540
10 Quantum Statistics
or, in units of eV, as
F 3.19 7 eV ,
compared with 3.24 eV given in the CRC Handbook of Chemistry and Physics. The
corresponding Fermi temperature, T F , is then
T F 37,000 K .
We see from this example that the Fermi temperature is typically of the order
of several tens of thousands of degrees kelvin, which means that for a typical solid
metal sample, the thermal energy k B T will be only a small fraction of the Fermi
energy. For this reason, the distribution f (() behaves essentially like a step function,
so that approximating it as
f (() ≈ f 0 (() =
1 ,
, << F
0 ,
, >> F
,
(10.4.25)
is actually an excellent approximation even at room temperature. The Pauli Exclusion Principle thus implies that a FD gas has a large mean energy even at absolute
zero.
The result (10.4.4) for U 0 implies that the contribution of the conduction
electrons to the heat capacity, C V , is zero, which differs sharply from the value
3
2 k B per electron that would be predicted by classical equipartition arguments [see,
for example, Sect. 5.5.2]. Physically, this is explained by the fact that F is so high
that only a very small fraction of all electrons will lie within k B T of the top of
the distribution, that is, near vacant quantum states to which they may be excited
thermally in order to contribute to C V . For this reason, only a very small number
of the conduction electrons can contribute to C V , and hence the experimental heat
capacity associated with the conduction electrons will be very nearly zero.
A zero contribution from the conduction electrons to the heat capacity of a metal
will strictly be true only when we have precisely the 0 K distribution. How, then,
will C V behave for f (() differing slightly from f 0 (()? To have some idea of what
to expect, we shall pursue the physical argument a little further. Those electrons
whose energies lie no more than k B T below F can be thermally excited to at most
k B T above F , since electrons deeper below the Fermi surface cannot be thermally
excited (as there will be no empty levels into which they can be excited). The number
of electrons, N
ce , so excited will thus be
N
ce
k B T
F
N ce ,
(10.4.26)
and their average excitation energy will be k B T , so that their contribution, U , to
the internal energy will be
10 Quantum Statistics
or, in units of eV, as
F 3.19 7 eV ,
compared with 3.24 eV given in the CRC Handbook of Chemistry and Physics. The
corresponding Fermi temperature, T F , is then
T F 37,000 K .
We see from this example that the Fermi temperature is typically of the order
of several tens of thousands of degrees kelvin, which means that for a typical solid
metal sample, the thermal energy k B T will be only a small fraction of the Fermi
energy. For this reason, the distribution f (() behaves essentially like a step function,
so that approximating it as
f (() ≈ f 0 (() =
1 ,
, << F
0 ,
, >> F
,
(10.4.25)
is actually an excellent approximation even at room temperature. The Pauli Exclusion Principle thus implies that a FD gas has a large mean energy even at absolute
zero.
The result (10.4.4) for U 0 implies that the contribution of the conduction
electrons to the heat capacity, C V , is zero, which differs sharply from the value
3
2 k B per electron that would be predicted by classical equipartition arguments [see,
for example, Sect. 5.5.2]. Physically, this is explained by the fact that F is so high
that only a very small fraction of all electrons will lie within k B T of the top of
the distribution, that is, near vacant quantum states to which they may be excited
thermally in order to contribute to C V . For this reason, only a very small number
of the conduction electrons can contribute to C V , and hence the experimental heat
capacity associated with the conduction electrons will be very nearly zero.
A zero contribution from the conduction electrons to the heat capacity of a metal
will strictly be true only when we have precisely the 0 K distribution. How, then,
will C V behave for f (() differing slightly from f 0 (()? To have some idea of what
to expect, we shall pursue the physical argument a little further. Those electrons
whose energies lie no more than k B T below F can be thermally excited to at most
k B T above F , since electrons deeper below the Fermi surface cannot be thermally
excited (as there will be no empty levels into which they can be excited). The number
of electrons, N
ce , so excited will thus be
N
ce
k B T
F
N ce ,
(10.4.26)
and their average excitation energy will be k B T , so that their contribution, U , to
the internal energy will be