Chemistry Reference
In-Depth Information
particles in practice. However, it provides the basis for the deduction of further
average values.
In plasma chemistry, the characteristics of particles of the same kind is generally
of interest. The determination of plasma properties can be described in an analogous
way by the reduced one-particle distribution function
f
α
(
)
x
,
v
,
t
for each particle
species α. Note that the velocity
m
α
replaces the particle momentum without
loss of generality here and in the forthcoming for a given mass
m
α
of the particle
species α. The value of this distribution function
f
α
at a given position
v
=
p
/
x
and velocity
v
in the 6D phase space made up of three ordinary and three velocity coordinates
represents the phase-space density. This means that the number
dN
α
of particles of
species α that can be found in the volume element
d
3
x
around the position
x
and
in the surrounding
d
3
v
of velocity
v
is given by
f
α
(
x
,
v
,
t
)
d
3
xd
3
v
. Consequently, the
integral
f
α
(
x
,
v
,
t
)
d
3
xd
3
v
=
N
α
(
t
)
(9.4)
over the whole phase space yields the total number
N
α
of particles of species α.The
one-particle distribution function
f
α
is related to the
N
-particle distribution function
ρ by the integration over all but one volume element
d
3
x
k
d
3
v
k
according to
N
α
ρ
f
α
(
x
,
v
,
t
)
≡
f
1
(
x
1
,
v
1
,
t
)
=
(
x
1
,
v
1
,
...
,
x
N
,
v
N
,
t
)
d
3
x
2
d
3
v
2
...
d
3
x
N
d
3
v
N
, (9.5)
where α denotes the particle species to which the selected particle 1 belongs.
Now the problem is reduced to seven dimensions (including time) for each
species. Before discussing the corresponding continuity equation that generates
f
α
,
a further reduction of the problem should be mentioned. If the information about the
velocity of the particles (in the sense of the average value of the velocities of all par-
ticles of species α on a microscopic scale) is no longer of interest, the distribution
f
α
can be averaged over the velocity. This integration over
d
3
v
yields the particle density
f
α
d
3
v
n
α
(
x
,
t
)
=
(
x
,
v
,
t
)
(9.6)
of species α in the configuration space. The description of densities for every species
as a function of time in the plasma system corresponds to the fluid or hydrodynamic
treatment that will be discussed in Section 9.1.4. The hierarchy of plasma equations
is displayed in Figure 9.2.
9.1.2 K
INETIC
E
QUATION
The following introduction focusses on the distribution function of one particle
species and its determination. It is shown that distribution functions for all species
finally represent the basis for the description of all interesting processes in a plasma
such as transport, ionization, energy dissipation, particle conversion rates, and so
on. The general concept of a classical kinetic description that yields the distribution
function is based upon the separation of the particle trajectory into the movement
of particles without interaction with others and the interaction process itself. At this
