Clebsch−Gordan coefficients for *X ⊗ *X in diamond O7h−Fd3m and rocksalt O5n−Fm3m

Abstract

By using the method described in the previous paper, based on properties of space group irreducible representations as induced from ray representations of subgroups, Clebsch−Gordan coefficients are calculated for *X ⊗ *X in diamond O7h−Fd3m and rocksalt O5h−Fm3m structures. Tables of coefficients for these stars are presented. An example of explicit calculation of the coefficients is given for these symmorphic and nonsymmorphic groups with multiplicity included in the former.

Selection rules for the double space group O1h

“…Analysis of scattering processes involving photons, phonons or magnons in crystalline solids generally requires the appropriate selection rules to be worked out. Recently attention has been directed to the calculations of the Clebsch-Gordan coefficients of the space group representations [103][104][105][106][107][108][109][110], Birman et al [107][108][109][110] have shown that the elements of the first order, one excitation, scattering tensor are precisely certain Clebsch-Gordan coefficients or prescribed linear combinations ; the elements of the second order, two-excitation process are a particular sum of products of Clebsch-Gordan coefficients.…”

Selection rules for the double space group of the beta-Wolfram structure

“…Recently much attention has been directed to the Clebsch-Gordan coefficients of the space group representations [28][29][30][31][32][33][34][35].…”

Clebsch-Gordan coefficients for scattering tensors in ZnO and other wurtzite semiconductors

“…The Raman scattering tensors in cubic Cu 2 O ($O_h^4 $ ) were analyzed by Birman 6. The mathematical theory of the CGCs was derived by Berenson and Birman 7 who calculated CGCs for diamond ($O_h^7 $ ) and rocksalt ($O_h^5 $ ) 8. Birman et al 9 have shown that the matrix elements of the Luttinger–Bir–Picus ${\bf k} \cdot {\bf p}$ effective Hamiltonian can be constructed as products of the CGCs times symmetrized tensorial field quantities.…”