10. Appendix

Solution to Problem 6.7 633
axes. For E E0 the constant energy curve is shown schematically in the
following figure.
k 2
2 2 1/2
2 k 1 2
1
k 1
2 2
E=E o+k 1– k 2
The DOS is proportional to:
dSk
|∇kE|
kB (dk1)
0
2 (dk2) 2
k2 1 k2 2
Using the result that:
dx
√
x2 a2 log
e x √ x2 a2
k2 B (E E0)
kB
0
dk2
|k1|
kB
0
dk2
k2 2 (E E0)
We obtain the DOS as proportional to:
loge kB
loge E E0 ∼ C loge E E0
The dependence on E of the first term in the above expression can be neglected
since k 2 B ≫ (EE0). As a result, it can be approximated by a constant
C. The dependence of the DOS is, therefore, determined by the second term
which diverges logarithmically at E E0.
The results we have obtained in this problem show that the energy dependence
of the DOS is strongly dependent on the functional relation between
the energy E and the wave vector k which, in turn, depends strongly on the
dimensionality of the k-space and real space under consideration.
Solution to Problem 6.7
(c) A good starting point to solve this problem would be [6.44], the classic paper
by R.J. Elliott, on the intensity of optical absorption by excitons. For direct
excitons the transition probability per unit time (P) for an incident radiation