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.