Geoscience Reference

In-Depth Information

5

4

3'

3

3

2'

2

2

1

1

0

0.5

1.0

1.5

2.0

2.5

3.0

PARTICLE SIZE (in units of
l
)

Fig. 3.18

Dependence of matching distance
R
(
a
) on particle radius
a
. The scale of both axes

is l

2D=
v
T
, the ion mean free path.
Curve 1
shows the function
R
(
a
) for the potential-free

condensation.
Curves 2 and 2
0
are the matching distances for the attractive potential at different

Coulomb lengths, l
c
D

D

3 (
dotted line
).
Curves 3 and 3
0
display the same

l (
solid line
)andl
c
D

for the repulsive potential

This consideration shows that the value of
R
is of the order of the mean free

path
l
. If we assume that the interaction contains a long-range potential, such as the

Coulomb potential, which is sufficiently strong at distances of the order of
l
or more,

and is short range potential effective at shorter distances (of the order of
a
), then it is

possible to find a general expression for
R
. The behavior of the short-range potential

at short distances is of no importance for the derivation described next. It can be

even singular at
r
D
a
.

It is clear what to do. We introduce an effective particle radius a

that is defined

from a specially formulated asymptotic condition imposed on
n
(
r
).