Chemistry Reference
In-Depth Information
where
F
a
indicates the external force on the particle. Consequently, the number of
particles in the small phase space volume
d
3
xd
3
v
changes by
(
ˆ
x
,
ˆ
f
α
(
x
,
v
,
t
)
d
3
xd
3
v
−→
f
α
v
,
t
)
d
3
xd
3
ˆ
v
ˆ
f
α
m
·∇
v
f
α
d
3
F
a
∂
f
α
∂
=
(
x
,
v
,
t
)
+
t
+
v
·∇
x
f
α
+
t
+···
ˆ
xd
3
ˆ
v
.
(9.8)
Since the size of the phase space volume does not change during the collisionless
motion of particles because of Liouville's theorem, that is,
d
3
d
3
xd
3
v
,the
total temporal change of the distribution function
f
α
follows in the limit of
xd
3
ˆ
v
ˆ
=
t
−→
0:
m
α
·∇
v
f
α
d
3
xd
3
v
.
f
α
d
3
xd
3
v
F
a
∂
d
dt
f
α
∂
(
)
=
t
+
·∇
x
f
α
+
x
,
v
,
t
v
(9.9)
As long as the motion of particles is collisionless, the total change over time of the
particle number and of the distribution function equals zero, that is,
df
α
0. On
the other hand, the interaction between the particles causes alterations of momentum
and kinetic energy of the particles and, depending on the type of collision, can result
in changes of the particle number in the corresponding phase space volume. The
consideration of this total change of the distribution function in time due to collisions
leads to the basic kinetic equation [2-5,7]
/
dt
=
coll
,
F
a
m
α
·∇
v
f
α
∂
f
α
∂
df
α
dt
t
+
v
·∇
x
f
α
+
=
(9.10)
which describes the distribution function
f
α
for the particle species α.The
right-handsideof(9.10)isreferredtoasthecollisionintegralandincludesallcollision
processes that result in a change of the distribution function. Its peculiarity specifies
the type of the kinetic equation. An example for a collision integral of charge carriers
in an electric potential
(
x
,
v
,
t
)
is given in Section 9.2.1.1.
Most processing plasmas are weakly ionized so that collisions between charged
particles can be neglected while the short-range interaction between charged and
neutral particles and of neutral particles with each other are dominant. The kinetic
equationforsuchplasmaswithpredominantbinarycollisionprocessesistheso-called
Boltzmann equation:
F
a
m
α
·∇
v
f
α
∂
f
α
∂
df
α
dt
bi
t
+
v
·∇
x
f
α
+
=
.
(9.11)
coll
The explicit form of its collision integral is given in Section 9.1.3. In fully ionized
plasmas, the long-range interaction between charge carriers, that is, Coulomb col-
lisions, have to be included into the kinetic description. The corresponding kinetic
equation is the Fokker-Planck equation. If collisions between particles of the plasma
become unimportant, the plasma is referred to as collisionless. Then, the right-hand
side in (9.10) vanishes and the so-called Vlasov equation is obtained.
