Fall Term 2002
Introduction to Plasma Physics I
22.611J, 6.651J, 8.613J
Problems for study
1. Collision Operator Properties. The Coulomb collision operator for like-particle collisions in
Landau form is given by,
¶
µ
Z
¡ ¢
∂
∂
∂
0
3 0
−
f (v) f v0
C (f, f ) = Γ
· d v U(v − v ) ·
0
∂v
∂v ∂v
with,
0
U(v − v ) =
Γ =
µ
¶
(v − v0 ) (v − v0 )
I−
|v − v0 |2
2πq 4 ln Λ
m2
1
|v − v0 |
Prove the conservation laws for the collision operator:
R
Particle Conservation
0 =R d3 vC(f, f )
Momentum Conservation 0 =R d3 vmvC(f, f )
Energy Conservation
0 = d3 v 12 mv 2 C(f, f )
2. Two Species Collisions: Show that the conservation laws also hold for collision between two
species, but that the sum over species is now required:
R
Particle Conservation
0 = d3 vCαβ (fα , fβ )
R 3
(fα , fβ ) + mβ vCβα (fβ , fα )]
0 = d v [mα vC
R αβ
Momentum Conservation
3
£ 0 = d vmα vCαα (fα , fα )
¤
R
0 = d3 v 12 mα vR2 Cαβ (fα , fβ ) + 12 mβ v2 Cβα (fβ , fα )
Energy Conservation
0 = d3 v 12 mα v 2 Cαα (fα , fα )
Note that particle’s are conserved by each collision operator, while momentum and energy is
only conserved as a species sum. Make a statement of the physical meaning of the two sepa
rate conservation laws for each of momentum and energy. Recall that the collision operator
between species, α, and, β, is given by,
¶
µ
Z
¡ ¢
mβ ∂
∂
∂
0
3 0
· d v U(v − v ) ·
Cαβ (fα , fβ ) = Γαβ
fα (v) fβ v0
−
0
∂v
mα ∂v ∂v
2
2
4
2πZα Zβ e ln Λ
Γαβ =
mα mβ
and the tensor, U, defined in terms of particle velocities, has the same definition as above.
1
3. Temperature Equilibration: Show that temperature equilibration proceeds according to a term
of the form,
r
∂ 3
me 2
nTe = −ν ei
n (Te − Ti )
∂t 2
mi π
by taking the energy moment,
Z
1
d3 v me v 2 Cei (femax , fimax )
2
You may use the expanded form of, Cei , obtained by assuming a Maxwellian distribution for
the ions,
L
E
(fe ) + Cei
(fe )
Cei ' Cei
with the Lorentz collision operator,
ve3
L (fe )
v3
¢ ∂
1 ∂ ¡
1 − µ2
2 ∂µ
∂µ
L
Cei
(fe ) ≡ ν ei
L ≡
and the energy exchange operator,
E
(fe )
Cei
¸
·
Ti ∂ 1 ∂
me 3 1 ∂
+
fe
= ν ei
v
2mi e v2 ∂v me v2 ∂v v ∂v
The electron-ion collision frequency defined as,
ν ei ≡
4πne4 ln Λ
me ve3
p
and we have used the convention, ve = Te /me , for the thermal velocity. n.b. different from
convention in Landau problem to make expressions cleaner. . .
What is the relative rate of angle scattering vs. thermalization for electrons scattering off
ions?
2