Environmental Engineering Reference
In-Depth Information
This result must be combined with the condition for plasma quasineutrality
N
e
D
Zn
,
where
n
is the total number density of particles. Combining these equations to
remove the electron number density, we find the average charge of the particles to
be
(
ln
"
2
Zn
#
)
.
m
e
T
2
3/2
r
0
T
e
2
W
0
T
Z
D
(1.67)
π
„
2
1.2.9
Thermoemission of Electrons
For high temperatures or large particle size,
Ze
2
/(
r
0
T
) becomes small. For exam-
ple, at
r
0
1200
would be required to make this parameter become unity. If the parameter is small,
then it follows from (1.66) that
D
10
μ
mand
T
D
2000 K, the very large particle charge of
Z
D
2
m
e
T
2
3/2
exp
.
W
0
T
N
e
D
(1.68)
π
„
2
This formula describes the equilibrium density of electrons above a flat surface
if the electric potential of the particle is small compared with a typical thermal
energy. Therefore, the conditions near the particle and far from it ar
e i
dentical.
Then the average particle charge is determined from the relation
N
e
D
Zn
,where
the number densities of electrons and of particles are both known.
Equation (1.68) allows us to obtain a simple expression for the electron current
from the surface of a hot cathode. In the case of equilibrium between electrons
and a hot surface, the electron current from the surface is equal to the current
toward it. Assuming unit probability for electron attachment to the surface upon
their contact, we obtain that the electron current density
i
toward the surface is
equal to the electron current density from the surface:
e
1/2
3
exp
.
em
e
T
2
4
T
W
0
T
i
D
N
e
D
(1.69)
2
π
m
e
π
2
„
This result is known as the Richardson-Dushman formula [61-65] describing the
electron current density for electron emission by a hot surface. This type of emis-
sion is called thermoemission of electrons. For the analysis of gas discharge prob-
lems, it is convenient to rewrite the Richardson-Dushman formula (1.69) for the
thermoemission current density in the form
A
R
T
2
exp
,
W
T
i
D
(1.70)
wherein the Richardson parameter
A
R
, according to (1.69), has the value [66]
60 A/(cm
2
K
2
). Table 1.5 contains the parameters in (1.69) for real metals. In this
table,
W
is the metal work function,
T
b
is the metal boiling point, and
i
b
is the
current density at the boiling point.
