Environmental Engineering Reference
In-Depth Information
a small parameter 1
Φ
(
R
). Using the last formula for the above example of an
argon dusty plasma (
T
e
D
1eV,
T
i
D
400 K,
r
0
D
1
μ
m), we obtain
l
D
36
μ
m
10
10
cm
3
,and
l
for the number density of electrons and ions
N
0
D
D
59
μ
mfor
10
9
cm
3
. Comparing these values with those in Section 6.2.1 for screening
by free ions, we conclude that trapped ions dominate in screening the particle field
if the number densities of plasma electrons and ions is below
N
0
N
0
D
10
10
cm
3
.
We also can use these formulas for the criterion for plasma rareness if plasma
ions do not screen the particle field. As for free ions, this criterion has the form
N
0
D
N
,where
l
(
N
)
D
R
0
. For the parameters in the example under considera-
tion (
T
e
D
1eV,
T
i
D
400 K), we have from this
N
0
r
0
0.03 cm
1
,
10
6
cm
3
for particle radius
r
0
which corresponds to
N
D
m.
In considering the case where trapped ions dominate in shielding of the parti-
cle field, we also give the expressions for the number density of free and trapped
ions in the particle field according to (6.30) and (6.44). Using a simplified formula,
U
(
R
)
3
D
1
μ
z
(
R
)
e
2
/
R
, for the potential energy of the particle field and (6.45) for the
current charge
z
(
R
), we obtain for the number density of free
N
i
(
R
) and trapped
N
tr
(
R
)ionsatdistance
R
from the particle [82]
D
t
1
"
1
9/2
#
R
l
2
4
R
0
π
N
i
(
R
)
D
N
0
C
,
R
2
R
2
p
2
r
0
R
0
Φ
(
R
)
1
.
R
l
N
tr
(
R
)
D
N
i
(
R
)
(6.46)
In the case (6.36) where free ions dominate in screening of the particle field, we
have the following expressions for the number density of free and trapped ions on
the basis of (6.30) and (6.36) [82]:
t
1
"
1
#
2
R
l
5/2
4
R
0
π
N
i
(
R
)
D
N
0
C
,
R
2
R
2
p
2
r
0
R
0
Φ
(
R
)
1
.
R
l
N
tr
(
R
)
D
N
i
(
R
)
(6.47)
On the basis of these limiting cases one can give a general algorithm for determi-
nation of the parameters of a self-consistent field accounting for screening by both
free and trapped ions. Indeed, the screening charges due to free
Q
i
and trapped
Q
tr
ions are given by the relations
Z
Z
l
l
N
i
(
R
)
R
2
dR
,
Q
tr
N
tr
(
R
)
R
2
dR
,
Q
i
D
4
π
D
4
π
p
r
0
R
0
r
0
