Fundamentals of Plasma Physics

394 Chapter 13. Fokker-Planck theory of collisions

The current density will be

J=niqiui+neqeue. (13.75)

It is convenient to transform to a frame moving with the electrons, i.e., with the velocity
ue. In this frame the electron mean velocity will be zero and the ion meanvelocity will be
urel=ui−ue.The current density remains the same because it is proportional tourel
which is frame-independent. The distribution functions in this frame willbe

fi(v) =

ni

π^3 /^2 (2κTi/mi)^3 /^2

exp

(

−mi(v−urel)^2 / 2 κTi

)

fe(v) =

ne

π^3 /^2 (2κTe/me)^3 /^2

exp

(

−mev^2 / 2 κTe

)

. (13.76)

Since the ion thermal velocity is much smaller than the electron thermal velocity, the
ion distribution function is much narrower than the electron distributionfunction and the
ions can be considered as a mono-energetic beam impinging upon the electrons. Using
Eq.(13.57), the net force on this ion beam due to the combination of frictional drag on
electrons and acceleration due to the electric field is

mi

∂urel

∂t

=

neq^2 iq^2 elnΛ

4 πε^20 μe

{

∂

∂v

[

v−^1 erf

(√

me

2 κTe

v

)]}

v=urel

+qiE. (13.77)

In steady-state the two terms on the right hand side balance each other inwhich case

E=−

neqie^2 lnΛ

4 πε^20 μe

{

∂

∂v

[

v−^1 erf

(√

me

2 κTe

v

)]}

v=urel

. (13.78)

Ifurelis much smaller than the electron thermal velocity, then Eq.(13.64a) canbe used to
obtain

E=

neqie^2 lnΛ

3

√

ππε^20 me

urel

(2κTe/me)^3 /^2

=

Ze^2 m^1 e/^2 lnΛ

3

√

ππε^20

J

(2κTe)^3 /^2

(13.79)

whereurel=J/neeandμ−e^1 =1/mi+1/me≃ 1 /mehave been used. Equation (13.79)
can be used to define the electrical resistivity

η=

Ze^2 m^1 e/^2

3

√

ππε^20

lnΛ

(2κTe)^3 /^2

(13.80)

so that

E=ηJ. (13.81)

If the temperature is expressed in terms of electron volts thenκ=eand the resistivity is

η=1. 03 × 10 −^4

ZlnΛ

T

3 / 2

e

Ohm-m. (13.82)