PHY251 Homework Set 11
Reading:
Chapter 12, 14 (all sections except 6)
Homework:
Chapter 12, Problems 32,
Chapter 14, Problems 2, 6(Li, Al, Cu only!),9(Li, Al, Cu
only!),10,17,18,31
Hints and Solutions
Problem XII.32
(15 points)
Consider a freeelectron gas at a temperature
T
such that
kT
<<
E
F
. Write down an
expression for the electron number density
N
/
V
for electrons that have an energy in
excess of
E
F
. Show, by making the change of variables (
E

E
F
)/
kT
=
x
, that the
number density is proportional to
T
. Calculate an expression for
N
/
V
under these
circumstances, making use of the fact that
0
∫
∞
(e
x
+ 1)

1
dx
= ln 2.
Hints:
In working out the integral over E the integrand is such that (
x
+
E
F
)
½
2245
E
F
½
.
Solution:
From notes or book:
n
e
(
E
>
E
F
) = (
N
e
/
V
) =
E
F
∫
∞
n
e
(
E
)
dE
=
E
F
∫
∞
g
(
E
)
dE
F
FD
(
E
;
T
) =
= (8
π
√
2
m
e
3/2
/
h
³)
E
F
∫
∞
√
E
dE
(e
x
+ 1)

1
2245
(8
π
√
2
m
e
3/2
/
h
³)
√
E
F
E
F
∫
∞
kT dx
(e
x
+ 1)

1
=
= (8
π
√
2
m
e
3/2
/
h
³)
kT
√
E
F
ln2 = [
√
2
m
e
3/2
/(
π
²
h
³)]
kT
√
E
F
ln2 = (6.85 nm

3
)
kT
√
E
F
ln2.
In turn,
E
F
can be expressed in terms of the total electron density of the
metal...
Problem XIV.2
(10 points)
Assuming that an electron under the influence of a drag force starts from rest at
t
=0,
show that the solution of Equation (141) is
v
(
t
) = (
eE
τ
/
m
) [1

e

t
/
τ
].
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 Spring '01
 Rijssenbeek
 Physics, Electron, Work, li, EF EF kT

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