470 Chapter 16. Non-neutral plasmas
whereLis the axial length of the plasma. The second term above can be directly evaluated
as
Re∫t−∞dt2il∣
∣
∣φ ̃l∣
∣
∣
2
e^2 ωit
(ω−lω 0 (r))= Re∫t−∞dt
2il(ωr−lω 0 (r)−iωi)
(ωr−lω 0 (r))^2 +ω^2 i∣
∣
∣ ̃φl∣
∣
∣
2
e^2 ωit=
∫t−∞dtl 2 ωi
(ωr−lω 0 (r))^2 +ω^2 i∣
∣
∣φ ̃l(t=0)∣
∣
∣
2
e^2 ωit=
l
(ωr−lω 0 (r))^2 +ω^2 i∣
∣
∣ ̃φl∣
∣
∣
2
e^2 ωit→
l
(ωr−lω 0 (r))^2∣
∣
∣ ̃φl∣
∣
∣
2
asωi→ 0 (16.62)Thus, in the limitωi→ 0 ,
δWl = −qL
4 B∫ 2 π0dθ∫a0dr[(
1
(ω−lω 0 (r))+
lω 0 (r)
(ω−lω 0 (r))^2)
dn 0
drl∣
∣
∣φ ̃l∣
∣
∣
2 ]
= −
πqL
2 B∫a0drω
(ω−lω 0 (r))^2dn 0
dr
l∣
∣
∣φ ̃l∣
∣
∣
2. (16.63)
For thel=1mode this can be evaluated using Eq. (16.50) to give
ω=−
q
ε 0 Ba^2∫rp0n 0 (r′)r′dr′δWl=1 = −πqL
2 B∫a0drω
(ω−ω 0 (r))^2dn 0
dr[
Sr(
ω−ω 0 (r)
2 ω)] 2
= −
πqLS^2
8 ωB∫a0dr
dn 0
drr^2= −
πqLS^2
8 ωB∫a0dr(
− 2 rn 0 (r)+
d
dr[
r^2 n 0 (r)]
)
=
πqLS^2
4 ωB∫rp0rn 0 (r)dr (16.64)wheren 0 (r) = 0forrp < r ≤ahas been used. Finally, using the middle line of
Eq.(16.51) this becomes
δWl=1=−ε 0 πLa^2 S^2
4(16.65)
and so thel= 1dioctron mode is seen to be a negative energy mode. This result can
also be derived by considering the change in system energy which results when the non-
neutral plasma is attracted to a fictitious image charge chosen to maintain the wall as an
equipotential when the plasma moves off axis (see assignments).
Forl ≥ 2 , the negative energy nature of the diocotron mode can be understood by
considering the change in the electrostatic energy stored in an isolatedparallel plate
capacitor when the distance between the parallel plates is varied. The stored energy is
W=CV^2 /2=Q^2 / 2 CandQis constant because the capacitor is isolated. Increasing the