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)