Selection Rules Connecting Different Points in the Brillouin Zone
Selection rules for indirect radiative transitions and for intervalley scattering are investigated for Ge and Si. Comparison with experimental results of Haynes and of Benoit \`a la Guillaume supports (1) the present picture of the band structure of Ge with a conduction band minimum at the zone boundary, (2) the assignment of ${L}_{{2}^{\ensuremath{'}}}$ as the symmetry of the $\mathrm{LA}$ phonon at the zone boundary in the [111] direction. The absence of the $\mathrm{LA}$ phonon in the Haynes radiative emission experiment still requires explanation. To obtain the required selection rules we take the product of two irreducible representations $i$ and $j$ that belong to different wave vectors k and ${\mathbf{k}}^{\ensuremath{'}}$ The resulting character product is expressed in a form appropriate to a third group ${G}_{{\mathrm{k}}^{\ensuremath{'}\ensuremath{'}}}$ at k\ensuremath{''}=k+k\ensuremath{'}. If the elements of ${G}_{{\mathrm{k}}^{\ensuremath{'}\ensuremath{'}}}$ are applied to k to generate a star and $N(C)$ is the number of points of the star invariant (or equivalent) under any element $R$ in the class $C (\mathrm{of} {G}_{{\mathrm{k}}^{\ensuremath{'}\ensuremath{'}}})$ then the character product is $N(C)〈{\ensuremath{\chi}}^{i}(R){\ensuremath{\chi}}^{j}(R)〉$ i.e., $N(C)$ times the product of the characters averaged over those $R$ in $C$ which belong to ${G}_{{\mathrm{k}}^{\ensuremath{'}}}$ and ${G}_{{\mathrm{k}}^{\ensuremath{'}\ensuremath{'}}}$ (i.e., those whose characters can be found in the tables at k\ensuremath{'} and k\ensuremath{'}\ensuremath{'}).