Plasma Radiation

Liu Jian

2006.11.24

ljssspku@gmail.com

Outline

Introduction Basic Electrodynamics Results Radiation Transport Radiation Emission

Plasma Bremsstrahlung Electron Cyclotron Radiation Synchrotron Radiation

Radiation Scattering Incoherent Thomson Scattering

Introduction

Introduction

Accelerated charged particles are sources of electromagnetic radiation. Bremsstrahlung emission: caused by electric micr

o-fields. Scattered radiation: external radiation fields intera

cting with the plasma. Cyclotron radiation & synchrotron radiation: charg

ed particles moving in magnetic fields, depending on the energy range of particles.

Introduction

The interaction of radiation with plasma—emission, absorption, scattering and transport.

These are keys to understanding many effects in both laboratory and natural plasmas.

We shall limit our discussion of radiation to plasmas to thermal equilibrium, with few exceptions.

Non-thermal emission is in many instances still relatively poorly understood.

Basic Electrodynamics Results

Electrodynamics of Radiation Fields

According to the Maxwell equations:

together with the Lorentz gauge condition:

We can get the retarded potentials:

jAtc

02

2

22 ]

1[

0

2

2

22 ]

1[

q

tc

01

tc

A

rd

rr

trjtrA

),(

4),( 0

rdrr

trqtr

),(

4

1),(

0

with

we find

where These expressions are the Liénard-Wiechert potentials. Feynman used the retarded potentials to express the ele

ctric field in this form:

))(()(),( 00 trrtretrj

crrtt /

))((),( 0 trretrq

ctRttRvcR

cvetrA

/)(

0

4),(

ctRttRvcR

ectr

/)(04

1),(

)()( 0 trrtR

)()( 0 trtv

where n is the unit vector from the source to the field point; ret denotes that the expression within the square brackets must be evaluated at the retarded time

The first term represents the Coulomb field of the charge at its retarded position.

The second is the correction, being the product of the rate of change of this field and the retardation delay time R/c.

The final term contains the radiation electric field

retdt

nd

cR

n

dt

d

c

R

R

netrE

2

2

2220

1)(

4),(

ctRtt /)(

1 RE rad

where cv /

Power Radiated By An Accelerated Charge

the Poynting vector determines the instantaneous flux of energy.

The power P, radiated per unit solid angle Ω is

where

Then

In the ultra-relativistic limit (β→1) the effect of the denominator is dominant in determining the radiation pattern; the dipole distribution familiar from the non-relativistic limit deforms with the lobs inclined increasingly forward as in the figure above.

In the non-relativistic limit g→1 and we recover

the dipole distribution:

where θ is the angle between and n. Larmor’s formula for the power radiated in all direction

s follows on integrating over solid angle:

The corresponding relativistic expression is:

3

0

22

6 c

ved

d

dPP

32

22

0

2

)1(

])()[(

6

c

eP

Much of our discussion will focus on the

distinct characteristics of radiation from

particles accelerated in the plasma micro-

electric fields and in any magnetic fields present.

Where plasmas are subject to external electro-magnetic fields the incident radiation is scattered, with the scattering governed by the Thomson cross-section

Where the classical electron radius

2

3

8eT r

)4/( 20

2 mcere

Frequency Spectrum of Radiation From An Accelerated Charge

We consider next how the radiated energy is distributed in frequency.

let then Introducing the Fourier transform of a(t):

The energy radiated per unit solid angle is:

dtta

d

dW 2)(

2)(ta

d

dP

dtetaa ti

)(

2

1)(

da

d

dW 2)(

Thus

Finally, using the conclusion before we can find in the radiation zone :

The results summarized before provide a basis for the forma-lism needed to describe radiation emitted by charged particles.Much of the rest of the chapter is taken up with the characteri-stics of emission from particles moving in particular fields.Emission is only part of the story. We shall see absorption in Chapter11. Then we first summarize some ideas central to radiation transport in plasmas.

Transporting

Radiation Transport In Plasma

The general problem of radiation transport in plasmas is complicated.

Fortunately, for our purposes it does not need to be discussed in detail.

For simplicity, we ignore scattering and take account of emission and absorption in the transport equation. This is strictly valid only under conditions of local thermodynamic equilibrium (LTE).

Geometric Optics Assumption

We can deal with these problems in terms of geometric optics.

If denotes the spectral density of energy of the radiation flux, then by energy conservation in steady state, we have

According to the principal assumption in geometric optics: the properties of the medium vary slowly with position. So one may picture the radiation being transported along rays.

The net radiation flux across an element is:

is the intensity of the radiation and s denotes displacement along the ray. Its importance in radiationtransport is due to the fact that it can be measured more or less directly. It is defined by:

Intensity of The Radiation

In general, the intensity is a function both of direction and position in the medium.

When it is a function of position alone, the radiation is said to be isotropic.

Suppose the plasma through which the radiation is passing is loss-free, isotropic but slightly inhomogeneous, so that the ray suffers bending. Then by energy conservation:

supposing no reflection of energy at the interface takes place.

Now from Snell’s law, is constant (where n is the refractive index) along the

ray. Then

leads to

so that At frequencies much greater than the plasma

frequency, and we can get along a ray path.

The result for an anisotropic plasma ismore complicated.

Emission

Plasma Bremsstrahlung

The spectral range of bremsstrahlung is very wide, from just above the plasma frequency into the X-ray continuum for typical plasma temperatures.

Bremsstrahlung results from electrons undergoing transitions between two states of the continuum in the field of an ion (or atom).

In place of a full quantum mechanical treatment we opt instead for a semiclassical model of bremsstrahlung which turns out to be adequate for most plasmas.

let us make an estimate of plasmabremsstrahlung from a simple model in which an electron moves in the Coulombfield of a single stationary ion of charge Ze. Then

and substituting in Larmor’s formula, the power radiated by the electron is given by

If we take the spatial distribution of the plasma electrons about the ion to be uniform, then the contribution to the bremsstrahlung from all electrons in encounters with this test ion is found by summing the individual contributions to give,

The cut-off at is needed to avoid divergence. A value for may be chosen in a number of ways and plasma bremsstrahlung is not sensitive to this choice.

For present purposes we take , the de Broglie wavelength, the distance over which an electron may no longer be regarded as a classical particle.

,where is the electron temperature.

Thus If denotes the ion density, the total bremsstrahl

ung power radiated per unit volume of plasma

We see that the power radiated as bremsstrah-lung is proportional to the product of electron and ion densities and to .Thus any high Z impurities present will contribute bremsstrahlung losses disproportionate to their concentrations.

Note that since electron–electron collisions do not alter the total electron momentum they make no contribution to bremsstrahlung in the dipole approximation.

Classical Picture

The exact classical treatment of an electron moving in the Coulomb field of an ion is a standard problem in electrodynamics.

The classical emission spectrum can then be massaged to agree with the quantum mechanical spectrum by multiplying by a correction factor, the Gaunt factor.

Most of the bremsstrahlung is emitted at peak electron acceleration, i.e. at the distance of closest approach to the ion.

Provided the energy radiated as bremsstrahlung is a negligibly small fraction of the electron energy (we

treat the ion as stationary) the electron orbit is

hyperbolic and the power spectrum dP(ω)/dω from a test electron colliding with plasma ions of density ni may be shown to be

impact parameter for 90◦ scattering the incident velocity of the electron is a dimensionless factor, known as the Gaunt fa

ctor, which varies only weakly with ω. Collisions described by a small impact parameter produce

hard photons; less energetic photons come from distant encounters, with correspondingly large impact parameters.

The Emission Coefficient

The emission coefficient is the power radiated per unit volume per unit solid angle per unit (angular) frequency and, in the low frequency limit, is given by

and is the Maxwellian-averaged Gaunt factor. In our case

Quantum Mechanical Picture

While the classical description of bremsstrah-lung is useful in the low frequency range, at high frequencies a quantum mechanical formulation is needed.

For present purposes it is enough to treat the electron as a wave packet.

We can get

Recombination Radiation

free–bound transitions leads to recombination radiation.

The final state of the electron is now a bound state of the atom.

This event involving electron capture is known as radiative recombination and the emission as recombination radiation.

In certain circumstances, recombination radiation may dominate over bremsstrahlung.

Inverse Bremsstrahlung:free-free absorption

The process inverse to bremsstrahlung, free–free absorption, occurs when a photon is absorbed by an electron in the continuum.

For a plasma in local thermal equlibrium, we may then appeal to Kirchhoff’s law to find the free–free absorption coefficient.

In the Rayleigh–Jeans limit this gives:

Inverse Bremsstrahlung:free-free absorption

Absorption of radiation by inverse bremsstrahlung is most effective at high densities, low electron temperature and for low frequencies.

The mechanism is important for the efficient absorption of laser light by plasmas.

We expect absorption to be strongest in the region of the critical density , since this is the highest density to which incident light can penetrate.

In the neighbourhood of the critical density

so that free–free absorption is sensitive to the wavelength of the incident laser light.

Plasma Corrections to Bremsstrahlung

Up to now we have ignored plasma effects in discussing bremsstrahlung emission and its transport through the plasma.

Transport: for an isotropic plasma, the emission coefficient is valid only for frequencies

The bremsstrahlung emission described before was determined on the basis of binary encounters between electrons and ions. However collisions in plasmas are predominantly many-body rather than binary.

Bremsstrahlung As Plasma Diagnostic

Bremsstrahlung emissivity through its dependence on electron temperature, plasma density and atomic number clearly has potential as a plasma diagnostic.

In practice the picture is less clear.

Electron Cyclotron Radiation

We consider next radiation by an electron moving in a uniform, static magnetic field.

Electron Cyclotron Radiation

Using this we can calculate the power radiated by an electron per unit solid angle per unit frequency interval.

In the weakly relativistic limit electron cyclotron emission (ECE) has potential as a diagnostic.

Synchrotron Radiation

Synchrotron radiation from hot plasmas: electron energies ranging from some tens to few hundreds KeV

Synchrotron radiation By Ultra-relativistic Electrons: electron energies are ultra-relativistic.

Scattering

Scattering of Radiation By Plasmas

A plane monochromatic electromagnetic wave incident on a free electron at rest is scattered.

The scattered wave having the same frequency as the incident radiation;

The scattering cross-section is defined by the Thomson cross-section.

For scattering by electrons the Thomson cross-section has the value ;scattering by ions, being at least six orders of magnitude smaller, rarely matters in practice.

Thank you!