1
Theory of Solids HW12 Solution
By Qifeng Shan
Ashcroft and Mermin’s book
161
Let
h
(
k
) be any oneelectron property whose total density is
,
)
(
)
(
4
3
k
k
k
g
h
π
d
H
(16.33)
where g is the electronic distribution function. If, for example,
h
(
k
) is the electronic
energy,
ε
(k), then H is the energy density u; if
h
(
k
) is the electronic charge,
e, then
H
is
the charge density,
ρ
. The value of the density
H
in the neighborhood of a point changes
because electrons move into and out of the neighborhood, some as the result of the
semiclassical equations of motion, and some as a result of collisions. The change in
h
due
to collisions is
coll
3
coll
)
(
4
t
g
h
π
d
dt
dH
k
k
(16.34)
(a) Show from (16.8) that (d
H
/ d
t
)
coll
vanishes provided that all collisions conserve
h
(i.e.,
provided that there is only scattering between levels
k
and
k
’ with
h
(
k
) =
h
(
k
’)).
(b) Show that if (16.8) is replaced by the relaxationtime approximation (16.9), then (d
H
/
d
t
)
coll
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 Fall '09
 Shan
 Solid State Physics, Fundamental physics concepts, dk, coll dt coll

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