Fundamentals of Plasma Physics

56 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD

one-dimensional Poisson’s equation

d^2 ̄φ

dx^2

=−

e

ε 0

(ni(x)−ne(x)) (2.110)

shows that in order for ̄φto have convex curvature, it is necessary to haveni(x)> ne(x)
everywhere. This condition will now be used to estimate the inflow velocity of the ions at
the location where they enter the sheath from the bulk plasma.
Ion energy conservation gives
1
2

miu^2 (x) +e ̄φ(x) =

1

2

miu^20 (2.111)

which can be solved to give

u(x) =

√

u^20 − 2 e ̄φ(x)/mi. (2.112)

Using the ionflux conservation relationn 0 u 0 =ni(x)ui(x)the local ion density is found
to be
ni(x) =

n 0

(

1 − 2 eφ ̄(x)/mu^20

) 1 / 2. (2.113)

The convexity requirementni(x)>ne(x)implies
(
1 − 2 e ̄φ(x)/miu^20

)− 1 / 2

>ee

̄φ(x)/κTe

. (2.114)

Bearing in mind that ̄φ(x)is negative, this can be re-arranged as

1 +

2 e| ̄φ(x)|

miu^20

<e^2 e|

φ ̄(x)|/κTe

(2.115)

or
2 e|φ ̄(x)|
miu^20

<

2 e|φ ̄(x)|

κTe

+

1

2

(

2 e|φ ̄(x)|

κTe

) 2

+

1

3!

(

2 e| ̄φ(x)|

κTe

) 3

+... (2.116)

which can only be satisfied for arbitrary|φ ̄(x)|if

u^20 > 2 κTe/mi. (2.117)

Thus, in order to be consistent with the assumption that the probe is more negative than
the plasma to keepd^2 φ/ ̄dx^2 negative and hence ̄φconvex, it is necessary to have the
ions enter the region of non-neutrality with a velocity slightly larger than the so-called “ion
acoustic” velocitycs=

√

κTe/mi.

The ion current collected by the probe is given by the ionflux times the probe area, i.e.,

Ii = n 0 u 0 qiA

= n 0 cseA. (2.118)

The electron current density incident on the probe is

Je(x) =

∫∞

0

dvqevfe(v,0)

=

n 0 qee−qe ̄φ(x)/κTe

√

π 2 κTe/me

∫∞

0

dvve−mv

(^2) / 2 κTe
= n 0 qe

√

κTe

2 πme

e−e|

φ ̄(x)|/κTe

(2.119)