Download - Quasilinear Theory with Spontaneous Emission
Quasilinear Theory with Spontaneous Emission
An Important Concept
Cyclotron resonance is not an efficient process
• Only a narrow spectrum of waves can interact with the beam particles.
0p z zk v
0 0p z A zk v v v
Resonant Interaction
k
00
pkv
2
8kE
The purpose of introducing the concept of spontaneous emission is
two fold:
• The concept itself is important in plasma physics.
• Induced and spontaneous emissions of radiation are essential in radio astronomy.
• One should know that in quasilinear theory spontaneous emission of waves can be more important than “collisions”.
We must remember:
• Conceptually in plasmas particles and waves are inseparable.
• Particles are accelerated by absorbing wave energy.
• Then waves are emitted by accelerated particles.
• In a plasma in thermodynamic equilibrium the rate of wave absorption is equal to the rate of wave emission.
However, theoretically
• Our understanding of spontaneous radiation is limited to resonant emission.
• It is not known whether non-resonant quasilinear process can or cannot lead to spontaneous emission.
• It is a totally new topic area.
One may imagine:
• If the plasma deviates slightly from an equilibrium state, the two rates are no longer in equilibrium.
• For instance, a weak instability sets in and some “waves” gain energy.
• “In principle” the plasma may adjust itself so that a new equilibrium state is attained. But there is difficulty!
In Plasmas
• We may have spontaneous emission or induced emission.
• We can have induced absorption.• But spontaneous absorption does not
make sense (within the context of quasilinear or linear theory).
• That is why “heating” is difficult.
The Usual Quasilinear Theory
• Only considers induced emission of waves.
• In general it is desirable to include spontaneous emission in the theory.
• What does “spontaneous emission” mean.
Let us consider the wave kinetic equation.
• In the quasilinear theory discussed so far the wave kinetic equation take the form
• The growth rate may be conceived as an emission coefficient.
• The term on the right-hand side denotes
“induced emission” of waves.
222k
k k
EE
t
Theory includes spontaneous emission
• The wave kinetic equation becomes
• The last term represents spontaneous emission of waves. It is discussed in the literature.
• The meaning of “spontaneous is clear.
222k
k k k
EE S
t
Let us consider the case of Landau damping
• If the plasma is in thermodynamic equilibrium and the plasma is in a stationary state, we expect
• The following relation is anticipated
• The emissivity depends on temperature. • The relation yields thermal fluctuations.
222 0k
k k k
EE S
t
22k k kS E
QLT with Spontaneous Emission
22
2
,
4 ,
Re ( , ) /
s
kss s
s s
F t
t
EF te Fdm k m
v
vk k k v k
v k v
2 22 2
2
( )2 8
Re ( , ) /k s s s
k k
E n e FE d d
t k
k v
k vk
Astrophysical Radio Emission
• Intense radiation is mostly produced via induced processes rather than spontaneous processes.
• Spontaneous emission usually require large population of energetic electrons.
• On the other hand our understanding of induced emission process is still very limited.