Quantum Statistics
235
≈ 3
f
kT
R
ε
Here, we have used Eq. (7.87) for dN/dε and Eq. (7.89). A more detailed
calculation gives
1
e
v
C =
2
2
f
kT
R
π
ε
(7.95)
Since kT/ε f is quite small at ordinary temperatures, the electronic specific
heat also is small and the total specific heat is described quite well by the Debye
theory. It should, however, be noted that the Debye specific heat at low
temperatures is proportional to R(T/θ)
3
[see Eq. (7.66)] so that at sufficiently
low temperatures the electronic specific heat becomes dominant. At low
temperatures, the total specific heat is given by
C v =
2
4
3
12
( / )
5
2
f
kT
R T
R
π
π
θ +
ε
(7.96)
and the observed nonzero limit of C v /T as T → 0, for metals such as copper,
indicates the presence of the linear electronic contribution. Experimentally, in
the case of copper, C v /T for T → 0 is about 0.7 × 10
–3
J/mol/K
2
whereas the value
predicted for copper (ε f ≈ 7 eV), by Eq. (7.96), is about 0.54 × 10
–3
J/mol/K
2
. The
difference is a measure of the deviation of the model from the real situation.
Electrical and thermal conductivities: Some general characteristics of the
electrical and thermal conductivities of metals can be discussed in terms of the
free-electron theory of metals. This discussion will be based on the assumptions
that (i) the conducting electrons move with the velocity v f = (2ε f /m)
1/2
, which is
reasonable since most of the conducting electrons will be in states close to the
Fermi level, (ii) the electrons have a mean free path of λ and that they carry
information over a distance of λ (λ ≈ 500 Å) .
In the presence of an external electric field E, the electrons acquire an average
drift velocity v which is equal to half of the average acceleration εE/m multiplied
by the interval λ/v f between two collisions. Therefore, the current is
1
2
en (eλE/mv f ) where n is the electron density. This satisfies Ohm’s law since
v f , being large, is essentially independent of E. The electrical conductivity is
then
σ = e
2
nλ/2mv f
(7.97)
For calculating thermal conductivity, it is noted that since the electrons
carry information over a distance of λ, the energy carried across an area by the
235
≈ 3
f
kT
R
ε
Here, we have used Eq. (7.87) for dN/dε and Eq. (7.89). A more detailed
calculation gives
1
e
v
C =
2
2
f
kT
R
π
ε
(7.95)
Since kT/ε f is quite small at ordinary temperatures, the electronic specific
heat also is small and the total specific heat is described quite well by the Debye
theory. It should, however, be noted that the Debye specific heat at low
temperatures is proportional to R(T/θ)
3
[see Eq. (7.66)] so that at sufficiently
low temperatures the electronic specific heat becomes dominant. At low
temperatures, the total specific heat is given by
C v =
2
4
3
12
( / )
5
2
f
kT
R T
R
π
π
θ +
ε
(7.96)
and the observed nonzero limit of C v /T as T → 0, for metals such as copper,
indicates the presence of the linear electronic contribution. Experimentally, in
the case of copper, C v /T for T → 0 is about 0.7 × 10
–3
J/mol/K
2
whereas the value
predicted for copper (ε f ≈ 7 eV), by Eq. (7.96), is about 0.54 × 10
–3
J/mol/K
2
. The
difference is a measure of the deviation of the model from the real situation.
Electrical and thermal conductivities: Some general characteristics of the
electrical and thermal conductivities of metals can be discussed in terms of the
free-electron theory of metals. This discussion will be based on the assumptions
that (i) the conducting electrons move with the velocity v f = (2ε f /m)
1/2
, which is
reasonable since most of the conducting electrons will be in states close to the
Fermi level, (ii) the electrons have a mean free path of λ and that they carry
information over a distance of λ (λ ≈ 500 Å) .
In the presence of an external electric field E, the electrons acquire an average
drift velocity v which is equal to half of the average acceleration εE/m multiplied
by the interval λ/v f between two collisions. Therefore, the current is
1
2
en (eλE/mv f ) where n is the electron density. This satisfies Ohm’s law since
v f , being large, is essentially independent of E. The electrical conductivity is
then
σ = e
2
nλ/2mv f
(7.97)
For calculating thermal conductivity, it is noted that since the electrons
carry information over a distance of λ, the energy carried across an area by the