198 Chapter 6. Cold plasma waves in a magnetized plasma
Perhaps the most important result of this era was a peculiar, but useful, reformulation
by Appleton, Hartree, and Altar^3 (Appleton 1932) of theω>>ωpi,ωcilimit of Eq.(6.49).
The intent of this reformulation was to expressn^2 in terms of its deviation from the vacuum
limit,n^2 =1.An obvious way to do this is to defineξ=n^2 − 1 and then re-write Eq.(6.47)
as an equation forξ,namely
A(ξ^2 +2ξ+1)−B(ξ+1)+C=0 (6.66)or, after regrouping,
Aξ^2 +ξ(2A−B)+A−B+C=0. (6.67)
Unfortunately, when this expression is solved forξ,the leading term is− 1 and so this
attempt to find the deviation ofn^2 from its vacuum limit fails. However, a slight rewriting
of Eq.(6.67) as
A−B+C
ξ^2
+
2 A−B
ξ+A=0 (6.68)
and then solving for 1 /ξ, gives
ξ=2(A−B+C)
B− 2 A±
√
B^2 − 4 AC
. (6.69)
This expression does not have a leading term of− 1 and so allows the solution of Eq.(6.49)
to be expressed as
n^2 =1+2(A−B+C)
B− 2 A±
√
B^2 − 4 AC
. (6.70)
In theω>>ωci,ωpilimit whereS,P,Dare given by Eq.(6.64), algebraic manipulation of
Eq.(6.70) (cf. assignments) shows that there exists a common factor in thenumerator and
denominator of the second term. After cancelling this common factor, Eq.(6.70) reduces
to
n^2 =1−
2
ω^2 pe
ω^2(
1 −
ω^2 pe
ω^2)
2
(
1 −
ω^2 pe
ω^2)
−
ω^2 ce
ω^2sin^2 θ±Γ
(6.71)
where
Γ=
√√
√
√ω^4 ce
ω^4sin^4 θ+4ω^2 ce
ω^2(
1 −
ω^2 pe
ω^2) 2
cos^2 θ. (6.72)Equation (6.71) is called the Altar-Appleton-Hartree dispersion relation (Appleton 1932)
and has the desired property of showing the deviation ofn^2 from the vacuum dispersion
n^2 =1.
We recall that the cold plasma dispersion relation simplified considerably when either
θ= 0orθ=π/ 2 .A glance at Eq.(6.71) shows that this expression reduces indeed to
n^2 =R,Lforθ= 0.Somewhat more involved manipulation shows that Eq.(6.71) also
reduces ton^2 +=Pandn^2 −=RL/Sforθ=π/ 2.
(^3) See discussion by Swanson (1989) regarding the recent addition of Altar to this citation