In a fluid composed of electrically charged constituent particles, each pair of particles interact through the Coulomb Force ,
|
The associated length λ
D ≡ 1/''k
0'' is called the
Debye Length . The Debye length is the fundamental length scale of a classical plasma.
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,
:
.
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 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 .
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
:
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 ''k
0'', the Debye or Fermi-Thomas wave vector.
