Physics 240B, Spring 2008
Homework 1 Solutions
Jesse Noffsinger
February 14, 2008
Solutions are in general not the original work of the author.
Problem 1)
Chapter 6 of Kittel is very helpful for this problem. Portions not worked out explicitly here
appear in the recommended texts.
a)
The total number of electrons up to wavevector
k
is:
N
e
= 2
×
πk
3
(2
π/L
)
3
=
V
3
π
2
k
3
(1)
The density of states,
D
(
k
) is:
D
(
k
) =
dN
dk
=
V
π
2
k
2
(2)
while:
ǫ
=
¯
h
2
k
2
2
m
(3)
Yielding
D
(
ǫ
) =
dN
dǫ
=
V
2
π
2
parenleftbigg
2
m
¯
h
2
parenrightbigg
3
/
2
ǫ
1
/
2
(4)
b)
The total number of electrons is:
N
=
V
3
π
2
parenleftbigg
2
mǫ
F
¯
h
2
parenrightbigg
3
/
2
→
ǫ
1
/
2
F
=
3
N
ǫ
F
bracketleftBigg
V
π
2
parenleftbigg
2
m
¯
h
2
parenrightbigg
3
/
2
bracketrightBigg
−
1
(5)
Entering this expression for
ǫ
1
/
2
F
into Eqn (4) gives:
D
(
ǫ
F
) =
3
N
2
ǫ
F
(6)
1
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c)
C
v
=
∂U
∂T
(7)
The total energy of the sytem being
U
=
integraldisplay
ǫf
(
ǫ,T
)
D
(
ǫ
)
dǫ
(8)
which depends on temperature only through the FermiDirac distribution.
The Sommerfield
expansion [Eqn 2.70 in Ashcroft and Mermin] can be applied, or a direct following of Kittel’s
derivation gives:
C
v
=
π
2
3
k
2
b
TD
(
ǫ
F
)
(9)
d)
If we associate an energy
±
μH
with spin up/down electrons, we can find the paramagnetic
susceptibility of the free electron gas, defined as
χ
=
∂M
∂H
(10)
The total magnetic moment per volume is
M
=
μ
(
n
+
−
n
−
), arising from the difference in
the number of spin up and spin down electrons.
M
=
1
2
μ
integraldisplay
[
f
(
ǫ
−
μH
)
−
f
(
ǫ
+
μH
)]
D
(
ǫ
)
dǫ
(11)
The 1/2 accounts for the spin in the standard density of states formula.
Taking
μH
to be
small with respect to the electronic energies [Ziman pg.332], and the energy derivative of the
FermiDirac distribution to be a delta function peaked at
ǫ
F
:
M
=
μ
2
HD
(
ǫ
F
)
(12)
or
χ
=
μ
2
D
(
ǫ
F
)
(13)
e)
The thermal conductivity is defined as the negative ratio between the temperatue gradient and
the total flux of thermal energy.
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 Spring '08
 Cohen
 Electron, Work, Fundamental physics concepts, free electron, effective mass, Kittel, FermiDirac distribution

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