Chemistry Reference
In-Depth Information
with respect to the directions
v
/
v
of the electron velocity
v
using spherical harmonics
[4,32,33]
k
∞
v
v
f
e
f
k
(
(
x
,
v
,
t
)
=
x
,
v
,
t
)
⊗
(9.48)
k
=
0
or Legendre polynomials if the direction of the force and the expected inhomogeneity
are parallel to a fixed space direction. Because of the orthogonality of the spherical
harmonics, this approach reduces the Boltzmann equation to a set of equations for the
(tensorial) expansion coefficients
f
k
of rank
k
where the expansion is frequently trun-
cated after the second term, that is,
k
1. Here,
f
0
is the isotropic part of the velocity
distribution and the further coefficients represent contributions to the distribution
anisotropy. Different FDM codes have been developed to solve the resulting set of
equationsforstationarydischargeplasmaswithzero,one,ortwospatialdimensionsas
well as for time-dependent plasmas with zero or one spatial dimensions [15,33,34].
As an example for the solution of the spatially 1D electron Boltzmann equation
by means of a multiterm method, the calculated spatial evolution of the isotropic
distribution
f
0
(
=
in the cathode-fall region of an abnormal H
2
-Ar-N
2
DC glow
discharge in a deposition reactor is shown where the magnitude of the velocity was
replaced by the kinetic energy
U
z
,
U
)
2 [35]. The solver BOLSIG+ that provides
steady-state solutions of the electron Boltzmann equation is freely available [36]
Figure 9.5.
=
m
e
v
2
/
10
6
10
6
10
5
10
5
10
4
10
4
10
3
10
3
10
2
10
2
10
1
10
1
10
0
10
0
0
0.6
0.7
50
100
0.5
Potential energy
0.4
150
0.3
200
0.2
250
0.1
0
Cathode
FIGURE 9.5
Spatial behavior of the isotropic distribution
f
0
(
z
,
U
)
in the cathode-fall region
of a glow discharge in a gas mixture with 71.5% H
2
, 15.3% Ar, and 13.2% N
2
obtained by a
10-term Boltzmann calculation.
