Environmental Engineering Reference
In-Depth Information
From the above formulas, we have that clusters that are found in thermodynamic
equilibrium with an ionized gas may be positively and negatively charged. In par-
ticular, according to the Saha formula we have for the ratio of the number densities
for singly positively charged
N
C
and singly negatively charged
N
clusters
2
exp
T
e
n
1/3
,
m
e
T
e
2
3/2
exp
,
N
N
D
Δ
2
N
e
W
T
e
D
π
„
2
Δ
D
I
0
(1)
C
EA
2
W
.
(6.65)
For example, in the case of copper we have
I
0
(1)
D
7.73 eV, EA
D
1.23 eV, and
/(
T
e
n
1/3
) is small for
large clusters and high electron temperatures. Hence, the sign of the mean cluster
charge is determined by the sign of the function ln
W
D
4.4 eV, which gives
Δ
D
0.16 eV, and the ratio
Δ
(
T
e
). This is a monotonic func-
tion of the electron temperature, so at high electron temperatures clusters are pos-
itively charged, and at low electron temperatures they have mean negative charge.
The transition electron temperature
T
is determined by the equation
(
T
)
D
1.
In particular, in the case of copper and for large clusters it is
T
D
2380 K for
10
11
cm
3
,
T
D
10
12
cm
3
,and
T
D
N
e
D
1
2640 K for
N
e
D
1
2970 K
10
13
cm
3
. As is seen, the temperature
T
of zero cluster charge
increases with increasing number density of plasma electrons.
Thus, a metal cluster is positively charged at high temperatures and negatively
charged at low temperatures. As a demonstration of this fact, Figure 6.17 gives
the average charge of tungsten clusters in a plasma as a function of the cluster
temperature, and also the cluster temperature
T
at which this cluster becomes
neutral on average, that is,
for
N
e
D
1
1. Table 6.2 contains the parameters in (6.65)
for some parameters of a cluster and plasma. In addition, according for (6.62), the
cluster charge is proportional to
n
1/3
, and Table 6.2 contains the proportionality
factor for the parameters considered.
(
T
)
D
6.3.2
Conversion of an Atomic Vapor into a Gas of Clusters
Being guided by a cluster plasma consisting of a buffer gas and addition of metal
in the form of metal clusters, we consider the first stage of formation of a cluster
plasma from a supersaturated atomic metal vapor. This allows one to ignore the
processes of cluster evaporation, whereas the vapor condensation consists in the
formation of nuclei of condensation, and cluster growth results from atom attach-
ment to nuclei of condensation. In the case of metal vapor condensation the nuclei
of condensation are diatomic metal molecules which are formed in the three body
process. One can assume in the next stage of the growth process that the energy
excess in atom attachment to metal molecules is transferred into oscillation de-
grees of freedom, and then this energy is transferred to the buffer gas atoms in
collisions. Therefore, we have the following scheme for the cluster growth process
