Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S16 / 2 / 4

Using the replacement prescribed in the question, $k_{\mathrm{B}}T\to E_F$, givesThe number of electrons in a Debye sphere is consequentlyA useful check eliminates $n_e$: with $r=\hslash {\omega }_{\mathrm{pe}}/E_F$ and $k_F=(3{\pi }^2{n}_e{)}^{1/3}$, $k_F{\lambda }_D=\sqrt{2}/r$ and $N_D=8\sqrt{2}/(9\pi r^3)$.

Having substantially fewer than one electron in this sphere shows that the classical many-particle Debye-screening picture is not applicable. It does not exclude collective plasmons in a degenerate electron gas. A proper zero-temperature Thomas-Fermi screening calculation uses $\partial n/\partial \mu =3n/(2E_F)$ and gives ${\lambda }_{\mathrm{TF}}^2=2{ϵ}_0{E}_F/(3n_e{e}^2)$, rather than literally replacing $T$ by $T_F$. The boxed count follows the source's stated prescription and exposes its physical limitation.