 Thermal-shell estimate of electronic heat capacity

For a three-dimensional free electron model, the spin-summed density of states at the Fermi energy is $g(E_F)=3N/(2E_F)$. A shell estimate treats $N_{\mathrm{act}}\sim g(E_F)k_{B}T$ electrons as thermally active and assigns each an excitation energy of order $k_{B}T$. For the illustrative choice $3k_{B}T/2$, $\Delta U\sim (9/4)N{k}_B^2{T}^2/E_F$ and differentiation gives $C_V\sim (9/2)N{k}_{B}T/T_F$. The numerical coefficient is heuristic: the controlled Sommerfeld expansion instead givesBoth estimates express Pauli exclusion principle suppression of the classical heat capacity when $T\ll T_F$. A shell argument establishes scaling rather than a universal prefactor.