Environmental Engineering Reference
In-Depth Information
Z
Z
in the Fuks formula (6.7), we obtain for the rate of attachment of posi-
tively charged ions (electrons) to a negatively charged cluster
!
D
N
0
Ze
2
4
π
J
D
h
1
exp
Tr
0
i
.
(6.9)
Ze
2
T
In the limit
Ze
2
/(
r
0
T
)
1 this formula is transformed into the Langevin formu-
la [64]:
Ze
2
D
T
4
π
J
D
D
4
π
ZeK
N
0
.
(6.10)
Let us find the charge
Z
of a cluster in an ionized gas with positive ions and
electrons as charged atomic particles. Equalizing the rates of attachment of positive
ions and electrons to the cluster surface, we obtain the following expression for the
equilibrium cluster charge:
r
0
T
e
2
ln
D
C
N
C
r
0
T
e
2
ln
K
C
N
C
K
N
Z
D
D
N
D
,
(6.11)
where
N
C
,and
N
are the number densities of positive ions and electrons far
from the cluster. In a quasineutral plasma,
N
C
D
N
, the currents of positive ions
and electrons given by (6.7) and (6.9) are equal. For the cluster equilibrium charge
this yields
r
0
T
e
2
ln
D
D
D
r
0
T
e
2
ln
K
K
Z
D
.
(6.12)
We also consider the case of a nonequilibrium plasma consisting of electrons and
positive ions if
T
e
T
i
. Then (6.11) takes the form
r
0
T
e
e
2
ln
K
e
K
i
Z
D
,
where
K
e
and
K
i
are the mobilities of electrons and ions in a buffer gas. Note that in
a gas discharge plasma the electron distribution function differs from the Maxwell
one, and the electron temperature is not a characteristic of a Maxwell distribution
of electrons, but it is introduced as
T
e
eD
e
/
K
e
.
In deriving the Fuks formula, we assume the electric field of the cluster
E
D
D
Ze
/
r
0
is weak near the cluster surface, which gives
Ze
2
r
0
λ
eE
λ
T
.
Since
Ze
2
/
r
0
r
0
, which is the criterion for a dense
ionized gas. In addition, the Fuks formula (6.7) is valid under the criterion
T
, this condition gives
λ
e
2
T
r
0
.
(6.13)
