Environmental Engineering Reference
In-Depth Information
ions
P
tr
p
2
(1
N
tr
(
R
)
D
N
i
(
R
)
.
P
tr
)
Using the conservation laws for the energy and orbital momentum
of a
captured
ion, we find that ion capture in a closed orbit is possible at
R
p
r
0
R
0
,andthe
probability
P
tr
of ion capture in a closed orbit is
r
1
Z
cos
θ
0
p
r
0
R
0
,
r
0
R
0
R
2
P
tr
(
R
,
ε
)
D
p
tr
(
R
,
ε
)
D
d
cos
θ
D
cos
θ
D
,
R
0
0
where
θ
0
is the boundary angle in Figure 6.5 at which
r
min
D
r
0
. This gives for the
number density of trapped ions in the region given in (6.42)
1
!
,
R
N
i
(
R
)
R
2
p
2
r
0
R
0
r
1
r
1
p
R
0
r
0
.
r
0
R
0
R
2
r
0
R
0
R
2
N
tr
(
R
)
D
C
One can include in this formula the probability for an ion to be removed from the
particle field if the interaction potential is relatively small. The probability
P
tr
of
capture in a closed orbit is zero at the boundary of the region of the action of the
particle field,
R
D
l
, and we account for this fact by an additional factor 1
R
/
l
.
As a result, we obtain for the number density of trapped ions [82]
1
!
N
i
(
R
)
R
2
p
2
r
0
R
0
r
1
r
1
1
,
r
0
R
0
R
2
r
0
R
0
R
2
l
R
N
tr
(
R
)
D
C
p
R
0
r
0
.
R
(6.44)
We now analyze the screening of the particle field if it is created by trapped ions.
Using like for free ions the current charge
z
(
R
) inside a sphere of radius
R
,replac-
ing in (6.33) the number density of free ions by that of trapped ions, and repeating
the operation in deduction (6.36), we obtain for the current charge
z
(
R
)
2
4
p
3
5
s
e
2
T
i
16
p
π
9
1
2
N
0
R
5/2
r
0
R
0
R
l
z
(
R
)
D
j
Z
j
Φ
(
R
)
"
1
9/2
#
2
R
l
Dj
Z
j
,
(6.45)
where
1
!
,
r
1
r
1
r
0
p
R
0
1.05
j
2/9
1
2
r
0
R
0
R
2
r
0
R
0
R
2
Z
j
Φ
(
R
)
D
C
l
D
.
N
0
Φ
(9
l
/11)
(
R
), and take it at a distance where the integrand
has a maximum. One can find a small correction to the result by expansion over
We assume a weak dependence,
Φ
