In a fluid composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force,
In reality, these longrange 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 shortrange "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 positivelycharged "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, positivelycharged 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 DebyeHückel approximation, we maintain the system in thermodynamic equilibrium, at a temperature T high enough that the fluid particles obey MaxwellBoltzmann statistics. At each point in space, the density of electrons with energy j has the form
In the FermiThomas 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 noninteracting electrons, whereas the fluid which we are studying contains a Coulomb interaction. Therefore, the FermiThomas approximation is only valid when the electron density is high, so that the particle interactions are relatively weak.