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Electron-Electron Bremsstrahlung in a Hot Plasma E Electron-Electron Bremsstrahlung in a Hot Plasma E. Haug Lehrstuhl für Theoretische Astrophysik der Universität Tübingen (Z. Naturforsch. 30 a, 1546-1552 [1975] ; received November 7, 1975) Using the differential cross section for electron-electron (e-e) bremsstrahlung exact to lowest- order perturbation theory the bremsstrahlung spectra and the total rate of energy loss of a hot electron gas are calculated. The results are compared to the nonrelativistic and extreme-relativistic approximations available and to previous work. The e-e contribution to the bremsstrahlung emis- sion of a hydrogenic plasma is found to be not negligible at temperatures above 108 K, especially in the short-wavelength part of the spectrum. 1. Introduction 2. Bremsstrahlung Spectra The observation of the X-radiation emanating The number of photons emitted per unit time, per from a hot Maxwellian plasma is very useful in unit volume, per unit energy interval by an electron obtaining quantitative estimates on the physical gas of uniform number density ne at temperature T parameters such as electron temperature and density. is given by The basic requirement for the deduction of these Pee(K,T) (2.1) quantities is the knowledge of bremsstrahlung cross sections. Whereas there ajp no problems concerning 2 3 -1 »e c f dW d p2 /(Pl) /(p2) VlPi ^lizl da the X-radiation arising from electron-ion (e-i) colli- sions, the cross section for electron-electron (e-e) where c is the velocity of light, ^ and e2 are the total bremsstrahlung was known only in nonrelativistic electron energies, and do/dk is the cross section for (quadrupole) and in extreme-relativistic approxima- generating a photon of energy k in the collision of tions 2. In order to get the bremsstrahlung spectra two electrons with the momenta px and p2; the in the intermediate energy region, one had to inter- energies , £2> and k are in units of the electron polate between the two limiting cases 3' 4. Another 2 rest energy m c , the momenta px and p2 are in difficulty is due to the fact that the cross section for units of m c, and (px p2) =£1£2-Pt 'P2 is the in- e-e bremsstrahlung should be known for arbitrary variant product of the four-momenta (f^Pi) and directions of the electron velocities while it is usually (e2, p2). given in a special frame of reference, e.g., the center- The relativistic Maxwell-Boltzmann equilibrium of-mass system or the rest system of one of the distribution of electrons at temperature T is de- colliding particles 2. Now a covariant formula for scribed by 6 the cross section of e-e bremsstrahlung differential a~t!x with respect to photon energy and angle is available /(P) = (2.2) which is exact to lowest-order perturbation theory 5. 4>TZTK2 (1/T) Using this expression it is possible to compute the where r = k\\ Tjm c~ (kn = Boltzmann constant) is the e-e bremsstrahlung spectra of a hot plasma and to non-dimensional temperature and the function K2 is specify the range of validity and the accuracy of the the modified Bessel function of the second kind. approximation formulae. Additionally it makes it f(p) is normalized to unity, feasible to calculate the total energy emitted by an 3 electron gas of temperature T. This energy can ex- //(p)d p = l. (2.3) ceed the total bremsstrahlung rate generated by In (2.1) the quantity V (Pi P2)' — €2 originates electron-proton (e-p) collisions in a hydrogenic from the invariant expression for the flux of incident plasma at very high temperatures. The results are particles7> 8, and the factor | prevents from the compared to those obtained by Maxon 4 employing double counting of each electron. the interpolation method. The cross section do/dk depends, apart from £x, Reprint requests to Dr. E. Haug, Lehrstuhl für Theoreti- sche Astrophysik der Universität Tübingen, D-7400 Tübin- e2, and k, only on the angle £ between the vectors gen, Auf der Morgenstelle, Gebäude C. Pi and P-2. Since the distribution /(p) is a function E. Haug • Electron-Electron Bremsstrahlung in a Hot Plasma 1547 of the absolute value p = j p ] = j/f2 — 1 of the elec- This formula has been derived by Maxon and Cor- tron momentum, most of the angle integrations in man 12. The integral (2.1) can be performed easily. Replacing the inte- l gration variable cos £ by /u = (px p2) = £x £2 ~ (2.11) px p2 cos one obtains cne2 (2.4) P,e(k, T) = can be computed numerically. The values for [2 x K2(\/T)]' 0.025 kjx 5 are given in Table 1. do For temperatures up to r=108K the nonrela- f dej fde2e~ & + £>»xf dju Vju2 - 1 dk tivistic formula (2.10) is a very good approxima- tion to Equation (2.4). The values of P™(k,r) are The integration intervals in (2.4) follow from the slightly too low throughout the spectrum. At k^ T kinematics of the e-e process. All the combinations = 10keV corresponding to r«^1.16-108K the of the variables £j, e2, and f t are allowed which relative error is less than 1% for the photon energies satisfy the condition 9 above 1 keV. k< /*- 1 The dipole spectrum emitted by a nonrelativistic + e2 ~ V(£i + £a)2 - 2 (^ + 1) ' (2'5) hydrogenic plasma at temperature T is given by 13- Since the computation of do/dk from the differential cross section d2o/dkdQ available5 requires two 16 o ^e ^p (k, T) = a r c (2.12) angle integrations in case of arbitrary directions of o~ r~ the momenta pt and p2 , one has to carry out a • V2J (tzx) e~kl2x K0(k/2 x) total of 5 integrations to obtain Pee (k,T). For "low" temperatures an approximate where np is the proton number density and K0 the formula for Pee(k,x) can be derived. Fedyushin 10 modified Bessel function of the second kind. At tem- and Garibyan 11 have calculated the cross section for peratures kßT> 20 keV the contributions of higher the e-e quadrupole emission valid in the center-of- multipoles to the bremsstrahlung spectrum become mass system of the colliding electrons for nonrela- important. They are most simply included by em- tivistic energies £A — 1 1, £2 — 1 ^ 1. This expres- ploying the correction factor 1 sion , generalized to arbitrary frames of reference, R = l/(B-vk) (2.13) is given by 5 derived by Quigg 14 who has given a table with the do NR 4 ar, 4A: \ parameters B and Dasa function of r. Then dk 15 k Pep(k, T) R-P?*(k,x) (2.14) where oi = e2/hc^ 1/137 is the fine-structure con- stant, r0 = e2/m c2 is the classical radius of the elec- In the nonrelativistic approximation the ratio of e-e tron, and the function F has the form (quadrupole) to e-p (dipole) bremsstrahlung emis- 3 x2 12 (2 — x)* — 7 x2 (2 — x)2 — 3 x* l + Vl-x F(x)- 17- VI-x + — rs In (2.7) (2-x)2 (.2-x)3 V* Using the ordinary Maxwell distribution /NR(P) = {2 7ZX)~3I2 e~P'l2r (2.8) Equation (2.1) reduces to dOvr P** (k, r) = i ne2 c (2 * T) "3 / d3Pl / d3p2 | Pl - p2 | exp { - (p2 + p22) /2 r) . (2.9) It is convenient to introduce the new variables 12 p=p^ — p> and P = p1+p2. Then the integration over P can be carried out, and the result is Pe? (k,x)=^-ar2c n2 (n T) ~3/2 k [ e " */<«> dx . (2.10) 15 Ja; E. Haug • Electron-Electron Bremsstrahlung in a Hot Plasma 1548 1 Table 1. Function I{kfr) = J e~*/(«) dx . 0 k/r 0.025 0.050 0.075 0.10 0.20 0.30 0.40 0.5 I(k/r) 1.311 • 105 2.910•104 1.193 • 104 6290 1299 497.8 245.9 139.5 k/r 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 I (k/r) 86.44 56.89 39.12 27.82 20.32 11.51 6.917 4.341 2.816 k/r 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 I(k/r) 1.874 1.273 0.8802 0.6172 0.4380 0.3141 0.2273 0.1658 k/r 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 I(k/r) 0.1218 0.09001 0.06690 4.997 10-2 3.749-10 2 2.825-10 2 2.136 -10 "2 1.621 -lO"2 sion at a particular photon energy k can be defined 12 the photon energy h v = m c2 k = 60 keV. The relative as (setting nc error made by using (2.10) increases from « 8% at h r = 60 keV to « 30% at h v = 300 keV. It is PlR(k, r) V™{k/r) = (2.15) seen that the contribution of e-e bremsstrahlung to R T 'ep ) the emission of a hydrogenic plasma is no more 1 WC/J negligible at such high temperatures, especially in (k\r)- 10 V2 e-k'2tK0(k/ 2r) the short-wavelength part of the spectrum. Al //>' = 300 keV, e.g., the e-e emission amounts to about At a given temperature r the ratio y""(K[TV^ik/r)) in- creases considerably towards the high-energy part 70% of the e-p emission.
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