Fundamentals of Plasma Physics

390 Chapter 13. Fokker-Planck theory of collisions

and so, using Eq.(13.44),

hF(v)=

nFmT

μ

(

mF

2 πκTF

) 3 / 2 ∫

exp

(

−mFv′^2 / 2 κTF

)

|v−v′|

dv′. (13.54)

The velocity integral in Eq.(13.54) can be evaluated using standard means (see assign-
ments) to obtain

hF(v)=

nFmT

μv

erf

(√

mF

2 κTF

v

)

(13.55)

where

erf(x)=

2

√

π

∫x

0

exp(−w^2 )dw (13.56)

is the Error Function.
Thus, Eq.(13.52) becomes

∂uT

∂t

=

niq^2 Tq^2 ilnΛ

4 πε^20 m^2 T

mT

μi

{

∂

∂v

[

v−^1 erf

(√

mi

2 κTi

v

)]}

v=u 0

+

neq^2 Tqe^2 lnΛ

4 πε^20 m^2 T

mT

μe

{

∂

∂v

[

v−^1 erf

(√

me

2 κTe

v

)]}

v=u 0

(13.57)

whereμ−i,e^1 =m−i,e^1 +m−T^1.
This can be further simplified by noting (i) quasi-neutrality implies

niZqi+neqe=0 (13.58)

whereZis the charge of the ions, (ii) the masses are related by

mT

μi,e

=1+

mT

mi,e

, (13.59)

and (iii) the velocity gradient of the Error Function must be in the direction ofu 0 because
that is the only direction there is in the problem. Using these relationships and realizing
that both the left and right sides are in the direction ofu,Eq.(13.57) becomes

∂u

∂t

=

nee^2 lnΛ

4 πε^20

q^2 T

m^2 T







Z

(

1+

mT

mi

)

d

du









erf

(√

mi

2 κTi

u

)

u









+

(

1+

mT

me

)

d

du









erf

(√

me

2 κTe

u

)

u















. (13.60)

Let us define

ξi,e=

√

mi,e

2 κTi,e

u (13.61)