Geoscience Reference
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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
ˇ j U.r / j
For nonsingular attractive potentials (r D a), this equation permits us to obtain
the rather general result
D 1 C j U.a/ j
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
a 4
r 2 .r 2
U.r/ D
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
For metallic particles, 2 is
r ˇe 2
r 2
a 2 D 1 C
Substituting this expression into Eq. 3.183 gives
r 2ˇe 2
D 1 C
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 .
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