Boyd T.J.M., Sanderson J.J. The Physics of Plasmas

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9.2 Electrodynamics of radiation ﬁelds 327

Fig. 9.2. Polar diagram of the instantaneous power radiated by an accelerated charged

particle; θ

max

denotes the angle at which peak power is radiated.

where

S · n =

16π



|n ×{(n −β

β) ×

β}|



ret

(9.12)

Note that (dP(t)/d) d denotes the radiated power measured by the observer at

time t due to emission by the charge at time t



. It is often useful to consider power

as a function of the retarded time t



.Then

dP(t



)

d

= (S · n)R



= g(S ·n)R

16π

|n ×{(n −β

β) ×

β}|

(9.13)

Relativistic effects appear both in the numerator and, through g, in the denominator.

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 lobes inclined increasingly forward as in Fig. 9.2.

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

d

16π

sin

θ (9.14)

where θ is the angle between

β and n.

328 Plasma radiation

Larmor’s formula for the power radiated in all directions follows on integrating

over solid angle, i.e.

P =



d

d =

˙v

6πε

(9.15)

The corresponding relativistic expression for the radiated power may be found by

replacing (9.15) by its covariant form, giving

P =

6πε



d p

dτ

d p

dτ



6πε

[(

β)

− (β

β ×

β)

]

(1 − β

)

(9.16)

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 electromagnetic ﬁelds the incident

radiation is scattered, with the scattering governed by the Thomson cross-section

8π

(9.17)

where the classical electron radius r

= (e

/4πε

9.2.2 Frequency spectrum of radiation from an accelerated charge

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

spectrum is represented in terms of frequencies at a detector it is natural to revert

to using time t (the observer’s time). Then

dP(t)

d

16π



|n ×{(n −β

β) ×

β}|



ret

≡|a(t)|

(9.18)

The energy radiated per unit solid angle dW/d is found by integrating (9.18) over

time, giving

d



∞

−∞

|a(t)|

dt (9.19)

Introducing the Fourier transform of a(t)

a(ω) =

√

(2π)



∞

−∞

a(t)e

iωt

dt (9.20)

and using Parseval’s theorem allows us to represent the energy radiated per unit

solid angle as

d



∞

−∞

|a(ω)|

dω (9.21)

9.2 Electrodynamics of radiation ﬁelds 329

Thus

d



∞

[|a(ω)|

+|a(−ω)|

]dω = 2



∞

|a(ω)|

dω (9.22)

The energy radiated per unit solid angle per unit angular frequency interval is then

d dω

= 2|a(ω)

| (9.23)

Finding a(ω) from (9.20) is simpliﬁed if we change the variable of integration from

t to t



, so removing the evaluation of the term in square brackets at the retarded

time; then

a(ω) =



32π



1/2



∞

−∞

exp



iω





R(t



)





n ×{(n −β

β) ×

β}





(9.24)

Since we wish to determine the spectrum in the radiation zone (r  r

in Fig. 9.1),

n is effectively time-independent and R(t



)  r − n ·r



) so that

W (ω, n)

d dω

16π





∞

−∞

exp



iω





−

n · r



)





n ×{(n −β

β) ×

β}





(9.25)

Thus the energy radiated per unit solid angle per unit frequency interval is deter-

mined as a function of ω and n once r



) is prescribed.

For purposes of calculation, we cast (9.25) in a slightly different form. Using the

representation (see Exercise 9.1)

n ×{(n −β

β) ×

β}





n × (n ×β

β)



we integrate (9.25) by parts to ﬁnd, in the radiation zone,

W (ω, n)

d dω

16π





∞

−∞

exp



iω





−

n · r



)



[n × (n ×β

β)]dt



(9.26)

The results summarized in this section provide a basis for the formalism needed

to describe radiation emitted by charged particles. Much of the rest of the chapter

is taken up with the characteristics of emission from particles moving in particular

ﬁelds. Emission is of course only part of the story. Plasmas in thermal equilib-

rium that emit radiation absorb it as well and details of absorption mechanisms

are crucial for the radiative heating of plasmas as we shall see in Chapter 11.

Before discussing the emission of radiation we ﬁrst summarize some ideas central

to radiation transport in plasmas.

330 Plasma radiation

9.3 Radiation transport in a plasma

Fig. 9.3. Pencil of radiation refracted across an element of plasma.

The general problem of radiation transport in plasmas is complicated but fortu-

nately for our purposes 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 equi-

librium (LTE), a concept to be introduced later in this section.

The equation of radiative transfer can be thought of as an expression of energy

conservation in terms of geometric optics. If F

denotes the spectral density of the

radiation ﬂux then, by energy conservation in steady state,

∇·F

= 0 (9.27)

Note that F

has dimensions of power per unit area per angular frequency interval.

The principal assumption in geometric optics is that the properties of the medium

vary slowly with position; that is, the scale length of variations is very much greater

than the wavelength of radiation in the medium. In this approximation, one may

picture the radiation being transported along rays. Figure 9.3 shows an element of

plasma of cross-section dA, thickness ds and n is the unit normal outwards from

the surface. The net radiation ﬂux across this element is

· n = dF

cos θ = I

(s) cos θ d (9.28)

(s) is the intensity of the radiation and s denotes displacement along the ray. Its

9.3 Radiation transport in a plasma 331

importance in radiation transport is due to the fact that it can be measured more or

less directly. I

(s) is deﬁned by

(s) = I

(s) cos θ d dω d A

where dP

is the time-averaged power in the spectral range dω crossing an area dA

within a cone of solid angle d. I

is expressed in units of watts per square metre

per steradian per unit angular frequency. 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 and

isotropic but slightly inhomogeneous so that a ray, on passing through the element

of plasma shown in Fig. 9.3, suffers bending. Then, by energy conservation,

+ dI

) cos θ

d

dω dA − I

cos θ

d

dω dA = 0 (9.29)

supposing no reﬂection of energy at the interface takes place. Now from Snell’s

law, n sin θ is constant (where n is the refractive index) along the ray. Then, since

d

sin θ

dθ

sin θ

dθ

sin θ

cos θ





cos θ

(9.29) leads to

ω2





cos θ

d

= I

ω1

cos θ

d

so that

= const. (9.30)

along a ray path in a slowly varying inhomogeneous, isotropic, transparent

medium. At frequencies much greater than the plasma frequency, n

 1 and (9.30)

simpliﬁes to I

= const. along a ray path. The result for an anisotropic plasma is

more complicated since Snell’s law is no longer obeyed in general but holds only

for waves propagating in certain directions relative to the magnetic ﬁeld.

Next we relax the requirement that the plasma be both source-free and loss-free.

Within the geometrical optics representation we introduce absorption and emission

terms on the right-hand side of (9.27). Let α

be the absorption coefﬁcient per

unit path length in the plasma, so that a radiative ﬂux I

dA d suffers a loss

dA d ds in travelling a distance ds. Similarly we introduce an emission co-

efﬁcient 

deﬁned so that 

dA d ds is the emission from the elemental volume

into solid angle d in the direction of the ray. Then,

∂ I

∂s

− α

+ 

(9.31)

332 Plasma radiation

Now, ∂ I

/∂s is the rate of change of I

due to the change in refractive index along

a ray path; from (9.30)

∂ I

∂s

so that





= 

− α

(9.32)

This is the radiative transport equation.

To solve the transfer equation, we introduce a source function



(9.33)

and deﬁne an optical depth τ

τ =



dτ =−



ds (9.34)

in which the minus sign denotes that the optical depth is measured back into the

plasma along the ray path. Then (9.32) reads

dτ





− S

(9.35)

Integrating along the ray path in the plasma between points A and B one has,

(A)

−τ(A)

(B)

−τ(B)



τ(B)

τ(A)

(τ )e

−τ

dτ (9.36)

In practice it may be permissible to ignore curvature of the ray path where changes

in refractivity are negligible. Where A, B are points on a plasma–vacuum boundary

as in Fig. 9.4 then τ(A) = 0, τ(B) = τ

, the total optical depth of the plasma, and

n(A) = 1 = n(B). Thus, neglecting reﬂection at the boundaries, the emergent

intensity is

= I

inc

−τ



(τ )e

−τ

dτ (9.37)

The ﬁrst term on the right-hand side takes account of absorption of the incident

radiation while the second represents contributions from sources within the plasma,

again allowing for absorption of radiation in transit from its origin to the point A.

When I

inc

= 0,



(τ )e

−τ

dτ (9.38)

Two important limiting cases of this result correspond to τ

 1, when the plasma

is said to be optically thin and the opposite limit τ

 1, when it is optically

9.3 Radiation transport in a plasma 333

Fig. 9.4. Ray path through the plasma.

thick. In the optically thin limit, absorption along a ray path is negligible so that

the emergent intensity is simply the sum of contributions along the ray, i.e.,







(s)ds (9.39)

In the optically thick limit τ

→∞so that after integrating (9.38) by parts



)

(9.40)

In other words the intensity affords a direct measure of the source function. Be-

tween these two limits one has to solve the radiative transfer equation to determine

the intensity of the radiation observed at a detector.

A medium in thermal equilibrium that is perfectly absorbing emits radiation that

is Planckian, i.e. the intensity I (ω) = B(ω) where

B(ω) =

hω

8π

[exp(

hω/k

T ) − 1]

−1

(9.41)

is the black body intensity (for a single polarization) and

h = h/2π where h is

Planck’s constant. In the classical limit (9.41) reduces to the Rayleigh–Jeans form

B(ω) =

8π

(9.42)

334 Plasma radiation

Then from (9.40) we ﬁnd



8π

(9.43)

which deﬁnes a radiation temperature T

By and large radiation emitted by laboratory plasmas, unlike that from stellar

sources, does not correspond to a black body spectrum. We have to abandon the

notion of global thermal equilibrium for something less complete, local thermody-

namic equilibrium (LTE), a concept that lends itself to about as many deﬁnitions as

it ﬁnds application! Broadly speaking, homogeneous plasmas can be assumed to be

in an LTE state when collision processes are dominant. The radiation ﬁeld is locally

Planckian with temperature T

. Only under LTE conditions is the source function



/α

= B(ω). In reality of course laboratory plasmas are rarely homogeneous

and this imposes additional restrictions on the validity of LTE.

It is still possible to describe the radiation ﬁeld locally by a temperature T

even when the temperature is globally non-uniform. This means that the region

concerned has to be sufﬁciently local for the temperature to be considered uniform

while at the same time extensive enough for thermodynamics to be valid. The LTE

approximation breaks down when the source function is no longer a local function

of electron temperature but depends on the radiative ﬂux from other regions of the

plasma.

9.4 Plasma bremsstrahlung

We turn next to the principal sources of radiation from fully ionized plasmas,

bremsstrahlung and, with magnetic ﬁelds present, cyclotron or synchrotron ra-

diation. We shall deal with these separately, since the spectral characteristics in

each case are quite distinct. The spectral range of bremsstrahlung is very wide,

extending from just above the plasma frequency into the X-ray continuum for

typical plasma temperatures. By contrast the cyclotron spectrum is characterized

by line emission at low harmonics of the Larmor frequency. Synchrotron spectra

from relativistic electrons display distinctive characteristics as we shall see later

on. Moreover, whereas cyclotron and synchrotron radiation can be dealt with

classically, the dynamics being treated relativistically in the case of synchrotron

radiation, bremsstrahlung from plasmas has to be interpreted quantum mechan-

ically, though not usually relativistically. Bremsstrahlung results from electrons

undergoing transitions between two states of the continuum in the ﬁeld of an

ion (or atom). Oppenheimer (1970) has described bremsstrahlung graphically as

the shaking off of quanta from the ﬁeld of an electron that suffers a sudden

jerk.

9.4 Plasma bremsstrahlung 335

In place of a full quantum mechanical treatment we opt instead for a semi-

classical model of bremsstrahlung which turns out to be adequate for most plasmas.

Classically we can think of bremsstrahlung in terms of the emisssion of radiation

by an electron undergoing acceleration in the ﬁeld of a positive ion. The classical

emission spectrum can then be massaged to agree with the quantum mechanical

spectrum by multiplying by a correction factor, the Gaunt factor.

To see what is involved let us make an estimate of plasma bremsstrahlung from

a simple model in which an electron moves in the Coulomb ﬁeld of a single

stationary ion of charge Ze. Then

v|=

4πε

and substituting in Larmor’s formula (9.15), the power radiated by the electron is

given by

6πε



4πε



(9.44)

If we take the spatial distribution of the plasma electrons about the ion to be uni-

form, then the contribution to the bremsstrahlung from all electrons in encounters

with this test ion is found by summing the individual contributions to give,

P =

8π Z

3(4πε

)



∞

min

8π Z

3(4πε

)

min

The cut-off at r = r

min

is needed to avoid divergence. A value for r

min

may be

chosen in a number of ways and we shall see later that plasma bremsstrahlung is not

sensitive to this choice. For present purposes we take r

min

 λ

deB

, the de Broglie

wavelength, the distance over which an electron may no longer be regarded as a

classical particle. For a thermal electron

deB

∼

h/(mk

)

1/2

where T

is the electron temperature. Thus

P 

8π Z

3(4πε

)





If n

denotes the ion density, the total bremsstrahlung power radiated per unit

volume of plasma, P

, is then

8π



4πε







= 5.34 × 10

−37

1/2

(keV) Wm

−3

(9.45)

336 Plasma radiation

We see that the power radiated as bremsstrahlung is proportional to the product

of electron and ion densities and to Z

. 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.

9.4.1 Plasma bremsstrahlung spectrum: classical picture

The exact classical treatment of an electron moving in the Coulomb ﬁeld of an

ion is a standard problem in electrodynamics. 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 n

may be shown to

dP(ω)

dω

16π

√



4πε



G(ωb

/v) (9.46)

where b

= Ze

/4πε

is the impact parameter for 90

◦

scattering, v the incident

velocity of the electron and G(ωb

/v) a dimensionless factor, known as the Gaunt

factor, which varies only weakly with ω.

Most of the bremsstrahlung is emitted at peak electron acceleration, i.e. at the

distance of closest approach to the ion. Collisions described by a small impact

parameter produce hard photons; less energetic photons come from distant encoun-

ters, with correspondingly large impact parameters. Denoting the impact parameter

by b, collisions producing hard photons correspond to b  b

and the electron orbit

is approximately parabolic. In the opposite limit b  b

, the electron trajectory is

more or less linear and in reality it is only in this limit that a classical picture of

bremsstrahlung is justiﬁed. For an electron following the linear trajectory shown

in Fig. 9.5, r

(t) = (vt)

+ b

. The components of the acceleration normal and

parallel to the trajectory are,

˙v

⊥

(t) =

4πε

[(vt)

+ b

]

3/2

˙v



(t) =

4πε

[(vt)

+ b

]

3/2

Integrating (9.26) over the solid angle allows us to express the energy radiated per

unit frequency interval in the non-relativistic limit as

dW (ω)

dω

6π





∞

−∞

v(t)e

iωt

