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

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9.7 Scattering of radiation by plasmas 357

the incident frequency) and along the direction of propagation of the scattered wave

(up-shifting the frequency).

The next step is determining the scattered power. Making use of the property of

the delta function expressed in Exercise 9.1 we may invert (9.85) to get

(r, t) =

P · E

i(k

·r−ω

(9.87)

The corresponding Poynting ﬂux is

cε





|P · E

The power per unit solid angle per unit frequency in the scattered radiation is then

d dω

= r

S

|P · e|

δ(ω

− ω

) (9.88)

where S

 is the mean incident Poynting ﬂux and e = E

/|E

Provided correlations between plasma electrons and ions are unimportant the to-

tal scattered power is found by adding the contributions from individual electrons.

We shall see in Section 9.8 that this is permissible provided kλ

 1. For present

purposes it is enough to note that for kλ

 1, the phase difference between the

radiation ﬁelds from a test electron and from electrons in its Debye cloud is large

and in consequence the radiation ﬁelds of the test electron and its Debye cloud are

incoherent. Under these conditions the total scattered power is found by averaging

(9.88) over the electron distribution.

At this point we need to exercise some care. In computing the total power

in the radiation scattered by electrons in some volume element dr we add the

contributions from individual electrons which means that we must measure time

at the source, not at the detector. As we have already seen in Section 9.2.2 this

has the effect of introducing a factor dt/dt



≡ g. Shefﬁeld (1975) has given an

instructive physical argument for this ﬁnite transit time effect, the need for which

was identiﬁed by Pechacek and Trivelpiece (1967). The scattered power from a

distribution of electrons f (v) in a scattering volume V is then

d dω

= r



S

|P · e|

f (v)g

δ(ω − k · v)dr dv (9.89)

showing both g factors, one from (9.85), the other from the transit time effect.

In the non-relativistic limit these distinctions disappear and (9.89) is greatly

simpliﬁed by discarding g

and taking the polarization dyadic out of the integrand

since it is velocity-independent in this limit. Then (9.89) reduces to

d dω



|(nn − I) · e|



S

dr







(9.90)

358 Plasma radiation

where f

) is the projection of the velocity distribution along k, i.e.

) =



f (v

⊥

)dv

⊥

For a Maxwellian distribution the differential scattering cross-section per unit vol-

ume per unit frequency interval is then

d dω

= r

S(k,ω)sin

θ (9.91)

where θ = cos

−1

(n · e) and the scattering ‘form factor’ S(k,ω) is given by the

Gaussian

S(k,ω)=



2πk



1/2

exp



−

m ω



(9.92)

In principle the form factor affords a direct measure of the electron distribution

function projected on to k. In practice mapping the distribution function from

measurements of scattered radiation is not usually sufﬁciently accurate to be

useful. It does however serve to discriminate non-Maxwellian distributions from

Maxwellian. Thomson scattering measurements do allow electron temperature and

electron density to be determined in equilibrium plasmas. The height of the spec-

trum determines the electron density while the half width, or more exactly the full

width at half maximum (FWHM), ω, given by

ω = 4ω



ln 2



1/2

sin

(9.93)

provides a direct measure of electron temperature.

9.7.2 Electron temperature measurements from Thomson scattering

Thomson scattering of laser light is a widely used diagnostic in fusion plasmas,

providing good spatial resolution. The scattered power is proportional to σ

where L denotes the length of plasma traversed by the light beam. Given the

smallness of σ

this means that only a minute fraction of incident photons will

be scattered. For a density n

∼ 10

−3

and taking L ∼ 0.01 m, the fraction of

photons scattered is only about 10

−10

. Various factors reduce this yet further.

Aside from the requirement that the source of the incident radiation be both

monochromatic and of high brightness, a light scattering diagnostic needs optics

to carry the laser light into the plasma and dispose of it in a beam dump, optics

to collect the scattered light and separate out the different wavelengths, and a

detector. Allowing for the geometry of the collection optics, the efﬁciency of the

optical system and the quantum efﬁciency of the detector means that in practice

9.7 Scattering of radiation by plasmas 359

Fig. 9.13. Thomson scattering measurements of electron temperature in the T3-A tokamak

(after Peacock et al. (1969)).

the fraction of incident photons collected may be no more than 10

−15

. A key issue

in light scattering experiments is the need to prevent stray light from reaching the

detector, given that the ratio of the scattered power that is detected to the incident

power is so small. One way round this problem is by wavelength discrimination

since stray light, as distinct from scattered light, is not shifted in wavelength. Other

precautions include an arrangement of bafﬂes to reduce light scattered from the

walls and entry port of the input beam and the beam dump. Apart from quantum

noise from both the detector and amplifying electronics, plasma bremsstrahlung

and recombination radiation will be present as background noise at the detector.

Moreover unlike the scattered light, this emission is generated over the plasma as

a whole, not simply from the scattering volume.

A landmark in Thomson scattering experiments was the measurement of electron

temperature in the tokamak T3-A by Peacock et al. (1969). Prior to this experi-

ment, measurements of the plasma energy had been made using a diagmagnetic

diagnostic which is subject to a number of limitations. Thomson scattering estab-

lished beyond doubt that temperatures of the order of 1 keV were reached in early

tokamak experiments. Results from these measurements are shown in Fig. 9.13.

360 Plasma radiation

9.7.3 Effect of a magnetic ﬁeld on the spectrum of scattered light

The effect of a magnetic ﬁeld on the scattering form factor (Salpeter (1961), Hag-

fors (1961) and Dougherty, Barron and Farley (1961)) appears only when the wave

vector k is almost orthogonal to the magnetic ﬁeld B. With this alignment the

spectrum is modulated at integral multiples of the electron cyclotron frequency. As

the orthogonality of k with B weakens, the component of electron motion along the

magnetic ﬁeld gives rise to a Doppler broadening of the gyro-resonances. Once the

Doppler line width 2(k · b)V

, where V

is the electron thermal velocity, exceeds

the spacing between the resonances, 

, the resonances will be smeared out. So a

necessary condition for observing magnetic ﬁne structure is 2kV

cos(

k · b)  

In practice spatial variation in the magnetic ﬁeld over the scattering volume may

also result in demodulation.

Magnetic modulation of the spectrum was ﬁrst detected by Evans and Car-

olan (1970) in an experiment using a relatively dense (n

∼ 10

−3

), cool

≈ 20 eV) theta-pinch plasma as a source. With the scattering vector k almost

perpendicular to B the ﬁne structure shown in Fig. 9.14 was resolved. The regularly

spaced peaks show a separation approximately equal to the electron cyclotron

frequency, measured independently by Faraday rotation and corresponding to a

magnetic ﬁeld of about 1.5 T. Forrest, Carolan and Peacock (1978) subsequently

made use of the sensitivity of the depth of modulation at the cyclotron frequency

to the orthogonality of k and B to measure the direction of the poloidal magnetic

Fig. 9.14. Magnetic modulation of scattered light (after Evans and Carolan (1970)).

9.8 Coherent Thomson scattering 361

ﬁeld in a tokamak. For the relatively lower densities and much higher temperatures

of tokamak plasmas the ﬂux of scattered light in individual harmonics is too weak

to allow harmonic structure in the scattered light spectrum to be resolved. This

difﬁculty was overcome using a multiplexing technique due to Shefﬁeld (1975),

in which the free spectral range of a Fabry–Perot interferometer is set equal to

the cyclotron harmonic frequency. The sensitivity of the modulation to the orthog-

onality condition allowed the pitch of the ﬁeld line to be determined to within

0.15

◦

9.8 Coherent Thomson scattering

In treating incoherent Thomson scattering the expression for the total power was

found by summing the contributions from individual electrons, a procedure per-

missible as long as kλ

 1. This condition ensures that correlations between

plasma particles are unimportant. The next task is to determine the scattering form

factor when collective effects have to be taken into account; this happens when

kλ

≤ 1. The summation over the plasma particles has now to be carried out mak-

ing proper allowance for correlations. The need to allow for correlations became

apparent from observations of ionospheric backscatter. Gordon (1958) suggested

that it should be possible to detect Thomson scattering of radar pulses from the

ionosphere. Bowles (1958) found that while the scattered power was in broad

agreement with that predicted, the bandwidth of the scattered signal proved to be

much less than the Doppler width determined by thermal electrons. It appeared

that although the radar pulse was scattered by electrons, their behaviour in turn

was governed by ion dynamics, a result borne out by subsequent studies of the

scattering of radiation by plasma (see Dougherty and Farley (1960), Fejer (1960)

and Salpeter (1960)).

9.8.1 Dressed test particle approach to collective scattering

A helpful way of dealing with coherent Thomson scattering makes use of the con-

cept of test particles. Before dealing with this let us ﬁrst consider what needs to be

done. When a test charge is introduced into a plasma we know that plasma electrons

and ions react to its presence by forming a shielding cloud around it, resulting in the

test particle being ‘dressed’ by this cloud of particles. We saw in Section 7.8 how

to describe this response in terms of the plasma dielectric function. Our objective

is to determine the spectrum of density ﬂuctuations which governs the form factor

by superposing the ﬁelds of all dressed test particles in the plasma. In other words

each electron and ion in turn takes on the role of test particle and the individual

contributions are added up. Since particle correlations have been allowed for by

362 Plasma radiation

means of dressed test particles, these particles themselves are then uncorrelated

so that no further consideration of correlations is needed. A distinction is drawn

between test electrons and test ions. For electrons we have to take account of the

electron itself as well as the electron cloud dressing it. Test ions on the other hand

need not be counted as such since their contribution to scattering is insigniﬁcant,

so that only the electron cloud contributes in the case of ions.

Using the results from Section 7.8 a Klimontovich density deﬁned as N

≡



(r, v) dv, given by

(k,ω) =

2π

electr ons





1 −

1 + χ

+ χ



δ(ω − k · v

)

2π

ions





1 + χ

+ χ



δ(ω − k · v

) (9.94)

in which χ

denotes the susceptibility of species α. Since we equate coherent

scattering from the plasma with incoherent scattering from a gas of non-interacting

dressed ions and electrons, the spectral power density is now determined by form-

ing the quantity |N

(k,ω)|

. Introducing the scattering form factor

S(k,ω)= lim

T →∞

V →∞

(2π)

|N

(k,ω)|



(9.95)

and evaluating (9.95) using (9.94) gives

S(k,ω) =

2π



electr ons





1 −

1 + χ

+ χ



δ(ω − k · v

)

ions





1 + χ

+ χ



δ(ω − k · v

)

5

(9.96)

where the square of the delta functions has been represented in the same way as

in Exercise 9.6. Evaluating the summations of the delta functions in (9.96) using

the representation of the Klimontovich distribution in (7.1) results in contributions

(V/k)



K αk

(ω/k)



= (V /k) f

αk

(ω/k), where α = e, i and the subscript k de-

notes the projection of the velocity distribution along k deﬁned in Section 9.7.1.

Then the scattering form factor reduces to

S(k,ω)=

2π



1 −

1 + χ

+ χ







+ Z



1 + χ

+ χ







(

(9.97)

9.8 Coherent Thomson scattering 363

It remains to evaluate the susceptibilities and hence determine the form factor

explicitly. In the case of a Maxwellian plasma it is straightforward to show that

pα

√

2π







exp(−v

/2V

)

ω − kv





w(ξ

) (9.98)

where V

= (k

)

1/2

is the species thermal speed, ξ

= ω/(

√

2kV

) and

w(ξ) =

√



ζ e

−ζ

dζ

ξ − ζ

is related to the plasma dispersion function deﬁned in Section 7.3.2. For an appro-

priate contour C, w(ξ) may be evaluated to give

w(ξ) = 1 − 2ξ e

−ξ



dζ + i

√

πξe

−ξ

(9.99)

From (9.98) it is evident that for kλ

 1, χ

 1 and consequently the form

factor is determined by the O(1) contribution from the ﬁrst term in (9.97). This

is just the incoherent Thomson scattering discussed in Section 9.7.1. However for

kλ

 1, χ

 1 and contributions to S(k,ω) from coherent scattering become

important. For χ

 1 the entire electron term may be ignored and the sole

contribution to the form factor comes from the ion component.

The spectrum of scattered radiation reﬂects the collective effects exhibited by the

form factor. For an unmagnetized plasma the resonant denominator corresponds

to electron plasma waves in the region of ω = ω

and to ion acoustic waves

in the low frequency range ω ∼ kc

, where c

is the ion acoustic speed. For

both features the shape of the resonance is determined by Landau and collisional

damping. Electron plasma waves are strongly Landau damped unless kλ

 1;

in this case one expects a resonant feature at the Langmuir frequency. Under

conditions where Landau damping becomes signiﬁcant the electron resonance will

be broadened correspondingly. The ion feature in turn reﬂects the characteristics

of ion acoustic waves which are governed by both ions and electrons. In particular

the ion feature will be weak unless T

 T

to ensure that ion Landau damping is

not severe.

The fact that the two contributions to the spectrum are well separated allows

a simpliﬁcation to be made in the general expression for S(k,ω) (see Salpeter

364 Plasma radiation

Fig. 9.15. Salpeter shape function.

(1960)). In the ion term Salpeter set χ

∼ 1/(kλ

)

and in the electron term χ

∼ 0.

Then (see Exercise 9.14)

S(k,ω)=

(2π)

1/2



αe

(ξ

) +

(2π)

1/2



1 + k





αi

(ξ

) (9.100)

in which the shape function



(ξ) =

exp(−ξ

)

|1 + α

w(ξ)|

(9.101)

has the same functional form for both the electron and ion features and

= Z



1 + k



The shape function is represented in Fig. 9.15. The scattering parameter α

is an

important index in scattering from plasmas; α

 1 corresponds to incoherent

Thomson scattering and α

 1 to cooperative scattering. In terms of the scattering

angle θ = cos

−1

(

− 2k

cos θ + k

)

1/2



4πλ

sin θ/2

(9.102)

where λ

is the wavelength of the incident radiation.

9.9 Coherent Thomson scattering: experimental veriﬁcation 365

9.9 Coherent Thomson scattering: experimental veriﬁcation

It is clear from (9.102) that by varying the scattering angle θ one can pass from

a regime of incoherent Thomson scattering (α

 1) to one of coherent (or col-

lective) Thomson scattering for which α

> 1. Alternatively, for a ﬁxed scattering

geometry one could sweep through α

= 1 by switching to longer wavelength

light. In practice this is not usually an option. Substituting values of λ

typical

of a moderately dense laboratory plasma with electron temperature of 1 keV, and

choosing λ

= 1.06 µm corresponding to neodymium laser light, it follows from

(9.102) that to observe coherent Thomson scattering one has to look close to the

forward direction. In this case the shrinking solid angle sets a limit in practice.

Moreover, stray light problems are exacerbated for small θ. For realistic choices

of scattering angle and laser wavelength, the condition for coherent Thomson

scattering translates into a condition that the electron density n

 10

−3

Both low and high frequency features in the scattered light spectrum have diag-

nostic potential. The height of electron line provides a measure of electron density

while from the ion resonance we may in principle deduce the ratio of electron to

ion temperatures and hence determine T

if T

is known from other measurements.

However, as we shall see, the presence of impurity ions may introduce ambiguities

into this measurement.

The ﬁrst identiﬁcation of the ion feature in a laboratory plasma was made by

DeSilva, Evans and Forrest (1964) from studies of ruby laser light scattered by a

hydrogen arc plasma. The electron density, needed to characterize α

, was mea-

sured independently from Stark broadening of the H

line. By detecting scattered

light at two angles it was possible to isolate both incoherent (α

 1) and coherent

(α

> 1) Thomson regimes.

Coherent Thomson scattering diagnostics have been used to advantage in labo-

ratory plasmas, notably in laser-produced plasmas where the high electron density

eases the other constraints, despite difﬁculties over and above those already out-

lined in Section 9.7.2. Problems may arise on account of the sensitivity of the ion

feature to a number of effects, for example becoming asymmetric due to electron

drift velocities and the presence of impurities in the plasma. In thermal plasmas,

the plasma lines are usually weak features in the spectrum and hence difﬁcult to

resolve. These difﬁculties notwithstanding, various groups, for example Baldis,

Villeneuve and Walsh (1986), Baldis et al. (1996), Labaune et al. (1995, 1996)),

have used coherent Thomson scattering to characterize ion acoustic waves and

Langmuir waves in laser-produced plasmas.

The distribution worldwide of a number of powerful radar backscatter facilities

has allowed a range of parameters characterizing the ionospheric plasma to be

determined from measurements of Thomson scattering. These include not only

366 Plasma radiation

Fig. 9.16. Radar backscatter spectrum from the ionosphere, taken from an altitude of

300 km on 31 March 1971. The solid line represents the Salpeter spectrum ﬁtted to the

data points (courtesy of J.M. Holt).

electron density and temperature but ion density, ion mass, composition of the

plasma, mean drift velocity and the ion–neutral collision frequency. Figure 9.16

plots radar backscatter data as a function of the frequency shift with the line show-

ing the predicted spectrum. The dashed line denotes the spectral shift resulting

from the mean motion of the ionospheric plasma. This spectrum was recorded

from a height of 300 km, where O

is the only ion of signiﬁcance and this helped

minimize deviations from the Salpeter spectrum due to other ions being present.

Above this height, protons and He

ions from the solar wind increasingly affect

the composition of plasma in the high ionosphere. In lower regions plasma compo-

sition is governed by the photochemistry of the ionosphere. The effects of plasma

composition on the scattering form factor were ﬁrst discussed by Moorcroft (1963).

9.9.1 Deviations from the Salpeter form factor for the ion feature: impurity ions

Impurity ions are present to some extent in all laboratory plasmas. Given their

greater mass and consequently lower thermal velocities, impurities serve to en-