Fundamentals of Plasma Physics

3.9 Wave-particle energy transfer 117

and so the rate at which energy is transferred from the wave to the particles is

dW

dt

= −

mω^3 b

k^2

sinθ

[

ω

ωb

+α

]

≃−

mω^2 bv 0

k

sinθ (3.238)

Integration of Eq.(3.237) gives the unperturbed orbit solution

θ(τ) =θ 0 +ατ. (3.239)

This first approximation is then substituted back into Eq.(3.236) to get the corrected form

dθ

dτ

=α+

cos(θ 0 +ατ)−cosθ 0

α

(3.240)

which may be integrated to give the corrected phase

θ(τ) =θ 0 +ατ+

sin(θ 0 +ατ)−sinθ 0

α^2

−

τ

α

cosθ 0. (3.241)

From Eq.(3.241) we may write

sinθ= sin[(θ 0 +ατ) + ∆(τ)] (3.242)

where

∆(τ) =

sin(θ 0 +ατ)−sinθ 0

α^2

−

τ

α

cosθ 0 (3.243)

is the ‘perturbed-orbit’ correction to the phase. If consideration is restricted to times where
τ <<|α|, the phase correction∆(τ)will be small. This restriction corresponds to

(ωbt)^2 <<|kv 0 −ω|t (3.244)

which means that the number of wave peaks the particle passes greatly exceeds the number
of bounce times. Since bounce frequency is proportional to wave amplitude, thiscondition
will be satisfied for all finite times for an infinitesimal amplitude wave. Because∆is
assumed to be small, Eq.(3.242) may be expanded as

sinθ= sin(θ 0 +ατ)cos∆+sin∆cos(θ 0 +ατ)≃sin(θ 0 +ατ)+∆cos(θ 0 +ατ)(3.245)

so that Eq. (3.238) becomes

dW

dt

=−

mω^2 bv 0

k

[sin(θ 0 +ατ) + ∆cos(θ 0 +ατ)]. (3.246)

The wave-to-particle energy transfer rate depends on the particle initial position. This
is analogous to the earlier sawtooth potential analysis where it was shownthat whether
particles gain or lose average velocity depends on their injection phase.It is now assumed
that there exist many particles withevenly spacedinitial positions and then an averaging
will be performed over all these particles which corresponds to averaging over all initial
injection phases. Denoting such averaging by〈〉gives
〈
dW
dt

〉

= −

mω^2 bv 0

k

〈∆cos(θ 0 +ατ)〉

= −

mω^2 bv 0

k

〈[

sin(θ 0 +ατ)−sinθ 0

α^2

−

τ

α

cosθ 0

]

cos(θ 0 +ατ)

〉

(3.247)