Electronic Heat Capacity

At low temperatures the electronic contribution to heat capacity dominates over the lattice (Debye) contribution. This is a direct consequence of the Pauli exclusion principle — only electrons near the Fermi level can be thermally excited.

Sommerfeld Theory

The electronic heat capacity per unit volume is:

$C_{\text{el}} = \gamma T$

where γ is the Sommerfeld coefficient:

$\gamma = \frac{\pi^2 k_B^2}{3} D(E_F)$

For a free-electron gas, the density of states at the Fermi level is:

$D(E_F) = \frac{3n}{2 E_F}$

Substituting gives:

$\gamma = \frac{\pi^2 n k_B^2}{2 E_F}$

Fermi Energy (Free Electron Model)

$E_F = \frac{\hbar^2}{2 m_e} \left(3\pi^2 n\right)^{2/3}$

Molar Sommerfeld Coefficient

Experimentally, γ is often quoted in mJ/(mol·K²):

$\gamma_{\text{mol}} = \gamma \cdot V_{\text{mol}} \times 1000 \quad [\text{mJ·mol}^{-1}\text{K}^{-2}]$

where V_mol = N_A / n is the molar volume.

Physical Insight

At room temperature C_el ≈ γ × 300 K is much smaller than the Dulong–Petit lattice value of 3R ≈ 25 J/(mol·K), explaining why electrons do not contribute 3/2 k_B per electron as classical theory predicted.

Implement

def sommerfeld_gamma_J_m3_K2(n_m3):
    # Returns γ in J/(m³·K²)
    ...

def electronic_heat_capacity_J_m3K(n_m3, T_K):
    # Returns C_el = γ T in J/(m³·K)
    ...

def sommerfeld_coefficient_mJ_mol_K2(n_m3):
    # Returns γ in mJ/(mol·K²)
    ...