524 APPENDIX B. BOLTZMANN TRANSPORT THEORY
Elastic Collisions
Elastic collisions represent scattering events in which the energy of the electrons remains un-
changed after the collision. Impurity scattering and alloy scattering discussed in Chapter 8 fall
into this category. In the case of elastic scattering the principle of microscopic reversibility
ensures that
W(k,k
′
)=W(k′
,k) (B.25)i.e., the scattering rate from an initial statekto a final statek
′
is the same as that for the reverse
process. The collision integral is now simplified as
∂f
∂t)
scattering=
∫ [
f(k′
)−f(k)]
W(k,k′
)d^3 k′(2π)^3=
∫ [
g(k′
)−g(k)]
W(k,k′
)d^3 k′(2π)^3(B.26)
The simple form of the Boltzmann equation is (from equation B.17)
−∂f^0
∂Ekvk·eE =∫
(gk−gk′)W(k,k′
)d^3 k′=
−∂f
∂t)
scattering(B.27)
The relaxation time was defined through
gk =(
−∂f^0
∂E)
eE·vk·τ=
−∂f
∂t)
scattering·τ (B.28)Substituting this value in the integral on the right-hand side, we get−∂f^0
∂Ekvk·eE=−∂f^0
∂EkeτE·∫
(vk−vk′)W(k,k′
)d^3 k′
(B.29)or
vk·E=τ∫
(vk−vk′)W(k,k′
)d^3 k′
·E (B.30)and
1
τ
=
∫
W(k,k′
)[
1 −
vk′·E
vk·E]
d^3 k′
(B.31)In general, this is a rather complex integral to solve. However, it becomes considerably sim-
plified for certain simple cases. Consider, for example, the case of isotropic parabolic bands and
elastic scattering. In figure B.4 we show a geometry for the scattering process. We choose a
coordinate axis where the initial momentum is along thez-axis and the applied electric field is