Fundamentals of Plasma Physics

388 Chapter 13. Fokker-Planck theory of collisions

θ^2 .Thus, we obtain

∆vrel∆vrel|allb,φ = v^3 rel∆tfF(vF)dvF

∫^2 π

0

dφ

λ∫D

bπ/ 2

θ^2

(

ˆxxˆcos^2 φ+ˆyyˆsin^2 φ

)

bdb

= v^3 rel∆tfF(vF)dvFπ(ˆxxˆ+ˆyyˆ)

∫λD

bπ/ 2

θ^2 bdb

=∆tfF(vF)dvF (ˆxˆx+ˆyˆy)

q^2 Tq^2 FlnΛ

4 πε^20 μ^2 vrel

. (13.36)

Sincevreldefines thezdirection, we may write

xˆxˆ+ˆyyˆ=I−

vrelvrel

v^2 rel

(13.37)

whereIis the unit tensor. Thus,

∆vrel∆vrel|allb,allφ=∆t

q^2 Tq^2 FlnΛ

4 πε^20 μ^2

fF(vF)dvF

(

vrel^2 I−vrelvrel

v^3 rel

)

. (13.38)

Using Eq.(13.12) this can be transformed to give

∆vT∆vT|allb,allφ=∆t

qT^2 q^2 FlnΛ

4 πε^20 m^2 T

fF(vF)dvF

(

v^2 relI−vrelvrel

v^3 rel

)

. (13.39)

The tensor may be expressed in a simpler form by noting

∂

∂vT

∂vrel

∂vT

=

∂

∂vT

vrel

vrel

=

I

vrel

−

vrel

vrel^2

vrel

vrel

=

vrel^2 I−vrelvrel

v^3 rel

. (13.40)

Inserting Eq.(13.40) in Eq.(13.39) and then integrating over all the field particles shows
that
〈
∆vT∆vT
∆t

〉

=

q^2 TqF^2 lnΛ

4 πε^20 m^2 T

∫

fF(vF)dvF

∂

∂vT

∂

∂vT

|vT−vF|. (13.41)

However,vTis constant in the integrand and so∂/∂vTmay be factored from the integral.
Dropping the subscriptTfrom the velocity and changingvFto be the integration variable
v′this can be re-written as
〈
∆v∆v
∆t

〉

=

q^2 TqF^2 lnΛ

4 πε^20 m^2 T

∂

∂v

∂

∂v

∫

|v−v′|fF(v′)dv′. (13.42)

It is convenient to define the Rosenbluth potentials

gF(v)=

∫

|v−v′|fF(v′)dv′ (13.43)

hF(v)=

mT

μ

∫

fF(v′)

|v−v′|

dv′ (13.44)