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Electric field screening

Screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases (classical plasmas) and conduction electrons in metals. In astrophysics, electric field screening is important because it makes electric fields largely irrelevant. However, because the fluids involved have charged particles, they can generate and are affected by magnetism which is a very relevant and complex area of astrophysics.

In a fluid composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force,

$\mathbf{F} = \frac{q_1 q_2}{4\pi\epsilon_0 \left|\mathbf{r}\right|^2}\hat{\mathbf{r}}$ .

This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the Coulomb force diminishes with distance as 1/r², the average number of particles at each distance r is proportional to r², assuming the fluid is fairly isotropic. As a result, a charge fluctuation at any one point has non-negligible effects at large distances.

In reality, these long-range effects are suppressed by the flow of the fluid particles in response to electric fields. This flow reduces the effective interaction between particles to a short-range "screened" Coulomb interaction.

For example, consider a fluid composed of electrons. Each electron possesses an electric field which repels other electrons. As a result, it is surrounded by a region in which the density of electrons is lower than usual. This region can be treated as a positively-charged "screening hole". Viewed from a large distance, this screening hole has the effect of an overlaid positive charge which cancels the electric field produced by the electron. Only at short distances, inside the hole region, can the electron's field be detected.

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Electrostatic screening

The first theoretical treatment of screening, due to Debye and Hückel (1923), dealt with a stationary point charge embedded in a fluid. This is known as electrostatic screening.

Consider a fluid of electrons in a background of heavy, positively-charged ions. For simplicity, we ignore the motion and spatial distribution of the ions, approximating them as a uniform background charge. This is permissible since the electrons are lighter and more mobile than the ions, and provided we consider distances much larger than the ionic separation. In condensed matter physics, this model is referred to as jellium.

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 the first of Maxwell's equations, which gives

$- \nabla^2 [\Delta\phi(r)] = \frac{1}{\epsilon_0} [Q\delta(r) - e\, \Delta\rho(r)]$ .

To proceed, we must find a second independent equation relating Δρ and Δφ. There are two possible approximations, under which the two quantities are proportional: the Debye-Hückel approximation, valid at high temperatures, and the Fermi-Thomas approximation, valid at low temperatures.

Debye-Hückel approximation

In the Debye-Hückel approximation, 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

$\rho_j (r) = \rho_j^{(0)}(r) \; \exp\!\left[\frac{e\phi(r)}{k_B T}\right]$

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

$e \Delta\rho \simeq \epsilon_0 k_0^2 \Delta\phi$

where

$k_0 \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{\rho e^2}{\epsilon_0 k_B T}}$

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

Fermi-Thomas approximation

In the Fermi-Thomas approximation, we maintain the system at a constant chemical potential and at low temperatures. (The former condition corresponds, in a real experiment, to keeping the fluid in electrical contact at 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 and the potential energy -eφ. Since the chemical potential is kept constant,

Δμ = Δ T - e Δφ = 0

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

$\rho = 2 \frac{1}{(2\pi)^3} \frac{4}{3} \pi k_F^3 \quad , \quad E_F = \frac{\hbar^2 k_F^2}{2m}$ .

Perturbing to first order, we find that

$\Delta\rho \simeq \frac{3\rho}{E_F} \Delta E_F$ .

Inserting this into the above equation for Δμ yields

$e \Delta\rho \simeq \epsilon_0 k_0^2 \Delta\phi$

where

$k_0 \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{3e^2\rho}{\epsilon_0 E_F}} = \sqrt{\frac{m e^{2} k_{f}}{\epsilon _{0} \pi ^{2} \hbar ^{2}}}$

is called the Fermi-Thomas screening wave vector.

It should be noted that we used a result from the free electron gas, which is a model of non-interacting electrons, whereas the fluid which we are studying contains a Coulomb interaction. Therefore, the Fermi-Thomas approximation is only valid when the electron density is high, so that the particle interactions are relatively weak.

Screened Coulomb interactions

Our results from the Debye-Hückel or Fermi-Thomas approximation may now be inserted into the first Maxwell equation. The result is

$\left[ \nabla^2 - k_0^2 \right] \phi(r) = - \frac{Q}{\epsilon_0} \delta(r)$

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

$\phi (r) = \frac{Q}{4\pi\epsilon_0 r} e^{- k_0 r}$

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 Fermi-Thomas wave vector. Note that this potential has the same form as the Yukawa potential.

Electric field screening

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Contents

Electrostatic screening

Debye-Hückel approximation

Fermi-Thomas approximation

Screened Coulomb interactions

See also

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