Fundamentals of Plasma Physics

128 Chapter 4. Elementary plasma waves

1 2 3

/k T
 0 /m

≈^1

k^2 D^2 

≈−

p^2 


^2

Figure 4.1: Susceptibilityχas a function ofω/k

√

κTσ 0 /mσ.

Since the ion-to-electron mass ratio is large, ions and electrons typically have thermal
velocities differing by at least one and sometimes two orders of magnitude. Furthermore,
ion and electron temperatures often differ, again allowing substantially different electron
and ion thermal velocities. Three different situations can occur in atypical plasma de-
pending on how the wave phase velocity compares to thermal velocities. These situations
are:

1. Case whereω/k >>

√

κTe 0 /me,

√

κTi 0 /mi

Here both electrons and ions are adiabatic and the dispersion relationbecomes

1 −

ω^2 pe

ω^2

(

1+3

k^2

ω^2

κTe 0

me

)

−

ω^2 pi

ω^2

(

1+3

k^2

ω^2

κTi 0

mi

)

=0. (4.29)

Sinceω^2 pe/ω^2 pi=mi/methe ion contribution can be dropped, and the dispersion

becomes

1 −

ω^2 pe

ω^2

(

1+3

k^2

ω^2

κTe 0

me

)

=0. (4.30)

To lowest order, the solution of this equation is simplyω^2 =ω^2 pe. An iterative solution

may be obtained by substituting this lowest order solution into the thermal term which,

by assumption, is a small correction becauseω/k >>

√

κTe 0 /me.This gives the