Fundamentals of Plasma Physics

386 Chapter 13. Fokker-Planck theory of collisions

the respective relative velocities before and after the collision, then

vrel 1 = vrelzˆ

vrel 2 = vrelcosθˆz+vrelsinθˆb. (13.22)

Thus

∆vrel = vrel 2 −vrel 1

= vrel(cosθ−1)ˆz+vrelsinθˆb

≃−

vrelθ^2

2

ˆz+vrelθ(ˆxcosφ+ˆysinφ) (13.23)

whereφis the angle between the impact parameter and thexaxis and the scattering angle
θis assumed to be small. The fact that thezcomponent is negative reflects the slowing
down of the particle in its initial direction of motion.
The cross section associated with impact parameterband the range of azimuthal impact
angledφisbdφdb.In time∆tthe incident particle moves a distancev∆tand so the vol-
ume swept out for this cross-section isbdφdbv∆t.The number of relevant field particles
encountered in time∆twill be the densityfF(vF)dvFof relevant field particles multi-
plied by this volume, i.e.,fF(vF)dvFbdφdbv∆tand so the change in relative velocity for
all possible impact parameters and all possible azimuthal angles for agivenvFwill be

∆v|allb,allφ=v∆tfF(vF)dvF

∫

dφ

∫

∆vbdb. (13.24)

The limits of integration for the azimuthal angle are from 0 to 2 πand the limits of integra-
tion of the impact parameter are from the 90^0 impact parameter to the Debye length. On
writing∆vin component form, this becomes

∆vrel|allb,allφ=vrel∆tfF(vF)dvF

∫ 2 π

0

dφ

∫λD

bπ/ 2

vrel

(

θcosφ,θsinφ,−θ^2 / 2

)

bdb.

(13.25)
Thexandyintegrals vanish upon integration overφand so

∆vrel|allb,allφ = −zvˆ^2 rel∆tfF(vF)dvFπ

∫λD

bπ/ 2

θ^2 bdb

= −zvˆ^2 rel∆tfF(vF)dvFπ

∫λD

bπ/ 2

(

qTqF

2 πε 0 bμvrel^2

) 2

bdb

= −zˆ∆tfF(vF)dvF

q^2 TqF^2

4 πε^20 μ^2 v^2 rel

lnΛ (13.26)

whereΛ=λD/bπ/ 2.
Using Eq.(13.12) the above expression can be transformed to give the test particle ve-
locity change to be

∆vT|allb,allφ = −ˆz∆tfF(vF)dvF

q^2 TqF^2

4 πε^20 μmTv^2 rel

lnΛ

= −∆t

qT^2 q^2 FlnΛ

4 πε^20 μmT

(vT−vF)

|vT−vF|^3

fF(vF)dvF. (13.27)