Environmental Engineering Reference
In-Depth Information
a third particle in the recombination process may be a bound electron. Under equi-
librium between internal and plasma electrons, the Saha formula (6.59) is valid,
but the electron release process is connected to internal electrons.
The equilibrium (6.58) between free electrons and bound electrons in a metal
cluster corresponds to a long mean free path of electrons in a gas compared with
the cluster radius in accordance with criterion (6.5) and the ionization potential
I
Z
(
n
) of a large cluster for
n
1. In this case removal of an electron from the
cluster surface outside the region of action of the cluster Coulomb field requires
that this interaction is overcome, which gives
Ze
2
r
0
C
I
Z
(
n
)
D
I
0
(
n
) ,
(6.60)
where
I
0
(
n
) is the ionization potential of a neutral cluster consisting of
n
atoms.
This relation allows one to transform the Saha formula (6.59) to the form
A
exp
,
A
m
e
T
e
2
exp
.
3/2
Ze
2
r
0
T
e
P
Z
P
Z
1
D
2
N
e
I
0
T
e
D
(6.61)
π
„
2
From this formula we find the Saha formula (6.59) is reduced to the Gauss form
for a large cluster charge
Z
1:
P
Z
(
n
)exp
"
#
,
Z
)
2
(
Z
P
Z
(
n
)
D
2
Δ
2
(
ln
"
2
N
e
#
)
,
m
e
T
e
2
3/2
r
0
T
e
e
2
I
0
(
n
)
T
e
r
0
T
e
e
2
2
Z
D
Δ
D
.
(6.62)
2
π
„
One can use a roughmodel for the cluster ionization potential assuming it to be a
monotonic function of the number
n
of cluster atoms. We represent the ionization
potential
I
0
(
n
) of a neutral metal as
const
n
1/3
I
0
(
n
)
D
W
C
,
where
W
is the metal work function, that is, the electron binding energy for a
macroscopic cluster. Including in the consideration the atomic ionization potential
I
0
(1)
I
and transferring to the ionization potential of a charged cluster according
to (6.60), we have in this model for the cluster ionization potential
D
Ze
2
r
0
C
I
W
n
1/3
I
Z
(
n
)
D
W
C
.
(6.63)
Evidently, within the framework of this model we have for the binding energy of a
negatively charged cluster
EA
W
n
1/3
I
1
(
n
)
D
W
C
,
(6.64)
where EA is the electron affinity of an atom.
