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

ε_{b}

, and electrostatic potential

ϕ

. An Electron's potential energy is

- e ϕ

, so the local kinetic Fermi energy becomes

μ + e ϕ

. For a three-dimensional spin-degenerate parabolic band at zero temperature,

n (ϕ) = \frac{1}{3 π ^{2}} [\frac{2 m ( μ + e ϕ )}{ℏ ^{2}}]^{3/2}, δ n ≃ e g (E_{F}) ϕ,

(1)

where

g (E_{F}) = d n / d μ

is the spin-summed density of states per volume per energy. The induced charge is

ρ_{ind} = - eδ n = - e^{2} g (E_{F}) ϕ

. Combining it with Poisson equation gives

(\nabla^{2} - k_{TF}^{2}) ϕ = - \frac{ρ _{ext}}{ε _{b}}, k_{TF}^{2} = \frac{e ^{2} g ( E _{F} )}{ε _{b}} .

(2)

For a point external charge

Q

, spherical symmetry and decay at infinity select

ϕ (r) = \frac{Q}{4 π ε _{b} r} e^{- r / λ_{TF}}, λ_{TF} = \frac{ε _{b}}{e ^{2} g ( E _{F} )} .

(3)

Thus the bare

1/ r

potential is cut off exponentially. Fourier transformation makes the same result transparent:

ϕ (q) = ρ_{ext} (q) / [ε_{b} (q^{2} + k_{TF}^{2})]

, so the static longitudinal dielectric factor is

1 + k_{TF}^{2} / q^{2}

relative to the background. For a free Electron band,

g (E_{F}) = 3 n / (2 E_{F})

and

λ_{TF}^{2} = 2 ε_{b} E_{F} / (3 n e^{2})

Figure 1.
Local Thomas-Fermi screening and optical/acoustic modes of an illustrative diatomic chain
. **Details**.

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 μ

. 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.