3.1. st at es i n t he Kronecker product s 3.2. X st at es i n t he Kronecker product s [G]( W) [G](j W). In this case, the irreps of Fk = F [G]( W) [G](j W). The irreps of FX D(x) may be ( f ) (x) i i 4h m are ordinary irreps of the point group D(1). The char obtained directly from Table 1 by multiplying [D(1)]3 on 2d 2d acters of the Kronecker product of the projective irreps [D(1)]37 (for the singlevalued projective irreps [D(x)]15) 2d 2d [FW ]1 [FW ]1 [D(1)]1 [D(2)]1 obtained from Table (1) (2) and [D(1)]12 (for the doublevalued projective trreps 2d 2d 2d are given in Table 4 with the characters of those ordinary [D(x)]67), respectively (see Table 7). 2d irreps of the little cogroup D(1) which appear in the 2d As seen from Table 2 for k(p) = k(i) = W, decomposition (p) = (i) = 1, the space of eight functions (27) with n = l = 1, 2, t(p) = t(i) = 2 transforms according to some [D(1)]1 [D(2)]1 = a1 + a2 + e. (38) 2d 2d small rep of the little cogroup D(x). Four function (27) 4h with n = l = 1, t(p) = t(i) = 2 transform according to the This rep of D(1) induces into F = Oh the rep 2d +  ± ± ±  projective rep +. This decomposition of the induced 1 2 3 4 5 4 rep is obtained using the Frobenius reciprocity theorem (see (D(1))1 (D(1))1 =(D(x))1 +(D(x))3 +(D(x))5 (39) 2d 2d 2d 2d 2d Table 5). of the point group D(x) = D(1) with the factor system Table 7. Characters of single and doublevalued projective irreps 2d 2d of the little cogroup D(x) with the factor system corresponding corresponding to the little cogroup D(x). The functions (27) 2d 4h with n = l = 1 and n = l = 2 can be considered as basis to the little cogroup D(x) 4h functions of the projective rep of the little cogroup D(x) 4h D2d E S4x S1 C2x Uyz Uyz y z induced by the rep (39) of D(x) with the factor system 4x 2d corresponding to the little group D(x). This induction can [D(x)]1 1 1 i i 1 1 1 i i 4h 2d be made using the Frobenius theorem. [D(x)]2 2 1 i i 1 1 1 i i 2d [D(x)]3 3 1 i i 1 1 1 i i 3d [D(x)]4 4 1 i i 1 1 1 i i 2d Table 8. Kronecker products [D(1)]i [D(1)]j in terms of the pro2d 2d [D(x)]5 5 2 0 0 2 0 0 0 2d jective irreps of D(x) (with the factor system of the little co 2d [D(x)]6 6 2 2i 2i 0 0 0 0 2d group D(x), Table 7) and subduction of the projective irreps 4h [D(x)]7 7 2  2i  2i 0 0 0 0 2d of the little cogroup D(x) onto the point group D(x) 4h 2d W1 W2 W3 W4 W5 W6 WThe same procedure may be used for all possible W1 1,3,5 2,4,5 6 7 6 7 6, Kronecker products [D(1)]i [D(2)]j (i, j = 1  7) both for 2d 2d W2 2,4,5 1,3,5 7 6 7 6 6, single and doublevalued irreps. As a result, one obtains Table 5 where the subduction of single and doublevalued W3 6 7 1 2 3 4 W4 7 6 2 1 4 3 ordinary irreps of Oh on D(1) are also given. For example, 2d (W) W5 6 7 3 4 1 2 the direct product [G]3 [G]( W) has the component W6 7 6 4 3 2 1   + (rep [Oh]Kr is induced from the ordinary irrep b1 W7 6, 7 6, 7 5 5 5 5 1,2,3,1 3 of D(1)). After induction (15), one obtains Table 6 that 2d [D(x)]i( GX ) X1 X2 X3 X4 X(x) gives directly the components of the Kronecker products 4h [D(x)]i D(x) 5 5 2, 3 1, 4 6, involved. 4h 2d 3 Физика твердого тела, 2003, том 45, вып. 1380 V.P. Smirnov, R.A. Evarestov, P. Tronc It gives symmetry of phonons at the symmetry points of the Table 9. X states in the Kronecker products [G]( W ) [G](j W ) i BZ. For example, the electric dipole transitions are allowed W1 W2 W3 W4 W5 W6 Wfrom the initial electronic W3 state to the intermediate Wand W7 states (when spinorbit interaction is taken into W1 1,2,3,4 1,2,3,4 5 5 5 5 5(2) account, see Table 5). From these states, with assistance W2 1,2,3,4 1,2,3,4 5 5 5 5 5(2) of the phonons of symmetry W1, the transitions are allowed +  ± W3 5 5 4 3 3 4 1, in the final and X states of symmetry,,, X5 and 6 7 W4 5 5 3 4 4 3 1, 2 ± ± ±,, 2, 2X5 (see Table 5, 8). 6 7 W5 5 5 3 4 4 3 1, W6 5 5 4 3 3 4 1, W7 5(2) 5(2) 1, 2 1, 2 1, 2 1, 2 3(2), 4(2) 5. Conclusions Remark. Numbers (m) in parentheses mean that the preceding irrep enters m times in the product. Our approach to the selection rules in crystals is based on the projective irreps of point groups and consists of three At the same time, they are the basis functions of the steps. small rep of the little group GX(x) contained in the basis of 1) At first, one finds the wave vector selection rules. The the Kronecker product (26) and which, due to the relation results may be given in the form of tables where the rows and columns are numbered by wave vectors of the direct (X [G]WW) = ( FX ) GX G, (40) (x) (x) product factors. Any row (column) of this table contains the representatives of all the irreducible stars of the Kronecker determines all the Xcomponents in the Kronecker proproduct. duct (26). For example, the singlevalued irreps [GX]i 2) Next, it is sufficient to fix one row (the first wave i = 1, 2, 3, 4 are contained in (26) (see Table 8 where all vector) of this table and then consider only columns (the the Kronecker products [D(1)]i [D(1)]j and the subduction 2d 2d second wave vector) giving on the intersection with the of single and doublevalued projective irreps of D(x) on chosen row the wave vectors of different irreducible stars. 4h D(x) are given). After induction (15), one obtains Table 9 Each wave vector is related to some little cogroup. Two 2d that gives directly Xcomponents of the Kronecker products cogroups correspond to the two factors in the Kronecker involved. product and the third — to the resulting one. The intersection of two former cogroups is also a subgroup of the resulting cogroup (the corresponding wave vectors 4. Selection rules for electrical dipole satisfy the wave vector selection rule). The Kronecker transitions product of the projective irreps of these cogroups taken on elements of their intersection is a small projective rep The symmetry of the dipole operator is the vector rep with the needed factor system of the resulting cogroup and of O7: =[G ]4 =. Since the vector k(p) = 0, can be decomposed on the irreducible components, if the v h projective irreps of the latter are known. k( f ) = k(i) (the so called direct transitions, X X, m l L L, W W etc are only allowed). The symmetry of 3) At last, the induction procedure from the projective rep allowed final states for W W transitions is pointed out of the two initial cogroups intersection to the resulting little in Table 5 by the entries of the columns containing b2 cogroups is realized in order to find the definitive selection and e ( D2d = b2 + e) in the row corresponding to rules for allowed transitions (subduction coefficients of the symmetry of the initial state. For example, the direct Kronecker products). The Frobenius reciprocity theorem transition is allowed from the initial state of symmetry W3 may be used at this stage, if the projective rep of the coto the final states of symmetry W6 and W7. groups intersection is decomposed into the irreducible ones. In the case of phononassisted electric dipole transitions, The suggested approach seems to be the most easytothese selection rules have to be supplemented with the use, if comparing to the traditional subgroup [1–4] and selection rules where the operator has the symmetry of the full grup [5] methods. It does not depend either on the phonon participating in the transition. In silicon crystal Si choice of the coordinate system origin and of the kstar atoms occupy the site a of symmetry Td. The symmetries vectors in the description of space groups and their small of phonons in this crystal are given by the rep of the space irreps, or on the form of presentation of irreps of space group G = O7 induced (indrep) by the vector rep t2 of the h groups (small irreps of little groups [9,10] or pequivalent site symmetry group Td [8,11]. The short symbol of this projective irreps of little cogroups [11]). Our approach may indrep is be easily supplemented to the computer program generating the irreps of space groups given at Bilbao Crystallographic (4, 5+), X(1, 3, 4), L(1+, 2, 3+, 3), W (1, 2, 2). Server [12,13]. Физика твердого тела, 2003, том 45, вып. A point group approach to selection rules in crystals References [1] R.J. Elliott, R.J. Loudon. J. Phys. Chem. Solids 15, 146 (1960). [2] M. Lax, J. Hopfield. Phys. Rev 124, 115 (1961). [3] C.J. Bradley, A.P. Cracknell. The mathematical theory of symmetry in solids. Clarendon, Oxford (1972). [4] A.P. Cracknell, B.L. Davis, S.C. Miller, W.F. Love. Kronecker products Tables. Vol. 1–4. Plenum, N.Y. (1979). [5] J.L. Birman. Theory of crystal space groups and infrared and Raman lattice processes of insulating crystals. Handbuch der physik. Band XXV/2b. Springer–Verlag, Berlin (1974). [6] G.L. Bir, G.E. Pikus. Symmetry and straininduced effects in semiconductors. John Wiley & Sons, N.Y. (1974). [7] S.L. Altmann. Induced Representations in Cyrstals and Molecules. London (1977). [8] R.A. Evarestov, V.P. Smirnov. Site Symmetry in Crystals: Theory and Applications. Vol. 108. of Springer Series in Solid State Sciences, 2nd ed. Berlin (1997). [9] S.C. Miller, W.F. Love. Tables of Irreducible Representations of Space Groups and CoRepresentations of Magnetic Space Groups. Pruett, Boulder (1967). [10] The irreducible representations of space groups / Ed. by J. Zak. Benjanin, Elmsford. N.Y. (1969). [11] O.V. Kovalev. Representations of the crystallographic space groups: irreducible repersentations, induced representations and corepresentations 2nd ed. Gordon and Breach, Philadelphia, PA (1993). [12] E. Kroumova, C. Capillas, A. Kirov, M.I. Aroyo, H. Wondratschek, S. Ivantchev, J.M. Madariaga, J.M. PerezMato. Bilbao Crystallographic Server, www.cryst.ehu.es. [13] S. Ivantchev, E. Kroumova, J.M. Madariaga, J.M. PerezMato, M.I. Aroyo. J. Appl. Cryst. 33, 1190 (2000). Физика твердого тела, 2003, том 45, вып.
