Geoscience Reference
In-Depth Information
3.5.3.1
Potential Driven Condensation
If the potential U.r/does not have a singularity at r
D
a and behaves monotonously,
then the expressions for J and F.r/can be considerably simplified. In this case the
angular momentum L.r/ reaches the minimum at the particle surface, r
D
a.
The function ‰ is ‰.x/
D
x
ˇU.a/. Equations
3.119
,
3.120
,and
3.115
then
reduce to
Repulsion (U.r/>0)
F.r/
D
e
ˇU.r/
;
(3.121)
r
1
p
e
ˇU.r/
3
2
;ˇ
.U.a/
U.r//r
2
1
a
2
r
2
G.r/
D
(3.122)
r
2
a
2
x
s
˛1
e
s
ds is the incomplete gamma function.
Here .˛;x/
D
".a/
D
J.a/=J
0
.a/
D
e
ˇU.a/
(3.123)
Attraction U.r/<0
1
p
e
ˇjU.r/j
.3=2;ˇ
j
U.r/
j
/
F.r/
D
(3.124)
r
1
p
e
ˇU.r/
3
2
;ˇ
U.r/r
2
U.a/a
2
1
a
2
r
2
G.r/
D
(3.125)
r
2
a
2
and
".a/
D
1
C
ˇ
j
U.a/
j
(3.126)
3.5.3.2
Condensation of Polar Molecules
The foregoing theory can be readily applied for calculating the flux of polar
molecules toward a charged particle. Although the interaction potential depends on
the orientation of the polar molecule, we can ignore this dependence and consider
the dipoles directed to the particle center. In this case the interaction potential is
de
2
r
2
;
U.r/
D
(3.127)
