Mechanisms of Electron Losses: Electron-Ion Recombination

The ionization processes were considered as a source of electrons and positive ions, e.g., as a source of plasma generation. Conversely, the principal loss mechanisms of charged particles, the elementary processes of plasma degradation, will now be examined. Obviously, the losses together with the ionization processes determine a balance of charge particles and plasma density. The variety of channels of charged particle losses can be subdivided into three qualitatively different groups.

The first group includes different types of electron-ion recombination processes, in which collisions of the charged particles in a discharge volume lead to their mutual neutralization. These exothermic processes require consuming the large release of recombination energy in some manner. Dissociation of molecules, radiation of excited particles, or three-body collisions can provide the consumption of the recombination energy.

Electron losses, because of their sticking to neutrals and formation of negative ions, form the second group of volumetric losses, electron attachment processes. These processes are often responsible for the balance of charged particles in such electronegative gases as oxygen (and, for this reason, air); CO₂ (because of formation of O−); and different halogens and their compounds. Reverse processes of an electron release from a negative ion are called the electron detachment.

Note that although electron losses in this second group are due to the electron attachment processes, the actual losses of charged particles take place as a consequence following the fast processes of ion-ion recombination. The ion-ion recombination process means neutralization during collision of negative and positive ions.

Finally, the third group of charged particle losses is not a volumetric one like all those mentioned previously, but is due to surface recombination. These processes of electron losses are the most important in low pressure plasma systems such as glow discharges. The surface recombination processes are usually kinetically limited not by the elementary act of the electron-ion recombination on the surface, but by transfer (diffusion) of the charged particles to the walls of the discharge chamber.

The mhd equations

So far we have applied the arguments of classical fluid dynamics to obtain a closed set of equations for the plasma fluid variables but, except for the introduction of Joule heating, we have taken almost no account of the fact that a plasma is a conducting fluid. This we do now by specifying the force per unit mass F. Except in astrophysical contexts, where gravity is an important influence on the motion of the plasma, electromagnetic forces are dominant. For a fluid element with charge density q and current density j we then have

ρF = qE + j × B (3.32)

where the fields E and B are determined by Maxwell’s equations (2.2)–(2.5). Equations (2.6) and (2.7) for q and j are not suitable in a fluid model. However, our first objective is to obtain a macroscopic description of the plasma in which the fields are those induced by the plasma motion. Thus, we now introduce the basic assumption of MHD that the fields vary on the same time and length scales as the plasma variables. If the frequency and wavenumber of the fields are ω and k respectively, we have ωτ_H ~1 and kL_H ~1, where τ_H and L_H are the hydrodynamic time and length scales. A dimensional analysis then shows that both the electrostatic force qE and displacement current ε₀μ₀∂E/∂t may be neglected in the non-relativistic approximation ω/k << c. Consequently, (3.32) becomes

ρF = j × B (3.33)

and (2.3) is replaced by Ampere’s law

j = (1 / μ0) grad x B (3.34)

Now, Poisson’s equation (2.4) is redundant (except for determining q) and just one further equation for j is required to close the set. Here we run into the main problem with a one-fluid model. Clearly, a current exists only if the ions and electrons have distinct flow velocities and so, at least to this extent, we are forced to recognize that we have two fluids rather than one. For the moment we side-step this difficulty by following usual practice in MHD and adopting Ohm’s law

j = σ(E + u × B) (3.35)

as the extra equation for j. The usual argument for this particular form of Ohm’s law is that in the non-relativistic approximation the electric field in the frame of a fluid element moving with velocity u is (E + u × B). However, this argument is over-simplified, unless u is constant so that the frame is inertial, and later, when we discuss the applicability of the MHD equations, we shall see that the assumption of a scalar conductivity in magnetized plasmas is rarely justified. The status of (3.35) should be regarded, therefore, as that of a ‘model’ equation, adopted for mathematical simplicity.

This closes the set of equations for the variables ρ, u, P, T, E, B and j but before listing them it is useful to reduce the set by eliminating some of the variables. Although in electrodynamics it is customary to think of the magnetic field being generated by the current, in MHD we regard Ampere’s law (3.34) as determining j in terms of B. Then Ohm’s law (3.35) becomes

E = (1/σμ₀) grad × B − u × B (3.36)

so determining E. Finally, substituting (3.36) in (2.2), treating σ as a constant, and using (2.5), we get the induction equation for B

∂B/∂t = (1/σμ₀) grad²B +grad× (u × B) (3.37)

Since we have eliminated j and E, this is now the only equation we need add to the set derived at the end of the last section for the fluid variables.