Chemistry Reference
In-Depth Information
where
Z
α
and
e
0
are the particle charge number and elementary charge. If time-
dependent electromagnetic fields are present, then Maxwell's equations
e
0
α
∇
x
·
D
(
x
,
t
)
=
Z
α
n
α
(
x
,
t
)
(9.34)
∂
∂
∇
x
×
H
)
=
J
t
D
(
(
)
+
(
)
x
,
t
x
,
t
x
,
t
(9.35)
)
=−
∂
∂
∇
x
×
E
t
B
(
x
,
t
(
x
,
t
)
(9.36)
∇
x
·
B
(
x
,
t
)
=
0
(9.37)
e
0
α
Z
α
j
α
are the dielectric displacement,
magnetizing field, and total current density carried by the charged particles, respec-
tively, and the term
have to be satisfied, where
D
,
H
, and
J
=
∂
D
/∂
t
denotes the displacement current. In vacuum, the relations
D
ε
0
E
and
H
=
B
μ
0
hold for the external sources with the vacuum permittivity ε
0
and permeability of free space μ
0
. Then, the electric field satisfies Poisson's equation:
=
/
e
0
ε
0
E
x
(
x
,
t
)
=−
Z
α
n
α
(
x
,
t
)
,
(
x
,
t
)
=−∇
x
(
x
,
t
)
(9.38)
α
and relation (9.35) turns into
c
2
∂
1
t
E
∇
x
×
B
μ
0
J
(
)
=
(
)
+
(
)
x
,
t
x
,
t
x
,
t
,
(9.39)
∂
/
√
ε
0
μ
0
is the speed of light in vacuum. The set of equations has to be
completed by an equation system for the external electric discharge circuit.
where
c
=
1
9.1.5 N
UMERICAL
M
ETHODS
The theoretical description of transport phenomena in plasma applications can be car-
ried out by quite different modeling approaches. As mentioned earlier, the Liouville
equation (9.3) is the most general equation and it is valid in all situations. However,
it is so complex from a computational point of view that it cannot be solved for real
plasma applications. A fully kinetic description of the transport of charge carriers has
become possible by means of particle models. Their accuracy is mainly limited by
the number of particles that can be treated in the simulation. This modeling approach
is computationally much more time consuming than the solution of hydrodynamic
equations. The latter are the most restricted equations with respect to their range of
applicability. However, fluid (or hydrodynamic) models are in principle computation-
ally fast and have been used for many practical engineering simulations. In addition,
so-called hybrid methods have been established that use the simpler hydrodynamic
equations as soon as possible and employ the more complex kinetic equations only
where necessary. A scheme of basic numerical modeling approaches discussed in the
following text is represented in Figure 9.3.
