Geoscience Reference
In-Depth Information
and
L
2
m
.r/
Z
Z
dL
2
p
L
2
m
.r/
L
2
:
e
ˇE
dE
G.r/
D
A.r/
(3.118)
E
0
.E
0
/
L
2
Here the energy E
0
is defined by the condition L
2
0 and
ˇm
2
3=2
m
2
r
A.r/
D
:
On integrating over L
2
in Eqs.
3.117
and
3.118
gives
Z
dx
e
ˇx
h
p
x
ˇU.r/
.r
r/
C
r
p
‰.x/
.r
r/
i
1
p
a
F.r/
D
E
0
.E
0
/
(3.119)
and
dx
e
x
"r
x
r
2
‰
ˇU.r/
.r
r
/
#
Z
1
p
a
2
G.r/
D
(3.120)
ˇE
0
.E
0
/
We must remember that r
depends on
x
.
3.5.3
Nonsingular Potentials
In this section we consider the molecular fluxes onto the particle surface assuming
that the interaction potentials are nonsingular, that is, they remain finite at r
D
a.
Most widespread potentials are nonsingular, for example, the Coulomb potential
or the interaction of polar molecules with a charged particle. The common feature
of these potentials is their monotonic behavior that provides the minimum of L.r/
to locate at r
D
a. The consideration of nonsingular potentials is much simpler
than singular ones having a singularity at the particle surface. Examples of such
potentials are also well known, for example, the potential of image force which, as
we see next, plays an important role in considering the particle charging.
Two simplest examples illustrate our approach: the free condensation of
molecules onto the particle surface (see Sect.
3.3
) and the capture of polar molecules
by charged particles.
