10. Appendix

654 Appendix B
Solution to Problem 7.6
Figure 7.26(b) shows the scattering geometry used by Henry and Hopfield in
determining the polariton dispersion of GaP.
(a) From this figure it is clear that the minimum value of the polariton wave
vector q for a given value of kL and the scattering angle ı is: qmin kL sin ı.
(b) From Pythagoras Theorem one obtains that q2 a2 q2 min .ora [q2
q2 min ]1/2 . Since ks a kL cos ı one obtains the result: ks kL cos ı a
kL cos ı [q2 q2 min ]1/2 .
(c) Using the result in (b) we can write:
kL ks kL (kL cos ı [q 2 q 2 min ]1/2 )
kL(1 cos ı) q[1 (qmin/q) 2 ] 1/2
Since ˆL hckL and ˆS hckS (where c is the speed of light inside the
crystal) the Raman frequency ˆL ˆS hc(kL ks) ˆL(1 cos ı)
hcq[1 (qmin/q) 2 ] 1/2 .Forq ≫ qmin one obtains the dependence of ˆL ˆS on
q as hcq. This is the reason why the series of broken curves in Fig. 7.26(a) are
“nearly” parallel straight lines for different values of ı. Forq near qmin these
lines are not really straight lines since ˆL ˆS depends on q as [q 2 q 2 min ]1/2 .
Solution to Problem 7.8
(a) The solution to this part of the problem can already be found in Fig.
7.28. The results of translating the various Feynman diagrams into perturbation
terms can be found in (7.50). The correspondence between the diagrams
and the terms in (7.50) is simply: (a) ⇔ term 1; (b) ⇔ term 2; (c) ⇔ term 3
etc. Students are urged to work through each diagram to learn the “art” of
translating Feynman diagrams into perturbation theory expressions. In addition,
notice how some of the terms are related by permutation of the timeorder
of the vertices. This is a very convenient way to generate all the diagrams
by first starting with a few diagrams and then use them to generate
all the remaining diagrams via permutation of the order in which the vertices
occur in time.
(b) To write down all the Feynman diagram contributing to a phonon-assisted
optical transition in an indirect bandgap semiconductor, let us start with the
schematic diagram in Fig. 6.16. The Feynman diagram corresponding to the
process labeled as (1) in Fig. 6.16 is: