Geoscience Reference
In-Depth Information
3.6.2
Solving the Kinetic Equation
The most general solution to Eq.
3.23
is
f
s
.r;E;L/
D
B
s
.E;L;R/
L
2
m
.r/
L
2
L
2
m
.r/
(3.161)
with .x/being the Heaviside step function and L
m
the maximal admissible angular
momentum. It is
L
2
m
.r/
D
L
2
.r/
for repulsive and nonsingular attractive potentials at
E
> 0. At
E
< 0 (attractive
potentials), the ions cannot occur in the states with L>L
a
at r;r
(see Fig.
3.16
).
Then
L
m
.r/
D
L.r/ .r
r
/
C
L
a
.r
r/;
(3.162)
If, however, R;r
, then a part of the trajectories that are able to reach the
particle surface (with L<L
a
) is populated as a result of ion diffusion. In this
case (Fig.
3.17
),
L
m
.r/
D
L.R/:
The function B
s
.E;L;R/ is defined by the boundary conditions f
1
.a/
D
0
and has the form
B
s
.E;L;R/
D
C.R/e
ˇ.EU.R//
.E
U.R//Œ.L
L
a
/
C
ı
s;1
.L
a
L/;
(3.163)
where the constant C.R/ is defined by the boundary condition at
r
D
R
n
fm
.R/
D
n
c
.R/ and ı
i;k
is the Kroneker delta.
It is easy to see that the distribution in this form contains no outgoing ions at
r
D
a
, because the factor .L
L
a
/ in the first term does not permit trajectories
crossing the point
r
D
a
. Hence, f
1
.a/
D
0. The factor .E
U.R// reflects the
fact that no ions with the lower energy U.R/ can appear in the free molecule zone
(such ions could appear only as a result of collisions with the carrier gas molecules).
The structure of the expression in the square braces on the right-hand side of
Eq.
3.163
provides the independence of the s
D
1 component of the distribution
on L
a
, which means that all ions with L<L.r/can move toward the particle
center (but not all can reach the particle surface). The outgoing component f
1
does
not contain the contribution from the trajectories crossing the particle surface.
