Chemistry Reference
In-Depth Information
2. The
multi-term approximation
for stronger electric fields by use of surface
spherical harmonics with Legendre polynomials
P
j
f
e
(
v
e
)
=
f
e
(
v
e
)
·
P
j
(
cos ϑ
)
.
(3.129)
j
3. The
electron beam approximation
for runaway electrons in the case of very
strong electric field strength, only. All the electrons have nearly the same
velocity
U
,low
cross section and the dominating forward scattering in collisions with the
background gas. The high anisotropic electron velocity distribution can be
approximated by the Dirac delta function
v
e
due to the strong acceleration within the potential drop
f
e
(
v
e
)
∼
v
e
(
U
)
·
a
·
δ
(
v
0
−
v
e
)
(3.130)
with the mean energy loss factor
a
due to the scattering.
Beside the different amount of the electric field strength which influences the
anisotropy of the electron velocity distribution function, the consideration of all
the relevant inelastic electron collisions with their specific collision cross section is
a great challenge in modeling of molecular nonthermal plasmas.
On the other hand, the
velocitydistributionfunctionofheavyspecies
in the quasi-
neutral plasma can be well described by a Maxwellian velocity distribution function
with separate temperatures for the neutrals and ions, e.g.,
T
n
=
T
gas
and
T
+
. Usually
the temperature of the ions exceeds the neutral gas temperature due to heating in
the external electric field. The difference in the heavy particle temperature may be
expressed by the approximation
m
n
·
v
D
,
A
+
T
+
=
T
gas
+
.
(3.131)
3
·
k
B
This formula estimates the result of collisions between neutral gas particles with the
mass
m
n
and positive ions
A
+
with the drift velocity
v
D
,
A
+
.
3.5 AVERAGED QUANTITIES, TRANSPORT EQUATIONS,
AND RATE COEFFICIENTS
3.5.1 W
EIGHTED
A
VERAGE OF
P
HYSICAL
Q
UANTITIES
At known velocity distribution function
f
A
the weighted average of a physical
quantity
A
(
v
)
A
1
n
·
A
=
(
v
)
·
f
A
(
v
)
·
d
3
v
(3.132)
is achieved by integration over all velocities providing the local averaged value
A
.
