= Neutral one-dimensional metal equilibrium density
{title2=$n_e^*=24\alpha/(\pi^2a_B)$}
A spin-degenerate one-dimensional gas has $k_F=\pi n_e/2$ and kinetic energy $N_eE_F/3$. For ions $+Ze$ spaced by $2R$, charge neutrality gives $n_e=Z/(2R)$. If the model assigns electrostatic energy $-\alpha Z^2e^2/(4\pi\epsilon_0R)$ per ion, the total is $-\alpha Ze^2n_e/(2\pi\epsilon_0)+Z\pi^2\hbar^2n_e^2/(24m_e)$. 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.
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