Free electron model (${ϵ}_{\mathbf{k}}={\hslash }^2|\mathbf{k}{|}^2/(2m)$)

The free electron model approximates conduction electrons by noninteracting particles with the displayed parabolic dispersion relation, filling one-particle states according to Fermi-Dirac distribution. A uniform positive background supplies charge neutrality. Periodic boundary conditions give one spatial state per $k$-space volume $(2\pi {)}^d/V_d$ in $d$ dimensions; the physical spin degeneracy is two. The model captures basic state counting and degeneracy but omits the periodic lattice potential, interaction corrections and detailed electronic band structure.

A spin-degenerate one-dimensional gas has $k_F=\pi n_e/2$ and kinetic energy $N_e{E}_F/3$. For ions $+Ze$ spaced by $2R$, charge neutrality gives $n_e=Z/(2R)$. If the model assigns electrostatic energy $-\alpha Z^2{e}^2/(4\pi {ϵ}_{0}R)$ per ion, the total is $-\alpha Z{e}^2{n}_e/(2\pi {ϵ}_0)+Z{\pi }^2{\hslash }^2{n}_e^2/(24m_e)$. Minimizing gives the displayed density, independent of $Z$ within this specific effective energy; the ion spacing still depends on $Z$. This is not a universal equilibrium density for real one-dimensional materials.

The low-temperature heat capacity of a three-dimensional free electron gas is $({\pi }^2/2)N_{e}kT/T_F$. Expressed per mole of metal atoms with $z$ model conduction electrons each, this becomes the displayed result. Expressed per mole of conduction electrons, set $z=1$. A thermal-window counting argument predicts the $T/T_F$ scaling but gives only an order-one coefficient; it must still retain the number of electrons when changing the molar basis.