10. Appendix

656 Appendix B
All the diagrams have corresponding diagrams in which the holes are scattered
by phonons rather than electrons.
Solution to Problem 7.10
This problem is another illustration of the power of group theory. The yellow
exciton in Cu2O is electric-dipole forbidden. However, optical excitation via
an electric quadrupole transition of representation ° 5 (or ° 25 ) is allowed. This
opens up the possibility of a resonant Raman process in which the incident
photon excites the yellow exciton via an electric quadrupole transition, the
exciton is scattered via the creation or annihilation of a phonon and finally a
scattered photon is emitted via an electric dipole transition of representation
° 4 (or ° 15 ). According to the Matrix Element Theorem the phonon in this
case should have the same representation as the direct product of ° 5 and
° 4 (see also the solution to Problem 7.5). The character table for the space
group of Cu2O (O4 h ) is nonsymmorphic (see Problem 3.1) but its factor group
isomorphic to the factor group of the space group of diamond (O7 h ). Thus we
can use the characters for the space group of diamond in Table 2.16 to obtain
the characters of ° 5 ⊗ ° 4 :
{E} {C2} {S4} {Ûd} {C3} {i ′ } {i ′ C2} {i ′ S4} {i ′ Ûd} {i ′ C3}
° 5 3 1 1 1 0 3 1 1 1 0
° 4 3 1 1 1 0 3 1 1 1 0
° 5 ⊗ ° 4 9 1 1 1 0 9 1 1 1 0
By inspection we can show that the direct product: ° 5 ⊗° 4 is reducible to the
direct sum: ° 1 ⊕ ° 3 ⊕ ° 4 ⊕ ° 5 . When combined with the results of Problem
3.1 we find that all the odd-parity phonons in Cu2O should become Ramanactive
via this quadrupole-dipole transition mechanism whenever the incident