(lect.4 (Collision Processes

Collision Processes
Collisions mediate the transfer of energy and momentum between various
species in a plasma, and as we shall see later, allow a treatment of highly
ionized plasma as a single conducting fluid with resistivity determined by
electron-ion collisions.
Collisions which conserve total kinetic energy are called elastic. Some
examples are atom-atom, electron-atom, ion-atom (charge exchange) etc. In
inelastic collisions, there is some exchange between potential and kinetic
energies of the system. Examples are electron-impact ionization/excitation,
collisions with surfaces etc. In this section, we consider both types of collision
process, with particular emphasis on Coulomb collisions between charged
particles, this being the dominant process in a plasma.
3.1 Mean free path and cross-section
To properly treat the physics of collisions we need to introduce the concept
of mean free path - a measure of the likelihood of a collision event. Imagine
electrons impinging on a box of neutral gas of cross-sectional area A.
If there are nn atoms m?3 in volume element Adx , the total area of atoms in
the volume (viewed along the x-axis) is nnAdx? where ? is the cross-section
and nnAdx? ? A, so that there is no “shadowing”. The fraction of particles
making a collision is thus nnAdx?/A - the fraction of the cross-section blocked
by atoms. If ? is the incident particle flux, the emerging flux is ?’ ? ?(1 -
nn?dx) and the change of ? with distance is described by
d?
dt ? ?nn??
The solution is
? ? ?0 exp??x/?mfp?
?mfp ? 1/nn?, where ?mfp is the mean free path for collisions characterised by
the cross-section ?. The physics of the interaction is carried by ?, the rest is
geometry. The mean free time between collisions, or collision time for
particles of velocity v is ? ? ?mfp/v and the collision frequency is ? ? ??1 ?
v/?mfp ? nn?v.
Averaging over all of the velocities in the distribution gives the average
collision requency
? ? nn?v
where we have allowed for the fact that ? can be energy dependent as we
shall see below.
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3.2 Coulomb collisions
Coulomb collisions between free particles in a plasma is an elastic process.
Let us consider the Coulomb force between two test charges q and Q:
F ? qQ
4??0r2 ? C
r2
This is a long range force and the cross-section for interaction of isolated
charges is infinity! It is quite different from elastic “hard-sphere” encounters
such as that which can occur between electrons and neutrals for example. In
a plasma, however, the Debye shielding limits the range of the force so that
an effective cross-section can be found. Nevertheless, because of the nature
of the force, the most frequent Coulomb deflections result in only a small
deviation of the particle path before it encounters another free charge. To
produce an effective 90? scattering of the particle (and hence momentum
transfer) requires an accumulation of many such glancing collisions. The
collision cross-section is then calculated by the statistical analysis of many
such small-angle encounters.
Consider the Coulomb force on an electron