In a fluid composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force,
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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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 &delta(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
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
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,
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.
Our results from the Debye-Hückel or Fermi-Thomas approximation may now be inserted into the first Maxwell equation. The result is
Electrostatic screening
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
where kB is Boltzmann's constant. Perturbing in φ and expanding the exponential to first order, we obtain
where
The associated length λD ≡ 1/k0 is called the Debye length. The Debye length is the fundamental length scale of a classical plasma.Fermi-Thomas Approximation
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 EF. The Fermi energy is related to the density of electrons by
Perturbing to first order, we find that
Inserting this into the above equation for Δμ yields
where
is called the Fermi-Thomas screening wave vector.Screened Coulomb Interaction
which is known as the screened Poisson equation. The solution is
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.