8.2 Analysis of the warm plasma electrostatic dispersion relation 235tion Eq.(5.51) derived in Chapter 4 to the situation of a magnetized warm plasma. This
means that Eq.(8.29) should revert to Eq.(5.51) in the limit thatB→ 0 .The Bessel iden-
tity
∑∞
n=−∞In(λ)=eλ (8.30)together with the conditionαn→α 0 ifB→ 0 show that this is indeed the case.
Furthermore, Eq.(8.29) should also be the warm-plasma generalization of the cold,
magnetized plasma, electrostatic dispersion
kx^2 S+kz^2 P=0; (8.31)demonstration of this correspondence will be presented later. Equation (8.30)can be used
to recast Eq.(8.29) as
D(ω,k)=1+∑
σe−k(^2) ⊥r (^2) Lσ
k^2 λ^2 Dσ
∑∞
n=−∞In(
k^2 ⊥r^2 Lσ)
[1+α 0 σZ(αnσ)]=0. (8.32)UsingI−n(z)=In(z), the summation overncan be rearranged to give
D(ω,k)=1+∑
σe−k(^2) ⊥r (^2) Lσ
k^2 λ^2 Dσ
I 0
(
k^2 ⊥r^2 Lσ)
(1+α 0 σZ(α 0 σ))+∑∞
n=1In(
k^2 ⊥r^2 Lσ)
[2+α 0 σ{Z(αnσ)+Z(α−nσ)}]
.
(8.33)
In order to obtain the lowest order thermal correction, it is assumed thatα 0 =ω/k‖vTσ>>
1 in which case the large argument expansion Eq.(5.73) can be used to evaluateZ(α 0 ).In-
voking this expansion and keeping only lowest order terms, shows that
1+α 0 Z(α 0 ) = 1+α 0{
−
1
α 0[
1+
1
2 α^20+...
]
+iπ^1 /^2 exp(−α^20 )}
= −
1
2 α^20+iα 0 π^1 /^2 exp(−α^20 ). (8.34)It is additionally assumed that|αn|=|(ω−nωcσ)/k‖vTσ|>> 1 forn = 0;this cor-
responds to assuming that the wave frequency is not too close to a cyclotron resonance.
With this assumption the large argument expansion of the plasma dispersion function is
also appropriate for then =0terms, and so one can write
2+α 0 [Z(αn)+Z(α−n)] = 2+α 0
[
−
1
αn−
1
α−n
+i√
π(
e−α(^2) n
+e−α
(^2) −n)
]
= 2−
ω
ω−nωcσ−
ω
ω+nωcσ+iα 0√
π(
e−α(^2) n
+e−α
(^2) −n)
= −
2 n^2 ω^2 cσ
ω^2 −n^2 ω^2 cσ+iα 0√
π(
e−α(^2) n
+e−α
(^2) −n)