= Thomas-Fermi screening
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{wiki=Thomas–Fermi_screening}
In the long-wavelength static response of a degenerate conductor, a small electric potential $\phi$ changes the electron density by $\delta n=e g(E_F)\phi$, where $g(E_F)$ is the spin-summed <density of states> per volume per energy. The induced charge is therefore $-e^2g(E_F)\phi$. With background <permittivity> $\varepsilon_b$, <Poisson equation> becomes
$$
(\nabla^2-k_{\rm TF}^2)\phi=-\rho_{\rm ext}/\varepsilon_b,\qquad
k_{\rm TF}^2=e^2g(E_F)/\varepsilon_b.
$$
A point charge $Q$ then has $\phi(r)=Qe^{-k_{\rm TF}r}/(4\pi\varepsilon_b r)$. Larger electronic density of states gives stronger screening. This local response approximation does not capture the microscopic oscillatory tail of <Friedel oscillations>.https://www.damtp.cam.ac.uk/user/tong/em/el6.pdf[Sections 7.7.3–7.7.5: Thomas-Fermi screening and its nonlocal corrections]{ref}
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