174 Chapter 5. Streaming instabilities and the Landau problem
to an instability. However, curves of the form sketched in Figs.5.6(c) and (d) do have
Q(z)taking on positive real values and so do correspond to unstable solutions. Marginally
unstable situations correspond to whereCQcrosses the positive realQaxis, sinceCQis a
mapping ofCzwhich was the set of marginally unstable frequencies.
Let us therefore focus attention on what happens whenCQcrosses the positive realQ
axis. Using the Plemelj formula on Eq.(5.96) it is seen that
ImQ=q^2
mε 0
π[
∂f 0
∂v]
v=ωr/k(5.97)
and, on moving alongCQfrom a point just below the realQaxis to just above the real
Qaxis, ImQgoes from being negative to positive. Thus,[∂f 0 /∂v]v=ωr/kchanges from
being negative to positive, so that on the positive realQaxisf 0 is aminimumat some value
v=vmin(here the subscript “min” means the value ofvfor whichf 0 is at a minimum and
notvitself is at a minimum). A Taylor expansion about this minimum gives
f(v)=f[vmin+(v−vmin)]=f(vmin)+0+(v−vmin)^2
2f′′(vmin)+... (5.98)Sincef(vmin)is a constant, it is permissible to write
∂f 0
∂v=
∂
∂v[f 0 (v)−f 0 (vmin)]. (5.99)This innocuous insertion off(vmin)makes it easy to integrate Eq. (5.96) by parts
Q(z = vmin)=q^2
mε 0
P
∫∞
−∞dv∂
∂v
[f(v)−f(vmin)]
(v−vmin)+iπ[
∂
∂vf(v)]
v=vmin
=
q^2
mε 0P
∫∞
−∞dv1
(v−vmin)^2
[f(v)−f(vmin)]=
q^2
mε 0∫∞
−∞dv1
(v−vmin)^2[
0+
(v−vmin)^2
2f′′(vmin)+...]
;
(5.100)
in the second line advantage has been taken of the fact that the imaginary partis zero by
assumption, and in the third line the ‘P’ for principle part has been dropped because there
is no longer a singularity atv=vmin.In fact, since the leading term off(v)−f(vmin)is
proportional to(v−vmin)^2 , this qualifying ‘P’ can also be dropped from the second line.
The requirement for marginal instability can be summarized as:f(v)has a minimum at
v=vmin, and the value ofQis positive, i.e.,
Q(vmin)=q^2
mε 0∫∞
−∞dv[
f(v)−f(vmin)
(v−vmin)^2]
> 0. (5.101)
This is just a weighted measure of the strength of the bump inflocated to the right of the
minimum as shown in Fig.5.6(e). The hatched areas with horizontal linesmake positive
contributions toQ, while the hatched areas with vertical lines make negative contributions.