Fundamentals of Plasma Physics

17.7 The strongly coupled regime: crystallization of a dusty plasma 497

Ion dynamics are qualitatively different because all ions approaching the dustgrain are
accelerated, leading to the situation that no zero velocity ions exist near the dust grain. In
particular, an ion starting with infinitesimal inward velocity at infinity whereψ= 0has
nearly zero energy, i.e.,W≃ 0. The energy conservation equation for this slowest ion is

miv^2 /2+eφ(r)=0 (17.63)

and so the velocity for this slowest ion has the spatial dependence

vmin=

√

−

2 eφ(r)

mi

. (17.64)

The ion density is thus

ni(r) =

∫∞

vmin

fi(r,v)d^3 v

= ni 0

(

mi

2 πκTi

) 3 / 2

exp

(

−

qiφ(r)

κTi

)∫∞

vmin

exp

(

−

mv^2 / 2

κTi

)

d^3 v

= ni 0

4

√

π

exp(ψ)

∫∞

√ψexp

(

−ξ^2

)

ξ^2 dξ. (17.65)

This can be expressed in terms of the Error Function

erfz=

2

√

π

∫z

0

exp(−ξ^2 )dξ (17.66)

and, in particular, using the identity

4

√

π

∫∞

z

exp

(

−ξ^2

)

ξ^2 dξ=

2

√

π

∫∞

z

dξexp(−ξ^2 )−

2

√

π

∫∞

z

dξ

d

dξ

(

ξexp(−ξ^2 )

)

(17.67)

it is seen that (Laframboise and Parker 1973)

ni

ni 0

=exp(ψ)

(

1 −erf

√

ψ

)

+

2

√

π

√

ψ. (17.68)

The error function has the small-argument limit

lim

z→ 0

erfz≃

2 z

√

π

(17.69)

and so forψ<< 1
ni
ni 0

=1+ψ (17.70)

which is identical to the Boltzmann result given byfluid theory.
However, because
lim
z→∞
e−z

2

(1−erfz)=0, (17.71)

whenψ>> 1 the ion density has a non-Boltzmann dependence

ni

ni 0

=

2

√

π

√

ψ (17.72)