Fundamentals of Plasma Physics

6.2 Dielectric tensor 183

What does all this algebra mean? Equation (6.25) states that forθ=0the dispersion re-
lation has two distinct roots, each corresponding to a natural mode (or characteristic wave)
constituting a self-consistent solution to the Maxwell-Lorentz system. The definitions in
Eqs.(6.12) and (6.26) show that

R=1−

∑

σ

ω^2 pσ

ω(ω+ωcσ)

, L=1−

∑

σ

ω^2 pσ

ω(ω−ωcσ)

(6.28)

so thatRdiverges whenω=−ωcσwhereasLdiverges whenω=ωcσ.Sinceωcσ=
qσB/mσ,the ion cyclotron frequency is positive and the electron cyclotron frequencyis
negative. Hence,Rdiverges at the electron cyclotron frequency, whereasLdiverges at the
ion cyclotron frequency. Whenω→∞, bothR,L→ 1 .In the limitω→ 0 ,evaluation of
R,Lmust be done very carefully, since

ω^2 pσ

ωcσ

=

nσq^2 σ

ε 0 mσ

mσ

qσB

=

nσqσ

ε 0 B

(6.29)

so that

ω^2 pi

ωci

=−

ω^2 pe

ωce

. (6.30)

Thus

lim

ω→ 0

R,L =1−

1

ω

[

ω^2 pi

(ω±ωci)

+

ω^2 pe

(ω±ωce)

]

=1−

ω^2 pi+ω^2 pe

ωciωce

≃ 1 −

neq^2 e

ε 0 me

mi

qiB

me

qeB

=1+

ω^2 pi

ω^2 ci

=1+

c^2

v^2 A

(6.31)

wherevA^2 =B^2 /μ 0 ρis the Alfvén velocity. Thus, at low frequency, bothRandLare
related to Alfvén modes. Then^2 =Lmode is the slow mode (largerk) and then^2 =R
mode is the fast mode (smallerk). Figure 6.1 shows the frequency dependence of the
n^2 =R,Lmodes.