526 APPENDIX B. BOLTZMANN TRANSPORT THEORY
This weighting factor(1−cosα)confirms the intuitively apparent fact that large-angle scatter-
ings are much more important in determining transport properties than small-angle scatterings.
Forward-angle scatterings(α=0), in particular, have no detrimental effect onσorμfor the
case of elastic scattering.
Inelastic Collisions
In the case of inelastic scattering processes, we cannot assume thatW(k,k
′
)=W(k′
,k).Asa
result, the collision integral cannot be simplified to give an analytic result for the relaxation time.
If, however, the system is non-degenerate, i.e.,f(E)is small, we can ignore second-order terms
infand we have
∂f
∂t∣
∣∣
∣
scattering=
∫[
gk′W(k′
,k)−gkW(k,k′
)]d (^3) k′
(2π)^3
(B.34)
Under equilibrium we have
fk^0 ′W(k′
,k)=fk^0 W(k,k′
) (B.35)or
W(k′
,k)=fk^0
fk^0 ′W(k,k′
) (B.36)Assuming that this relation holds for scattering rates in the presence of the applied field, we have
∂f
∂t∣∣
∣
∣
scattering=
∫
W(k,k′
)[
gk′fk^0
f^0 k′−gk]
d^3 k′(2π)^3(B.37)
The relaxation time then becomes
1
τ=
∫
W(k,k′
)[
1 −
gk′
gkfk^0
fk^0 ′]
d^3 k′(2π)^3(B.38)
The Boltzmann is usually solved iteratively using numerical techniques.
B.2 AVERAGINGPROCEDURES...........................
We have so far assumed that the incident electron is on a well-defined state. In a realistic
system the electron gas will have an energy distribution andτ, in general, will depend upon the
energy of the electron. Thus it is important to address the appropriate averaging procedure forτ.
We will now do so under the assumptions that the drift velocity due to the electric field is much
smaller than the average thermal speeds so that the energy of the electron gas is still given by
3 kBT/ 2.
Let us evaluate the average current in the system.
J=
∫
evkgkd^3 k
(2π)^3