Simple analytic form of the relativistic Thomson scattering spectrum

Volume 79A, number 5,6

PHYSICS LETTERS

27 October 1980

SIMPLE ANALYTIC FORM OF THE RELATIVISTIC THOMSON SCATTERING SPECTRUM A.C. SELDEN Cuiham Laboratory (EURATOM/UKAEA Fusion Association), Abingdon, Oxon 0X14 3DB, UK Received 17 July 1980 Revised manuscript received 4 August 1980

A precise analytic result in re’ativistic scattering theory obtained by Zhuravlev and Petrov is presented in a form suitable for routine analysis of Te data for fusion plasmas.

Thepurpose of this note ~ is to draw attention to a simple yet accurate analytic formula for the relativistic Thomson scattering spectrum which can be used for routine statistical analysis of laser scattering data from high temperature plasmas (100 eV ~ T~~ 100 keV for 90°-scattering).The formula is easily derived from a result of Zhuravlev and Petrov [1] ,who integrated the relativistic scattering integral analytically. Numerical comparison with the computationalresults of Matoba et al. [2] shows only ~0.1% difference in the normalised scattering spectra when Te = 20 keV (table 1); the curves still agree within ~1% at 100 keY. In contrast, existing polynomial expansions have errors of 8% (second order) and 13% (first order) at 20 keV and need to be replaced by a more accurate expression for use with Thomson scattering diagnostics of fusion plasmas. The scattered power per unit wavelength, i.e. the wavelength spectral density function, can be written as the simple product of one analytic function of

Table

1 Comparison of spectral density functions for Te 6 = 90°. SM(e) ___________________

+0.6 +0.5 +0.4 +0.3 +0.2 +0.1

0.0 —0.1 —0.2

—0.3 —0.4

_______

0.045 2

0.0818 0.1457 0.2530 0.4237 0.6744 1.0000

0.0819 0.1458 0.2532 0.4238

1.3432 1.5697 1.5037

1.3433 1.5703

—0.6 0.7

0.0069

—o.s

20

keV and

SZh(e)

0.045 1

1.0764 0.4946 0.1109

=

____________

0.0013 0.0008 0.0007 0.0005 0.0002 0.0000 0.0000

0.6745 1.0000

0.0001

0.0004. 0.0008

1.5049 1.0782

0.0016 0.0026

0.4959 0.1113 0.0069

0.0036 0.0039

Here 2~ mec2/kTe, c(a) is a normalising constant, 0 is the scattering angle, and = (X 5/X~) 1 measures the relative shift in wavelength of the scattered light, where X~and A 5 are the incident and scattered wavelengths, respectively. This expression for S(e, 0) is similar in form to the various approximations currently in use, but its numerical values are far more precise. Ex—

(e, 0) with the exponential of another,0)], as follows: (1) 1(, 0)exp[—2csB(e, S(e, O) c(a)A where A(e,0)(l

+)~[2(1 —cos0)(1

÷)+2]h/2,

(2a)

B(e, 0) = {i + e2/[2(1 —cosO)(1 + e)] }1/2 ~1, (2b) c(a) = (a/ir)1/2(1 ~ + ~—5a~ i-..) whena~1 (2c) —

A more detailed description wifi be given in a Culham laboratory report.

givesofaneq. expression identical with pansion of the3)r.h.s. (1) to first order (retaining terms in , ae Sheffield’s high temperature correction [3] to the classical gaussian scattering spectrum, viz. s(’)(e, 0) = {1 e + ae3/[4 sin2(0/2)] } S(0)(, 0), (3) —

405

Volume 79A, number 5,6

PHYSICS LETTERS

Table 2 Relativistic blue shift versus Te, 6 (keV)

CC

3.77 7.80 16.75 27.10 39.21

—0.05 —0.1 —0.2 —0.3 —0.4

53.56

—0.5

71.00 93.27

—0.6 —0.7

=

Table 3 Density correction factor versus electron temperature, 0

90°

Te

+ ~(2)

—0.0517 —0.107 —0.229 —0.37 —0.54 —0.73

—0.0499 —0.0995 —0.196 —0.28 —0.35 —0.39

—

—

—

—

where S(0)(e, 8) is the zero-order scattering spectrum used for low temperature plasmas (Te < 100 eV). The formula, eq. (1), can be used to find the relativistic blue shift e~of the scattering spectrum as a function of temperature (table 2). This can also be expanded in the form ~ = 41) + 42) + of which the first term is identical with Sheffield’s original result, i.e. 41) = —(7/a) sin2(0/2), and the second term is ...,

(4) c = 7 sin (0/2)[l + 7 sin (0/2)] /2a showing a small positive correction to the first-order shift. Values of 41) and e~ are tabulated alongside the “exact” values in table 2 for comparison. There is a discrepancy in electron density at the highest temperatures (table 3) owing to the difference of a transverse term in the two forms of the scattering integral employed in the literature [1,2] This can be incorporated as a correction factor q(a) in the calibration for a given scattering angle, and does not affect ,

.

406

27 October 1980

=

90°.

q(~~) 1 2

0.9968 0.9942 0.9853 0.9713 0.9446 0.8753 0.7891

5 10 20 50 100 -

_________

the measurement of Te in practice for arbitrary plasma density. I am grateful to George Magyar for first bringing the work of Zhuravlev and Petrov to my notice, and to all my other colleagues, both on JET and at Cuiham laboratory, for many useful discussions and for their encouragement, which has been much appreciated. I am also pleased to acknowledge the invaluable help provided by Dr. Matoba s group in communicating their computer results to me, without which no such exact comparison would have been possible. .

,

.

References [11 V.A. Zhuravlev and G.D. Petrov, Soy. J. Plasma Phys. 5 (1979) 3. [2] T. Matoba, T. ltagaki, T. Yamauchi and A. Funahashi, [3] ~