Plasma Physics and Technology
Problem sheet #10
Kinetic theory II
1. (F. F. Chen 7.2) An electron plasma wave with 1 cm wavelength is
excited in a 10 eV plasma with n = 1015 cm−3 . The excitation is then
removed and the wave Landau damps away. How long does it take for
the amplitude to fall by a factor of e?
2. (Exam 2014/2015) Consider a one dimensional electron distribution
function of the form
2
np
u
nb
(u − ub )2
f0 (u) = 1/2 exp − 2 + 1/2 exp −
,
π vt
vt
π vb
vb2
resulting from the injection of an electron beam of average speed
qub and
density nb on a Maxwellian plasma of density np , where vt = kBmT is
the thermal electron speed. Suppose as well that vt ∼ vb and ub vt ,
nb np , and neglect ion motion.
R +∞
(a) Calculate −∞ f0 (u)du to verify that the distribution function is
correctly normalised.
(b) Sketch f0 (u) and show where do you expect that unstable waves
may exist.
(c) In which interval of phase speeds would you search for unstable
waves? [Suggestion: determine where the two components of f0
give the same contribution]
(d) Determine the frequency, wave number and growth rate for the
fastest growing mode.
3. (F. F. Chen 7.3) An infinite, uniform plasma with fixed ions has an
electron distribution function composed of (1) a Maxwellian distribution of “plasma electrons” with density np and temperature Tp at rest
in the laboratory frame, and (2) a Maxwellian distribution of “beam
electrons” with density nb and temperature Tb centered at ~v = V ~ux . If
nb np , plasma oscillations in the x-direction are Landau damped. If
nb is large, there will be a two-stream instability. The critical density
for the onset at the instability can be estimated by setting the slope of
the total distribution function to zero, as follows:
(a) write expressions for fp (v) and fb (v), using the abbreviations v =
B Tp
B Tb
and b2 = 2km
;
vx , a2 = 2km
(b) assuming that the value of the phase velocity vϕ will be the value
of v at which fb (v) has the largest positive slope, find vϕ and
fb0 (vϕ );
(c) find fp0 (vϕ ) and set fp0 (vϕ ) + fb0 (vϕ ) = 0;
(d) para V b show that the beam
density is given approxi
critical
√
2
mately by nnpb = 2e TTpb Va exp − Va2 .
4. (Exam 2015/2016, Gardner’s theorem) We want to study the propagation of Langmuir waves starting from Vlasov’s equation. As it has
been shown in class, if we assume immobile positive ions (mi → ∞),
the dispersion relation can be written in the form
2 Z +∞
ωpe
∂g
1
(k, ω) = 1 − 2
du = 0
k −∞ ∂u u − ω/k
where g(u) is the unidimensional distribution function
Z +∞ Z +∞
1
g(u) =
f0 (u, vy , vz )dvy dvz .
n0 −∞ −∞
(a) Justify that if g is Maxwellian and the wave phase speed is much
larger than the electron thermal speed we can, on a first approximation accounting only for the contribution of the electrons of the
body of the distribution, neglect the pole on the integral. Obtain
the dispersion relation in this case
R +∞
[Suggestion: recall that for a Maxwellian and u vϕ , −∞ g(u)/(u−
vϕ )2 du ' 1/vϕ2 + 3vt2 /vϕ4 ]
(b) In fact ω can be complex. There are unstable modes if the imaginary part of the frequency is positive. We want to show Gardner’s
theorem, establishing that a single-humped velocity distribution is
always stable.
The proof can be made by
contradiction. Consider ω =
ωr + iγ in the expression from
g(u)"
a), where ωr and γ are the
real and imaginary parts of the
frequency, respectively. Assuming γ > 0 the integral in
the dispersion relation can be
made along the real axis u,
v0"
since the pole is above that
axis.
i. Show that the dispersion relation can be written in the form
∂g
ωr
2 Z +∞
ωpe
u
−
∂u
r (k, w) = 1 − 2
k =0
k −∞ u − ωr 2 + γ 2
k
2
ωpe
γ
i (k, ω) = − 2
k k
Z
+∞
−∞
u−
∂g
∂u
ωr 2
k
k
+
γ 2
k
=0
ii. Show that
2
ωpe
1+ 2
k
Z
+∞
−∞
∂g
∂u
(v0 − u)
2
2 = 0 ,
u − ωkr + γk
where v0 is the value of u corresponding to the hump in the
distribution function (see figure).
[Suggestion: consider the linear combination r − i (kv0 −
ωr )/γ]
iii. Show that the expression from the previous question can never
be satisfied and conclude about the stability of single-humped
distributions.
u"