Fundamentals of Plasma Physics

396 Chapter 13. Fokker-Planck theory of collisions

(a) Show that the Rosenbluthhpotential can be approximated as

hF(v) =

mT

μ

∫

fF(v′)

|v−v′|

dv′

≃

menF

vμ

where

1

μ

=

1

me

+

1

mF

.

What is the form ofhF(v)for beam electrons interacting with background

plasma (i) electrons and (ii) ions?

(b) By approximating

(

v^2 − 2 v·v′+v′^2

) 1 / 2

=v

(

1 −

v·v′

v^2

+

v′^2

2 v^2

)

where the beam velocityvis much larger than the background species velocity

v′, show that the Rosenbluthgpotential can be approximated as

gF(v) =

∫

|v−v′|fF(v′)dv′

≃

(

v+

1

v

3 κTF

2 mF

)

nF.

Hint: use symmetry arguments when considering the term involvingv·v′.

(c) Assume that the beam velocity isz-directed and letFT=

∫

d^2 v⊥fTso thatFT

describes the projection of the 3-D beam distribution onto thezaxis of velocity

space. Sincev=vzandvy=vx≈ 0 show that it is possible to write

hF(v) ≃

menF

vzμ

gF(v) ≃

(

vz+

1

vz

3 κTF

2 mF

)

nF.

Show that the Fokker-Planck equation can be written as

∂FT

∂t

≃

∑

F=i,e

nFq^2 TqF^2 lnΛ

4 πε^20 m^2 T









∂

∂vz

(

FT

mT

μv^2 z

)

+

1

2

∂^2

∂vz^2

(

FT

∂^2

∂vz^2

(

vz+

3 κTF

2 mFvz

))









≃

∑

F=i,e

nFq^2 TqF^2 lnΛ

4 πε^20 m^2 T

∂

∂vz

[

FT

mT

μv^2 z

+

3

2 v^3 z

κTF

mF

∂FT

∂vz

]

.

(d) Taking into account charge neutrality andvz>>vTishow that this can be recast

as

∂FT

∂t

≃

nee^4 lnΛ

4 πε^20 m^2 e

∂

∂vz

[

FT

2+Z

v^2 z

+

3

2 v^3 z

κTe

me

∂FT

∂vz

]