Coulomb-to-kinetic energy ratio of an electron gas ($E_C/E_F\sim r_s$)

In three dimensions, let $a=(3/4\pi n{)}^{1/3}$ be the mean-spacing radius and $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$ 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 $k_{B}T$.