Neutral one-dimensional metal equilibrium density ($n_e^{\ast }=24\alpha /({\pi }^2{a}_B)$)

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.