16.5 Diocotron modes 469erty and Levy 1970)
w ̄ =∫t−∞dt〈
E 1 ·
(
J 1 +ε 0∂E 1
∂t)〉
=
ε 0
2E 12 +
∫t−∞dt〈E 1 ·J 1 〉. (16.56)Unlike the uniform plasma analysis in Section 7.4, here both plasma non-uniformity and
wall boundary conditions must be taken into account. The change in system energyW ̄ due
to establishment of the diocotron wave is found by integratingw ̄over volume and is
W ̄ =
∫
d^3 r(
ε 0
2(
∇·φ 1 ∇φ 1 −φ 1 ∇^2 φ 1)
+
∫t−∞dt n 1 qrω 0 (r)E 1 θ)
= q∫
d^3 r(
1
2
n 1 φ 1 +rω 0 (r)∫t−∞dt n 1 E 1 θ)
(16.57)
where Eq.(16.55) and the perfectly conducting wall boundary conditionφ 1 (a) = 0have
been used. Using Eq.(7.62) and (16.32), the energy for a givenlmode can be written as
Wl=
q
2Re∫
d^3 r(
1
2
n ̃lφ ̃∗
le2 ωit+rω
0 (r)∫t−∞dtn ̃lE ̃lθ∗e^2 ωit)
(16.58)
In order to evaluate this expression, it is necessary to express allfluctuating quantities
in terms of the oscillating potentialφ ̃l. Using Poisson’s equation in Eq.(16.35) it is seen
that
n ̃l =−l ̃φl
rB(ω−lω 0 (r))dn 0
dr(16.59)
and the complex conjugate of the linearized azimuthal electric field is
E ̃ 1 ∗θ=(
−il ̃φl
r)∗
=il ̃φ∗
1
r. (16.60)
The wave energy is thus
δWl = −ql
2 B
Re∫
d^3 r
∣
∣
∣φ ̃l∣
∣
∣
22 r(ω−lω 0 (r))dn 0
dre^2 ωit+rω 0 (r)∫t
−∞dtil∣
∣
∣ ̃φl∣
∣
∣
2
e^2 ωit
r^2 (ω−lω 0 (r))dn 0
dr
= −
qL
4 BRe∫ 2 π0dθ∫a0dr
e^2 ωit
(ω−lω 0 (r))
+ω 0 (r)∫t
−∞2ile^2 ωitdt
(ω−lω 0 (r))
dn 0
drl∣
∣
∣ ̃φl∣
∣
∣
2