13.3 Electrical resistivity 393and the slowing down time becomes
τs=3
√
ππε^20
nee^2 lnΛm^2 T
q^2 T1
(
Z
(
1+
mT
mi)(
mi
2 κTi) 3 / 2
+
(
1+
mT
me)(
me
2 κTe) 3 / 2 ).
(13.70)
The temperature terms scale as the inverse cube of the thermal velocity and so if the
ion and electron temperatures are the same order of magnitude, then the ion contribution
dominates. Thus, the slowing down is mainly done by collisions with ions and the slowing
down time is
τs≃3
√
ππε^20
nee^2 lnΛm^2 T
q^2 T(2κTi/mi)^3 /^2Z(
1+
mT
mi). (13.71)
The beam takes a longer time to slow down in hotter plasmas.
Intermediate case: beam faster than ions, slower than electrons
In this case, the slowing down equation becomes
∂u
∂t=−
nee^2 lnΛ
4 πε^20q^2 T
m^2 T(
Z
(
1+
mT
mi)
1
u^2+
(
1+
mT
me)(
me
2 κTe) 3 / 2
4 u
3√
π)
(13.72)
and the slowing down time isτs=4 πε^20 m^2 TneqT^2 e^2 lnΛ(
Z
(
1+
mT
mi)
1
u^3 +4
3
√
π(
1+
mT
me)(
me
2 κTe) 3 / 2 ). (13.73)
13.3 Electrical resistivity
If a uniform steady electric field is imposed on a plasma this electric field will accelerate
the ions and electrons in opposite directions. The accelerated particles will collide with
other particles and this frictional drag will oppose the acceleration so that a steady state
might be achieved where the accelerating force balances the drag force. The electric field
causes equal and opposite momentum gains by the electrons and ions and therefore does not
change the net momentum of the entire plasma. Furthermore, electron-electron collisions
cannot change the total momentum of all the electrons and ion-ion collisions cannotchange
the momentum of all the ions. The only wave for the entirety of electrons to lose momentum
is by collisions with ions.
Let us assume that an equilibrium is attained where the electrons and ionshave shifted
Maxwellian distribution functions
fi(v) =ni
π^3 /^2 (2κTi/mi)^3 /^2exp(
−mi(v−ui)^2 / 2 κTi)
fe(v) =ne
π^3 /^2 (2κTe/me)^3 /^2exp(
−me(v−ue)^2 / 2 κTe