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)