At
T
= 0 the Fermi-Dirac distribution has every quantum state with
ε<µ
occupied,
f
F
(
ε
) = 1 for
ε < µ
, and every state with
ε > µ
unoccupied,
f
F
(
ε
) = 0 for
ε > µ
.
ε
=
µ
=
ε
F
is the Fermi surface.
12
ε
ε
F
D(
)
ε
f
(
)
ε
F
T=0
At
T
= 0 the product
f
F
(
ε
)
D
(
ε
) cuts off at
ε
=
ε
F
and drops to zero.
The distribution of quantum sates in
k
-space at
T
= 0 is the interior of a solid ball of
radius
k
F
:
12
Actually the plot of
f
F
(
ε
)
above is produced assuming
µ
is independent of
T
. As we have seen this is not a good
assumption for a classical ideal gas (the classical case is best visualised by the high
T
curves in
f
B
), but is often a good
assumption for low
T
, in particular when
k
B
T<<µ
. As we shall see for a real Fermi gas
µ
has a slight
T
dependence
near
ε
F
and the curves for
f
F
(
ε
)
at different
T
do not all cross at exactly the same point.
62

k
k
k
k
F
y
z
x
For
T >
0 some electrons are thermally excited above
ε
, leaving unoccupied states
below
ε
. The excited and unoccupied states lie in a band of width
k
B
T
around
ε
F
.
ε
ε
F
ε
D(
)
ε
Equal areas
k
T
B
f
(
)
F
T>0
The above graph of
f
F
(
ε
)
D
(
ε
) as a function of
ε
shows the number of electrons with energy
ε
: the total area under the curve is
N
, the number of electrons in the crystal. The chemical
potential can be determined as a function of
T
and
N
by the condition
N
=
integraldisplay
∞
0
f
F
(
ε
)
D
(
ε
)
dε
=
V
2
π
2
parenleftbigg
2
m
¯
h
2
parenrightbigg
3
2
integraldisplay
∞
0
ε
1
2
parenleftBig
e
ε
−
μ
k
B
T
+ 1
parenrightBig
dε.
(33)
63

Heat capacity of a free electron gas
Classically, an ideal gas of
N
particles has heat capacity
13
C
V
=
3
2
Nk
B
,
(34)
and specific heat
c
V
=
C
V
V
=
3
2
N
V
k
B
=
3
2
n
e
k
B
,
(35)
where
n
e
=
N
V
is the number of electrons per unit volume.
14
The observed
c
V
in metals
is only about
∼
1% of (35), a phenomenon which perplexed nineteenth century physicists
but which, from a more modern perspective, can be qualitatively be understood as being
due to the Pauli exclusion principle. Only those electrons within a distance
∼
k
B
T
of the
Fermi surface are free to contribute to the specific heat, electrons any deeper than
k
B
T
below the Fermi surface have their dynamics ‘frozen’ by the exclusion principle: there are
no unoccupied energy states nearby and so these electrons have no degrees of freedom to
contribute to the specific heat.
To get a quantitative expression for the specific heat due to free electrons in a metal
we first need the internal energy
U
(
T
) =
integraldisplay
∞
0
εf
F
(
ε
)
D
(
ε
)
dε
(36)
.
The heat capacity is then
∂U
∂T
vextendsingle
vextendsingle
V
but one subtlety is that we need to calculate
U
(
T
) at
constant
N
, while the right hand side of (36) is a function of
µ
which in turn depends
on
T
and
N
through (33). It is not possible to perform the integral (36) analytically so
we resort to an approximation, but it is a very good approximation.
Define the
Fermi
temperature
,
T
F
by
k
B
T
F
=
ε
F
.

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- Spring '19
- Crystallography, The Land, Cubic crystal system, Reciprocal lattice, primitive cell, lattice systems, BCC Lattice Cell