10. Appendix

Equating the two expressions for Nn/Nm one obtains now:
u(Ó) Anm/{Bmn exp[(En Em)/kBT] Bnm}.
Solution to Problem 7.5 649
This expression becomes equal to Planck’s Radiation Law when one assumes
that:
Bmn Bnm and Anm
Bnm
Solution to Problem 7.5
8applehÓ3 n 3 r
c 3
According to Problem 3.7 the wurtzite crystal structure possesses 4 atoms
per primitive unit cell (double the number in the zincblende structure). As
a result, there are 9 zone-center optical phonon modes with symmetries:
°1 ⊕2°3 ⊕°5 ⊕2°6 (or A1 ⊕2B1 ⊕E1 ⊕2E2 using the C6v point group notation).
One should note that the E modes are doubly degenerate and not all of these
optical phonons are Raman-active.
To determine the symmetry of the Raman-active phonon modes we note
that, unlike absorption, Raman scattering involves two electromagnetic (EM)
waves: one incident and one scattered wave. If we want to annihilate a photon
and generate a phonon, as in optical absorption, then the phonon must
have the same symmetry as the photon (which belongs to the same representation
as a vector). We have already obtained this result as one of the many
applications of group theory (see Section 2.3.4). In Raman scattering we annihilate
the incident photon and create a scattered photon with the generation
or annihilation of a phonon. The symmetry of the phonons involved is going
to be the same as in the case where two photons are annihilated to generate
one phonon (i.e. a two-photon absorption process). The reason is that the matrix
elements describing the photon creation and annihilation processes are the
same except for a term involving the photon occupancy factor Np (see Section
7.1). Thus the representation of the phonon (or phonons) involved in Raman
scattering must be contained within the direct product of the representations of
two vectors.
For example, a vector in zincblende-type crystals belongs to the °4 irreducible
representation. Thus the representation of the Raman-active phonons
must belong to the direct product:
°4 ⊗ °4 °1 ⊕ °2 ⊕ °4.
Although the zone-center optical phonons in zincblende-type crystals have °4
symmetry, Raman scattering from phonon modes with the °1 and °2 symmetry
can also be observed in the two-phonon spectra of zincblende-type crystals.
See, for example, Fig. 7.22 for the various symmetry components of the twophonon
Raman spectra of Si.