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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 n
n
atoms m
3
in volume element Adx , the total area of atoms in
the volume (viewed along the x-axis) is n
n
Adx
where is the cross-section
and n
n
Adx
A, so that there is no shadowing. The fraction of particles
making a collision is thus n
n
Adx
/A - the fraction of the cross-section blocked
by atoms. If
is the incident particle flux, the emerging flux is (1 -
n
n
dx) and the change of with distance is described by
d dt
n
n The solution is

0
exp
x/
mfp
mfp
1/n
n
, 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
n
n
v.
Averaging over all of the velocities in the distribution gives the average
collision requency
n
n
v
where we have allowed for the fact that
can be energy dependent as we
shall see below.
1 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 0
r
2 C
r
2
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 as it follows the unperturbed path
shown in the picture, only the perpendicular component matters because the
parallel compoent of the force reverses direction after q passes Q. Thus F
/F
b/r or
F
Cb
r
3 Cb
x
2
b
2 3/2
where b is the impact parameter for the interaction. Since x (the parallel
coordinate) changes with time, we integrate along the path to obtain the net
perpendicular impulse delivered to q
m F dt
Cb dx
x
2
b
2 3/2
dt
dx
Now
2
dx
x
2
b
2 3/2 x
b
2
x
2
b
2 1/2
1/b
2
, x

1/b
2
, x

so that 2C
m
b
We consider a statistical average over a random distribution of such small
angle collisions. For a random walk with step length
s, the total
displacement after N steps is
s
N
s (i.e. (s)
2
(
s)
2
) where N is
the number of steps. The total change in velocity is thus 2
N
2
Now integrate over the range of impact parameters b to estimate the number
of glancing collisions in time t. Small angle colisions are much less likely
because of the geometrical effect
N
b n2bdbt
We combine the three equations above and integrate over impact parameter: 2 8
nC
2
t
m
2
b min
b max
db
b
b
min
is the closest approach that satisfies the small deflection hypothesis. We
obtain this be setting
v v
b
min 2C
m 2 2qQ
4 0
m 2
Outside D
the charge Q is not felt. We thus take b
max

D
and b min
b max
db
b
ln
D
b
min
ln
ln
is called the Coulomb logarithm and is a slowly varying function of
electron density and temperature. For fusion plasma ln
6 - 16. One
usually sets ln
10 in quantitative estimations.
We are finally in a position to find the elapsed time necessary for a net 90 deflection i.e. (
v)2 v
2
. 2

2 8
nC
2
t
m
2 ln and 90
1/t
8
nq
2
Q
2
ln 16 2 0
2
m
2 3
For electron-ion encounters, q
-e and Q Ze so
3
90ei

ei n
i
Z
2
e
4
ln 2 2 0
2
m
2 3
and ei
n
i ei
v implies that ei Z
2
e
4
ln 2 2 0
2
m
2 4
Lets review this derivation
Perpendicular impulse 1/vb
Angular deflection v/v 1/v
2
Random walk to scatter one radian (v/v 1) 1/()
2
1/v
4
Integrate over b ln .
The dependence
1/v
4
has very important ramifications. A high temperature
plasma is essentially collisionless. This means that plasma resistance
decreases as temperature increases. In some circumstances populations of
particles can be continually accelerated, losing energy only through
synchrotron radiation (for example runaway electrons in a tokamak).
3.3 Energy transfer in electron-ion collisions
A pervading theme in plasma physics is m
e
m
i
. This has consequences
for collisions between the species in that we expect very little energy transfer
between the species. To illustrate this, consider such a collision in the
centre-of-mass frame (a direct hit).
Collison between ion and electron in
centre of mass frame.
m 0
MV
0
0 m MV momentum
m 0
2
MV
0
2
m
2
MV
2
energy
eliminate V, V
0
:
m 0
2
1 m
M m
2
1 m
M
4
0
and V
V
0
Now translate to frame in which ion M is at rest (initially). In this frame
theinitial electron energy is
E
e0 1
2 m
0
V
0 2 1
2 m
0
2
1 m
M
2
and the final ion energy is
E
i 1
2 M2V
0 2
2M m
2 0
2
M
2 Their ratio is given by
ion (final)
electron (initial) 2m
2 0
2
1
2
m 0
2
M 1 m
M
2 4m
M
and thus
e
i,
E
4m
e
E
e
m
i
e
e,
E E
e
i
i,
E E
i
For glancing collisions the energy transfer between ions and electrons is even
less.
Coulomb collisions result in very poor energy transfer between electrons and
ions. The rate of energy transfer is roughly (m
e
/m
i
) slower than the e-i
collision frequency. On the other hand, energy transfer rate and collision
frequency are the same for collisions between ions ii

ei
m
e
/m
i
: ii ei m
e
3 e
3
m
i
2 i
3
m
e
m
i 1/2
m
e e
2
m
i i
2 3/2
m
e
m
i 1/2
for T
e
T
i
That is, E
ei m
i
m
e ei E
ii

ii
m
i
m
e 1/2 ei E
ee

ee

ei
5