Screened Coulomb interactions

Let ρ denote the number density of electrons, and φ the electric potential. At first, the electrons are evenly distributed so that there is zero net charge at every point. Therefore, φ is initially a constant as well.

We now introduce a fixed point charge Q at the origin. The associated charge density is Qδ(r), where δ(r) is the Dirac delta function. After the system has returned to equilibrium, let the change in the electron density and electric potential be Δρ(r) and Δφ(r) respectively. The charge density and electric potential are related by Poisson's equation, which gives

$${\displaystyle -\nabla ^{2}[\Delta \phi (r)]={\frac {1}{\varepsilon _{0}}}[Q\delta (r)-e\Delta \rho (r)],}$$

where ε₀ is the vacuum permittivity.

To proceed, we must find a second independent equation relating Δρ and Δφ. We consider two possible approximations, under which the two quantities are proportional: the Debye–Hückel approximation, valid at high temperatures (e.g. classical plasmas), and the Thomas–Fermi approximation, valid at low temperatures (e.g. electrons in metals).

Debye–Hückel approximation

Main article: Debye–Hückel theory

In the Debye–Hückel approximation,^[3] we maintain the system in thermodynamic equilibrium, at a temperature T high enough that the fluid particles obey Maxwell–Boltzmann statistics. At each point in space, the density of electrons with energy j has the form

$${\displaystyle \rho _{j}(r)=\rho _{j}^{(0)}(r)\;\exp \left[{\frac {e\phi (r)}{k_{\mathrm {B} }T}}\right]}$$

where k_B is Boltzmann's constant. Perturbing in φ and expanding the exponential to first order, we obtain

$${\displaystyle e\Delta \rho \simeq \varepsilon _{0}k_{0}^{2}\Delta \phi }$$

where

$${\displaystyle k_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {\rho e^{2}}{\varepsilon _{0}k_{\mathrm {B} }T}}}}$$

The associated length λ_D ≡ 1/k₀ is called the Debye length. The Debye length is the fundamental length scale of a classical plasma.

Thomas–Fermi approximation

Main articles: Thomas–Fermi screening and Lindhard theory

In the Thomas–Fermi approximation,^[4] named after Llewellyn Thomas and Enrico Fermi, the system is maintained at a constant electron chemical potential (Fermi level) and at low temperature. The former condition corresponds, in a real experiment, to keeping the metal/fluid in electrical contact with a fixed potential difference with ground. The chemical potential μ is, by definition, the energy of adding an extra electron to the fluid. This energy may be decomposed into a kinetic energy T part and the potential energy −eφ part. Since the chemical potential is kept constant,

$${\displaystyle \Delta \mu =\Delta T-e\Delta \phi =0.}$$

If the temperature is extremely low, the behavior of the electrons comes close to the quantum mechanical model of a Fermi gas. We thus approximate T by the kinetic energy of an additional electron in the Fermi gas model, which is simply the Fermi energy E_F. The Fermi energy for a 3D system is related to the density of electrons (including spin degeneracy) by

$${\displaystyle \rho =2{\frac {1}{(2\pi )^{3}}}\left({\frac {4}{3}}\pi k_{\mathrm {F} }^{3}\right),\quad E_{\mathrm {F} }={\frac {\hbar ^{2}k_{F}^{2}}{2m}},}$$

where k_F is the Fermi wavevector. Perturbing to first order, we find that

$${\displaystyle \Delta \rho \simeq {\frac {3\rho }{2E_{\mathrm {F} }}}\Delta E_{\mathrm {F} }.}$$

Inserting this into the above equation for Δμ yields

$${\displaystyle e\Delta \rho \simeq \varepsilon _{0}k_{0}^{2}\Delta \phi }$$

where

$${\displaystyle k_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {3e^{2}\rho }{2\varepsilon _{0}E_{\mathrm {F} }}}}={\sqrt {\frac {me^{2}k_{\mathrm {F} }}{\varepsilon _{0}\pi ^{2}\hbar ^{2}}}}}$$

is called the Thomas–Fermi screening wave vector.

This result follows from the equations of a Fermi gas, which is a model of non-interacting electrons, whereas the fluid, which we are studying, contains the Coulomb interaction. Therefore, the Thomas–Fermi approximation is only valid when the electron density is low, so that the particle interactions are relatively weak.

Result: Screened potential

Our results from the Debye–Hückel or Thomas–Fermi approximation may now be inserted into Poisson's equation. The result is

$${\displaystyle \left[\nabla ^{2}-k_{0}^{2}\right]\phi (r)=-{\frac {Q}{\varepsilon _{0}}}\delta (r),}$$

which is known as the screened Poisson equation. The solution is

$${\displaystyle \phi (r)={\frac {Q}{4\pi \varepsilon _{0}r}}e^{-k_{0}r},}$$

which is called a screened Coulomb potential. It is a Coulomb potential multiplied by an exponential damping term, with the strength of the damping factor given by the magnitude of k₀, the Debye or Thomas–Fermi wave vector. Note that this potential has the same form as the Yukawa potential. This screening yields a dielectric function ${\displaystyle \varepsilon (r)=\varepsilon _{0}e^{k_{0}r}}$.

Classical physics and linear response

A mechanical ${\displaystyle N}$-body approach provides together the derivation of screening effect and of Landau damping.^[2][5] It deals with a single realization of a one-component plasma whose electrons have a velocity dispersion (for a thermal plasma, there must be many particles in a Debye sphere, a volume whose radius is the Debye length). On using the linearized motion of the electrons in their own electric field, it yields an equation of the type

$${\displaystyle {\mathcal {E}}\Phi =S,}$$

where ${\displaystyle {\mathcal {E}}}$ is a linear operator, ${\displaystyle S}$ is a source term due to the particles, and ${\displaystyle \Phi }$ is the Fourier-Laplace transform of the electrostatic potential. When substituting an integral over a smooth distribution function for the discrete sum over the particles in ${\displaystyle {\mathcal {E}}}$, one gets

$${\displaystyle \epsilon (\mathbf {k} ,\omega )\,\Phi (\mathbf {k} ,\omega )=S(\mathbf {k} ,\omega ),}$$

where ${\displaystyle \epsilon (\mathbf {k} ,\omega )}$ is the plasma permittivity, or dielectric function, classically obtained by a linearized Vlasov-Poisson equation,^[6]: §6.4 ${\displaystyle \mathbf {k} }$ is the wave vector, ${\displaystyle \omega }$ is the frequency, and ${\displaystyle S(\mathbf {k} ,\omega )}$ is the sum of ${\displaystyle N}$ source terms due to the particles.^{[2]: Equation 20}

By inverse Fourier-Laplace transform, the potential due to each particle is the sum of two parts^[2]: §4.1 One corresponds to the excitation of Langmuir waves by the particle, and the other one is its screened potential, as classically obtained by a linearized Vlasovian calculation involving a test particle.^[6]: §9.2 The screened potential is the above screened Coulomb potential for a thermal plasma and a thermal particle. For a faster particle, the potential is modified.^[6]: §9.2 Substituting an integral over a smooth distribution function for the discrete sum over the particles in ${\displaystyle S(\mathbf {k} ,\omega )}$, yields the Vlasovian expression enabling the calculation of Landau damping.^[6]: §6.4

Quantum-mechanical approach

In real metals, the screening effect is more complex than described above in the Thomas–Fermi theory. The assumption that the charge carriers (electrons) can respond at any wavevector is just an approximation. However, it is not energetically possible for an electron within or on a Fermi surface to respond at wavevectors shorter than the Fermi wavevector. This constraint is related to the Gibbs phenomenon, where Fourier series for functions that vary rapidly in space are not good approximations unless a very large number of terms in the series are retained. In physics, this phenomenon is known as Friedel oscillations, and applies both to surface and bulk screening. In each case the net electric field does not fall off exponentially in space, but rather as an inverse power law multiplied by an oscillatory term. Theoretical calculations can be obtained from quantum hydrodynamics and density functional theory (DFT).