Pergamon J. Quant. Spec~rosc. Radiat. Tran$er Vol. 58. No. I. pp. 141-143. 1997

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NOTE

COMMENT ON “BROADENING OF THE RAMAN RESONANCE IN PHOTON SCATTERING IN PLASMAS”

CARLOS A. IGLESIAS Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550, U.S.A

(Recehed 22 August 1996)

Abstract-A recent examination of photon scattering in the solar interior by Tsytovich et al. [JQSRT 55, 787 (1996)] is shown to interpret incorrectly past calculations of the Raman resonance and Doppler shift contributions. c! 1997 Elsevier Science Ltd

Photon scattering plays an important role in the radiative opacity of the solar interior. Furthermore, plasma corrections to the single electron Thomson scattering result are significant.’ Recently, Tsytovich et al.2 (hereafter TBDF) proposed further opacity reductions associated with the Raman resonance. It is shown that the effects are already included in past work and a possible explanation for the confusion is offered. For simplicity, a hydrogen plasma is assumed.

The relevant quantity for opacity calculations is the scattering transport cross section given in Eq. (45) of TBDF,

where the probability per unit time that an incident photon with energy fro in the direction k be scattered with energy tie’ in the direction k’ is given by3

WE’kk. = (1 + P’) &* _ q.v) (277Ye4 i

(1 + x&q, Sr)l’, 3 = electron

Zm,oo' 18(q, n)l’ l&(4, W’9 o! = proton (2)

with II, the electron number density, fr the velocity distribution for species a, q = k’ - k and R = CO’ - o the momentum and energy transfer respectively, p = cos0 with 0 the angle between the initial and final photon direction, and m, the electron mass. The susceptibilities are such that the dielectric response function is given by

(3)

and represent plasma effects on the photon scattering process. The term q.v in Eq. (2) is known as the Doppler shift since it depends on the velocity of the plasma particle.

Equation (1) is usually written in terms of the dynamic structure factor which describes the power spectrum of the electron density fluctuations.“.’ The connection can be established since for weakly-coupled plasmas (assumed by TBDF) the dynamic structure factor is given

by”

qsz - q.v) (4)

141

142 Note

With Eq. (4) and the relation k’c = CO’ (neglects dispersion), the transport cross section in Eq. (1) can be rewritten as

X. o”(w) = & dw’

I s

fl &(q, 0)

0 -I dp(l - ~)(l + P’) $

0

giving the desired resultM Following TBDF, thus restricting the discussion to high frequencies (i.e., o>>o,, with opt the

plasma frequency), then ho is much greater than the plasma excitation energies for which S,(q, Cl) has appreciable strengths and Eq. (5) simplifies to

e4 +I

= 2m2c4 s d/41 - PN + p2We(q) ~,

(6)

(7)

where the exact sum rule.4.5

x

Se(q) = L dQS,(q, Q) , (8)

was used with S,(q) the electron static structure factor. Substituting into Eq. (7) the weak-coupling approximation,4.5

s,(q) x k,’ + q2 2kf

with kT the temperature in energy units and

k,Z = 4ze2nJkT, (10)

yields the well known result, a:(w), in Eq. (47) of TBDF. Although the weakly-coupled limit has been assumed throughout the discussion, Eqs. (6) to (8) are more general.U

The discussion leading to Eq. (7) shows that the Doppler shift q-v is included through the definition in Eq. (4) and the resonance in S&q, Cl) at Q NN + o, (as well as any broadening) is automatically contained through the sum rule in Eq. (8). Consequently, contrary to the claims by TBDF, G:(W) has both the Doppler shift and the Raman resonance contributions. For consistency, improvements to a:(o) should be added systematically (e.g., corrections to S,(q) beyond the weakly-coupled, non-relativistic classical limit). Note that results for weakly-coupled quantum plasmas (appropriate for the solar interior) based on Eq. (7) are available’ and were implemented in opacity calculations.’ Furthermore, calculations of Eq. (6) including some neglected frequency dependent terms that reduce to unity at high frequencies (e.g., dispersion) lead to a small increase in solar opacities.*

It follows that the predicted additional reduction in the solar interior opacity by TBDF may not be realized. A possible source of confusion is the association of the static structure factor with the dynamic structure factor at zero frequency. That is,

X(q) = X(q, n = O)(.? (11)

which is incorrect but may explain the misleading comments surrounding Eq. (47) of TBDF. Curiously, Tsytovich et al9 earlier claimed that photon scattering near the Raman resonance, including spontaneous and stimulated contributions, had negligible impact on solar opacities.

Finally, the separation by TBDF of the photon scattering into electron and ion contributions is specious and only possible under certain approximations (e.g., weak-coupling limit). Note that photons do not scatter from a uniform medium (except in the forward direction), rather they mostly scatter from the electron densitvfluctuations which depend in an essential manner on the ion motion

Note 143

through the self-consistent interactions between electrons and ions. 4.5 Photon scattering by the ion density fluctuations is smaller by the factor (m,/mJ2 with m, the ion mass and is usually neglected.

Acknowledgements-It is a pleasure to recognize valuable discussions with David B. Boercker. Work performed under the auspices of the Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

REFERENCES

I. Boercker, D. B., Ap.J., 1987, 316, L95. 2. Tsytovich, V. N., gingham, R., de Angelis, I-J. and Forlani, A., JQSRT, 1996, 55, 787. 3. There is a typographical error in Eq. (3) of TBDF: ~~‘(4, D) should read Ic(q, Q)(-‘. 4. Bekefi, G., Radiation Processes in Plasmas. Wiley, New York, 1966. 5. Sitenko, A. G., Electromagnetic Fluctuations in Plasma. Academic Press, New York, 1967. 6. Williams, B., Compton Scattering. McGraw-Hill, London, 1977. 7. Iglesias, C. A. and Rogers, F. J., Ap. J., 1991, 371, 408; Ap. J., 1996, 464, 943; Seaton, M. J.. Yan Y..

Mihalas, D. and Pradhan A., MNRAS, 1994, 266, 805. 8. Iglesias, C. A. and Rose, S. J., Ap. J. Letters, 1996, 466, Ll15. 9. Tsytovich, V. N., gingham. R. and de Angelis, U.. J. Plusma. Phys.. 1995, 53, 335.