THE UNIVERSITY OF MICHIGAN 2764-3-T STUDIES IN RADAR CROSS SECTIONS XLII - ON MICROWAVE BREMSSTRAHLUNG FRQM A COOL PLASMA by M. L. Barasch August 1960 Report No. 2764-3-T on Contract DA 36-039 SC-75041 The work described in this report was partially supported by the ADVANCED RESEARCH PROJECTS AGENCY, ARPA Order Nr. 120-60, Project Code Nr. 7700. Prepared For The Advanced Research Projects Agency and the U. S. Army Signal Research and Development Agency Ft. Monmouth, New Jersey

THE UNIVERSITY OF MICHIGAN 2764-3-T ASTIA Availability Notice Qualified requestors may obtain copies of this report from ASTIA.

THE UNIVERSITY OF MICHIGAN 2764-3-T STUDIES IN RADAR CROSS SECTIONS I "Scattering by a Prolate Spheroid", F. V. Schultz (UMM-42, March 1950), W-33(038)-ac-14222. UNCLASSIFIED. 65 pgs. II "The Zeros of the Associated Legendre Functions P m(1') of Non-Integral Degree", K. M. Siegel, D. M. Brown, H.E. Hunter, H.A. Alperin and C. W. Quillen (UMM-82, April 1951), W-33(038)-ac-14222. UNCLASSIFIED. 20 pgs. III "Scattering by a Cone", K. M. Siegel and H.A. Alperin (UMM-87, January 1952), AF-30(602)-9. UNCLASSIFIED. 56 pgs. IV "Comparison between Theory and Experiment of the Cross Section of a Cone", K.M. Siegel, H.A. Alperin, J.W. Crispin, Jr., H.E. Hunter, R.E. Kleinman, W. C. Orthwein and C.E. Schensted (UMM-92, February 1953), AF-30(602)-9. UNCLASSIFIED. 70 pgs. V "An Examination of Bistatic Early Warning Radars", K. M. Siegel (UMM-98, August 1952), W-33(038)-ac-14222. SECRET. 25 pgs. VI "Cross Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies", R. R. Bonkowski, C. R. Lubitz and C. E. Schensted (UMM-106, October 1953), AF-30(602)-9. SECRET - Unclassified when appendix is removed. 63 pgs. VII "Summary of Radar Cross Section Studies under Project Wizard", K. M. Siegel, J.W. Crispin, Jr. and R. E. Kleinman (UMM-108, November 1952), W 33(038)-ac-14222. SECRET. 75 pgs. VIII "Theoretical Cross Section as a Function of Separation Angle between Transmitter and Receiver at Small Wavelengths", K. M. Siegel, H.A. Alperin, R.R. Bonkowski, J.W. Crispin, Jr., A.L. Maffett, C.E. Schensted and I.V. Schensted (UMM-115, October 1953), W-33(038)-ac-14222. UNCLASSIFIED. 84 pgs. IX "Electromagnetic Scattering by an Oblate Spheroid", L. M. Rauch (UMM-116, October 1953), AF-30(602)-9. UNCLASSIFIED. 38 pgs. X "Scattering of Electromagnetic Waves by Spheres", H.Weil, M. L. Barasch and T.A. Kaplan (2255-20-T, July 1956), AF-30(602)-1070. UNCLASSIFIED. 104 pgs. 111

THE UNIVERSITY OF MICHIGAN 2764-3-T XI "The Numerical Determination of the Radar Cross Section of a Prolate Spheroi K. M. Siegel, B. H. Gere, I. Marx and F. B. Sleator (UMM-126, December 1953), AF-30(602)-9. UNCLASSIFIED. 75 pgs. XII "Summary of Radar Cross Section Studies under Project MIRO'', K.M. Siegel, INL.E. Anderson, R.R. Bonkowski and W.C. Orthwein (UMM-127, December 1953), AF-30(602)-9. SECRET. 90 pgs. XIII "Description of a Dynamic Measurement Program", K. M. Siegel and J. M. Wolf (UMM-128, May 1954), W-33(038)-ac-14222. CONFIDENTIAL. 152 pgs XIV "Radar Cross Section of a Ballistic Missile", K. M. Siegel, M. L. Barasch, J.W. Crispin, Jr., W.C. Orthwein, I.V. Schensted and H. Weil (UMM-134, September 1954), W-33(038)-ac-14222. SECRET. 270 pgs. XV "Radar Cross Sections of B-47 and B-52 Aircraft", C. E. Schensted, J.W. Crispin, Jr. and K.M. Siegel (2260-1-T, August 1954), AF-33(616)-2531. CONFIDENTIAL. 155 pgs. XVI "Microwave Reflection Characteristics of Buildings", H. Weil, R. R. BonkowE T.A. Kaplan and M. Leichter (2255-12-T, August 1954), AF-30(602)-1070. SECRET. 148pgs. XVII "Complete Scattering Matrices and Circular Polarization Cross Sections for the B-47 Aircraft at S-band", A. L. Maffett, M. L. Barasch, W.E. Burdick, R. F. Goodrich, W.C. Orthwein, C. E. Schensted and K.M. Siegel (2260-6-T, June 1955), AF-33(616)-2531. CONFIDENTIAL. 157 pgs. XVIII "Airborne Passive Measures and Countermeasures", K. M. Siegel, M. L. Barasch, J. W. Crispin, Jr., R. F. Goodrich, A. H. Halpin, A. L. Maffett, W. C. Orthwein, C E. Schensted and C.J. Titus (2260-29-F, January 1956), AF-33(6d6)-2531. SECRET: 177 pgs. XIX "Radar Cross Section of a Ballistic Missile II", K. M. Siegel, M. L. Barasch, H. Brysk, J.W. Crispin, Jr., T. B. Curtz and T.A. Kaplan (2428-3-T, January 1956), AF-04(645)-33. SECRET. 189 pgs. XX "Radar Cross Section of Aircraft and Missiles", K. M. Siegel, W.E. Burdick J.W. Crispin, Jr. and S. Chapman (WR-31-J, March 1956). SECRET. 151 p XXI "Radar Cross Section of a Ballistic Missile III", K. M. Siegel, H. Brysk, J.W. Crispin, Jr. and R. E. Kleinman (2428-19-T, October 1956), AF-04(64 33. SECRET. 125 pgs. iv

THE UNIVERSITY OF MICHIGAN 2764-3-T XXII "Elementary Slot Radiators", R. F. Goodrich, A. L. Maffett, N.E. Reitlinger, C. E. Schensted and K. M. Siegel (2472-13-T, November 1956), AF-33(038)28634, HAC-PO L-265165-F31. UNCLASSIFIED. 100 pgs. XXIII "A Variational Solution to the Problem of Scalar Scattering by a Prolate Spheroid", F. B. Sleator (2591-1-T, March 1957), AF-19(604)-1949, AFCRC-TN-57-586, AD 133631. UNCLASSIFIED. 67 pgs. XXIV "Radar Cross Section of a Ballistic Missile - IV The Problem of Defense", M. L. Barasch, H. Brysk, J.W. Crispin, Jr., B.A. Harrison, T. B.A. Senior, K. M. Siegel, H. Weil and V. H. Weston (2778-1-F, April 1959), AF-30(602)1953. SECRET. 362 pgs. XXV "Diffraction by an Imperfectly Conducting Wedge", T. B.A. Senior (2591-2-T, October 1957), AF-19(604)-1949, AFCRC-TN-57-591, AD 133746. UNCLASSIFIED. 71 pgs. XXVI "Fock Theory", R. F. Goodrich (2591-3-T, July 1958), AF-19(604)-1949, AFCRC-TN-58-350, AD 160790. UNCLASSIFIED. 73 pgs. XXVII "Calculated Far Field Patterns from Slot Arrays on Conical Shapes", R. E. Doll. R. F. Goodrich, R.E. Kleinman, A. L. Maffett, C. E. Schensted and K.M. Siegel (2713-1-F, February 1958), AF-33(038)-28634 and 33(600)-36192; HAC-POs L-265165-F47, 4-500469-FC-47-D and 4-526406-FC-89-3. UNCLASSIFIED. 115 pgs. XXVIII "The Physics of Radio Communication via the Moon", M. L. Barasch, H. Brysk, B.A. Harrison, T.B.A. Senior, K. M. Siegel and H. Weil (2673-1-F, March 1958), AF-30(602)-1725. UNCLASSIFIED. 86 pgs. XXIX "The Determination of Spin, Tumbling Rates and Sizes of Satellites and Missiles", M.L. Barasch, W.E. Burdick, J.W. Crispin, Jr., B.A. Harrison, R. E. Kleinman, R.J. Leite, D. M. Raybin, T. B.A. Senior, K.M. Siegel and H. Weil (2758-1-T, April 1959), AF-33(600)-36793. SECRET. 180 pgs. XXX "The Theory of Scalar Diffraction with Application to the Prolate Spheroid", R.K. Ritt (with Appendix by N. D. Kazarinoff), (2591-4-T, August 1958), AF-19(604)-1949, AFCRC-TN-58-531, AD 160791. UNCLASSIFIED. 66 pgs. XXX~I "Diffraction by an Imperfectly Conducting Half-Plane at Oblique Incidence", T.B.A. Senior (2778-2-T, February 1959), AF-30(602)-1853. UNCLASSIFIED. 35 pgs.

THE UNIVERSITY OF MICHIGAN 2764-3-T XXXII "On the Theory of the Diffraction of a Plane Wave by a Large Perfectly Conducting Circular Cylinder", P.C. Clemmow (2778-3-T, February 1959), AF-30(602)-1853. UNCLASSIFIED. 29 pgs. XXXIII "Exact Near-Field and Far-Field Solution for the Back-Scattering of a Pulse from a Perfectly Conducting Sphere", V. H. Weston (2778-4-T, April 1959), AF-30(602)-1853. UNCLASSIFIED. 61 pgs. XXXIV "An Infinite Legendre Transform and Its Inverse", P.C. Clemmow (2778-5-T, March 1959). AF-30(602)-1853. UNCLASSIFIED 35 pgs. XXXV "On the Scalar Theory of the Diffraction of a Plane Wave by a Large Sphere", P. C. Clemmow (2778-6-T, April 1959), AF-30(602)-1853. UNCLASSIFIED. 39 pgs. XXXVI "Diffraction of a Plane Wave by an Almost Circular Cylinder", P. C. Clemmow and V. H. Weston (2871-3-T, September 1959), AF 19(604)-4933. UNCLASSIFIED. 47 pgs. XXXVII " Enhancement of Radar Cross Sections of Warheads and Satellites by the Plasma Sheath", C. L. Dolph and H. Weil (2778-2-F, December 1959), AF-30(602)-1853. SECRET. 42 pgs. XXXVIII "Non-Linear Modeling of Maxwell's Equations", J. E. Belyea, R. D. Low and K. M. Siegel (2871-4-T, December 1959), AF-19(604)-4993, AFCRC-TN-60-106. UNCLASSIFIED. 39 pgs. XXXIX "The Radar Cross Section of the B-70 Aircraft", R. E. Hiatt and T. B. A. Senior (3477-1-F, February 1960). North American Aviation, Inc. Purchase Order No. LOXO-XZ-250631. SECRET. 157 pgs. XL "Surface Roughness and Impedance Boundary Condition", R. E. Hiatt, T. B. A. Senior and V. H. Weston (2500-2-T, July 1960), AF 19(604)-4993, AF 19(604)-5470, AF 30(602)-1808, AF 30(602)-2099 and Autometric Corporation 33-S-101. UNCLASSIFIED. 96 pgs. XLI "Pressure Pulse Received Due to an Explosion in the Atmosphere at an Arbitrary Altitude, Part I, V. H. Weston, (2886-1-T, August 1960), AF 19(604)-5470. UNCLASSIFIED. XLII "On Microwave Bremsstrahlung From a Cool Plasma", M. L. Barasch, (2764-3-T, August 1960). DA 36(039)-sc-75041, UNCLASSIFIED.

THE UNIVERSITY OF MICHIGAN 2764-3-T PRE FACE This is the forty-second in a series of reports growing out of the study of radar cross sections at The Radiation Laboratory of The University of Michigan. Titles of the reports already published or presently in process of publication are listed on the preceding pages. When the study was first begun, the primary aim was to show that radar cross sections can be determined theoretically, the results being in good agreement with experiment. It is believed that by and large this aim has been achieved. In continuing this study, the objective is to determine means for computing the radar cross section of objects in a variety of different environments. This has led to an extension of the investigation to include not only the standard boundary-value problems, but also such topics as the emission and propagation of electromagnetic and acoustic waves, and phenomena connected with ionized media. Associated with the theoretical work is an experimental program which embraces (a) measurement of antennas and radar scatterers in order to verify data determined theoretically; (b) investigation of antenna behavior and cross section problems not amenable to theoretical solution; (c) problems associated with the design and development of microwave absorbers; and (d) low and high density ionization phenomena. K. M. Siegel vii

THE UNIVERSITY OF MICHIGAN 2764-3-T TABLE OF CONTENTS Summary 1 I. Introduction 2 II. Electron-Ion Bremsstrahlung from the Faster Electrons 4 III. The Impulse Approximation 9 IV. Electron-Neutral Bremsstrahlung 17 V. Evaluation of Results, Extension to Other Frequencies 21 VI. Use of Detailed Balance to Obtain Free-Free Absorption Coefficients from Bremsstrahlung Cross Sections 28 Acknowledgements 38 References 39 ix

THE UNIVERSITY OF MICHIGAN 2764-3-T SUMMARY Microwave Bremsstrahlung from, and free-free absorption in, a cool, partially-ionized plasma are treated. Electron-ion encounters are treated by the Born approximation and the classical impulse approximation, a Debyeshielded potential being used. Bremsstrahlung from electron-neutral collisions is treated by the Born approximation. The potential here is obtained by fitting a shielded Coulomb form to the Thomas-Fermi potential for distances less than about 7 atomic radii. For the plasma parameters chosen (T = 50000K, ne 1013/cm3, Z = 8) and microwave frequencies of the order of 50 KMc, it would appear that at the correspondingly low degree of ionization, the neutrals are most significant. An effective Z for the oxygen atoms is determined by matching the free-free absorption to Kramers' law. Its value, Z =. 17, compares reasonably with the results of previous investigators.

THE UNIVERSITY OF MICHIGAN 2764-3-T I. Introduction Among the classes of problems studied in the Radiation Laboratory have been those dealing with plasmas as either a source or absorber of radiation. One of- the specific radiation mechanisms considered has been Bremsstrahlung, while its inverse, free-free absorption, contributes to propagation losses in the plasma. Bremsstrahlung is a process in which a free electron is scattered by a potential into a free state of lower energy, the energy difference appearing as a radiated photon. In the inverse process, free-free absorption, a free electron in a potential absorbs a photon and makes a transition to a free state of higher energy. Most treatments of these effects have been for a fully ionized gas, in which the Coulomb potential of the positive ions was the potential of interest. Since we at the Radiation Laboratory are more frequently interested in plasmas of aerodynamic than of thermonuclear origin, the assumption of complete ionization is not useful for us, and we must consider the effect of electron encounters with neutral atoms too. (Since the orbital electron distribution of the atom is extended in space, a free plasma electron may penetrate it and thus "see" an incompletely shielded nuclear charge. ) For comparison with other work, we choose the neutrals to be oxygen atoms.

THE UNIVERSITY OF MICHIGAN 2764-3-T Since the work reported here is partially, but certainly not entirely, original in approach and technique, a brief review of the literature and its relation to the present work seems in order. An exact non-relativistic Bremsstrahlung cross section for pure Coulomb fields was given by Sommerfeld (Ref. 1). The result is in terms of hypergeometric functions, the numerical approximation of which in various cases has been discussed by many authors. Since we will consider the effect of Debye shielding on the Coulomb fields of the ions, and the atomic fields are also shielded, we cannot use the Sommerfeld result. The Born approximation has been applied to the shielded Coulomb potential by Dewitt (Refs. 2, 3), who discusses its limits of validity quite carefully. We follow him in its use for the faster part of the electron velocity distribution in electron-ion scattering. For the slower electrons we use a classical impulse approximation with the shielded potential; this has been used for the pure Coulomb potential by, for example, Roberts (Ref. 4). Bremsstrahlung from neutrals is here treated by matching a Thomas- Fermi potential roughly to a screened Coulomb form, and using the Born approximation for the faster electrons, neglecting the slower ones. This result compares reasonably with the "effective Z" obtained by Breen and Nardone (Ref. 5) for freefree absorption by oxygen atoms, using machine wave functions. Finally, the idea of using detailed balance to obtain the free-free absorption coefficient from the Bremsstrahlung cross section is hardly original (see, e. g., Reference 3); only the use of our cross sections and the detailed evaluation is new.

THE UNIVERSITY OF MICHIGAN 2764-3-T II. Electron-Ion Bremsstrahlung from the Faster Electrons Since the potential here, because of Debye screening, will depend on electron density ne and temperature T, we must pick specific values of these parameters. Such values, fairly realistic for an aerodynamic plasma of possible practical interest, are 13 3 ne = 1013/cm T = 5 x 103 OK We take, as an example, the plasma to be composed of oxygen atoms, 0 ions, and electrons. The validity of many of the approximations will depend on the choice of radiation frequency investigated. We are most interested in microwave frequencies near the plasma frequency, which is roughly 28 KMc. The computations will therefore be performed at v = 50 KMc They will subsequently be extended to 15, 35, and 125 KMc, to obtain some idea of frequency variation in this region. In this section we refer extensively to the work of Dewitt (Refs. 2, 3), who has applied the Born approximation to the Debye-screened Coulomb potential to treat Bremsstrahlung in a fully-ionized gas. He has investigated the validity of this approximation in detail. Let us refer to incident and scattered quantities Ze2 by the subscripts 1 and 2. It is useful to talk in terms of nl, 2 2 For the pure Coulomb field the Sommerfeld exact Bremsstrahlung cross section may be expanded in powers of n2 - nl; the first term of this expansion is the Born

THE UNIVERSITY OF MICHIGAN 2764-3-T approximation result. Thus for the pure Coulomb field, the Born approximation describes well the situation of low-frequency radiation (v1 - v2) from not-tooslow electrons. That is, n~ >) 1 is permissable as long as n2 - n1 << 1. For a screened potential, although there is no exact solution to compare it with, the Born approximation is shown by Dewitt to be even better for the low-frequency part of the spectrum than for the unscreened one, i. e. it is valid for smaller v1. Since we note that for v1 = T and v = 50 KMc, n1 - 8, and n2 - n1 -'- 10 we shall use it down to this value of v1. For faster electrons, it is of course even better, but we restrict ourselves to non-relativistic electrons, naturally. These limits of validity are quite crude, but since the Born approximation usually works better than it should, we shall use them. In what follows we use the following notation: P = momentum in energy units = mvc K = photon energy = hi 2 IL = mc 2 2 r= e /mc 2 c = e /Thc 16 222 2 go = 3 aZ rOm /Pi - 3 zor/P1 7= -hc/X X, the Debye length = T e written

THE UNIVERSITY OF MICHIGAN 2764-3-T 2 (- P1 P22)2 ~daer) = 0 (KP1) - n (p_2)2+2 pl +p )2+72] [(p- lp2)2+Ty 2] (2. 1) in which, of course, P2 is to be eliminated by conservation of energy, P1 -2 P 2P. (2. 2) Since the power is obtained by multiplying the cross section by the incident flux and photon energy and ion density ni and integrating over the electron velocity distribution, the contribution from this velocity range to the power/unit volume/ circular frequency interval may be written, where Z = 1 for the ions always, so n =n,. e 1 216 26 4 13/2 6 -P6/2kT P dw = n Z e 45 d e 3 In C 2 7kTm 2 (P1+ P2)2 + 2 (P-P2)2 + 2 ] i ~7lin'(p~_ p2)~ + ~ ~p~+p2)2+~2] [(p_ p2)2+2] P (2.3) It is not possible to introduce approximations to the bracketed term valid over the whole range of integration. P1 >'y is always valid, and thus (P1+P2)> > y. However, P1-P2 = Y at about Pic = 15c 4kTm, and since P1-P2 - Iu/P1, P1-P2 < y for greater P1. Since most of the contribution to this integral comes from P1 < PiCe, a fair approximation to the bracket is, if one is required, (2P1)\ 2p ln\ -~j 2(I~)2 Now for the very slowest electrons with P1 > 2Kg, the microwave radiation is the high-frequency limit of their spectrum. The Born approximation 6

THE UNIVERSITY OF MICHIGAN 2764-3-T fails here for the pure Coulomb field, but the Sommerfeld solution is reproduced excellently when the Born approximation is modified by the Elwert factor. Although its use cannot clearly be justified for the screened potential, we might hope the screening is weak enough to approximate the Coulomb case, and that it would be correct order-of-magnitude. The expression is dK P1 1 - e-2n dcB (K, P) = K B-E K P2 1 -2rn2 (2. 4) (P1+ P2)2 + y2 2'2 P1 P2 2' 2 2 2 -- (P - 1+P2) +2] [(PI P2+) 2 I The contribution from the very lowest velocities is described by the limit here P2/P1 O, P1 > > y, or dcrB (K, P1) 2 dK (2. 5) B-E 0 K (2.5) which differs from the Kramers' result in having a factor of 2 rather than 4These slowest electrons then contribute (again Z = 1 for 0 ). 2 32 2 6 4?r 3/2 -P 2/2pkT (2. 6) Pdw = n 3 Z e (n45 2 T dw Pd Pe This is valid for only a very small range, since for P1 = even 2c 22mK, P2/P1 = I3/2, which is certainly not the high-frequency limit. We would like now a cross section valid for electron velocities between the very lowest and the thermal range. The classical impulse approximation will

THE UNIVERSITY OF MICHIGAN 2764-3-T be used to furnish such a result. (In general, we expect that when the incident particle becomes too slow for a quantum-mechanical description of the scattering by the Born approximation, a classical description, in which the particle is regarded as having a definite orbit and is continuously subject to scattering forces, becomes increasingly valid.)

THE UNIVERSITY OF MICHIGAN 2764-3-T III. The Impulse Approximation Ideally, an exact classical trajectory treatment should be used to fill in the gap here. However, this cannot be carried out, ard we are forced to resort to the impulse approximation, which has been used elsewhere (Ref. 4) for Bremsstrahlung in a pure Coulomb field. Since a screened field causes less acceleration at large distances, the impulse approximation should be better for a given electron velocity here than for the Coulomb case. Now in a classical treatment we obtain a particle trajectory which is a function of the impact parameter s and the initial velocity v. For fixed values of these, the radiated power from one electron in dw is (Ref. 6) 8r e2 2 Ps, 8 e I a()1 dw, (3.1) where a(w) is the Fourier transform of the vector acceleration, and Ia(w)I = [ax(w2 + [ay(w)] 2. This is then multiplied by nidn(vo)vo and averaged over annuli of radius s. Finally, of course, we average over that part of the electron velocity spectrum for which the expression is valid. The impulse approximation consists in taking the acceleration which would be associated with an undeviated straight-line trajectory, i. e. acceleration but not displacement results from the presence of the scattering center, which is like the effect of an impulse. The geometry for the calculation is given by the following sketch:

THE UNIVERSITY OF MICHIGAN 2764-3-T -e v t _ r___Lx + Ze Then 2 2 2 2 r =s +v t x -vt y = s 0 0 1 2 e e2W a = -- VV(r) V(r) = -Ze =Ze m r 2 a z - W a zq (3.2) x m r ar y m r ar and, where the bar here indicates a Fourier transform on a, or a(w), Ze2v v it.a _ _________9_0 - e —W dt. (3.3) Since g(r) = r is even in t, we have Ze2 vo i \.,. dt a-x Vrm 0 t sin wt g(r) - e2 dt(3.4) a = cos wt g(r) r y'rm r Consider the integral in a, which we will call I (w). We change the integration variable to r from (3. 2), and integrate once by parts, obtaining cos w~ r2 e-r/X S o (3.5) O vr2 s 10

THE UNIVERSITY OF MICHIGAN 2764-3-T and with r = s cosh t, we have finally o (W) 3 v3 c sinh t e/coh dt Ko + (3.6) in which K is a modified Hankel function. Then i Ze20 w 2 SW 2 a = i Ko + [() ) (3.7) ~x 70mv2 o But aa 2 Ze s ___ IZ s1(w), (3. 8) _Y= ZeI[- (x) dx + g(r) dt g (3.9)?T m r 0 0 Now consider J(w) = K I(x) d =x xK +b2 dx, (3.10) _u 0'where S S a — bxk Vo 1 b2-3O Sa \ bw u K (u) du 0o a a K (a) a- 2+b2 K1 2+b22) (3.11) 11

THE UNIVERSITY OF MICHIGAN 2764-3-T We now need only F = g(r) dt =g(r r V 22 0 r v0 r2_s2which with the substitution r = s cosh t becomes -a cosh t 1 1Fs2 e o[ cosh2:t cosh t dt. (3.12) Consider the integral here, which is a function of a only; call it f(a). Then OD -a cosh t f'(a) = -a c he dt = -K a (a) (3.13) 0 so that a f(a) =- x K (x) dx- lim xKl(x) = aKl(a) (3.14) 0 x- OD and aKl(a) F v s2 (3.15) Then Y 7rmvZ 2co2 F2 2 a=Ze 2 /(y)+ (- ) K1 s ) ( ) j (3.16) a -- ZeV2- K s +( (3. 17) It should be noted that for (no shielding), these agree wih the It should be noted that for X & oo (no shielding), these agree with the corresponding quantities deduced by Roberts (Ref. 4). Then we have for the quantity J (u)j2, which we may call A2, 12

THE UNIVERSITY OF MICHIGAN 2764-3-T 2 2 [t2 2 W( 2gs A2 C 2K2(s) - 2 K (&s)1 (3.18) o with 2 C Ze 2 1 )2+( w )2 ir mvo vo Next we want to average over impact parameter s. We have no improvement to suggest over the usual procedure of taking the lower limit at Smin = DeBroglie' the distance within which the electron cannot be localized, so that it makes no sense to talk about closer approaches to the nucleus. Then we must evaluate the integral 00 2 =B =2r s ds C [Ki ( s) -w K s) 2 uc v o = 2r C + K (t) dt - t K (t) dt (3.19) in which we have used the symbol e for XDeBroglie to eliminate confusion with the screening radius, and x = & e. Let the first of these integrals be designated F(x), the second G(x). Now an integration by parts shows that F(x) = x K(x) Kl(x) - G(x), (3. 20) so we can concentrate on the second integral. As may be verified by differentiation, this is simply 13

THE UNIVERSITY OF MICHIGAN 2764-3-T x 2 2 x G(x) =2 [K,(x) - K (x)] = 2 [K2 (x)- (x)]. 2 0~J 2 o 2 2 (3.21) Then B =27r KC 2 9EK($ ) Kl($- 2 ic K_ (E) —K(s-2 _2 2 o( 22 2 o 2 2 (3.22) and d!~ (to) 8a2 16 Z2 6 dPo(w) 8r e ni Vo dne (vo) B 3 2 3 ni dne (vo) -w 3 c3 m v 1 (3.23) 72 1 owe 7K 717KO() Kl(7) - 2 K() - K K( K(r1) 2 (Vo o in which 2 2 r7 = I(- - > and E () + ( 2 and e= l)eBroglie In v In order to apply this result, we need a criterion for the validity of the classical description. This can be obtained in rough form by following Bohm (Ref. 7). We require that the size of a wave packet representing the electron be S the impact parameter, and that the momentum uncertainty involved in forming this packet be much smaller than that transferred during the collision. The impulse approximation should be better for the shielded than the pure Coulomb potential, so we will combine it with the criterion of Bohm, which is 14

THE UNIVERSITY OF MICHIGAN 2764-3-T 2s2 dx F(r) dx >> 1 (3. 24) -0O Inserting the impulse approximation for the trajectory into (3. 24) leads to an integral previously evaluated, and yields (Z = 1 for us) 4 Ze2 s s K1 ()> > 1 (3. 25) Thus, a limiting impact parameter is determined as a function of n1 or v1. For Vthreshold - (3. 25) is satisfied out to s/X= 5 or 6, which should kT / i include most of the effect of the potential, while for v =, s/X - 2. 5 is the limit. However, computation in both cases shows that integrating out to s = oo is justified because of the rapid decrease of the K functions. Incidentally, another requirement that the classical description be valid for this potential is that the relative variation of the potential over the size of the equivalent wave packet be small. That is, av'X. < 1,(3. 26) uDeBroglie V, (3. 26) or mvo. (3.27) Since r >/ s > XDeBroglie always in this description, and DeBrgie. 01 this will be satisfactory in general. The contribution from the range _K C v 4 should then be given by 15

THE UNIVERSITY OF MICHIGAN 2764-3-T FkT,/. Pdw 16 2 Z2e6dw PdPe P dw -3 n2 4 eda, 4?r (m)3/2 PdPe 3 e m4 C5 2 \ kT} + (/ w C)2 K o-) K1) - 1 [ 2] PC) - K KJ. (3.28) It may be noted that the ratio of the impulse approximation to the Kramers' result at the lower limit is approximately 1/7. At the upper limit, the agreement with the Born approximation result is much better, the ratio being 0. 83. Since the faster electron is deviated less from its trajectory, the superior agreement at the upper end may be interpreted as resulting from better validity of the impulse approximation, and is thus in agreement with expectation. It may also be noted that Dewitt gives a "classical low-frequency" expression (eq. 18 of Ref. 3) valid for weak shielding. This is applicable to only a very narrow velocity range, just about, for our w. At that limit, the ratio of the impulse cross section to his result is 1. 31, which is quite reasonable agreement. 16

THE UNIVERSITY OF MICHIGAN 2764-3-T IV. Electron-Neutral Bremsstrahlung As stated in the introduction, we base our calculations here on a screening radius derived from the Thomas-Fermi atom. While this is admittedly not very good for Z as small as 8, it is hoped that the model is still more physical than that of Nedelsky's (Ref. 8) frequently-quoted paper, which uses the potential Ze2 Ze2 V(r) r a a r (4.1) -O,r > a and must determine Z and a by recourse to experiment. We therefore take for the potential seen by an incident electron Z2 V(r) = - e (r) (4.2) r with 0 a solution of 1/2 d20 = 3/2 (4.3) dx2 0.885 ao 2 2 in which x = r/b, b = ~1/3. and ao is the Bohr radius, h /me 0(x) has been tabulated by Bush and Caldwell (Ref. 9). It may be fitted quite well by er/X, with X = 1. 33 b = 3. 13 x 109 cm for 0. We give here a plot -b x/X 1 of V(x) =- e and -p(x). X X Again, the Born approximation cross section may be used for P > clnkT, but different approximations are permitted with the stronger screening here. We write, then, for this part of the electron velocity distribution,

THE UNIVERSITY OF MICHIGAN 2764-3-T 2 2 P1 + P2-'-2P1 andPi + P2 - 2P, and P1- P2 2 ( 4.4) P1 + P2 P1 so that 16 6 2 47 m 32O -P2/2,/kT _Pdw = rnnn e 16 6 e2:4~~ 2m)/ c 4mnkT i (1 4P2) 2P2 2 _y2 -2 1p2/,y2 ) (4.5) { (2(1+4P2 Although, because the strong screening makes this case much different from the pure Coulomb field, we cannot justify the Born-Elwert result, we may use it to get some indication of the contribution of the slowest electrons. We need now the limit P2/P1 > 0, P1 < < 7, and find da (K) = 2 adK (W )4 (4.6) o K y which because of the shielding is quite a small result compared to the contribution from the thermal range. The classical impulse approximation is found not to be valid for neutrals, and no method has been found for treating the contribution of the intermediate part of the electron velocity distribution. However, there is no physical reason to expect any special phenomena to characterize this range, so that we still expect the contribution to the total power to be given essentially by (4. 5); the power rising with velocity from that given by (4. 6) to this value.

THE UNIVERSITY OF MICHIGAN 2764-3-T 0(x)/x e /x (xT = x/1. 33) 100.1 - ~~~~~~~~~~~~~~~~~~~~~~~~~.01. 0 1 2 3 4 5 7 3x = r/b COMPARISON OF THOMAS-FERMI AND SCREENED COULOMB FORMS OUT TO 7 "ATOMIC RADII" 19X

THE UNIVERSITY OF MICHIGAN 2764-3-T In fact, the extreme screening limit P1 - P2 < <y of the Born approximation holds up to velocities of the order of -. Since in this limit the Born approximation has the same form, o dK 4(P/y)4, as the Born-Elwert o K limit, differing only by the factor of 2, it seems plausible that a correct cross section in this velocity region - < v < - might have this form. In this case the ratio of power/cm in equal frequency ranges from thermal electrons to that from slow ones is essentially 2 mvth 4 Ptherm _ 2kT Vth V >> 1 (4. 7) slow Vsl so the slowest electrons can be neglected with respect to those at v= - Likewise, the high-energy tail P1 ^ y contributes little, and in fact, the integrand of (4. 5) peaks near vM = A, dropping off faster for v > vM than for v ( vM, but such that (4. 5) does give the essential contribution. 20

THE UNIVERSITY OF MICHIGAN 2764-3-T V. Evaluation of Results, Extension to Other Frequencies The integration indicated in (4. 5) for the Bremsstrahlung power from neutrals may be carried out, leading to a closed form. Consider In (4p2 (1 2 I a -P /2 MkT 21 + 2P I = \ PdPe -1- In (1 + 4 P ) - 72(i + 4p2/P 2 )' ~,ukT (5.1) With y = P2/2ptkT, 3 = 8/kT/y2, we find J1 2,kT j e iln (l + 3y) 1 + } dy (5. 2) and two integrations by parts yield finally I1,= 2 kT e1/ [ In(1+ 13/2) - e (1+13) Ei( — (5. 3) Here E i is the exponential integral, defined by G oa -t -Ei(-x) = _ t dt (5.4) x Numerically, -22 2 I1 = 4. 65 x 10 erg (5.5) and (4. 5) becomes, for the power/cm in dw from neutrals at 50 KMc, 16 6 2 4r -22 3 Pdw = nn- e z x4.6510 dwergs/cm sec _n e 3 M4C-3 -2kT3 nn dw 4.45 x 1029 ergs/sec cm, n being given in cm, (5.6) n n and dw in sec. Z = 8 has been taken. 21

THE UNIVERSITY OF MICHIGAN 2764-3-T For the ion Bremsstrahlung, the higher velocity electrons yielded the power expression (with n. = n ) Pdw =2 16 2 6 4 (r Tn 3/2 Pd- ne 3 m4c5 27r kT ) (5.7) 2 = k 1/2 x l -p2/2ekT,. 31 Px10 d2erg/cm se. (5. The integral i appearinrg here may alse be evaluated in close d form, and yields, with =(4kT/K) and e for22 2 2 (5.8) so that, with Z = 1 of course, and n =10t 3/c3, Pdw = 4.5x 10 dwerg/cm sec. (5.9) The integral arising from the impulse approximation must be evaluated numlerically. We find for it I o. 24 x 10- 10 erg (5.10) so that the slower electrons contribute to the ion Bremsstrahlung dr ---- 1.83 x 101 dr erg/cm3 sec (5.11) Let us now try to extend these results to a few other frequencies.

THE UNIVERSITY OF MICHIGAN 2764-3-T integrals, I1, I2 and I3 is based, as well as those entering directly into the forms of their integrands. First of all, it seems reasonable to take the Born approximation as valid for n2-n1 - 10. Then, if we use the same lower limit on P in I,, and I2 as before, we can use the Born approximation for ions and neutrals at frequencies such as 15 KMc and 35 KMc. Of course it is fine for higher frequencies, such as 125 KMc. The criterion 3. 25 for validity of a classical description (in the impulse approximation), is better satisfied at the low velocity end for lower frequencies, since the lowest v~ /'2h v /m, and we have the 1/v variation in 3. 25. For 125 KMc, it is still satisfied well enough. The variation of potential argument 3. 27 is likewise still reasonably good. Then the Born and impulse approximations will be used in the same velocity ranges as previously, for convenience in computation. (It is clear that new ranges of validity could be determined if desired, corresponding to the choices of v. ) Now the changes in form of the integrals must be investigated. The impulse approximation integral 13 is explicitly frequency-dependent and need simply be recomputed for the new values of w. The manner in which the Born approximation has been approximated must, however, be examined. We have the form.1 (P1+P2)2+y2 2'2 P1 P2 ur OC. - l(.n12) 2 (P2)2 + [(P1+P2)2 + 2] [(P1P2)2 + 2 ] (5. 12) 23

THE UNIVERSITY OF MICHIGAN 2764-3-T to approximate. Here 2 2 2 2 2 P1 -P2 = 2Ku, P2 = P1 -2K, (P + P2) = 2P1 -2K 2 2 22 + 2P14 (1 -' )' 4P1 - 4Ku - — 2 p4 p2 and 2 K 22 (P1 - P2)2 Pi h_ c 4.67 x 10 2 For ions, p < x10 4. 4 x 10 2 P1 xc FmkT P1 at 125 KMc and the same P = P in' varying as v /2 Inserting the expansions into (5. 11), we have 1 4P1 -4K -u — + 2 2 +P12(1 2_ K2 2 l 22+2 [4P1 -4Kp[ K+22 +7 1] 1 - p2 2 2 2 4 1 Pn 2 + k P1 4 P1 [ + 4 1( )2+I (- p )2 1- KM 2 ( +( +(KP)) Pi4 p, (5.14) 24

THE UNIVERSITY OF MICHIGAN 2764-3-T In view of the magnitudes of the parameters, as given, it is required to use a more detailed expansion than before, namely 1 4P4 1 1 1 o. aC In (5.15) 2 (K )2+ 2 2 2 2 which reduces only at higher frequencies and lower P1 values to the old form, which was 2n 2 a oc in ( K ) - - K2 )K (5.16) The integral I2 must be recomputed with this form for the new frequencies. Now let us look at I1, the Born approximation integral for the neutrals. Now y is very large, because of the severe screening. We have (-)2 —, 5. 5 x 10 at the lowest P1 / < 6.5x 10, the value at v =125 KMc. This leads to 1 I 4P 4P y2 [ln ( 72 1+ 4P2/72 (5.17) However, more care must be exercised before accepting this. Since P( is small, the bracketed quantity may be expanded as 25

THE UNIVERSITY OF MICHIGAN 2764-3-T 4P P 4 4 4p2 4p2 1 4 -"y2 (1- 8 Y2 - 7- (1 -'~ ) - -y8(2 (5. 18) so that lowest-order terms vanish, leaving those of order (P/y). Then we must be sure that terms in KM/'y have not been neglected; they are of the same order. We then write 1 (4P1 -4K+ 2) 2y 2) cr7 in - 2 2 2[2 ] (5.19) I 1 +24( )2 -45.20) n 1+ 4( ) 4 2 4( —- 4(Y)4+ 4 [(- -_ 4 _4 8-) - 4( 1+ 4 16 16p (5. 21) P1 4', as in (5. 18). Thus, the terms in K_/'y cancel when those leading to them are kept, and by luck we may use the expansion (5. 17), as previously. Therefore, 26

THE UNIVERSITY OF MICHIGAN 2764-3-T The results of the re-computations are: v = 15 KMc 35 KMc 50 KMc 125 KMc I2 3. 04x1018 3. 02x10 18 3. 00x1018 2. 81x1018 I3 1.48x10-18 1.33x10-18 1.25x10-18 1.05x10-18 Incidentally, the forms previously given for the contribution of the slowest electrons (the Born-Elwert limit) remain unaltered for both ion and electron Bremsstrahlung. Thus, finally, we conclude that with our approximations the ion Bremsstrahlung power varies only very slowly with frequency, and that from neutrals is frequency-independent, in the microwave range covered here. 27

THE UNIVERSITY OF MICHIGAN 2764-3-T VI. Use of Detailed Balance to Obtain Free-Free Absorption Coefficients from Bremsstrahlung Cross Sections In order to assess the importance of Bremsstrahlung at frequencies above the plasma frequencies, a knowledge of the free-free absorption coefficient is required, since radiation is absorbed in the plasma by this process. Now in Bremsstrahlung an electron of momentum P1 (in Heitler' s units) releases a photon of energy K (in range dK), and itself ends up with 2 2 2 momentum P2 (range /I/P2dK) such that P1 - P2 = 2Kg, where t = me. In free-free absorption, we can consider an electron of momentum P2 to absorb a photon of energy K to K + dK, so the electron ends up with P1 to P1 + P/P1 dK. Thus these two processes are inverse to each other. We have obtained Bremsstrahlung cross sections previously and want the free-free absorption coefficients. The idea immediately presents itself to use the detailed balance theorem of statistical mechanics to relate these two quantities. The detailed balance theorem requires that in equilibrium the probability of a transition from state 1 - state 2 of a closed system is equal to that from state 2 - state 1. (Note that this is detailed balance, a stronger statement than constancy of population of a state; the accounts balance between all pairs of states, not only in total income and outgo of population of each state. ) Then, basically, the detailed balance principle requires that 28

THE UNIVERSITY OF MICHIGAN 2764-3-T probability of Bremsstrahlung with P1 —(P2 P2, dK), photon with (K, K+ dK) emitted P2 number of electron states in /P dK at P2 x number of photon states in dK at K probability of free-free absorption of photon (K, K+dK), electron has P2- (P1,P + dK) P1 number of electron states in P- dK at P1 P1 (6.1) The denominators arise from the assumption that all degenerate states are equally probable within an energy level. The degeneracy exists because electron energy is a function of IP, not P, and is taken independent of spin, while photon direction of propagation and polarization do not affect the photon energy. Then, since the probability of entering any state varies inversely with the number of co-degenerate ones available to be entered, these numbers appear in the denominator. The quantities L/P dK are the ranges dP corresponding to 2 2 dK, obtained from conservation of energy, P1 - P 2 = 2K. Let us consider all process to take place in a volume V, which will cancel as it should in the final result. Then 8rp2 Number of electron states at P1 in dP1 = dK is dKV (6.2) P1 (hc)3 P1 Number of electron states at P2 in dP2 = 2 dK is dK V (6.3) (hc)3 P2 87r K2 dK Number of photon states at K in dK is.(hc) v (6. 4) 29

THE UNIVERSITY OF MICHIGAN 2764-3-T Next we want to relate the probabilities involved to the cross section of interest. There is a general relation between these two descriptions of a process. For consider a volume V in which there is a volume density nt of target systems, capable of making the transition 1- 2 when struck by another type of particle, the latter being present in density nR. Let o(l, 2) be the cross section for the transition, and P(1, 2) be the probability/unit time of the process. By the definition of cross section, we say _ events/unit time/target system c(l, 2) = incident particles/unit area/unit time Number of target systems in V P(1, 2) Number of target systems in V (6. 6) nR VR P(1, 2) - ~~~~~N R ~~~/Vv R(6 7) where there are N incident particles in V So R P(1, 2) = (6. 8) Finally, chose V such that NR or V = (n) (all results willbe independent of V in the end). Then P(1, 2) = vR (1l, 2) / V, which is the usual relation. 30

THE UNIVERSITY OF MICHIGAN 2764-3-T Then, for the Bremsstrahlung process P1 — P2, v lBr (P1, K) dK P( P(1- P2 ) (6.9) (the dK occurs because o(K, P1) obtained previously is a "cross section density" such that o(K, P1) dK is really the cross section for emission by an electron with P1 of a Bremsstrahlung photon in K to K + dK). But in the case of Bremsstrahlung it is the electrons which are the target systems, v1 still being the relative velocity, while the ions are the bombarding particles. So we must choose V = 1/n. as the volume within which our conceptual processes occur, and P(P1 -P2) = v ni B (K, P1) dK. (6. 10) For the free-free transitions, correspondingly, c cff (K,P2)dK P(P P1) (6. 11) Inserting all of these relations into the statement of detailed balance, we obtain B (KlPl) dKnivl cC 9ff(P2, K) dK 2-~P (6.12) V 2 dK V 8w K dK V x V dK V(hc)3 P2 V (hc)33 P whence (p(K, P=) ni v1(hc)3 PI (Pz, K)dK 2 () (6. 13) ff 8r8K c P2 31

THE UNIVERSITY OF MICHIGAN 2764-3-T in which gff(P2, K) dK is, as with Bremsstrahlung, to be interpreted as the absorption cross section. (We will use the symbol a for true absorption cross section from now on, dropping the dK). We may write gB(K, P1) = K-1 F(K, P1,X) for those cases P1 in which the Born approximation is valid, in which F is the quantity which appeared in brackets previously and was approximated in various ways. Then, with y =16/3 2 a r as before, o o.ff(P2,W) = - (_ e) (2T) F(K,P1,). (6.14)'f P2 ( We may multiply by the electron velocity distribution dn (P2) and integrate over P2 to obtain the absorption coefficient 00 -H)2/,92 OD -2 P2dP2e ac (w) = n n. 2r2 F(w,P, P2) e 1 \3fif? 22 cu rO2 C3 S, P d~a F(t, P1,1 2) (6. 15) where s = 2 kT. When the Born approximation is valid, F 2 (l-p2 2 2 +22] ~p2 (Pand in the tegration, P2)1 is to be expressed as a function of+ P2 by conservation of energy, P1 = P2 + 2,u h 32

THE UNIVERSITY OF MICHIGAN 2764-3-T This is precisely the result obtained by Dewitt in Reference 3; thus this conclusion is not original. However, it was felt desirable for the sake of clarity to include a detailed derivation of (6. 15) in this report. In the case of electron-neutral Bremsstrahlung and absorption, it is clear that the only change in the formulation is to replace ni by n, the density of neutrals. When the impulse approximation is used for electron-ion Bremsstrahlung, the quantity F is replaced by F = i ( )2 + (Jure )2 K (-)K()- - ) + 2(Pc )2 K2( 2 (6.17) which occurred in (3. 28). For practical purposes, we must again determine the range in P2 of validity of various approximations, for a given choice of K or w. For the neutrals, we find that the Born approximation which was used before for 2 2 P1 > pukT, correspondingly is used for P2 > MukT - 2Kwl. It may be written -~ 1 (p 2 + +2Kuj + y 1 2P2 _P22+ 2KA 2 22 2 2 2 F = -i- ln- + 2 \p,2 +2Kp - P2 +tY P2+ ) + y2 P2 )+ 2] (6.18) and again, in this range, approximated as 2 2 I [n (1 + 2 r) 22 2 2] (6..9)3 33

THE UNIVERSITY OF MICHIGAN 2764-3-T so that for the contribution of this part of the electron velocity spectrum to the absorption coefficient from neutrals we have a!() = (16 2a 22) ( c 3 4 1 3/2 1 (w) = n r -27 r MkT n( e3 r0 T( 2 /kT x kT ~~~~~~~1 Tl,7 ~2 kT 8 4.tkT 2 e ln (1 -T- LkT )_1 - e (1+ ) i - Eie(l+s28 jT 2 kT 8pLkT (6. 20) We note that, since Tiw/kT << 1 and -fw 4 7Y /8/u for the microwave frequencies we consider, the quantity in the large bracket is essentially frequency-independent and simplifies to the form used previously, and we do obtain in this range a Kramers-type absorption coefficient, - 1/3 The contribution of the lower part of the velocity spectrum would appear to be much smaller; in any case neglecting it yields a lower bound on a (w), which is of practical importance. Now if Kramers' law really held for all P2, we would have F = Ir/-3. We may define an effective Z by equating our a (o) with Z = 8 to that obtained from Kramers' law, in which the "effective Z" is assumed to appear. We had, evaluating (6. 20) 34

THE UNIVERSITY OF MICHIGAN 2764-3-T 2 16 2 2 c 3 4 4. 65 x10 22 (6.21) a'(W) nn Z -a r (6.21) n e 3 o (2kT)3/2 in which the numerical factor arising from the integration, containing /kT, has been explicitly evaluated rather than left as in (6. 20). Now Kramers' law yields oD 2 a)2 16 ( 2 2 c3A3 4 1 S PdP2/2y e kT ne 3 o f (2 3kT)3/2 PdPe (6. 22) Equating these two, regarding the Z in Kramers' law as Z effective, we obtain Z effective = 0.17. (6. 23) This compares reasonably with the result of Breen and Nardone, Z -,. 31 at 10, 000 R and 15, 000 i, for T = 80000K. The higher frequency radiation they consider should result from electrons which see a greater Z. Incidentally, the reasonable agreement of the effective Z with that of previous investigators may be regarded as a practical justification of the fitting of the potential for small r only. Although a formal analysis of the effect of underestimating the potential for r >7b, roughly, is difficult, it would appear that the potential has dropped off sufficiently by this radius so that, with the given velocity distribution, there are simply not enough scatterings giving rise to microwave Bremsstrahlung from large r to affect the power significantly. That is, since our effective Z differs from theirs by a factor of roughly 2, but should differ by some factor presumedly between 1 and 2, it would seem that a large error has not been made. 35

THE UNIVERSITY OF MICHIGAN 2764-3-T Now, for the ion contribution to free-free absorption, let us first look at the Born-approximation part. Here, where again P2 runs from ~ukT- 2Ki. to a, we find that F= ln(K)2p 2 +K22/P2 is the correct approximation, and a new integration is required, the integral being a function of frequency. For the classical part, since the function corresponding to F contains only a single momentum (classical scattering being a continuous process rather than a transition between states), we simply label this momentum P2. Rather than trying to derive a lower lower limit of integration, we note that the integrand drops off rapidly enough so that we ignore any contribution for smaller P. (That is, the integrand has dropped off so much by I2K/t that any extension to P = 0 makes negligible difference. ) Then this result stands unchanged, and must be integrated numerically as before. In view of the fairly smooth match when the Born and impulse approximations were joined at P = I little error is incurred by continuing to join them there, rather than at wukT - 2Kp., which procedure saves a new computation. We then obtain, combining the previous numerical results, for the free-free absorption coefficient of electrons in the field of ions, 4.52 15 kmc 18 4. 35 35 kmc 16 2 2 c 3 4 x 10-18 4. 26 at 50 kmc ae 3 oa r w ( /) ) (2IkT)32 3.86 125 kmc (6.21) 36

THE UNIVERSITY OF MICHIGAN 2764-3-T For that due to electrons in the field of neutrals, we had nn 2 16 a 2 2 (c3 4/ -22 (6.22) or (W) = ne Z 3 r r vT ((2~kT)3/2 X 4 65 X 10 (6.22) n e 3 o w c r (2 tkT)3/2 x4.65x10 for all these frequencies. In both these results, all quantities are in cgs units, a in cm For the low degree of ionization corresponding to T = 5000 K in air, the absorption due to neutrals should dominate strongly that due to ions. While we need not concern ourselves with a specific value of n and n., which would n 1 point too directly to a specific aerodynamic situation, we may comment that this free-free absorption may be extremely severe at the frequencies discussed, for reasonable practical values of n. 37

THE UNIVERSITY OF MICHIGAN 2764-3-T A CKNOWLEDGEMENTS I should like to thank Mr. Otto Ruehr of this laboratory for evaluating the integrals associated with the impulse approximation. Mr. Harold Hunter supervised the numerical computations leading to the results of Sections V and VI.

THE UNIVERSITY OF MICHIGAN 2764-3-T REFERENCES 1. Sommerfeld, A., Atombau und Spektrallinien, V. II, Vieweg, Braunschweig, (1939), Ch. 8. 2. Dewitt, H., "Scattering and Bremsstrahlung Energy Transfer in a Fully Ionized Gas Using the Debye Potential", UCRL, Livermore, California, Theo. PNU-56. 3. Dewitt, H., "The Free-Free Absorption Coefficient in Ionized Gases", UCRL, Livermore, California, UCRL-5377. 4. Roberts, C., "Radio-Frequency Radiation from Hypersonic Plasmas with Impressed Oscillating Electric Fields", Douglas Aircraft Company, Engineering Paper No. 847, 20 November, 1959. 5. Breen, R. and Nardone, M., "The Free-Free Continuum of Oxygen", Space Sciences Laboratory, Missile and Space Vehicle Department, General Electric Company, Philadelphia, Pennsylvania. 6. Landau and Lifshitz, The Classical Theory of Fields, Addison-Wesley, 1951. 7. Bohm, D., Quantum Theory, Prentice-Hall, 1951. 8. Nedelsky, L., "Radiation from Slow Electrons", P. R. 42, 641 (1932). 9. Bush, V., and Caldwell, S., "Thomas-Fermi Equation Solution by the Differential Analyzer, P. R. 38, 1898 (1931) 39

The University of Michigan, Ann Arbor, Michigan Ucasfe The University of Michigan, Ann Arbor, Michigao Unclassified Stdies in Rdr Cross.Sections-XLII Stdies in Rdar Cross Sectioos -XLII On Microwave Bremsatrahlung from a Cool Plasa1.reathuganfeere On Microwave Bremaatrahlong from a Cool Plasma 1. Bremaatrahlung and free-free M. L. Baraach asrto aclto o M. L. Baraach absorption calculation for T = 50000Kt, ne l 1ll /cm3' inT=500K e-10/c3i Report No. 2764-3-T, Auguat 1990, 39 pp- Illus U. S. Army Signal a plasma of electrons, Ot ions, Report No. 2764-3-T, Augusf1969, 39 pp- litus U. S. Army Sigal ad0aosa rqece Reserch nd DvelomentAgeny Cotrac DA 6-03 SC-5041and 0 atoms at frequencies Research and Development Agency Contract DA 36-039 SC-75041 esearh an Deveopmet Agecy Cntrac DA 6-0 1 SC-5041from 15 to 125 KtMc ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report fo 5t 2 ~ ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report 2.S ryMniRsac n 2. U. S. Army Signal Research and Micowae Bemstralun frm, nd reefre aborpionina col, Development Agency Microwave Bremastrahlung from, and free-free absorption in, a cool,.eeometAec Contract DA 39-039 SC-75041, partially-ionized plasma are treated. Electron-ion encounters ARAOrerNe12-0 partially-ionized plasma are treated. Electron-ion encounters are ARPA Order Nr. 120-90, treated by the Born approximation and the classical impulse approx-PrjcCoe70 treated by the Born approximation and the classical impulse approx- rjc oe70 mtoaDbesile oeta en sd rmsrhugfo imation, a Debye-shielded potential being used. Bremsstrahlung from ~electron-neutral collisions is treated by the Born approximation. The electron-neutral collisions is treated by the Born approximation. The potential here is obtained by fitting a shielded Coulomob form to the potential here is obtained by fitting a shielded Coulomb form to the Thomas-Fermi potential for distances less than about 7 atomic radii. Thomas-Fermi potential for distances less than about 7 atomic radii. 0 13 3 For the plasma parameters chosen (T =50 K, n to 13/cm3, Z =8) For the plasma parameters chosen (T = 5000'K, ne = 10 /cm, Z= 8) and microwave frquencies of the order of 50 KRc, it would appear and microwave frequencies of the order of 50 KMc, it would appear that at the correspondingly low degree of ionization, the neutrals are that at the correspondingly low degree of ionication, the neutrals are most significant. An effective Z for the oxygen atoms is determined most significant. An effective Z for the oxygen atoms is determined by matching the free-free absorption to Kramers' law. Its value, by matching the free-free absorption to Kramers' law. Its value, Z =. 17, compares reasonably with the results of previous investigators Z =.17, compares reasonably with the resnits of previous investigators. The University of Michigan, Ann Arbor, Michigan Unclassified The University of Michigan, An Arbor, Mchigan Ucasfe Studies in Radar Cross Sections - XLII ~~~~~~~~~~~~~~~~Studies in Redar Cross Sections - XLII On Microwave Bremsstrahlung from a Cool Plasma 1. Bremsstrahluntg and tree-free On Microwave Bremastrahiung from a Cool Plasma1.Bestaunadfr-re M. L. Barasch absorption calculation for ~~~~~~~~~~M. L. Barasch asrto aclto o T =50000K, ne = 1013/cm3, inT 500KO=11/c3i Report No. 2764-3-T, August 1960, 39 pp- tIlus U. S. Army Signal a plasma of electrons, 0& ions, Report No. 2764-3-T, August 1960, 39 pp- tilus U. S. Army SignalunOaostfrqecs Reserch nd evelpmen Agncy ontact A 36039SC-7041and 0 atoms at frequencies Research and Development Agency Contract DA 36-039 SC-75041 ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report fo 5t 2 ~ RAOdrN.106,PoetCd 70 nlsiidRpr 2. U. S. Army Signal Research and2.US.AmSinlRsacad Development Agency Microwave Bremastrablung from, and free-tr~ee absorption in, a cool, DvlpetAec Micrwav Brmssrahlng rom an fre-fre asortionina col, Contract DA 36-039 SC-75041. partially-ionized plasma 4re treated. Electron-ion encounters arotaceA3609SC701 partially-ionized plasma are treated. Electron-ion encounters a-re ARPA order Nr. 120-60, treated by the Born approximation and the classical impulse approxtreated by the Born approximation and the classical impulse approx- PoetCd 70iain ey-heddptnilbigue.Bestaln r imation, a Debye-shielded potenital being used. Bremastrablung from rjetCde70 imaction-nutaleoye-shiensid ptrential bein used Boreprxmastriong frmPoethoe70 electron-neutral collisions is treated by the Born approximation. Theelectron-neutralicollisions is treated bysthelBornCapproximation. The eeto-etacolsositraebyteBraprxmto.Tepotential here is obtained by fitting a shielded Coulomb form to the potential here is obtained by fitting a shielded Coulomb form to the Thomas-Fermi potential for distances less than about 7 atomic radii. Thomas-Fermi potential for distances less than about 7 atomic radii. coe(T=500 113 3 For the plasma parameters chosen (T = 5000 0K, ne = 1013 /c3, For th plasa paraetersK, n = mm =8 and microwave frequencies of the order of 50 KMc, it would appear 8 and microwave frequencies of the order of 50 KMc, it would appear taathecrspnigylwdreofoizion, the neutrals are that at the correspondingly low degree of ionization, the neutrals a-re mostatsignifcant.ponAngefeciv lo fogrete oxygenizatomi sdtrie most significant. An effective Z for the Oxygen atoms is determined bys matchficngtheA free-free absorpto theoxyrgers law.m its valeem by matching the free-free absorption to Kramers' law. Its value, by mat17,n copaes reasonablyrpwith the results ofapreIous ivalestgtos Z =. 17, compares reasonably with the results of previous investigators. Z=-1,cmae esnbywt h eut fpeiu netgt

The University of Michigan, Ann Arbor, Michigan Unclassified The University of Michigan, Ann Arbor, Michigan TheStUdniversitynof Michigdanr Crossnn Sectirbonsr, Michigan- | |Studies in Radar Cross Sections - XLII Studies in Radar Cross Sections- XLII On Microwave Bremsstrahlung from a Cool Plasma 1. Bremsstralung and free-free On Microwave Bremsstrahlung from a Cool Plasma M. L. Barasch absorption calculation for M. L. Barasch absorption calculation for T =5000~K, ne =1013/cm3, in T=5000K, ne1013/cm3, in Report No. 74-3-T, August 1960, 39 pp- Ilus U. S. Army Signal a plasma of electrons, O+ ions, Report No. 2764-3-T, August 1960, 39 pp- Illus U. S. Army gna a plasma of electrons, O+ ions, Research and Development Agency Contract DA 36-019 SC-75041 Aml and 0 atoms at frequencies Research and Development Agency Contract DA 36-039 SC-75041 andOatoms at frequencies ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report ARPA rderNr. 10-S, PrjectCode 700,Unclssifed Rportfrom 15 to 1ll KMc ARPA Order Nr. 120-SO, Project Code 7700, Unclassified Raport from 15 to 125 KMc 2. U. S. Army Signal Research and Microwave Bremsstrahlung from, and free-free absorption in, a cool, | Development Agency Microwave Bremsstrahlung from, and free-free absorption in, a cool, Development Agency partially-ionized plasma are treated. Electron-ion encounters are Contract DA 36-039 SC-75041, partially-ionized plasma are treated. Electron-ion encounters a Contract DA 36-019 SC-75041, treated by the Born approximation and the classical impulse approx- ARPA Order Nr. 120-60, treated by th Born approximation ad the classical impulse approx- ARPA Order Nr. 120-, imation, a Debye-shielded potentioal being used. Bremsstru lung from | Project Code 7700 imation, a Debye-shielded potential being used. Bremsstrahlung from Project Code 7700 eilcto-eat ionn, a D e aebyte-shieBrapoxelded potential becollisions is treated by the Born approximation. The electron-neutral collisions is treated by the Born approximation. The potential here is obtained by fitting a shielded Coulomb form to the potential here is obtained by fitting a shielded Coulomb form to the potential here is obtaned by fitting a shielded Coulomb form to theThomas-Fermi potential for distances less than about 7 atomic radii. Thomas-Fermi potential for distances less than about 7 atomic radii. For the plasma parametera chosen (T = 10000K n =113/c3,Z=) For the plasma parameters chosen (T = 5000~K, n = 10 13/cm3, Z =) l8 Fortheplsmaparmeerschsen( 500, e = 10/cm, =8)ll and microwave frequencies of the order of 50 KMc, it would appear and microwave frequencies of the order of 50 KMc, it would appear that at the correspondingly low degree of ionization, the neutrals are that at the correspondingly low degree of ionization, the neutrals are t snint A ee e fo the nt is re most significant. An effective Z for the oxygen atoms is determined most significant. An effective Z for the oxygen atoms is determined by matching the free-free absorption to Kramers law. It value, by mnatching the free-free absorption to Kramers' law. Its value, by matching the free-free absorption to Kramers' law. Its value, Z =.17, compares reasonably with the results of previous investigators Z =. 17, compares reasonably with the results of previous investigators. l r The University of Michigan, Ann Arbor, Michigan Unclassified The University of Michigan, Ann Arbor, Michigan Unclassified Studies in Radar Cross Sections-XLII Studies in Radar Cross Sections - XLII On Microwave Bremsstrahlung from a Cool Plasma 1. Bremsstrahlung and free-free On Microwave Bremsstrahlung from a Cool Plasma 1. Bremsstrahlung and free-free M. L. Barasch absorption calculation for M. L. Barasch absorption calculation for T=5000~K, ne = 1013/cm3, in T=5000~K, ne=1013/cm3, in Report No. 2764-3-T, August 1960, 39 pp- Illus U. S. Army Signal a plasma of electrons, ions, Report No. 264-3-T, August 1960, 39 pp- Ilus U. S. Army Signal a plasma of electrons, ions, Research and Development Agency Contract DA 36-039 SC-75041 to Research and DevelO atoms at frequenciy Contract DA 36-039 SC-75041 ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report rom 5 to 25 KMc ARPA Order oNr. 120-60, Project Code 7700, Unclassified Report from 15 to 125 KMc 2. U. S. Army Signal Research and. U. S. Army Signal Research and |Microwave Bremsstrahlung roarfrom, and free-free ab tbsorpti on in, a cool, Development Agency p artially-ionized plasma are treated. Electron-ion encounters are Contract DA 36-039 SC-75041, partially-ionized plasma r treated. Electron-ion encounters C DC ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~partially-ionized plasma are treated. Electron-ion encounters are AContrdeact105 A DA Order9NrC120-SO, treated by the Born approximation and the classical impulse approx- ARPA Order Nr. 120-60, treated by the Born approximation and the classical impulse approx- ARPA Order Nr. 120-60, |imation, a Debye-shielded potential being used. Bremsstrablung from Project Code 7700 imation, a Debye-shielded potential being used. Bremsstrablung from Project Code 7700 electron-neutral collisions is treated by the Born approximation. The electron-neutral collisions is treated by the Born approximation. The potential here is obtained by fitting a shielded Coulomb form to the potential here is obtained by fitting a shielded Coulomb form to the Thomas-Fermi potential for distances less than about 7 atomic radii. Thomas-Fermi potential for distances less than about 7 atomic radii. For the plasma parameters chosen (T =5000 0K, n = 10 13/cm3, Z =) For the plasma parameters chosen (T = 5000 0K, n = 13 /cm, Z =8) P e~~~~~~~~~~ and microwave frequencies of the order of 50 KMce, it would appear and microwave frequencies of the order of 50 KMc, it would appear tand microwave frequencies of the order of 50 KMc, it would appear that at the correspondingly low degree of ionization, the neutrals are that at the correspondingly low degree of ionization, the neutrals are most significant. An effective Z for the oxygen atoms is determined most significant. An effective Z for the oxygen atoms is determined by matching the tree-free absorption to Kramers law. Its value, by matching the free-free absorption to Kramers' law. Its value, Z = 17, compares reasonably with the results of previous investigators. EZ =. 17, compares reasonably with the results of previous investigators.

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