Geoscience Reference
In-Depth Information
Comparing this expression with the expansion of the square root on the right-
hand side of Eq.
3.178
gives
dx
e
x
p
x
1
C
r
2
a
2
Z
2
p
ˇ
j
U.r
/
j
x
2
D
0
For nonsingular attractive potentials (r
D
a), this equation permits us to obtain
the rather general result
2
D
1
C
2ˇ
j
U.a/
j
3.6.3.4
Charging of Neutral Particles
Let us consider the ion flux toward a neutral metallic particle. The incident ions
interact with the particle via the image potential
e
2
2a
a
4
r
2
.r
2
U.r/
D
(3.183)
a
2
/
It is seen that at a
/
l the potential is very weak, ˇU.l/
/
ˇe
2
a
3
=l
4
/
l
c
a
3
=l
4
,
D
e
2
=kT is the Coulomb length. Normally l
c
where l
c
/
l. Hence, we can use
Eq.
3.182
for calculating the charging efficiency.
The function
z
.a/ is known:
r
ˇe
2
2a
;
z
.a/
D
1
C
(3.184)
For metallic particles,
2
is
r
ˇe
2
ax
r
2
a
2
D
1
C
Substituting this expression into Eq.
3.183
gives
r
2ˇe
2
a
2
D
1
C
3.6.3.5
Charging Efficiencies
For charged particles the general result, Eq.
3.10
, should be used. The approxima-
tions Eqs.
3.11
and
3.12
allow us to express ˛.a/ in terms of the charging efficiency
˛
fm
.a;R/ found in the free molecule limit and the matching distance
R
.
