Geoscience Reference
In-Depth Information
not present far away from the particle. The function L
2
m
.r/ is defined as
L
2
m
.r/
D
L
2
.r/ .r
r
/
C
L
2
.r
r/
(3.106)
D
min
L
2
.r/
.
The dependence on
L
follows from the boundary condition (Eq.
3.105
):
Here L
2
B
s
.E;L/
D
Ce
ˇE
.E/Œ.L
L
/
C
ı
s;1
.L
L/
D
Ce
ˇE
.E/Œı
s;1
.L
L
/
C
ı
s;1
;
(3.107)
where
C
is the normalization constant of the Maxwell distribution, C
D
n
1
.ˇm=2/
3=2
, ˇ
D
1=kT ,andı
i;k
is the Kroneker delta. The angular momentum
L
separates the ion trajectories crossing the particle surface (L<L
) from those
passing aside the particle (L>L
). It is easy to see that the distribution in this form
contains no outgoing ions at
r
D
a
, because the factor .L
L
/ in the first term
does not permit for the trajectories crossing the point r
D
a. Hence, f
1
.a/
D
0.
3.5.2.2
Free Molecule Flux
The integration over L
2
in Eq.
3.29
is readily performed to give:
m
3
Z
L
2
J.a/
D
C
4
2
e
ˇE
dE:
(3.108)
In most practically important cases the angular momentum L.r/ reaches its
minimum at r
>a,i.e.L
D
L.r
/. The condition @
r
L
2
.r/
D
0 rewritten as
ˇ
ˇ
ˇ
ˇ
rDr
L
2
.r
/
mr
2
@U
@r
D
r
(3.109)
or
1
2
r
U
0
.r
/
C
U.r
/
D
E
(3.110)
defines r
as a function of
a
and
E
.
Let us introduce x
D
ˇE and
ˇL
2
2ma
2
D
r
2
a
2
.x
ˇU .r
//:
‰
D
(3.111)
The factor ‰ is seen to depend on
x
and
a
via r
and explicitly.
