= Solution
<Dielectric screening> is the rearrangement of a medium's charges in response to an applied charge. In a metal, mobile <Electrons> accumulate near a positive perturbation and reduce its long-range potential. <Thomas-Fermi screening> treats the <electron gas> locally as an equilibrium gas whose density responds to the local <electrostatic potential>, ignoring gradients of the density on the microscopic wavelength scale.
Take $e>0$, background <permittivity> $\varepsilon_b$, and <electrostatic potential> $\phi$. An <Electron>'s <potential energy> is $-e\phi$, so the local kinetic <Fermi energy> becomes $\mu+e\phi$. For a three-dimensional spin-degenerate parabolic band at zero <temperature>,
$$
n(\phi)=\frac1{3\pi^2}\left[\frac{2m(\mu+e\phi)}{\hbar^2}\right]^{3/2},\qquad \delta n\simeq e\,g(E_F)\phi,
$$
where $g(E_F)=dn/d\mu$ is the \b[<spin-summed density of states> per volume per <energy>]. The induced charge is $\rho_{\mathrm{ind}}=-e\delta n=-e^2g(E_F)\phi$. Combining it with <Poisson equation> gives
$$
\boxed{(\nabla^2-k_{\mathrm{TF}}^2)\phi=-\frac{\rho_{\mathrm{ext}}}{\varepsilon_b},\qquad k_{\mathrm{TF}}^2=\frac{e^2g(E_F)}{\varepsilon_b}.}
$$
For a point external charge $Q$, <spherical symmetry> and decay at infinity select
$$
\boxed{\phi(r)=\frac{Q}{4\pi\varepsilon_b r}e^{-r/\lambda_{\mathrm{TF}}},\qquad \lambda_{\mathrm{TF}}=\sqrt{\frac{\varepsilon_b}{e^2g(E_F)}}.}
$$
Thus the bare $1/r$ potential is cut off exponentially. <Fourier transformation> makes the same result transparent: $\phi(\mathbf q)=\rho_{\mathrm{ext}}(\mathbf q)/[\varepsilon_b(q^2+k_{\mathrm{TF}}^2)]$, so the static longitudinal dielectric factor is $1+k_{\mathrm{TF}}^2/q^2$ relative to the background. For a free <Electron> band, $g(E_F)=3n/(2E_F)$ and $\lambda_{\mathrm{TF}}^2=2\varepsilon_bE_F/(3ne^2)$.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2017-screening-and-phonons.png]
{title=Local Thomas-Fermi screening and optical/acoustic modes of an illustrative diatomic chain}
The local linear approximation requires a weak perturbation and slowly varying fields on the relevant microscopic scale. It omits nonlocal <Friedel oscillations> and atom-scale structure; a nonlinear local-density treatment is needed near a sufficiently strong charge. Finite <temperature> replaces $g(E_F)$ by the appropriate <compressibility> derivative $\partial n/\partial\mu$. The nondegenerate limit gives classical screening rather than the degenerate Thomas-Fermi length. A larger <density of states> means that a smaller potential can move more carriers, hence stronger screening.
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