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 simplification is permissible since the electrons are lighter and more mobile than the ions, provided we consider distances much larger than the ionic separation. In condensed matter physics, this model is referred to as jellium.
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
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
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
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.
Thomas–Fermi approximation
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,
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 EF. The Fermi energy for a 3D system is related to the density of electrons (including spin degeneracy) by
where kF is the Fermi wavevector. Perturbing to first order, we find that
Inserting this into the above equation for Δμ yields
where
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
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 Thomas–Fermi wave vector. Note that this potential has the same form as the Yukawa potential. This screening yields a dielectric function .