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Representations of the space group D¹⁹ ₄h, and the corresponding double space group Guccione, Rosalia Giuseppina 1960

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REPRESENTATIONS OF THE SPACE GROUP D 4 h , AND THE CORRESPONDING DOUBLE SPACE GROUP by ROSALIA GIUSEPPINA GUCCIONE Laurea, Universita d i Palermo, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1960 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f P W y s i c s  The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver 8, Canada. Date 10 S ^ j - v / t ^ w J ^ / u , I9<SQ  i i i ABSTRACT One of the e x i s t i n g methods of c a l c u l a t i n g the characters of i r r e d u c i b l e representations of space groups and double space groups i s described i n some d e t a i l , and applied to the case of 1 9 the non-symmorphic space group D;|h (14-j/amd) (which i s the space group of c r y s t a l s of white t i n ) . A complete l i s t of the characters of i r r e d u c i b l e represen-tations of t h i s space group and the corresponding double group i s given. Some useful r e l a t i o n s involving the characters are also included. i i ACKNOWLEDGEMENTS I should l i k e to thank Professor W. Opechowski for suggesting t h i s problem and f o r his valuable advice during the course of the research. I wish also to thank the National Research Council of Canada for f i n a n c i a l help i n the form of a Studentship. i v TABLE OF CONTENTS ACKNOWLEDGEMENTS ABSTRACT CHAPTER I CHAPTER II Introduction Space Groups and Their Properties Discussion of a Special Case J 9 '''"5$:; (Space Group D 4 h , White Tin) CHAPTER III Irreducible Representations of Space Groups Irreducible Representations of the .19 CHAPTER IV FIGURES I II III IV Space Group D^h Double Space Groups; Their Properties and Representations Irreducible Representations of the ^ e . „ „19T Double Space Group D^h C r y s t a l structure of white t i n Atomic positions i n the unit c e l l of the white t i n structure projected on the plane z=0 through the atom i n the center of the c e l l Symmetrical unit c e l l f o r white t i n Symmetrical unit c e l l f or the r e c i p r o c a l l a t t i c e of white t i n i i i i i 1 15 41 10 11 13 14 V TABLES I - X Characters of allowable representations of the groups of r\ A ,1, A ,X , M.N, V. V, y 34-39 XI-XIII Compatibility relations AO XIV-XXIII Characters of the specific representations of the double groups o f r , A . £ , A , X , M , f V V j W/ Y 49-56 XXIV-XXV Compatibility relations 56 BIBLIOGRAPHY 5 7 6 1-CHAPTER I Introduction t It i s a well known fact that representations of space groups play an essential part in the theoretical description of many solid state phenomena. The theory of la t t i c e vibrations, and that of electron band structure are probably the two most important examples of this fact. In the latter case the ex-istence of the electronic spin necessitates dealing with the representations of double space groups in addition to those of space groups. Although the general theory of the irreducible represen-tations of space groups was given by Seitz (1936) more than twenty years ago, and the characters for the space groups 0^ , 0 h 9 and 0£* (single, body-centered and face-centered cubic lattices) were calculated soon after by Bouckaert, Smoluchowski and Wigner (1936), there are many non-trivial space groups for which this has not yet been done. The work on irreducible representations of double space groups was started only a few years ago, by E l l i o t t (1954). The l i s t of space groups for which the characters of i r r e -ducible representations have been calculated and published i s given in the following table: -2-Space Group O j J (Pm3ra) 0,5 (Fm3m) 0 h 7 (Fd3m) 0^ (Im3m) (F4"3m) D6h (P63/mmc) D 3 (P3221) C^ v (C6me) Typical Crystal Structure Simple cubic Face-centered cubic Diamond Body-centered cubic Zincblende Hexagonal close packed Tellurium Graphite Wurzite Reference L. Bouckaert, R. Smoluchowski, and E. Wigner, Phys. Rev. 50. 58 (1936). T5)* R.J. E l l i o t t , Phys. Rev. 96, 280 (1954). "~ Bouckaert, Smoluchowski and Wigner, ibid. (D) E l l i o t t , ibid. C. Herring, J. Franklin Inst. 233, 525 (1942). W. DorTng and V. Zehler, Ann. Physik 13, 214 (1953). (D) E l l i o t t , ibid. Bouckaert, Smoluchowski and Wigner, ibid. (D) E l l i o t t , ibid. R.H. Parmenter, Phys. Rev. 100, 573 (1955). G. Dresselhaus, Phys. Rev. 100, 580 (1955). (D) Parmenter, ibid. (D) Dresselhaus, ibid. Herring, ibid. (D) E l l i o t t , ibid. Yu.A. Firsov, J. Exptl. Theoret. Phys. (USSR) 32. 1350 (1957). 7 5 ) Firsov, ibid. J.L. Carter, Ph.D. Thesis, Cornell Uni-versity, February 1953 (unpublished). (D) J.C. Slonczewski, Ph.D. Thesis, Rutgers University, June 1955 (unpublished). R.C. Case11a, Phys. Rev. 114, 1514 (1959). (D) Case11a, ibid. Space Group Typical Crystal Structure Reference G. Dresselhaus, Phys. Rev. 105, 135 (1957). (D) Dresselhaus, ibid. * (D) means that the characters of the double space group are given in the reference in question. In this thesis we outline a method, f i r s t used by Herring (1942) of calculating the characters of the irreducible re-presentations of a space group, and apply i t to the case of the space group D^ (in which white t i n crystalizes). We also adapt the method to the case of double space groups, and apply 19 t i t to the double space group D 4 n We shall now turn to a brief summary of the contents of the remaining chapters. In the f i r s t part of Chapter II we introduce the concept of space group and discuss some properties of space groups. 19 In the last part we discuss the space group D 4 h . In the f i r s t part of Chapter III, after having introduced the irreducible representations of a latti c e , we define the concept of the Brillouln zone (B-Z). We then give the defi-nition of the group G k of a vector k and outline the method of finding i t s irreducible representations when k ends inside or on the surface of the B-Z. The concepts of the kernel T k of a representation and factor group GPVT12 are introduced. Finally we outline the method to construct an irreducible represen-tation of a space group G from a given irreducible represen--.ci-tation of a group G . In the remaining part of the chapter 19 we apply the theory previously described to the case of D4Jl. We examine the groups G k of vectors k inside and on the 19 surface of the B-Z of D 4 n and we give their tables of charac-ters. We also consider points on lines of symmetry in order to give the so called "compatibility relations" for the group in question. In Chapter IV we introduce the concept of double space group and discuss i t s properties, and we construct the tables 19 t of characters of the double space group D 4 h . Some compatibility tables are included. CHAPTER II Space Groups and Their Properties Discussion of a Special Case (Space Group D||, White Tin) I I . 1 Let us consider the linear inhomogeneous trans-formations of the form: J ? = R T : + F I I . 1 . 1 where 7 and x' are position vectors of a point, R i s a real orthogonal three-dimensional matrix and t i s a real three-dimensional vector. The real orthogonal matrix R can be inter-preted as either a proper or improper rotation according as det(R) i s + 1 or - 1 respectively. Improper rotations are the inversion and operations representing a rotation followed by an inversion. The vector t? can be interpreted as a trans-lation, so that the transformation I I . 1 . 1 can be looked upon as a proper or improper rotation R followed by a translation t. Using Seitz' symbolism we w i l l denote the transformation I I . 1 . 1 by (Rlt). The product of two inhomogeneous transformations i s given by: (MtRs|']=(RS|Rt'.t) I I . 1 . 2 -6-The inverse of a transformation (R | t) i s : ^ I b l - ' . ^ - ' l - ^ ' t ) , I I . 1 . 3 and the conjugate of (R |t) i s : (S 11')"1 ( R | t ) ( S | t'J = ( V ' R S | s" '* b' + S"'t -S" 1 f) II.1.4 From the definition of the multiplication I I . 1 . 2 , of the i n -verse of an element I I . 1 . 3 and from the existence of the identity (E | 0 ) It follows that the set <$L of a l l real non-singular inhomogeneous transformations (R | t) i s a group. A subgroup R 3 of (R, i s the set of a l l "pure" rotations (R I 0 ) . Such subgroup is the group R3 of a l l the three-dimensional orthogonal matrices. Another subgroup of (R, Is the set of a l l "pure" translations (E | t) in the group. Such subgroup i s an invariant subgroup of (& * A crystallographic space group G is a special kind of subgroup of . A crystal-lographic space group G i s in fact a discrete subgroup of such that i t s pure translations are primitive and constitute an invariant subgroup of G; the primitive translations are pure translations of the form: ( E l t ^ U K i ^ • / * » t » + ' * ^ > ) 1 1 . 1 . 5 Here n^, n 2 and n 3 are integers and t^, t 2 and t 3 are three linearly independent translations, called basic primitive -7-translations. From the fact that the primitive translations of a space group G from an invariant subgroup of G, i t follows that i f (R | t) is an element of G then, whenever (E |t n) is a primitive translation, (E I Rt n) is also a primitive translation of G. In fact: ( E | ^ ) : ( R | t ) ( F | g ( ^ | t ) - 1 II. 1.6 Equation II.1.6 imposes some restrictions on both rotations and translations of a space group. • The rotational parts R of the elements (R I t) of a space group can only be proper ro-tations through integral multiples of 60° and 90° and improper rotations which are products of the mentioned rotations with the inversion. There are only 32 groups of rotations satis-fying these conditions, they are called point groups. The periodic structure generated by the primitive translations in a space group is called Bravais lattice. Eq. II. 1.6 means that the lattice in a space group must be invariant under the operations of the associated point group. The consequence is that only 14 lattices can exist, (see Koster, 1957). A lattice and a point group do not specify completely a space group. In fact a non-primitive translation r may appear in a space group but only in combination with a rotation other than the identity. We can therefore say that the most general element in a space group has the form: -8-(R|f(R.)+t) II. 1.7 where V(R) is either zero or a non-primitive translation and t is a primitive translation. Since only primitive trans-lations are associated with the identity rotation, r(E) = 0. Depending on whether the vector r(R) is-zero for every R, or i s not zero for some R a space group i s called symmorphic or non-symmorphic. A symmorphic space group contains the entire point group as a subgroup. A non-symmorphic space group does not contain the entire point group as a subgroup. How-ever, for any space group G the factor group G/T is isomorphic to the point group P of a l l the rotational parts of the operators of G. £ z P II.1.8 Such point group P i s said to belong to G. Examples of sym-1 q raorphic space groups are 0^ (simple cubic), 0^ (body-centered o cubic), (zincblende structure). Examples of non-symmorphic 7 4 space groups are 0 n f (diamond structure), Dgjj (hexagonal close-packed) . From the definition of space group of a crystal as a group of transformations under which the crystal is invariant and whose pure translations form an invariant subgroup i t follows that one can define a smallest volume, called unit c e l l , from which the entire crystal can be reproduced by translation - 9 -through the primitive translations. A unit c e l l can be defined in several ways. The simplest way i s to take as a unit c e l l the parallelopiped with edges t j , t 2 and tg. But such a unit c e l l has the disadvantage that i t does not necessarily go into i t s e l f under the operations of the point group which leaves the lattice invariant. A unit c e l l which has the symmetry of the point group P i s called symmetric^unit c e l l and can be defined in the following way: i t is the volume enclosed and bounded by the planes which form the perpendicular bisectors of a l l lines that extend from a given lattice point to a l l the re-maining lat t i c e points in the crystal. A concept which plays an important role in the theory of irreducible representations of a space group is that of re-ciprocal lattice. If t-p t 2 and tg are the three basic primitive trans-lations of a lattice T, then the basic primitive translations b l * b2 a n d b3 o f i t s r e c i P r ° c a l lattice are defined by the relations: An important property of the reciprocal lattice corre-sponding to a given direct lattice is that i t i s invariant under the same point group operations under which the direct l a t t i c e is invariant. This means that i f t^ and kj are primitive translations in the direct and reciprocal lattice respectively, and R i s a rotation belonging to the point group which leaves the d i r e c t l a t t i c e invariant, Rt^ and Rkj are again primitive translations of the di r e c t and r e c i p r o c a l l a t t i c e respectively. It follows: R.Ft.k- -.IKx(L-nta^er) II.1.10 F--RU- =2% *[i*\tgir) II.2 As we said in the introduction, we intend to study 19 the i r r e d u c i b l e representations of the space group D 4 n (white t i n ) , therefore we w i l l devote the rest of t h i s chapter to an analysis of i t s structure. The space l a t t i c e of white t i n (Fig. I) i s body-centered t t s tetragonal with a basis of two atoms at (0,0,0), (— — —) f f f2 2 associated with each l a t t i c e point, as shown i n Fig. II. Fig. I Cr y s t a l Structure of White Tin 11 Fig. II Atomic positions i n the unit c e l l of the white t i n structure projected on the plane z«0 through the atom i n the center of the c e l l . The symbols in parentheses denote the height of the atoms with respect to the plane z=0. For example, (-s,s) at the upper l e f t corner indicates that along the edge of the c e l l perpendicular to the plane z=0, there i s an atom at z^s and another at z=-s. Each atom has four nearest neighbours and twelve next nearest neighbours. The l a t t i c e constants are: 2s = 3.17 A 2t - 5.82 A . 19 The white t i n space group i s D 4 h . Its point group i s h 19 D4 * D4h i s n o t a s v m r a o r P f t i c space group. In fa c t i t con-tains non-primitive translations i n association with some of the elements of D 4 . A l l elements of D 4 which are also elements of the subgroup D g^- have no non-primitive trans-l a t i o n s associated with them. However, the remaining elements of the point group D 4 occur i n combination with a non-primitive -12-tra n s l a t i o n . The following operations belong to D 2 d: E i d e n t i t y S^ z, ro t a t i o n through +, ir/2 about the z-axis followed by a r e f l e c t i o n through the plane z= 0 . C 2 z r o t a t i o n through v about the z-axis C2x* C2y rotations through T about the x-and y-axis 0"dxy> c J^y r e f l e c t i o n s through the planes x=y and x=-y. The above operations appear i n the space group i n the form (R I t n ) where R i s a ro t a t i o n belonging to D 2 (j a n d * n i s a primitive t r a n s l a t i o n of the body-centered tetragonal l a t t i c e . The remaining operations of are: I inversion ^4z' ^ 4 z r o t a t i o n through + TT/2 about the z-axis **2xy» **2xy rotations through TT about the two l i n e s x=y z=0 , x= - y z=0 °h r e f l e c t i o n through the plane z=0 ^vxz' °vyz r e f l e c t i o n s through the v e r t i c a l planes y= 0 , x=0 . These operations appear i n the space group with the non-primi-t i v e t r a n s l a t i o n r : 13-We have said that white t i n has a body^centered tetragonal lattice. This type of Bravais la t t i c e can be regarded as generated by three of the eight vectors ex-tending from the center to the corners of a rectangular solid with a square basis. Using the coordinate system indicated in Figs. I and II, the three basic primitive translations can be taken to be: £ = VTt } + s £ tj r tfTt f - sic II.2.2 I, = \rrtT - sk Since s < (2? t, the symmetrical unit c e l l of white t i n is as shown in Fig. I l l . Fig. I l l Symmetrical Unit Cell for White Tin To find the reciprocal l a t t i c e of the b-c tetragonal lattice we use the relationship II.1.9 where t^, tT2 a n d t -14-are given by II.2 . 2 and b^, b 2 and b 3 are the unknowns. Solv-ing these three systems of equations one finds: X k s II.2.3 b - € £ r 3 _ t These are the primitive translations in a b-c tetragonal lattice with lattice constants {2ir/t, tfsfrr/t and 2ir/s. Since t > s, then f&ir/a > {2V/t. Prom this last relationship i t follows that the symmetrical unit c e l l in the reciprocal la t t i c e has a different shape from that in the direct lattice. Fig. IV (see Kosterjl957) Fig. IV Symmetrical Unit Cell for the Reciprocal Lattice of White Tin -15-CHAPTER III Irreducible Representations of Space Groups Irreducible Representations of the Space Group D( I I I . l It is well known (see, for example, Lomont, 1959) that there exists a systematic procedure for finding a l l the irreducible representations of a group G i f a l l the irreducible representations of an invariant subgroup H of G are given. This procedure was specialized to the case of space groups by Seitz (1936). However, he restricted himself to general considerations. Bouckaert, Smoluchowski and Wigner (1936) were the f i r s t to apply the general theory to specific space groups. The role of the invariant subgroup H of G when G i s a space group, i s played by the subgroup of primitive trans*-, lations T. Therefore in order to apply the general procedure mentioned above, We have f i r s t to discuss the irreducible representations of T. The lattice T i s an abelian group, therefore a l l i t s irreducible representations are one-dimensional. Its represen-tations (see Lomont) are of the form i JT-F JL III.1.1 where k i s a real vector and t i s a translation of T. From the definition of reciprocal l a t t i c e we know that i f K is a vector of the reciprocal l a t t i c e , then K*-? •= integer^^jr -16-and e Q = 1. It follows that the two representations e i k , t and e 1 ^ k + K q ^ ' t of a translation t are identical. To obtain one-to-one correspondence between the k-vectors and the irreducible representations of T, one introduces the concept of the Brillouin zone (B-Z). The B-Z is the symmetri-cal unit c e l l in the reciprocal lattice and i t has the following properties. To each point inside the B-Z corresponds a different irreducible representation of the group of pure translations. In fact,two points in the interior of the B-Z cannot differ by a primitive translation of the reciprocal lattice. Not to every point on the surface of the B-Z corresponds a different irreducible representation of the group of pure translations. In fact, every point on the surface i s equivalent to at least one other point on the surface. To each point k =• k + K_ outside the B-Z corresponds the same representation of the group of pure translations which corresponds to the point k inside the B-Z or on i t s surface. After having found in this way a l l the irreducible re-presentations of;the group of pure translations, we can now proceed to applying the systematic procedure for finding the irreducible representations of the space group. The f i r s t step in this procedure is to define for each irreducible representation of the group of pure translations (which means for each k in the B-Z) a group G k called the "group of the wave vector k". This group G i s defined as -17-the set of a l l elements (R | t) of G such that Rk - k + K q where K q is a reciprocal la t t i c e vector. This subgroup of G is i t s e l f a space group and of course contains T. Hence, T i s also an invariant subgroup of Gk. The second step consists in finding a l l the irreducible representations of G which have the property that the matrices representing pure translations are unit matrices multiplied by a factor of the form e i k , t . Here t i s any translation in T. Such representations are called "small" or "allowable" re-presentations of Gk. For points inside the B-Z the only value of K for which — — — Q Rk = k + K« i s K « 0. If P1* is the point group belonging to Ml G k and P (R) i s an irreducible representation of P1*, then an allowable representation of G k is given (see Koster, 1957) by: r ( R | t ) = e rW i n . 1 . 2 where (Rlt) belongs to G . For points on the surface of the B-Z one must distinguish between symmorphic and non-symmorphic space groups. For a k-vector on the surface of the B-Z of a symmorphic space group EIc^  III. 1 . 2 i s s t i l l valid. But i t i s not valid for a non-symmorphic space group. In fact, let us consider two elements (Rj I r' + t') and (RA I T" + t") of a group G k when G is a non-symmorphic space group. Here r' and 7 " are non-primitive translations and t' and t" are elements of T. -18-Their product i s (RjRjJ Rj T " + Rjt" + r* + t ' ) . Multiplying the matrices representing the two elements and using III.1.2 we obtain: III.1.3 • e r(R.R..j The matrix representing the product i s : UR" k.(t"*b'*T'*t" j e III.1.4 I 5 = e -19 ITT J • TT" Because for a non-symmorphic space group e J i s not, in general, equal to unity, i t follows that when a vector k i s on the surface of the B-Z, €:-<*.-j i l l . 1.2 does not give the allowable representations of G . Therefore a special procedure must be used. .But before we go into details about this special procedure, we describe the f i n a l step for finding the i r r e -ducible representations of the space group G from the allow-able representations of Gk. There is a well known theorem according to which i t i s possible to construct a representation of a group G from a given representation of the subgroup H of G.•• » The way of constructing a representation of G from a given representation of H i s as follows. Let g and h be the order of G and H ^ respectively. And let us consider the l e f t cosets of H. If Aj, A 2 ... A g and Bl» B2 *'* B h a r e * n e elements of G and H respectively, such cosets w i l l be: A^H, AgH, ... AnH with n = g/h. The product of these n cosets with an arbitrary but fixed element A of G w i l l be again a set of n cosets which is a permutation of the original set. If then one associates with each element A of G the corresponding permutation matrix, one obtains a representation of the group G by permutation matrices. Such matrices have the property that in each row and each column there i s only one element different from zero and this element is equal to unity. Now, let us consider an arbitrary but fixed element A - 2 0 -of G and an element of G which characterizes a coset of H. Their product w i l l be an element of G belonging to a coset, say the 1-th coset of H: AAj, = A]Bfc. Here i s a fixed element of H once A, A A and Aj have been fixed. Then we associate with the element A a matrix whose i-th, 1-th element is the element B k of the subgroup H. From what we said pre-viously i t follows that in each row and each column of the matrix representing the element A there i s only one element different from zero. Finally we replace in the matrix so ob-tained the elements B k of H by the matrices corresponding to these elements in the given representation of H. In this way one gets a representation of G from a given representation of H. If G i s a space group and H i s one of i t s subgroups Gk, one obtains by this method from a given allowable represen-tation of a subgroup G k a representation of G. Such represen-tation of G can be proved to be irreducible. Moreover, a l l the irreducible representations of G w i l l be obtained in this way (for details, see Lomont). A method for finding the allowable representations of a group G of a non-symmorphic space group when the vector k is on the surface of the B-Z has been given by HerringiimXThe same method i s applicable to vectors k ending inside the B-Z.) 7 He studied the non-symmorphic space groups: the space group 0 ^ (diamond structure) and the space group D6^ (close-packed hexagonal). We w i l l f i r s t explain his method and then we w i l l 19 apply i t to the space group D 4 h . -21-If (E I t) is an element of G such that k't = 2trx(integer), then any element (R i|t i)(E I t) of G k w i l l be represented, in an allowable representation, by the same matrix as the element (R j j t i ) . The set of primitive translations which satisfy the condition k't =» 2TTX (integer) constitutes a subgroup of T. This subgroup is called the kernel of the representation of G and w i l l be denoted by T k. Instead of considering the group Gk, Herring considered the factor group G^/T^. Since we are looking for those representations of G k which have the property of being allowable, the elements of G k / T k which correspond to the cosets of G k consisting of only primitive translations must also be ilc • tT represented by unit matrices multiplied by e . We now proceed to describing a method of finding the tables of characters of irreducible representations of the various groups according as the vector k ends in a "point of symmetry", on a "line of symmetry" or in a general point of the boundary of the B-Z. But f i r s t of a l l let us give some definitions. We w i l l say that a point on the boundary of the B-Z i s a "point of symmetry" i f the group G k of the vector k ending in k* i t contains more elements than the group G of any neighbour-ing vector k'. A "line of symmetry" is a line such that a l l the vectors k' k * terminating on i t have the same group G which contains more elements than the G of any k" near the line but not on i t . Points on the surface of the B-Z which are not points of - 2 2 -symmetry and which do not belong to a line of symmetry have a group G containing a reflection or a glide reflection in addition to the translation group. Let us now find the tables of characters for the group Gk/T of a point of symmetry. The elements of the factor group are cosets of the form (R | t ) T k where 7 can be 0 (a non-primitive translation "C ) or a primitive translation not belonging to the kernel ( t plus a primitive translation not belonging to T^). Prom now on we w i l l indicate a coset (R I t ) ! * simply by (R I t ) . These cosets (being elements of GVT*) are divided into classes. As usual, we w i l l say that two cosets (R | t) and (S It') belong to the same class i f there exists another coset (U 1 t") of G^ /T1* such that: ( u | t " ) " V l t ) ( u | t " ) : ( S | f ) III.1.5 To find the characters of the various classes we w i l l use Burnside's theorem (£og. III.1.10 below) and the condition where h^ is the number of elements in the i-th class, y^ . is i t s character and g is the order of the group G*/T^. But let us f i r s t derive Burnside's equations which give the characters of the various classes in a group G of elements Aj. Let a class CJL have elements: C± » (A^1',A2(il,... A^). It is known (see Lomont 1959) that every product CjC^ of two classes -23-of a group G can be decomposed into a sura of classes, i.e., Q C k C, HI.1.7 Since the matrix representing the class commutes with a l l the matrices representing the elements A of G, i t follows from Schur's lemma that M(C^) = "\\In where " j ^ i s a proportion a l i t y factor and I n is the unit matrix of dimension n. There-fore from III.1.7 we obtain: or W " h  I I 1 - 1 - 8 On the other hand in a representation of dimension n, the character of MCC^ i s ^ n and since the character of M(Ci) must also be equal to the sum of the characters of the matrices of the elements A]*-' ,A2** ,... A ^ ' i t follows: or •*7, III.1.9 Combining III.1.8 and III.1.9 we obtain: -24-1 • z III.1.10 where n i s the character of the i d e n t i t y element. Eqst. III. 1.10 are Burnside's equations. It i s not always necessary to consider and solve the complete set of equations III.1.6 and III.1.10 to f i n d the table of characters of a group cfc/Tr*. In f a c t , l e t (E I t j ) be a primitive t r a n s l a t i o n not belonging to T k* i f i t happens that f o r two cosets A * and A " belonging to the same class Cj (E\K)A'--A" III.1.11 then the character of the class Cj i s zero. The proof of t h i s statement i s as follows. From III.1.11 one gets: X ( E l M V X j • III.1.12 We are looking for those representations of G k for which the matrices corresponding to pure translations (E ) t) are of the form e i k , t I n . In other words % ( E | t x ) f 0, therefore III.1.12 gives X = 0. •J When such cases occur, the number of equations to be solved i s evidently reduced. -25-From the way the tables of characters are constructed one can conclude that a l l the Irreducible representations found are allowable i r r e d u c i b l e representations of G^/T^. Now, l e t us consider a vector k' ending on a symmetry l i n e of the surface of the B-Z. A symmetry l i n e goes always through a point of symmetry. We define the kernel T*5 of the group G as k' approaches the point of symmetry k. i s the set of a l l primitive translations t' such that e^'"*' as k*-*lc. Again one can construct the classes of G k /T^ as k'-»k and f i n d t h e i r characters by means of III.1.6, III.1.10 and III.1.12. The tables so obtained are the l i m i t s of the characters of G k'/T k' as k 1 — Ic. F i n a l l y , for a general point on the surface of the B-Z there can be at most two ir r e d u c i b l e representations of a group G . These are one-dimensional representations and can be constructed by inspection, (Herring 1942). As we said previously, from the allowable representations of the various groups G k one can construct a l l the ir r e d u c i b l e representations of the space group G. If q i s the dimension of an allowable representation of G , g i s the order of the point group P belonging to G and r i s the order of the point group P* belonging to G k, then the dimension of the correspond-ing representation of G i s ( g / r ) q . Of course i t i s desirable to have a c r i t e r i o n to check whether a l l the allowable representations of a group G and - 2 6 -hence a l l the irreducible representations of the space group G corresponding to the vector k have been found. Doring and Zehler (1953) stated that the following condition must be satisfied: For each group G K the sum of the squares of the dimensions of i t s allowable representations must be equal to the order of the point group III.2 As has been said, the Brillouin zone of a lattice is the symmetrical unit c e l l of i t s reciprocal lattice. The B-Z of the body-centered tetragonal latt i c e , which is the 19 lattice of the space group D 4 n , has been given in Fig. IV. There, points and lines of symmetry have been indicated. k As we said previously, the representations of a group G with k inside the B-Z are given by III.1.2. But i f one wants to write the tables of characters for points inside the B-Z, one can also consider for each vector k the factor group G V T * and, applying the same rules as for points on the surface of the B-Z, find a l l i t s allowable representations. We w i l l give tables of characters for representations of the various groups G K with the end point of the vector k wandering inside and on the surface of the B-Z. We w i l l start considering the groups of vectors k with the end point inside the B-Z. For a "general" vector k inside the B-Z, the group G K contains only primitive translations and T k consists of just *-* k one element t = 0. The representation of G i s a one by one 19 representation. The irreducible representation of D4ft -27-corresponding to k i s 16 by 16 because the order of' the point h 19 group D 4 belonging to D 4 h i s 16. r(0,0,0). When k = 0, G k i s the ent i r e space group G and the kernel i s the entire group of translations. Therefore G /T r => G/T. We know that for any space group G the factor group G/T i s isomorphic to the point group P belonging to G. 19 In case of the space group D^h , G/T i s isomorphic to the point h h group D 4 . Since a l l i r r e d u c i b l e representations of D 4 are known, a l l Irreducible representations of G/T are also known and they are allowable representations. See Table I. A(0,ky,0). This i s a general point on the y-axis which i s a l i n e of symmetry for the B-Z. The point group P A i s C 2 y . We w i l l give the l i m i t characters as A — T and as A -» X . As A(0,k y,0) — r, (0,0,0) the kernel T A i s the group of a l l t r anslations and G^/T i s isomorphic to the group C 2 v. As A — X(0,7r/^*t,0) the kernel T A contains a l l the pri m i t i v e translations n-^t^+ng^+^^a w i " t h n 1+n 2 aeven. The l i m i t i n g character of an operation (R | t) w i l l simply be the character of the same operation R i n an i r r e d u c i b l e represen-t a t i o n of the point group C 2 v times the l i m i t of e i A > t as A -* X. See Table II. Z.(k x,ky,0). This i s the general point along the l i n e of symmetry k x=k y, k^O. The point group P £ i s C 2 v. For f -» P and Z M ^ / f ^ t ,w/{2t, 0) the remarks made for A — r and A -* X are also v a l i d here. See Table I I I . A(0,0,k ?). This i s a general point of the l i n e of -28-symmetry x=y=0. The point group P is C 4 y. As A - » P , the kernel T A is the whole group of translations and the factor group G V T is isomorphic to C 4 y. See Table IV. Let us consider now the points on the boundary of the B-Z. X(O tir//^t f 0). This i s the intersection of the y-axis with the face y==-r//2't of the B-Z. The group P X is D 2 h > The kernel T: contains a l l the primitive translations n2t2+n2t2+n3t3 with n^+n2=even and n3=any integer. -Since X i s one of the most interesting points on the surface of the B-Z, we w i l l explain in detail the process of finding the irreducible representations of the factor group G X / T X . The elements of G X / T X are: (EI0), ( E l ^ ) , ( C 2 z i 0), ( C ^ l t - ^ , ( C 2 x l 0), ( C 2 x l t x ) , ( C 2 y l 0), ( C 2 y l t x ) , (I I T ) , (I It+tj), (o- hl-c-), C^hlTr+t!), < ° V x z l T > - < < 3 " v x z P + t 1 ) , ( c r v y 2 | r ) , ( 6 " „ y z | t r + t x ) , where the rotational parts are elements of D 2 h, tr i s given by II.2 . 1 , t j is given by II.2.2 and represents a l l primitive trans-lations not belonging to T**'. To find the classes of G*/T* one uses Eq. III.1.5. Only elements with the same rotational part can belong to the same class. In fact, let (S I t s ) and (R | t R ) be two different elements of G V T * 5 , then using Eq. III. 1.5 i t follows: -29-(sh iriR|tJ(S|t s )--(s- , l-S- 1 t 8 ) ( R s | R t , * t J I I I . 2 . 1 But in G X/T X a l l the rotational parts of the elements commute, therefore: The classes of G^/T^ are found to be: 16 G l 1 (E I 0) C2 1 (E I t x) C3 2 ( c 2 z i 0,ti) C4 2 ( c 2 x i o,t x) C 5 1 (C 2 y l 0) C6 1 ( c 2 y i t l ) c 7 2 (I +t x) C 8 2 (orh| r f r + t x ) C9 2 < ° " V W B I t + t i > C10 2 From III.1.12 i t follows that the characters of C„, C o C 0, C_, C.n are zero. Using III.1.6, III.1.7, and Y o 9 I v III.1.10 we can find.% , % 2 , % g , % Q t -30-W1+MI-K in.2.3 If n i s the dimension of an allowable Irreducible represen-tation of Gx/T*, the character X i o f 05 | °) i s ' X i ° n» an<-the character of (E | t j ) is ne 1 = -n ; i.e., X2 3 1 ~ n« From C 2C 5 » Cg i t follows: f ' ^ ) X f - X f or -)L-% / W / W /TV ^ On the other hand: CgCg =» Cg , therefore: X r Xe _ (r^) or- l XyXff-" / y i ' i ' " X - - " / v u l Sos we have: From Eq. III.2.3 i t follows: So we have the two solutions: Xj^Xs^-^^Xg*32 a n t* ^l = s X6 = !"*^2 = s""^5 0 = 2 ' F o r a 1 1 the characters of G X/T X see Table V. M(»r/ffft ,Tr//ITt ,0). This i s the intersection of the line kx^ky, kz=0 with the line k x - k y - i r / f 2 , t . The group -31-i s D' . The kernel T contains a l l the p r i m i t i v e trans-4 la t i o n s n^t-jL+n2t2+tt3t3 with n1+n2+n3=even. The r o t a t i o n a l parts of the elements i n G^/T15 do not commute, therefore i t i s possible that cosets with d i f f e r e n t r o t a t i o n a l parts belong to the same class. For example, the presence of C^zt C 4 2 * S 4 z ' S4z i n D 4 h m a k e s ( C 2 x l °>> <C2y' 0 )> ( C2x' * 1 > » ( C 2 y l *1> belong to the same class. For the table of characters f of G M/T M see Table VI. N(TT/2 f2"t,v/2flTt ,TT/2S) . The coordinates of t h i s point are given by the so l u t i o n of the following system of equations: J I - Y «. JL. y 4 JL z .JL _JL -0 • VT t * V7 b x s 2t l 2s1 - u X = - l r r ? vT t The group P1* i s C 2 f a. The kernel T N contains a l l the primitive translations n±tj+n2t2+n3*3 w h e r e n i i s ©ven, and n 2 and n 3 can be any integer. See Table VII. W(0,TT//2t,kz). This i s a general point on the l i n e ky^Tr/ Z l t , which Is a symmetry l i n e on the boundary of the B-Z. The group P W i s C 2 y. As W X(,0tv/{2ttQ) the kernel contains a l l p r i m i t i v e translations n^t^+ngtg+nglTg with nj+ng^even. See Table VIII. V ( T / / 2 " t , T r / ^ t , k z ) . This i s a general point on the -32-line kx«ky=-r/y"2\, which i s a line of symmetry of the B-Z. The point group P V i s C 4 v« As V -* MOr/lf^t,-r//2,t,0) the kernel TV contains a l l the primitive translations nl*l + n2^*2 + n3^3 w i t n nj+ng+ng'-even. See Table IX. Y(k x,7r/f2't,0). This is a general point on the i line ky^Tr/fSTt, kz=Q, which is a symmetry line on the boundary of the B-Z. The group P Y i s C^y. As Y — X(0,ir/V<rt,0) the kernel contains a l l the primitive translations ni*-i + n2^2 + n3^3 w i t J l n 1+n2 =° e v e n a n d n3 a n y i n t e S e r « As Y — M(ir/v^t,T/^rt,0) the kernel contains a l l the primitive translations n-^tj+ngtg+ngT^ with n^+ng+ng^even. See Table X. The reason why we have considered limit representations k of groups G of vectors k with their end-point on a symmetry line, (as for example the representations of G ^  as A -» T and A -*• X), i s that they are necessary for constructing the so-called compatibility relations, which* are of importance in physical applications. What the term "compatibility" means, is explained below. Fir s t we give without proof some statements on continuity properties*of representations. ^ ^ ' ^ i 0 ^ ^ ^ ^ ^ ^ ^ : r t } \ . - ^ ' On a line of symmetry the representation r t t of G i s a continuous function of k. If a line of symmetry terminates in a point k o of higher k k o symmetry, the group G i s a subgroup of G . If, among the matrices of a representation T k ° of G k°, -33-one considers only thevmatrices of those elements of G ° which occur also i n G k , we get a representation ^( s) 5° °f G k which i s c a l l e d by Lomont a "subduced" representation of G*. And, of course, as k k o the representation r k ° subduces a l i m i t representation r k ~ " k ° of G k. This subduced l i m i t representation of G k i s i n general decomposable into a d i r e c t sum of allowable i r r e d u c i b l e l i m i t representations of G k: r"* 1 c k * r w " u ° in.2.4 k k - » k If c t ° i s d i f f e r e n t from zero the representation 1^  ° i s sai d to be "compatible" with the representation T k o . Compatibility tables follow the tables of characters. Since each compatibility table r e f e r s to a f i x e d point of symmetry k o , no s p e c i a l symbols for l i m i t representations are necessary (that i s , we would write r k instead of r k ~ * k o ) . -34-TABLE I Characters of allowable representations of the group of T . r, r . ( C M ^ I * ; ( C W , C l y | 0 ) k v s ; ; i o ) K M fclxy >*o\iy\° rk r r r4 r, r g r,. H, 2 0 -2 0 0 2 0 -2 0 0 2 0 -2 0 0 -2 0 2 0 0 TABLE II Characters of allowable representations of the group of A . A ( r J A,(rj A,(r) z ^ n (E|0 (Crl° ( R l t ) ' ( E l t J A,(x) A,(x) A,00 A 4(x) - 3 5 TABLE I I I Characters of allowable representations of the group of Y.. W tip) ijr) zk(ry H*) V 1 ) £ w ( M ) -xlMfc) TABLE IV Characters of allowable representations of the group of A. ( C..M A,(r) A t(r) A,(r) A„(r) A , ( r ) 2 0 -2 0 0 -36-TABLE V Characters of allowable representations of the group of x, (E l " ) 2 2 -2 -1 0 0 0 0 2 -2 -2 I (l|t,r,t f) 0 0 0 0 0 0 0 0 -37-TABLE VI Characters of allowable representations of the group of 1 "* (Flo) 2 2 I 2 M M -2 -2 -I -2 0 0 0 0 2 2 -I -2 M M -2 -I - 2 2 [C^.c^o.t,] 0 0 0 0 0 0 2 -2 C l 5y|t ) / ( > C l ) < y |t.t, 0 -2 2 +t t) 0 0 0 0 0 0 0 0 0 0 0 0 ) o 0 0 0 ^ o U y ' ^ S y l ») 2 -I 0 0 ' C 1 G~ j „ ^ olxy / « « Y '.I -2 I 0 0 -33-TABLE VII Characters of allowable representations of the group of N . IV, Nt • N j Nk (t\0) I I I I (EU,) -I -I _ -I -I ( C t W y l T t t , ) -| | - | | ( I h ) I I "I -I 'I "I I I ( ^ U y ) O ) I "I "I I K x y K ) -I I I -I TABLE VIII Characters of allowable representations of the group of W . M 2 -39-TABLE IX Characters of allowable representations of the group of V . (Cui .c ;* 8 |f (c„|o) - L \4(M) 1 1 -1 -1 -2 L -u 0 -i 1 0 1 1 -2 -1 -1 2 -t - u 0 u I 0 -1 1 0 1 -1 0 TABLE X Characters of allowable representations of the group of Y . W ! X ( " ) M 2 (C„|£>,t,) 0 -40-TABLE XI Compatibility r e l a t i o n s between P and A,Z A . n r, n, r. r, n. * A , A t A j Au A j A , A 1 + A , £1 I , *u A . A , A , A , A , A 5 A r TABLE XII Compatibility r e l a t i o n s between X and A , W, Y. K - X A ^ A , A 1 + A 4 TABLE XIII Compatibility r e l a t i o n s between M and I, V, Y. y. X X -41-CHAPTER IV Double Space Groups; Their Properties and Representations 19 f Irreducible Representations of the Double Space Group D 4 h IV.1 It i s well known that under a ro t a t i o n R of the co-ordinate system to which a physical system i s referred, a wave function y of the system behaves d i f f e r e n t l y depending on whether ^ i s only a function of the s p a t i a l coordinates x, y and z, or i s also a function of the spin coordinate s. The operator 0 R corresponding to the r o t a t i o n R i s a point transformation i f the wave function y depends only upon the cartesian coordinates of the p a r t i c l e s i n the system. But i f y depends also upon the spin of the p a r t i c l e s i n the system, then the operator 0% i s s p l i t into two factors. Namely: 0 = U * Pfl IV. 1.1 R, R. * Here P R a f f e c t s only the position coordinates i n the wave function of the given system, U R a f f e c t s only the spin co-ordinate s. In the s p e c i f i c case of a c r y s t a l one i s interested i n wave functions describing electrons i n the c r y s t a l or, re-ducing the N-electron problem to the 1-electron problem, i n 1-electron wave functions. Therefore the spin coordinate s can only be +1 or -1. Hence the matrix u 2(R) representing the operator U R must -42-be a two by two matrix: u,(R) Actually, as i s well known, two matrices ± U 2(«,P,Y) operating in the spin space correspond to a-rotation R with Euler's angles a, p and <y. These matrices are unitary and unimodular (see for example Wigner, 1959). They are: 1 rMf): 2 1 & coo A-r2 IV.1.2 Considering both the spatial coordinates and the spin co-ordinates we can say that to each rotation R correspond two direct product matrices: IV. 1.3 To the inversion I in the product space correspond the two direct product matrices: IV. 1.4 -43-therefore, to an improper ro t a t i o n IR, which we know i s the product of a proper r o t a t i o n R with inversion I, w i l l correspond the .two matrices: t (R ) x p ( l f t j IV.1 .5 Since we are not interested i n the dependence of the wave functions on s p a t i a l coordinates, we can assume the electron being at the o r i g i n of the coordinate system. Hence the group U 2 of a l l unitary unimodular two-dimensional matrices u 2 i s a representation, which i s i r r e d u c i b l e , of the group R 3 . of a l l r e a l three-dimensional rotations. And since to each element of R 3 there correspond two elements i n U 2 i t i s customary to say that U 2 i s a double valued representation of R 3 > What has been s a i d for the group R_ can be repeated for the •5 group (R of a l l r e a l non-singular inhomogeneous transformations (R I t ) . As we s a i d i n Chapter I I , Section II.1, the group R 3 of pure rotations i s a subgroup of (R . If now we l e t the two matrices (± u 2(R)| 0) of U 2 correspond to an element R of R 3 and the two matrices (± u 2(R)| t) correspond to an element (R I t) of (k, , we w i l l say that the set Hg of matrices (± u 2(R)| t) and (± u 2(R)l 0) i s a double valued representation of <R . The m u l t i p l i c a t i o n of two elements (u2(R][) I t j ) and (u 2(R 2)| t 2 ) of U 2 i s defined by: -44 Let us consider now a space group G which we know is a discrete subgroup of (A. Since Ii 2 is a double valued re-presentation of , to each element of G correspond two matrices in Kg. The set of matrices of 'U 2 corresponding to the elements of G i s a subgroup G + of i s t h e n possibl to define a double space group in the following way: the "double" group G f of a space group G is an abstract group whose elements have the same multiplication table as the matrices of H 2 corresponding to the subgroup G of (SL . Rules which help to find the structure of a double space group from the known structure of the corresponding space group were obtained by E l l i o t t (1954). They were obtained by generalizing the rules derived by Opechowski (1940) for double point, groups. We shall state them again. It i s well known that to a class C Q of conjugate elements in R 3 there correspond two different classes Cn' and Cn' in U2. (A class C n of conjugate elements in Rg is made up of a l l rotations through an angle 2ir/n about a l l the possible axes). Exception to this rule i s the class of rotations through an angle TT. To such a class in R 3 corresponds only one class in U That the elements of C n and do not belong to a same class of U 2 means simply that there i s no element | of U 2 such i , * tt i tt that yn | = £ yn where yn and yn are elements of C n and C n respectively corresponding to «yn in C n. Let us consider now the group & whose elements have the form (R | t ) . To a class ^ n of conjugate elements in (R, there -45-correspond, i n general, two d i f f e r e n t classes £ n and 7 / n* i n Ug. Since in the group tig there i s no element £ which s a t i s f i e s the r e l a t i o n Y N $ = | yn" , there i s no r o t a t i o n a l part f of an element of 14 2 which would s a t i s f y the same r e l a t i o n . In other words the translations t do not a f f e c t the determination of classes i n t i g . Again an exception to t h i s rule i s the class £ 2 of (R, whose elements represent a r o t a t i o n through ir followed by a t r a n s l a t i o n . We s a i d previously that to a class C 2 of R 3 corresponds always only one class C 2 a C2' i n U 2. In other words there ex i s t s an element ^ of Ug such that y 2 | ~ J 7 2 i s s a t i s f i e d . As Opechowski ;(1940) has pointed out j i s a r o t a t i o n through TT. In the case of the group we can say that to a class tf2 o f corresponds only one class £ 2 3 £g in ^ L l 2 i f there Is an element ( | 2 I t) such that ( n , | t , ] ( f » I O = ( i . | t ) ( t ; i » 0 i v . 1 . 6 where <Y 2)|1) and (<y 2 | i ) are the elements corresponding to ( 7 2 I r ' ) of ^ 2 i n ^ 2 and respectively. Prom Eq. IV.1.5 i t follows that two conditions must be s a t i s f i e d : IV.1.7 -46-Both conditions are satisfied in the case of the group In fact (R contains a l l the elements with any real non-singular three-dimensional matrix and any real three-dimensional vector, and therefore i t contains also an element satisfying the con-ditions given by Eq. IV.1.6. We can therefore conclude that to a class £ 2 o f >^ corresponds always only one class in 1A2» Let us consider now a space group G. From the properties of the group i t follows: 1) To each class of G whose elements have a rotational part different from a twofold rotation correspond always two classes of G + 2) To a class ? 2 of G corresponds only one class ^ 2 a ^ 2 of G r i f there i s a transformation ( I 2| t) with | 2 either a twofold rotation about an axis perpendicular to at least one of the axes of the rotations in the elements of fcg or a reflection in a plane containing at least one of the Mentioned —* ' V' i —* axes and with t satisfying the relation (E- j 2 ) 1 = (E-7 2)t. As far as irreducible representations of a double space group are concerned, i t is possible to state a few rules similar to those existing for double point groups. 3) Each irreducible representation of G i s also an irreducible representation of G + . The proof i s just the same as that given by Opechowski for double point groups. If, as Opechowski did, we c a l l T '•specific" irreducible representation of a double space group G -47-an i r r e d u c i b l e representation of G which i s not an ir r e d u c i b l e representation of G, we can say exactly as Opechowski did for the double point groups: 4)A Hecessary and s u f f i c i e n t condition for an irr e d u c i b l e representation of G f being a s p e c i f i c i r r e d u c i b l e represen-t a t i o n of G + i s that the character, d i f f e r e n t from zero, of any element (R | t ) ' of G t be equal and of opposite sign to the character of the element (R I t ) " . The proof runs exactly the same way as f o r the double point groups. From 4) two rules follow: 4a) When to a class of a space group G correspond two classes i n the double space group G^ , i n a s p e c i f i c i r r e -ducible representation of G^the two classes have characters which are equal i n absolute value but of opposite sign. 4b) When to a class 6 2 °* a s P a c e group G corresponds only one class £ 2 i n G » i n a s p e c i f i c i r r e d u c i b l e represen-t a t i o n of G + the class <£ 2 has character zero. IV.2 Using the previous r u l e s to define the classes of a double space group and some of t h e i r characters, and using T. L\x.\% ** order of the double space group, and *•» . his z l we can get the tables of characters f o r the double space group D 4 h . See Tables XIV to XXIII. In the tables we w i l l denote the two classes of the double group of a vector k corresponding to a class (R|t) of the simple group of the vector k by (Rlt) and (R|t). (According to the notation introduced i n the text, one would -48-use (Rlt)* and (R It)" J> The wave functions of electrons, when one considers also t h e i r spins, are lin e a r combinations of products of s p a t i a l functions with spin functions. Since a spin function trans-forms as D^, a t o t a l function transforms as the d i r e c t product of a s i n g l e group representation with D|. This d i r e c t product can be decomposed i n terms of the s p e c i f i c representations of the double group. In the character of a r o t a t i o n through an angle 2ir/n i s 2 cos ir/n. We w i l l give a table of the d i r e c t products of the single group representations with for each point. Some compatibility tables are included. -49-TABLE XIV Characters of the s p e c i f i c representations of the double group of T . n, (EH i 2 -2 -2 -2 -2 Iff -iff Iff -iff' . VT ( c . „ c „ | o ) 0 0 0 0 (C„.Cl>,C,.,C„|i>) 0 0 0 0 (^ -j.*/ »'^-i2y »^"ixy ' ^ * i 5 y | ^ ) 0 0 0 0 a 2 -2 -2 -2 -2 2 I 4f -il? ft -i/r ifT !fT -if? 0 0 0 0 0 0 0 0 0 0 0 0 J. r . - v r v v r , , r.-iv -50-TABLE XV Characters of the s p e c i f i c representations of the double group of A • 1*1°) (c'lo) ( c l y , c i y | o ) (R|t)-(f|ti i -2 0 0 0 2 -2 0 0 0 A / l y - A ^ A ^ = A ^ D V i - A r TABLE XVI Characters of the s p e c i f i c representations of the double group of E . iJT) l (E|0) 2 2 -2 -2 ( ^ I * y 3 ^ t x y I'1') 0 0 0 0 0 0 (Mb]*(e|t«) -51-TABLE XVII Characters of the s p e c i f i c representations of the double group of A . A<(r) '"A7(r) 2 2 £ |0) -2 -2 . C „ . C 8 8 | 0 ) 0 0 ° J . y ^ d x y ' ^ o l . y ^ O l x y l 0 ) 0 0 A ^ : A , » D , . =A 7 -52-TA3LE XVIII Characters of the s p e c i f i c representations of' the double group of X. * 3 X, (E|o) I 2 (F|o) -2 -I (E|t ,) -2 -2 (E UI) 2 2 ( C ^ c ^ o ^ ) 0 0 ( C „ . C „ | o J t 1 ) 0 0 ( C y . C j o ) 0 0 (iU^*^) o o K .^U -^M o 0 i f c r « M i ) 2 -2 ( ^ h ^ l • (*«, !*) ^ V \ - X ^ ^ - - X 3 + X u -53- • TABLE XIX Characters of the s p e c i f i c representations of the double group of M . (E|o) k (l\o) -i+ (E|M + m <* Cc». - c*Jo) o ( c „ >c„|t,) 0 (c,x , c i y , c „ . c l y | o . M o ( l | * . t + t,) 0 ( I M ^ ) o (c,. y , c w y | r ) , ( c l J J Y i c l l t y | T * t , ) o ( C u r y , C l S y | r ) . ( C „ y . C ^ r ^ ) o K l * y '^olxy , ^ U y • ^oUyK) 0 -54-TA3LE XX Characters of the s p e c i f i c representations of the double group of N . (*|o) (i |r).(T|r^) ( M t ) - ( f l M * 7 IV, TABLE XXI Characters of the s p e c i f i c representations of the double group of W . ^(x) w » v,(x) vf(x) (E|O) I I I ) (l\o) -I -l -I - l (c„|o),(c„|t,) | i -l -I (^.Ul^l^M i - i i -I (R11)«(E| - X ( R l t ) -55-TABLE XXII Characters of the s p e c i f i c representations of the double group of V . V , ( M ) >|0) 2 2 » ) -2 -2 VTi-0 0 0 0 > 5WK-rl°) 0 0 -56-TABLE XXIII Characters of the s p e c i f i c representations of the double group of Y . I ' M ( f i " ) (U°).(e..K) ( f f j T ] . ( 5 t )*.!:,) K» , l T ) - ( g - « , l T * t , ) X M = y > ( M ) x M ^ N y ^ M £ ( x)=y f ( M ) - u - L - t -x(a|t) TABLE XXIV Compatibility r e l a t i o n s between r and A , I , A , A r A r Ac A , A , A , TABLE XXV Compatibility r e l a t i o n s between M a n d £ , V, Y. BIBLIOGRAPHY Bouckaert, L., Smoluchowski, R., and Wigner, E. Phys. Rev. 50, 58 (1936). Doring, W., and Zehler, V. Ann. Physik 13, 214 (1953). E l l i o t t , R.J. Phys. Rev. 96, 280 (1954). Herring, C. J. Franklin Inst. 233, 525 (1942). Koster, G.F. " S o l i d State Physics". Academic Press Inc., New York, 1957. Vol. 5, p. 174. Lomont, J.S. "Applications of F i n i t e Groups". Academic Press, New York and London, 1959. Opechowski, W. Physica 7, 552 (1940). Wigner, E. "Group Theory and i t s Application to the Quantum Mechanics of Atomic Spectra". Academic Press, New York and London, 1959. 

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