Chemistry Reference
In-Depth Information
w
=
(
m
α
v
+
m
β
v
β
)/(
m
α
+
m
β
)
, and the
scattering angle θ
= ∠
(
v
rel
/
v
rel
,
v
rel
/
v
rel
)
.
v
rel
−
The absolute value of the velocity
v
αβ
=
2
(
m
α
+
m
β
)
E
/(
m
α
m
β
)
depends
on the net change in internal energy
E
as a result of the collision process.
This energy change determines the type of collision process. For elastic colli-
sions that do not influence the internal state of the particles,
E
is equal to
zero. For inelastic collision processes, collisions of first kind, such as excitation
with
E
>
0, are distinguished from collisions of second kind (de-excitation) with
0.
Collision processes that change the particle number, that is, nonconservative
processes such as ionization and recombination, can be treated in a similar way.
Considerable simplifications of the collision integrals can be applied for the inter-
action of light particles such as electrons with much heavier particles (atoms or
molecules). Assumptions about the isotropy of the scattering processes and the energy
sharing after the collision allow further simplifications. The collision integral, tran-
sition probability, and differential cross section for the example of electron impact
ionization are explained in Section 9.2.1.1.
Equation (9.14) already yields the basis for a complete plasma description on
a microscopic level. If binary collisions are the dominant processes of the relevant
plasma components, then a coupled system of integro-differential equations of type
(9.14) has to be solved for each species with corresponding collision integrals for
every possible type of collision. Knowledge of the external forces
F
a
, of initial and
boundary conditions, and of the collision cross sections σ are prerequisites for the
solution.
E
<
9.1.4 M
OMENTS OF THE
K
INETIC
E
QUATION
The kinetic equation (9.10) and its solution, the distribution function
f
α
(
)
,
yields a microscopic description of the plasma component α. Because of its com-
plexity, it is not always possible to determine its complete solution. However,
detailed information about the velocity distribution of the particles is in many
cases not required. Thus, equations for the distribution function moments, that
is, for macroscopic properties, can be derived from the kinetic equation and can
be solved then. In general, the average value
x
,
v
,
t
)
relative to the velocity space with the normalization relation (9.6) is given by
[2-5,7]
g
(
v
)
of a particle property
g
(
v
g
n
α
(
x
,
t
)
g
(
v
)
=
(
v
)
f
α
(
x
,
v
,
t
)
d
3
v
.
(9.16)
The corresponding balance equation is obtained by multiplying the kinetic equation
(9.10) with
g
(
)
v
followed by an integration over the velocity space:
coll
d
3
v
.
(9.17)
g
g
g
g
F
a
m
·∇
v
f
α
d
3
v
)
∂
f
α
∂
df
α
dt
(
t
d
3
v
+
(
)
·∇
x
f
α
d
3
v
+
(
)
=
(
)
v
v
v
v
v
