Specific heat capacity of the free electron gas (Fermions)

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3.9 Specific heat capacity of the free electron gas (Fermions)

We again apply the Eq. (3.7) to (3.9) for the calculation of the particle number and the energy:

$N (T, V, μ) = \int_{- \infty}^{+ \infty} D (ϵ) f_{β μ} (ϵ) d ϵ$

(3.46)

and

$E (T, V, μ) = \int_{- \infty}^{+ \infty} ϵ D (ϵ) f_{β μ} (ϵ) d ϵ .$

(3.47)

$f_{β μ} (ϵ)$ is the Fermi statistics; obviously we find:

$N (T, V, μ) = \int_{- \infty}^{+ \infty} D (ϵ) f_{β μ} (ϵ) d ϵ = \int_{- \infty}^{ϵ_{F}} D (ϵ) d ϵ$

(3.48)

Multiplying both sides of Eq. (3.48) with $ϵ_{F}$ we get

$(\int_{- \infty}^{ϵ_{F}} + \int_{ϵ_{F}}^{+ \infty}) ϵ_{F} D (ϵ) f_{β μ} (ϵ) d ϵ = \int_{- \infty}^{ϵ_{F}} ϵ_{F} D (ϵ) d ϵ$

(3.49)

Since we calculate the derivation with respect to temperature, we can subtract a constant from the inner energy.
We get

$Δ U = \int_{- \infty}^{+ \infty} ϵ D (ϵ) f_{β μ} (ϵ) d ϵ - \int_{- \infty}^{ϵ_{F}} ϵ_{F} D (ϵ) d ϵ$

(3.50)

Combining the equations (3.49) and (3.50) we find

$Δ U = \int_{ϵ_{F}}^{+ \infty} (ϵ - ϵ_{F}) D (ϵ) f_{β μ} (ϵ) d ϵ - \int_{- \infty}^{ϵ_{F}} (ϵ - ϵ_{F}) (1 - f_{β μ} (ϵ)) D (ϵ) d ϵ .$

(3.51)

The first term describes the excitation of an electron from the energy $ϵ_{F}$ to $ϵ$ and the second term the excitation from $ϵ$ to $ϵ_{F}$ .
Finally we get

$c = \frac{d Δ U}{d T} = \int_{- \infty}^{+ \infty} (ϵ - ϵ_{F}) D (ϵ) \frac{\partial f_{β μ} (ϵ)}{\partial T} d ϵ .$

(3.52)

Only around the Fermi energy $\frac{\partial f_{β μ} (ϵ)}{\partial T}$ differs from zero; we therefor substitute $D (ϵ)$ by $D (ϵ_{F})$ and take

$x : = \frac{ϵ - ϵ_{F}}{k T},$

(3.53)

leading to

$c \approx k^{2} T D (ϵ_{F}) \int_{- \infty}^{+ \infty} \frac{x^{2} e^{x}}{{(e^{x} + 1)}^{2}} d x = \frac{π^{2}}{3} D (ϵ_{F}) k^{2} T,$

(3.54)

Since for free electrons

$N (ϵ) = c o n s t . * ϵ^{\frac{3}{2}},$

(3.55)

we get

$\frac{\partial N}{\partial ϵ} = c o n s t . * \frac{3}{2} ϵ^{\frac{1}{2}} = \frac{3}{2} \frac{N}{ϵ} .$

(3.56)

Therefore

$D (ϵ_{F}) = \frac{3 N}{2 ϵ_{F}} = \frac{3 N}{2 k T_{F}},$

(3.57)

and

$c = \frac{π^{2}}{2} N k \frac{T}{T_{F}} .$

(3.58)

For room temperature and a typical Fermi temperature of several $100 0^{o}$ K follows

$c \approx \frac{1}{100} N k .$

(3.59)

Thus at room temperature the heat capacity of electrons is not important.

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