10. Appendix

648 Appendix B
References
L. Liu: Effects of spin-orbit coupling in Si and Ge. Phys. Rev. 126, 1317 (1962).
M.Cardona: Modulation Spectrosocopy, Solid State Physics, Suppl. 11 (Academic, New
York, 1969) p. 66-73.
Solution to Problem 7.1
For simplicity, we assume that the two levels n and m (with energies En and
Em) are both non-degenerate, although it is rather straightforward to generalize
the result to the degenerate case. Let Nn and Nm be, respectively, the
occupancies of these two levels. First, for the sake of argument, let us assume
that there is no stimulated emission. If the rate of spontaneous emission from
level n to level m is Anm then the rate of depopulation of the level n via emission
is: NnAnm since Nn of the level n states are occupied. Similarly, if the
rate of absorption from level m to level n induced by a radiation field of frequency
Ó and energy density u(Ó) isuBmn then the rate of depopulation of
level m due to absorption is: uBmnNm. At thermal equilibrium the principle of
detailed balance (see p. 208) requires that the two rates be equal. This means:
NnAnm uBmnNm or Nn/Nm uBmn/Anm
At thermal equilibrium the ratio of the occupancies: Nn/Nm has to be equal to
exp[(En Em)/kBT] at temperature T according to the Boltzmann’s distribution
law (kB is Boltzmann’s constant). Equating these expressions for Nn/Nm
one obtains:
uBmn/Anm exp[(En Em)/kBT]
Combining these results one obtains:
u(Ó) [Anm/Bmn] exp[(En Em)/kBT]
which, after equating hÓ to (En Em), disagrees with Planck’s Radiation Law:
u(Ó)
8applehÓ 3 n 3 r
c 3 {exp[hÓ/(kBT)] 1}
except for kBT ≪ hÓ.
The way Einstein removed this disagreement between the classical result
and Planck’s Radiation Law is to postulate that, in addition to the spontaneous
emission processes between level n and m, there are stimulated emission
processes induced by u(Ó). If we denote the rate of stimulated emission as Bnm
then the only change we have to make to include the stimulated emission processes
is to replace the rate of depopulation of level n by: Nn(Anm uBnm).
Applying the principle of detailed balance again we obtain:
Nn/Nm uBmn/(Anm uBnm).