Electron gas

A collection of mobile electrons treated statistically. A dense electron gas in a metal generally requires Fermi-Dirac statistics.

A Wigner crystal is an ordered arrangement of electrons stabilized by their mutual repulsion when localization costs sufficiently little kinetic or thermal energy. In a degenerate low-density gas, the Coulomb-to-kinetic energy ratio of an electron gas increases with mean spacing. Classical electrons can also order at large Coulomb-to-thermal coupling, as in electron crystallization on liquid helium. A neutralizing background or confining environment is needed to discuss a stable electron assembly. Dimension, screening and disorder affect the transition.

The 1979 experiment of C. C. Grimes and colleagues studied a two-dimensional sheet of electrons above liquid helium. It reported crystallization at densities $3\times 10^8$ to $9\times 10^8{\mathrm{cm}}^{-2}$, with melting temperatures $0.35$ to $0.65\mathrm{K}$. The sheet was classical: Coulomb energy competed with thermal motion, unlike the kinetic-energy comparison used for a zero-temperature degenerate Wigner crystal. It supplies evidence for electron ordering while keeping these two physical regimes distinct.^[ref]

In three dimensions, let $a=(3/4\pi n{)}^{1/3}$ be the mean-spacing radius and $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$ the effective Bohr radius. The Coulomb scale is $E_C\sim e^2/(4\pi \epsilon a)$ and the Fermi energy scale is $E_F\sim {\hslash }^2/(m^{\ast }{a}^2)$, so $E_C/E_F\sim a/a_B^{\ast }=r_s$. Decreasing density favors repulsion relative to delocalization energy, motivating a Wigner crystal. This scaling does not determine the numerical quantum melting value of $r_s$. A classical nondegenerate gas instead compares Coulomb energy with $k_{B}T$.

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.