= Coulomb-to-kinetic energy ratio of an electron gas
{c}
{title2=$E_C/E_F\sim r_s$}
In three dimensions, let $a=(3/4\pi n)^{1/3}$ be the mean-spacing radius and $a_B^*=4\pi\varepsilon\hbar^2/(m^*e^2)$ the effective Bohr radius. The Coulomb scale is $E_C\sim e^2/(4\pi\varepsilon a)$ and the <Fermi energy> scale is $E_F\sim\hbar^2/(m^*a^2)$, so $E_C/E_F\sim a/a_B^*=r_s$. 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 $k_BT$.
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