Fundamentals of Plasma Physics

118 Chapter 3. Motion of a single plasma particle

Using the identities
〈sin(θ 0 +ατ)cos(θ 0 +ατ)〉= 0
〈sinθ 0 cos(θ 0 +ατ)〉=−^12 sinατ
〈cosθ 0 cos(θ 0 +ατ)〉=^12 cosατ

(3.248)

the wave-to-particle energy transfer rate becomes
〈
dW
dt

〉

=−

mω^2 bv 0

2 k

(

sinατ

α^2

−

τ

α

cosατ

)

=

mω^2 bv 0

2 k

d

dα

(

sinατ

α

)

. (3.249)

At this point it is recalled that one representation for a delta function is

δ(z) = lim

N→∞

sin(Nz)

πz

(3.250)

so that for|ατ|>> 1 Eq.(3.249) becomes
〈
dW
dt

〉

=

πmω^2 bv 0

2 k

d

dα

δ(α). (3.251)

Sinceδ(z)has an infinite positive slope just to the left ofz= 0and an infinite negative
slope just to the right ofz= 0, the derivative of the delta function consists of a positive
spike just to the left ofz= 0and a negative spike just to the right ofz= 0.Furthermore
α= (kv 0 −ω)/ωbis slightly positive for particles moving a little faster than the wave
phase velocity and slightly negative for particles moving a little slower. Thus〈dW/dt〉
is large and positive for particles moving slightly slower than the wave, while it is large
and negative for particles moving slightly faster. If the number of particles moving slightly
slower than the wave equals the number moving slightly faster, the energygained by the
slightly slower particles is equal and opposite to that gained by the slightly faster particles.
However, if the number of slightly slower particlesdiffersfrom the number of slightly
faster particles, there will be a net transfer of energy from wave to particles or vice versa.
Specifically, if there are more slow particles than fast particles, there will be a transfer of
energy to the particles. This energy must come from the wave and a more complete analysis
(cf. Chapter 5) will show that the wavedamps. The direction of energy transfer depends
critically on the slope of the distribution function in the vicinity ofv=ω/k,since this
slope determines the ratio of slightly faster to slightly slower particles.
We now consider a large number of particles with an initial one-dimensional distribution
functionf(v 0 ) and calculate the net wave-to-particle energy transfer rate averaged over
all particles. Sincef(v 0 )dv 0 is the probability that a particle had its initial velocity between
v 0 andv 0 + dv 0 , the energy transfer rate averaged over all particles is

〈

dWtotal

dt

〉

=

∫

dv 0 f(v 0 )

πmω^2 bv 0

2 k

d

dα

δ(α)

=

πmω^3 b

2 k^2

∫

dv 0 f(v 0 )v 0

d

dv 0

δ

(

kv 0 −ω

ωb

)

=

πmω^4 b

2 k^3

∫

dv 0 f(v 0 )v 0

d

dv 0

δ

(

v 0 −

ω

k

)

= −

πmω^4 b

2 k^3

[

d

dv 0

(f(v 0 )v 0 )

]

v 0 =ω/k

. (3.252)