Number of electrons in a Debye sphere from a plasmon energy ratio ($N_D=8\sqrt{2}/(9\pi r^3)$)

For a three-dimensional ideal degenerate electron gas, let $r=\hslash {\omega }_{\mathrm{pe}}/E_F$. If one heuristically defines a Debye length by replacing $k_{\mathrm{B}}T$ with the Fermi energy, then $k_F{\lambda }_D=\sqrt{2}/r$ and the associated electron count is the displayed expression. A count below one signals failure of the classical many-particle screening interpretation. It does not prohibit plasmons. The zero-temperature compressibility instead gives the Thomas-Fermi screening length squared $2{ϵ}_0{E}_F/(3n_e{e}^2)$.