a\s) p ( 2 ) ( l i t ) 1 1 exp(-ik»t) 1 2 2 exp(-ilk»t) 1 3 3 exp(-imxk«t) 1 4 4 exp(-im zk»t) 1 ( 2 x | s ) 14 i e x p ( - i k . ( t + a l 2+ v(m y))) - 1 2 3 exp(-ilk«(t + a^)) 1 32 exp(-im xk»t) 1 4 1 iexp(-im zk'(s - v(m z))) - 1 12 iexp( -ik»s_) - 1 21 iexp(-ilk» s.) - 1 34 exp(-imjfjc* (s. - a^ - a^ - v(rriy))) 1 4 3 exp(-im zk.(s - a^ - a^ + v(m y))) 1 1 3 i e x p ( - i k . ( t + a^ - v(m y))) - 1 2 4 exp(-ilk« (t -V- a^)) 1 31 iexp(-im xk.(s - v(m x))) -1 4 2 exp(-im k»t) 1 (76) Table ( 5 - 1 6 ) (continued) (x) k = f,b + Ah +• f b D 1 6 " - 1 - 1 ~2 3 - 3 2 h (als) D < 1 ) ( a \ s ) p ( 2 ) (Tit) 12 exp(-ik.t) 1 2 1 exp(-ilk»t) 1 34 iexp(-imxk. (t - a_i - =13 - v(my))) - 1 4 3 iexp(-im k«(t - a-. - a, - v(m ))) - 1 (rax|s) 1 3 exp(-ik. (t + 1 2 + £3) 1 2 4 iexp(-ilk_. (t + a.3 +v(my))) - 1 31 exp(-imxk«t) 1 4 2 iexp(-imzk«(s - v(mz))) - 1 (myI a) 1 1 iexp(-ik»s_) - 1 2 2 iexp (-ilk ^ s.) - 1 3 3 iexp(-imxk.'(s_ - a )^) - 1 4 4 iexp(-imzk» s_) - 1 (mzls) 1 4 exp(-ik«(t + a )^) 1 2 3 iexp (-ilk. (t 4- a± + v(m ))) - 1 32 iexp(-im k» (is - v(m ))) - 1 4 1 exp(-imzk»t) 1 (77) L^m (y) k = + ( a l s ) i j D J V ( a l s ) p ( 2 ) ( l i t ) 1 1 e x p ( - i k - t ) 1 2 2 e x p ( - i l k . t ) 1 3 3 exp(-im xk«t) 1 4 4 exp(-im yk»t) 1 ( 2 x | s ) 14 e x p ( - i k * (s 4- v(m y))) - 1 2 3 e x p ( - i l k . ( t + a j ) ) 1 32 exp(-irn xk«t) 1 4 1 exp(-iniyk. (s. - v(m y))) - 1 ( 2 y | s ) 1 3 exp(-ik«(s + v(m x))) - 1 2 4 e x p ( - i l k . ( t + fla)) 1 31 exp(-im xk_. (s. - v(m x))) - 1 4 2 exp (-iniyk. t ) 1 ( 2 z l s ) 1 2 exp(-ik«s_) - 1 2 1 exp ( - i l k . s) - 1 34 exp(-im xk«(s - a^ - v ( m z ) ) ) 1 4 3 exp(-im yk.(s - a^ - a 2 + v ( m z ) ) ) 1 (73) Tablv (5-17) (continued) (y) k = f b + f ^ D 16 2 h (als) u i . i ^ a l s ) p ( 2 ) ( l i t ) 1 2 exp (-ik-t) 1 2 1 exp [-ilk-t) 1 34 exp [-imxk«(t - a 2 - v(m z))) - 1 4 3 exp -inyk-(t - a 2 - v(m z))) - 1 (mjs) 1 3 e x p 1 -ik-(t + a 2 + a^)) 1 2 4 expl - i l k - ( s + v(m ))) - 1 31 exp l -imxk-t) 1 4 2 e x p l -imyk- (s - v(my))) - 1 14 exp( -ik-t) 1 2 3 e x p l - i l k - ( t + v(m z))) - 1 32 exp( -im xk-(s - v(m x))) - 1 41 exp( -imyk-t) 1 (mz|s) 1 1 exp ( -ik-s) - 1 2 2 exp ( -ilk-s) - 1 3 3 exp( -imxk- (s_ - a^)) - 1 4 4 exp ( -imyk -s_) - 1 (79) Table (5-13) (z) k = + f 2b_ 2 4- Jb, D; (als) i j DJ^UIs) p ( 2 ) (lit) 11 exp(-ik«t) 1 2 2 exp(-ilk.t) 1 3 3 exp(-imxk«t) 1 4 4 exp(-imyk«t) 1 ( 2 xls) 1 4 iexp(-ik« (s. + v(niy))) - 1 2 3 exp(-ilk.(t + ax)) 1 32 expt-inixk't) 1 4 1 iexp(-imyk« (s - v(my))) - 1 1 3 iexp(-ik«(s_ + v(mx))) - 1 2 4 exp(-ilk«(t + a^ )) 1 31 iexp(-imxk«(s - v(mx))) - 1 4 2 exp(-iniyk. t) 1 (2 z|s) 1 2 iexp(-ik« s.) - 1 2 1 iexp (-ilk. sj - 1 34 expt-inixk. (s - §2 - v(mz))) 1 4 3 exp(-imyk. (s_ - a^  - a 2 + v(mz))) 1 ( 8 0 ) (Table (5-1&) (continued) (z) k = + f 2b_ 2 + £b, D: (a |s) I.I >(a|s) p ( 2 ) ( l | t ) 12 exp (-ik.t) 1 21 exp r - i l k - t ) 1 34 iexp '-imxk «(t - a_2 - v(m z))) -1 43 iexp -imyk.(t - a 2 - v(m z))) -1 (m x |s) 13 expl - i k . ( t + a 2 + a^)) 1 24 iexpl - i l k - ( s + v(m y))) -1 31 expl -im k»t) 1 42 iexpl -im yk«(s - v(m y))) -1 (m y ls) 14 exp( - i k - t ) 1 23 iexp( - i l k - (t +v(m z))) -1 32 iexp( -im xk»(s - v(m x))) -1 41 exp( -iniyk-t) 1 (m z | s) 11 iexp( -ik-s.) -1 22 iexp( - i l k - s ) -1 33 iexp( - i m ^ . (s - a3)) - 1 44 iexp( -im yk.s) -1 (81) Table (5-19) (<*) k = 4- f 2b_ 2 +• f ^  D; (als) i j i.i (a|s) ( l i t ) 11 exp [-ik.t) 22 exp -i2 xk.t) 33 exp -i2 yk-t) 44 exp1 -i2 zk.t) 55 exp' - i l k - t ) 66 expi -im„k«t) 77 expi -imyk.t) 38 expi -im„k«t) (2 xls) 12 expl -ik. (t •+- a^)) 21 expl -i2 xk.t) 34 expl -i2yk«(t + a ± ) ) 43 expi -i2 zk-t 56 expl - i l k - ( t +• a x)) 65 expl -imxk't) 78 expl -im yk-(t + a x)) 87 expi -imgk»t) (82) Table ( 5 - 1 9 ) (continued) («) k = + f 2b_ 2 +• (als) i j 4) '(als) <2y)s) 1 3 exp -ik-(t + a2)) 2 4 exp M 2 x k ' ( t - a x - &3)> 31 exp - i 2 y k t) 4 2 exp - i2 z k- ( t - a1^- *-2 " a3)) 5 7 exp' -iTk.(t + a2)) 6 8 expi -imxk»(t - a^  - a3)) 7 5 expi - iniyk* t) 86 expi -imzk»(t - a^  + * 2 - a3)) 14 expi -ik- (t -f- a3)) 2 3 expi - i 2 k-(t - a.)) x~ ~2 32 expi - i2 y k- ( t - § 2 + a3)) 4 1 expi - i2 z k- t ) 5 8 expi -ilk* (t + a^ )) 6 7 expi -imxk-(t - ag)) 7 6 expi -iniyk - (t - a2 + a3)) 85 expi -im„k-t) (83) Table (5-19) (continued) («) k = + 4- f ^ (a|s) D ( l ) ( a | s ) i . i (T t i ) 15 exp(-ik» t ) 26 e x p ( - i 2 x k ' ( t - ag_ - S£ - a_3)) 37 exp ( - i2yk»(t - a 2 ) ) 48 exp ( - i2 zk»(t - a-j_ - a^)) 51 e x p ( - i l k « t ) 62 exp(-im xk .(t - a-^  - a 2 - a^))) 73 exp(-ira k . ( t - a 9 ) ) y c 84 exp(-im zk.(t - a^ - a^)) (m xl s) 16 e x p ( - i k ' ( t 4- a 2 4- a^)) 25 exp ( - i 2 xk»t) 38 e x p ( - i 2 y k . ( t 4- a 3 ) ) 47 exp ( - i 2 zk» (t 4- § 3 ) ) 52 exp(-ilk«(t 4- a 2 4- a^)) 61 exp(-imxk»t) 74 exp (-iniyk • (t a^)) 83 exp(-im2k» (t +- s^)) (84) Table (5-19) (continued) (<x) k = f.b. + f.b_ + D — — i — i 2 < $~ $ 16 2h (a la) i j D(l)(a(s) (my|s) 17 exp (-ik't) 28 exp ( - i 2 xk/t) 35 exp (-i2 k-t) y 46 exp [ - i 2 zk-t) 53 exp ; - i l k - t ) 64 exp -im xk»t) 71 exp1 -iniyk-t) 82 expi -im zk-t) (mz|s) 18 expl - i k - ( t + &i)) 27 expl - i 2 x k - t ) 36 expl - i 2 y k - ( t + a x)) 45 exp( - i 2 z k - t ) 54 exp( - i l k - ( t +- aj_)) 63 exp( -im xk.t) 72 exp( -imyk - (t + a^)) 81 exp( -im zk-t) BIBLIOGRAPHY Casher, A. and Gluck, M., Report of the I s r a e l i Technical  I n s t i t u t e , (unpublished), 1963 Hurley, A. C , "Ray Representations of Point Groups", P h i l . Trans. Roy. S o c , A 260. (1966) International Tables f o r X-Ray Crystallography, Kynoch Press, Birmingham, England, 1953 Jansen, L. and Boon, M., Theory of F i n i t e Groups. Applications  i n Physics, North Holland Publishing Co., Amsterdam, John Wiley & Sons, Inc., New York, 1967 Koster, G. F., "Space Groups and Their Representations", Soli d State Physics, J5_, 173, Academic Press Inc. Publishers, New. York, 1957 Kovalev, 0 . V., Irreducible. Representations of Space Groups, Kiev: Academy of Sciences, Ukrainian S. S. R., (in Russian), 1961, English t r a n s l a t i o n from Gordon and Breach Publishing Co., New York, 1965 Lomont, L. S., Applications of F i n i t e Groups, Academic Press Inc. Publishers, New York, 1959 (35) APPENDIX Rav Representations of Certain Point Groups The following two tables contain those i r r e d u c i b l e ray representations of the point groups D 2^ and which were needed to construct the allowable representations of the various l i t t l e groups. Table Al i s i d e n t i c a l with Hurley 1s ( 1 9 6 6 ) Table A l , and Table A 2 contains those parts o f Hurley»s Table A 4 which were needed. Table A l ° 2 v ° 2 h a - ( 2 ) a = ( 2 ) b =.m//a b = raia a 2 = e b 2 - e; ba - ab A 2 = E B 2 - E, D 2 a ^ ( 2 ) b = (2)j.a oc = ±1 BA = oc AB Arbitrary factors ± 1 for A and B oC = - l E A B AB 1 0 1 0 < Q ) 2 i 0 1 0 1 < r i)i2 0 1 0 - 1 (1)22 1 0 - 1 0 * ( r i > 2 0 0 0 ( 8 6 ) (86) T a b l e A2 D 2h a a = m a 2 •= e ^ = E b = m x a b 2 = e ; b a = a b B 2 = E; BA - AB c = m x a , ± b c 2 = e; c a = a c ; cb = be C 2 - E; CA =/5 A C ; CB = a r b i t r a r y f a c t o r s i l f o r A , B , and C <* =-1, 3 =1, y = i E A B c AB BC CA ABC < q ) u 1 1 1 0 1 0 0 0 0 0 0 1 0 1 1 - 1 0 0 0 1 0 1 - 1 1 * n.^22 1 - 1 1 0 - 1 0 0 0 a<ri> 2 0 2 0 0 0 0 0 « = 1, /3=1, =-1 1 1 0 1 0 0 1 0 ( Tihli 0 0 1 0 - 1 1 0 - 1 ( n . ) i2 0 0 1 0 1 1 0 1 ( n^22 1 - 1 0 1 0 0 - 1 0 2 0 0 2 0 0 0 0 ° < = 1 , / S = - 1 , tf=-l 1 0 1 1 0 1 0 0 (n.^21 0 1 0 0 1 0 - 1 1 0 1 0 0 - 1 0 1 1 (n.^2 1 0 - 1 - 1 0 1 0 0 X(rx) 2 0 0 0 0 2 0 0 

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