Fundamentals of Plasma Physics

5.4 Assignments 175

These contributions are weighted according to how far they are fromvminby the factor
(v−vmin)−^2.
The Penrose criterion extends the 2-stream instability analysis to an arbitrary distribu-
tion function containing finite temperature beams.

5.4 Assignments

1. Show that the electrostatic dispersion relation for electrons streaming through ions
   with velocityv 0 through stationary ions is

1 −

ω^2 pi

ω^2

−

ω^2 pe

(ω−k·v 0 )^2

=0.

(a) Show that instability begins when

(

k·v 0

ωpe

) 2

<

[

1+

(

me

mi

) 1 / 3 ]^3

(b) Split the frequency into its real and imaginary parts so thatω=ωr+iωi. Show

that the instability has maximum growth rate

ωi

ωpe

=

√

3

2

(

me

2 mi

) 1 / 3

.

What is the value ofkv 0 /ωpewhen the instability has maximum growth rate.

Sketch the dependence ofωi/ωpeonkv 0 /ωpe.(Hint- define non-dimensional

variablesǫ=me/mi,z=ω/ωpe,andλ=kv 0 /ωpe.Letz=x+iy and

look for the maximumysatisfying the dispersion. A particularly neat way to

solve the dispersion is to solve the dispersion for the imaginary part ofλwhich

of course is zero, since by assumptionkis real. Take advantage of the fact that

ǫ << 1 to find a relatively simple expression involvingy. Maximizeywith

respect toxand then find the respective values ofx,y,andλat this point of

maximumy.

2. Prove the Plemelj formula.


3. Suppose that
   E(x,t)=

∫

E ̃(k)eik·x−iω(k)tdk (5.102)

where

ω=ωr(k)+iωi(k)

is determined by an appropriate dispersion relation. Assuming thatE(x,t)is a real

quantity, show by comparing Eq.(5.102) to its complex conjugate, thatωr(k) must

always be an odd function ofk whileωimust always be an even function ofk.

(Hint- Note that the left hand side of Eq.(5.102) is real by assumption, and sothe right

hand side must also be real. Take the complex conjugate of both sides and replace