Fundamentals of Plasma Physics

498 Chapter 17. Dusty plasmas

which is much smaller than theexp(ψ)dependence predicted by the Boltzmann relation.
Poisson’s equation can be written non-dimensionally as

1

r ̄^2

∂

∂ ̄r

(

̄r^2

∂ψ

∂ ̄r

)

=

ni

ni 0

−

ne

ni 0

−

Zdnd

ni 0

=

ni

ni 0

−(1−α)

ne

ne 0

−α

nd

nd 0

(17.73)

where, following the normalization convention introduced earlier, ̄r=r/λdi.Upon substi-
tuting for the normalized densities, Poisson’s equation becomes

1

̄r^2

∂

∂r ̄

(

r ̄^2

∂ψ

∂ ̄r

)

︸ ︷︷ ︸

vacuum

= eψ

(

1 −erf

(√

ψ

))

+

2

√

π

√

ψ

︸ ︷︷ ︸

ions

−(1−α)exp

(

−

ψTi

Te

)

︸ ︷︷ ︸

electrons

−αexp

(

−Zψ ̄

)

︸ ︷︷ ︸

dust

. (17.74)

Becauseψbecomes large near the dust grain, this equation is highly non-linear and so
the linearization technique used on p. 8 for the conventional Debye shieldingderivation
cannot be invoked. Instead, an approximate solution to Poisson’s equation is obtained by
separating the collisionless region around the dust grain test particleinto three concentric
layers (regions) according to the magnitude ofψ:

Region 1 Region 2 Region 3

Definition ψd>ψ> 1 1 >ψ> 1 /Z ̄ 1 /Z>ψ ̄

Location adjacent to dust grain middle spherical layer outer layer

Ions non-Boltzmann, fast Boltzmann Boltzmann

Electrons Boltzmann Boltzmann Boltzmann

Dust zero density zero density Boltzmann

Dependence vacuum-like w/const. growing/decaying Yukawa decaying Yukawa

Poisson’s equation has the following approximations in the three regions

Region 1 :

1

̄r^2

∂

∂ ̄r

(

r ̄^2

∂ψ

∂r ̄

)

=0 (17.75)

Region 2 :

1

̄r^2

∂

∂ ̄r

(

r ̄^2

∂ψ

∂r ̄

)

= ψ+α (17.76)

Region 3 :

1

̄r^2

∂

∂ ̄r

(

r ̄^2

∂ψ

∂r ̄

)

=(1+αZ ̄)ψ. (17.77)

The reasons for these approximations are as follows: