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

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9.9 Coherent Thomson scattering: experimental veriﬁcation 367

Fig. 9.17. Effect of impurities on the scattering form factor.

hance the central part of the ion feature, so distorting the spectrum. In practice this

makes for difﬁculties in discriminating between a change in ionic composition and

one in which the temperature ratio T

changes.

It is straightforward to generalize the scattering form factor (9.97) to include

different species of ions. Evans (1970) considered the effects of increasing the

concentration of oxygen impurity ions (

) in a hydrogen plasma, ﬁnding a

central feature altering the spectrum from that for a pure hydrogen plasma. The

height of this feature increased with increasing relative abundance of oxygen ions.

For T

= 5 with 5% oxygen present, Fig. 9.17 shows that scattering from the

impurity ions dominates the ion feature, which is well-deﬁned for an impurity

368 Plasma radiation

Fig. 9.18. Ion feature in the coherent Thomson spectrum for a two-species plasma showing

contributions from Au and Be ions (after Glenzer et al. (1999)).

content of 1%. The effect of the impurity becomes even more marked at lower

ratios. Evans also examined the effect of increasing the charge Z of the im-

purity ion in a hydrogen plasma contaminated by 1% of

, where 0 < Z ≤ 15.

Increasing the effective charge of the impurity ion produced a scattered spectrum

broadly similar to that in which the impurity abundance is increased. In principle

it is feasible to deduce impurity concentrations from the scattered spectra, though

for most laboratory plasmas it would prove an unwieldy diagnostic.

Glenzer et al. (1999) have reported coherent Thomson scattering from dense

ICF plasmas with more than one ion species present. In particular, they formed a

plasma by irradiating targets in which the composition was controlled by coating

discs with multilayers of Au and Be of varying thickness. Figure 9.18 shows the

ion feature in the Thomson spectrum from a plasma containing 4% Au and 96%

Be with a pair of ion acoustic waves from each species clearly resolved.

The relative intensities of the ion components are determined by the damping of

these waves. For the spectrum in Fig. 9.18 gradients in plasma parameters are not

important since it corresponds to a time after the heating beam has been switched

off and in addition the low-Z blow-off plasma tends to be isothermal.

9.9 Coherent Thomson scattering: experimental veriﬁcation 369

9.9.2 Deviations from the Salpeter form factor for the ion feature: collisions

For dense, relatively cold plasmas Coulomb collisions become more important than

Landau damping in determining the line shapes. For the ion feature this change

appears whenever the mean free path for ion–ion collisions becomes comparable

to the wavelength of ion acoustic ﬂuctuations, i.e. for ν

/kc

≈ 1 where ν

is the

ion–ion collision frequency.

The effect of Coulomb collisions on the low frequency region of the spectrum

was determined by Kivelson and DuBois (1964) from the Balescu–Lenard kinetic

equation (see Section 12.2.1) and by Boyd (1966) using a ﬂuid model. The ﬂuid

model leads to a low frequency spectrum with two features. In addition to the

ion acoustic resonance there is another at zero frequency due to non-propagating

entropy ﬂuctuations. The width of the ion resonance is determined by the thermal

conductivity and viscosity coefﬁcients; that of the entropy ﬂuctuation resonance

depends only on thermal conductivity.

Mostovych and DeSilva (1984) measured the scattered light spectrum from a

dense low temperature source for which both density and temperature were well

characterized. Their scattered spectrum for an argon plasma is shown in Fig. 9.19.

The ﬁnite pulse length meant that the entropy ﬂuctuation contribution to the spec-

trum was not observed. Figure 9.19 plots the Lorentzian line from the ﬂuid model

using parameters from the experiment, showing generally satisfactory agreement.

Fig. 9.19. Effect of collisions on the scattering form factor. The data correspond to the

spectrum obtained by Mostovych and DeSilva (1984); the curve is the line shape from the

ﬂuid model (Boyd (1966)).

370 Plasma radiation

Subsequent work by Zhang, DeSilva and Mostovych (1989) succeeded in resolving

the contribution to the spectrum due to entropy ﬂuctuations.

Exercises

9.1 Section 9.2 summarizes some key results in the electrodynamics of radia-

tion from charged particles. Solutions to (9.1) are found in terms of the re-

tarded Green function (see Jackson (1975) and Boyd and Sanderson (1969)

for details).

In establishing Feynman’s result (9.7) one needs the delta function

property



∞

−∞

δ[h(x)]γ(x)dx =



γ(x



dh(x

)



in which the x

are the roots of h(x) = 0. Starting from (9.3) show that

E(r, t) =

4πε







n −β



ret

where g = 1 − n ·β

β. To cast this in Feynman form ﬁrst show that



n × (n ×β

β)

(E9.1)

Show that the expression (9.8) for E

rad

(r, t) follows from the





ret

term in (9.7). In the non-relativistic limit show that, at large distances

from the source, the radiation ﬁeld from a particle of charge e moving

with velocity

(t) is given by (e/4πε

0⊥

(

t − R/c

)

where

0⊥

is the

acceleration transverse to the line of sight.

Show that

n ×{(n −β

β) ×

β}

≡





n × (n ×β

β)



assuming that n is only weakly time-dependent.

9.2 Using (9.13), consider a charged particle executing instantaneously circu-

lar motion and show that

dP(t



)

d

16π

(1 − β cos θ)



1 −

(1 − β

) sin

θ cos

(1 − β cos θ)



Exercises 371

Compare this angular distribution with that found in the case in which

particle velocity and acceleration are collinear; in particular note that here

too the peak power is radiated in the forward direction.

[Hint: Choose a coordinate system with β

β instantaneously along Oz and

β along Ox; θ,φ are the usual polar angles.]

9.3 Establish the relativistic generalization of Larmor’s formula, (9.16).

[Hint: Rather than the lengthy exercise that a direct integration of (9.13)

involves, use the covariant form

P =

6πε



d p

dτ

d p

dτ



where p

is the momentum–energy four-vector and τ is the proper time.

On evaluating the scalar product of the four-vectors, the result follows

directly.]

For a particle with energy E moving in a uniform magnetic ﬁeld B show

that

=−KE

⊥

where K is a constant.

9.4 By integrating (9.38) successively in the optically thick limit (τ

→∞)

establish that the spectrum of the outgoing radiation is that of a black body

provided the temperature does not vary along the line of sight.

9.5 Establish (9.47) for the energy radiated by a classical electron and show

that the plasma bremsstrahlung emission coefﬁcient is described by (9.48).

For this you will need the Bessel function relationships

x[K

(x) + K

(x)] =−(d/dx)[xK

(x)K

(x)]

lim

x→0

(x) =−ln x lim

x→0

(x) = 1/x

Conﬁrm the results (9.55) and (9.56).

By comparing the free–bound emissivity 

to that from bremsstrahlung



show that



∼





exp





Under what conditions is 

negligible compared to 

Verify the expression (9.57) for the free–free absorption coefﬁcient.

372 Plasma radiation

9.6 To deduce (9.60) starting from (9.26) with the geometry of Fig. 9.7 use the

representation exp(ixsin y) =

∞

l=−∞

(x) exp(ily) so that

exp



iω



t −

n · r

(t)



∞



l=−∞





⊥

sin θ



×exp[i(ω −l − ωβ



cos θ)t]

From the geometry of Fig. 9.7 show that

n × (n ×β

β) =

x[−β

⊥

cos t cos

θ + β



sin θ cos θ ] +

y[β

⊥

sin t]

z[β

⊥

cos t sin θ cos θ − β



sin

θ]

≡ X

x + Y

y + Z

z (E9.2)

Use these results in (9.26) to show that

W (ω)

d dω

16π





∞

−∞

∞



l=−∞





⊥

sin θ



× exp[i(ω −l − ωβ



cos θ)t][X

+ Y

+ Z

]dt



Integrating over time and using Bessel recurrence relations, show that

d dω

8π



2π



∞



l=1



x(β



− cos θ)cot θ J

(x)

yiβ

⊥



(x)

z(cos θ − β



(x)



× δ(l − ω[1 −β



cos θ])

where T is the radiation emission time. To arrive at this expression the

term involving δ

(ω) has to be interpreted as lim

T →∞

(T/2π)δ(ω).

9.7 Integrate (9.60) over all directions to obtain (9.62). The algebra involved

is daunting if one goes about this directly. It is best to determine P

the guiding centre frame and then apply a Lorentz transformation to the

observer’s frame.

[Hint: The integration over θ uses properties of Bessel functions. The ar-

gument is due to Schott (1912). Making use of the representation J

(x) =



(2x sin α) cos 2lα dα leads to

d



8π



(2lβ

⊥

sin θ sin α)



⊥

cos 2α − 1



cos 2lα dα

Exercises 373

Then apply the result



π/2

(y sin θ)sin θ dθ =

1/2

(y) =

sin y

to obtain an expression for P

in the guiding centre frame. Finally Lorentz

transform to the observer’s frame to get (9.62).]

Use (9.16) to show that the total power radiated is

tot



6πε



⊥

1 − β



Check the assumption made in the development of the theory of cyclotron

radiation that the energy radiated is negligible compared with the total

energy of the radiating electron.

Establish (9.65) for the plasma cyclotron emissivity.

9.8 Show that the power radiated as cyclotron radiation compared with that

radiated as bremsstrahlung is

cyc

∼

2 × 10

1/2

(eV)B

9.9 Establish (9.74) for the synchrotron emissivity from moderately relativistic

electrons.

Using the physical model for synchrotron radiation outlined in Sec-

tion 9.6.2 show that the peak emission occurs at angles of θ

∼±γ

−1

for ultra-relativistic electrons (γ  1).

Establish (9.77), (9.78) for the power radiated by an electron in the ultra-

relativistic limit β

⊥

≡ β → 1 and l  1.

[Hint: The following results from the theory of Bessel functions are

needed:

(2lβ) =

√

3γ

1/3

(R)



(2lβ) =

√

3γ

2/3

(R)



2lβ

(x)dx =

√



∞

2l/3γ

1/3

(t)dt

2/3



3γ



−



∞

2l/3γ

1/3

(t)dt =



∞

2l/3γ

5/3

(t)dt]

374 Plasma radiation

Find approximate expressions for the radiated power in the limits

ω/ω

 1, ω/ω

 1 respectively. From (9.69) ﬁnd an expression for

the synchrotron emissivity in the ultra-relativistic limit.

Establish (9.80)

9.10 Estimate the electron energy needed to produce 4 keV photons assuming a

synchrotron source for X-ray emission from the Crab nebula.

9.11 Consider a cosmic radio source in isolation so that its population of high-

energy electrons is not renewed. Suppose the magnetic ﬁeld present is

estimated at 2 × 10

−9

T and use this to show that the energy  of the

electrons contributing to radio emission at 1 m wavelength is of the order

of 1 GeV.

Show that the lifetime of the radiating electrons is proportional to 

−1

and estimate this lifetime for 1 GeV electrons. How would you expect the

radio spectrum to change with time for this source?

9.12 The non-relativistic equation of motion of a charge in an electromagnetic

wave is

v = eE + ev × B +

6πε

where the last term is a small correction to the Lorentz equation

due to radiation. Show that the time average of the damping force is

(6πε

)

−1

/mc

)

where E

is the wave amplitude.

9.13 Show that in an equilibrium plasma for which the electron distribution

function is Maxwellian, the line proﬁle of Thomson scattered radiation is

determined by the form factor

S(k,ω)=



2πk



1/2

exp



−

mω



Use the experimentally determined half-width from Fig. 9.13 to compute

the plasma electron temperature.

9.14 Establish the Salpeter expression for the form factor in (9.100).

9.15 From a consideration of the form factor in a plasma in which an electron

drift relative to the ions is present, show that the effect of the drift is to

introduce an asymmetry into the ion feature. Interpret this result.

9.16 In Section 2.13.1 considerations of the relativistic dynamics of an electron

in an electromagnetic ﬁeld showed that the electron described a ﬁgure-of-

eight trajectory. Conﬁrmation of this has been provided in an elegant ex-

periment in which non-linear Thomson scattering was observed by Chen,

Maksimchuk and Umstadter (1998). Non-linear Thomson scattering gen-

erates harmonic spectra with characteristics that distinguish them from a

Exercises 375

Fig. 9.20. Angular pattern of second-harmonic Thomson scattered light (after

Chen, Maksimchuk and Umstadter (1998)).

number of other possible harmonic sources. The polar diagram in Fig. 9.20

shows the dependence of the intensity of second harmonic light (in arbi-

trary units) on azimuthal angle φ in degrees. Circles denote experimental

data points and the solid (dashed) lines show predicted dependences for

zero drift and a drift velocity v

= 0.2c respectively. Read the paper by

Chen, Maksimchuk and Umstadter for a full discussion of their results.

Non-linear plasma physics

10.1 Introduction

Linearization gives rise to such simpliﬁcation that in many cases it is pushed to its

limits and sometimes beyond in the hope that by understanding the linear problem

we may gain some insight into the non-linear physics. Perhaps the clearest example

of the progress that can be made by analysing linearized equations is in cold plasma

wave theory, but linearization, in one form or another, is almost universally applied.

For instance, the drift velocities of particle orbit theory are of ﬁrst order in the ratio

of Larmor radius to inhomogeneity scale length. In kinetic theory it is invariably

assumed that the distribution function is close to a local equilibrium distribution.

A question of fundamental importance is then, ‘How realistic and relevant are

linear theories?’ Some problems are essentially non-linear in that there is no useful

small parameter to allow linearization. Examples of these are sheaths, discussed in

Chapter 11, and shock waves. Primarily, our intention is to address the subsidiary

question: ‘Given that there is a valid linear regime, to what extent need we concern

ourselves with non-linear effects?’

Of course, if the linear solution predicts instability then we know that, in time, it

will become invalid because the approximation on which the linearization is based

no longer holds good. In such cases the aim might be to identify and investigate

non-linear processes that come into play and quench the instability. However, an

unstable linear regime is emphatically not a pre-requisite for an interest in non-

linear phenomena. There are many situations in which the linear equations give

only stable solutions but the non-linear equations are secular, i.e. under certain

conditions some solutions grow with time. Physically, this comes about because

the non-linear coupling of stable, linear modes generates new modes and, if these

are natural modes of the system, resonant growth of their amplitudes may occur.

Such parametric ampliﬁcation is widespread throughout physics and engineering

and is of particular interest in plasmas, which are well endowed with natural modes.

376