Compton scattering of photons from electrons in thermal (maxwellian) motion electron heating

s, h Affiitive Actios@@@porturdty Empbyes -, — - , - — -— — — s ~ - - - .-— — -— - —.- - - DKXAIMER - - - ~ reportwaspreparedasanaccmartof worksponsoredbyanagencyof the UnitedStatesGovernment Neitherthe UrdtedStatcaCovernrmntnor~y ~ncy t+epflnor anyof Sheir,employees, makesany wiriwntj,exprewor implied,or assumesanylegalliabilityor responsibility forthe accuracy,completeness, or usefulnessof anyinfo~tion, apparatus,product,or processdisclosed,or represe.nta that its w would not Infringeprivz~ely ownedrights Referencehereinto anyspcciflccommercialproduct,proccas,or serviceby tradename,trademark,manufacturer,or otherwise,doesnot neosaarilymrrstituteor implyits endorsement,recommendation, or favori!rg bythe Uniteds.ta~sGovernment or anyagencythereof, The viewsandoptions of authorsexpressedhereindonot necessarilystateor reflectthoseof the United StatssCovernmcntor anyagencythereof , — -—.— - LA-1OO65-MS UC-34 Issued: April 1984 ComptonScattering of Photonsfrom Electrons in Thermal (Maxwellian) Motion:Electron Heating JosephJ.Devaney —@ L& & : .)= xm.;”; !m?g’-+, = —=a m ! ‘–“ , Ty ; - , —- :’=l_., , ,., - ,, TABLE OF CONTENTS PAGE EXECUTIVE SUMMARY 00.0 0.0.0 vii ● ● ABSTRACT **.* * ** I INTRODUCTION II THE EXACT COMPTON SCATTERINGOF A PHOTON FROM A RELATIVISTIC MAXWELL ELECTRON DISTRIBUTION NUMERICALVERIFICATIONOF THE EXACT COMPTON SCATTERINGFROM A MAXWELL DISTRIBUTION 0.0.00 0.0 0.0000 10 THE WIENKE-LATHROPISOTROPICAPPROXIMATIONFOR THE COMPTON SCATTERINGOF A PHOTON FROM A RELATIVISTICMAXWELL ELECTRON DISTRIBUTION 000 00 .0 .00 0.0 11 NUMERICAL VERIFICATIONOF THE WIENKE-LATHROPISOTROPIC APPROXIMATIONOF COMPTON SCATTERING 14 COMPARISONOF THE WIENKE-LATHROPAPPROXIMATIONTO THE EXACT COMPTON SCATTERINGOF PHOTONS FROM A MAXWELLIAN ELECTRON GAS 14 WIENKE-LATHROPFITTED APPROXIMATIONTO COMPTON SCATTERING 14 VIII THE TOTAL COMPTON SCATTERINGCROSS SECTION AT TEMPERATURET 16 ● III ● Iv ● v m VII Ix ● APPLICATIONTO MONTE CARLO (OR OTHER) CODES: THE EXACT EQUATIONS 00 0000 0 0.0 0.0 17 APPLICATIONTO MONTE CARLO (OR OTHER) CODES: THE WIENKE-LATHROP ISOTROPICAPPROXIMATION 0 18 APPLICATION TO MONTE CARLO (OR OTHER) CODES: THE WIENKE-LATHROP FITTED APPROXIMATION 0 00 .00000.0 0.000 0.0000.0 26 THE MEAN SCATTERED PHOTON ENERGY, HEATING: THE EXACT THEORY 26 XIII THE MEAN SCATTERED PHOTON ENERGY, HEATING: THE WIENKE-LATHROP ISOTROPICAPPROXIMATION 00 0 000 29 ● x ● ● ● ● XI ● XII ● ● ● XIV ● ● ● THE MEAN SCATTEREDPHOTON ENERGY, HEATING: THE WIENKE-LATHROP ONE-PARAMETERFITTED APPROXIMATION 31 TABLE OF CONTENTs (coNT) PAGE xv XVI COMPARISONSOF THE MEAN SCATTEREDPHOTON ENERGY AND THE MEAN HEATING AS GIVEN BY THE EXACT, THE ISOTROPICAPPROXIMATION,AND THE FITTED APPROXIMATIONTHEORIES 31 A ScatteredPhoton Energy 31 B Heating 35 RECOMMENDATIONS 35 ACKNOWLEDGMENTS APPENDIX A 39 $-INTEGRATIONOF THE EXACT COMPTON DIFFERENTIAL CROSS SECTION 40 APPENDIX B &INTEGRATION OF THE ISOTROPICAPPROXIMATIONCOMPTON DIFFERENTIALCROSS SECTION 00 00 00 .0 43 REFERENCES 44 ● I , vi ● EXECUTIVE SUMMARY The Compton differentialscatteringof photons from a.relativisticMaxwell Distributionof electrons is reviewed and the theory and numerical values verified for applicationto particle transport codes We checked the Wienke exact covariant theory, the Wienke-Lathropisotropic approximation,and the WienlceLathrop fitted approximation Derivation of the approximationsfrom the exact theory are repeated The IUein-Nishinalimiting form of the equationswas verified Numerical calculations,primarily of limiting cases, were made as were comparisonsboth with Wienke’s calculationsand among the various theories An approximate (Cooper and Cummings), simple, accurate, total cross section as a function of photon energy and electron temperatureis presented Azimuthal integrationof the exact and isotropic cross sections is performed but rejected for practical use because the results are small differencesof large quantitiesand are algebraicallycumbersome The isotropic approximationis good for photons below MeV and temperatures below 100 keV The fitted approximationversion discussed here is generally less accurate but does not require integration,replacing the same with a table or with graphs We recommend that the ordinary Klein-Nishinaformula be used up to electron temperaturesof 10 keV (errors of — < 1.5% in the total cross section and of about 5% or less in the differentialcross section.) For greater accuracies,higher temperatures,or better specific detail and no temperature or photon energy limits, the exact theory is recommended However, the exact theory effectivelyrequires four multiple integrationsso that within its accuracy and temperatureand energy limits the Wienke-Lathropisotropic approximation is a simpler solution and is thereby recommendedas such The mean energy of a photon scattered from a Maxwell distributedelectron gas is calculatedby four methods: exact; the Wienke-Lathropisotropicand one-parameterfitted approximations;and the standard (temperatureT = O) Compton energy equation To about 4% error the simple Compton (T = O) equation is adequate up to 10-keV temperature Above that temperaturethe exact calculation is preferred if it can be efficientlycoded for practical use The isotropic approximationis a suitable compromisebetween simplicityand accuracy, but at the extreme end of the parameter range (T = 100 kev incident vii photon energy v’ = keV, scatteringangle = 180°) the error is as high as -28% For mid-range values like 10 to 25 keV, the errors are generallya percent or so but range up to about 8X (25 keV, 1800) The fitted approximation is generally found to have large errors and is consequentlynot recommended The energy deposited in the electron gas by the Compton scatteringof the photon, i.e., the heating, is only adequatelygiven by the exact expression for all parametersin the ranges —— < T < 100 keV and —— < v’ < 1000 keV For low depositionsthe heating is the differencebetween two large quantities Thus if one quantity is approximate,orders of magnitude errors can occur However, for scattered photon energy v >> T the isotropicapproximationdoes well (0.13% error for v’ = 1000 keV, = 180°, T = 10 keV, v = 790.7 keV; and 2.7% error for v’ = 1000 keV, Cl= 90°, v = 583.5 keV, T = 100 keV) The regular T= O Compton also does well for T —< 10 keV and v >> T (0.7% for v’ = 1000 keV, e = 180°, T = 10 keV, v = 790.7 keV; and 0.08% for v’ = 1000 keV, = 180°, T = keV, v = 795.9 keV) The fitted approximationis without merit for heating viii COMPTON SCATTERINGOF PHOTONS FROM ELECTRONS IN THERMAL (MAXWELLIAN)MOTION: ELECTRON HEATING by Joseph J Devaney ABSTRACT The Compton differentialscatteringof photons from a relativisticMaxwell distributionof electrons is reviewed The exact theory and the approximatetheories due to Wienke and Lathrop were verified for applicationto particle transport codes We find that the ordinary (zero temperature)Klein-Nishinaformula can be used up to electron temperaturesof 10 keV if errors of less than 1.6% in the total cross section and of about 5% or less in the differential cross section can be tolerated Otherwise, for photons below MeV and temperaturesbelow 100 keV the Wienke-Lathropisotropic approximationis recommended Were it not for the four integrationseffectivelyrequired to use the exact theory, it would be recommended An approximate (Cooper and Cummings), simple, accurate, total cross section as a function of photon energy and electron temperatureis presented I INTRODUCTION This report criticallyreviews the exact Compton differentialscattering of a photon from an electron distributedaccording to a relativisticMaxwell velocity distribution We base our study on the form derived by Wienke using field theoreticmethods.1-7 (ParticularlyEq (1) of Ref 1, whose derivation is presented in Ref 2.) Wienke was the first known to this writer to point out the simplicityand power of deriving the Compton effect for moving targets by the coordinatecovariant (i.e., invariant in form) techniques of modern field theory His derivation is equivalent to,’4but replaced, earlier methods8 which involved the tedious and obscure making of a Lorentz transformationto the rest frame of the target electron, applying the KleinNishina Formula, and making a Lorentz transformationback to the laboratory frame We also criticallyreview the Wienke-Lathropisotropicapproximation to the exact formula which selectivelysubstituteselectron averages into the exact formula so obviating integrationover the electron momenta and colatitude The electron directions in a Maxwell distributionare, of course, isotropic,hence the name chosen by Wienke and Lathrop We verify the theory for the exact expressionand the plausibilityarguments for the isotropicexpression We verify in detail numerical comparisons between the two theories at selected electron temperaturesand initial photon energies We rewrite the formulas in a form suitable for application,especially for the Los Alamos National LaboratoryMonte Carlo neutron-photoncode, MCNP.1O As a further approximation,Wienke and Lathrop have reduced the iso9 which tropic approximationto a one- or two-parameterfitted approximation, we also review As always, the choice between the methods is complexityversus accuracy and limitationsof parameter ranges We include a simple, accurate estimate of the total Compton cross section We give the mean scattered photon energy and the mean heating of the electron gas by the photon scattering Both quantitiesare given as a function of the photon scatteringangle, (3;the electron temperature;and the incident photon energy, v’ We compare these means, v> and I-D: as calculatedexactly, as calculatedwith the Wienke-Lathropisotropicand one-parameterfitted approximations,as well as with the unmodified,regular, T = O, Compton energy equation results Recommendationsare offered Because much of this report is devoted to derivationand verification, we recommend that a user-orientedreader turn first to the recommendationsof Section XVI, then for differentialcross sections, Sections IX to XI as desired, which give applicationstogether with referenceto Figs through 8, which show the accuracy of the cross-sectionapproximations For scattered photon energies and heating, refer first to Section xv for comparisonsand errors, particularlyFig 10 and Tables 11 and III, then as desired Sections XIII to XIV Refer to the Table of Contents for further guidance II THE EKACZ (XMPTON SCATTERINGOF A PNOTON FROM A REMTIVISTICMAXWELL ELECTRONDISTRIBUTION We will follow the notation and largely the method of Wienkel>9 and first choose the “natural”system of units in which h = c = and kT ~ T in keV Let the incoming photon and electron energies be v’ and s’, and the outgoing be v and c, with correspondingelectron momenta ~’ and ~, and photon momenta ~’ and ~, respectively The angle between $’ and ~ shall be (3;q is oriented relative to some fixed laboratory direction by azimuthal angle Oq The angle between ~’ and ~’ shall be a’ and that between ~ and $’ shall be a The azimuthal angle between the spherical triangle sides ~’ and T in order to yield fair to good heating numbers ACKNOWLEDGMENTS It is a pleasure to thank John Hendricks, Group X-6, for his interest, encouragement,and patience; Charles Zemach, T-DO, for help in field theory and for verifying the basic exact covariant theory; Bruce Wienke, C-3, for developing the basic theory and then generalizinghis isotropic approximation(with B Lathrop) for developinga fitted approximationand for countless calculations, graphs, and conversationshaving the objective of benefitingMonte Carlo applications;Art Forster, X-6, for suggesting alternativeintegrationmethods; Judi Briesmeister,X-6, and Marge Devaney, C-8, for brief reviews of integration as done in the Los Alamos Central Computing Facility (CCF); Tom Booth for suggestinghow to include the master library integrationsubroutineof TI-59 calculator;Peter Herczeg, T-5, for a lucid explanationof certain 6-function properties;Peter Noerdlinger,X-7, for criticism and suggestions;and Tyce McLarty for an HP15C coding suggestion 39 APPENDIXA ($-INTEGRATION OF TEE EXMX (XMPTONDIFFERENTIALCROSS SECTION We rewrite Eq (34) in the form da ‘~ iiH= –12 dK’ (K’ + m) ● / /K” + 2mK’ ● f(K’) ● da’ I,(Al) ~ where I is the $-integralwhich we shall analyticallyevaluate, (A2) SubstitutingEq (1) into Eq (32) we get - C ‘2=K cos co., - &sin at sin cos (A3) For conveniencedefine (A4) b E - K sin a, sin El (A5) (A6) as before p z cos ‘2 Then = a + b cos @ (A7) , and by Eq (32) v — v’ ‘1 : (1 - p) + K2 ‘1 = c + a + b cos @ ‘1 ‘ a’ + b cos @ ‘ (A8) where we have defined a! ~ c Observe that 40 + a = (v‘ /m) ( - W) + K - G Cos CKt Cos El (A9) I 27T It (functionof cos e) df3= J (functionof cos 13)df3 (A1O) I o SubstitutingEqs (A1O), (A8), (A6), and (33) into Eq (A2), we have n 12= d~ J ● ‘1 (C+K )[ o (1 - p)2 _ 2(1 - p)z + c ‘“K2 + 22 ‘1K2 ‘1 ‘2 ‘2 K2 — C+K (All) 21 Expanding in partial fractions and collecting terms of the same @ dependence gives 11 12= d$ ~[ o ‘1 C+K + ‘2 + (c +K2)2 ‘3 (C + K2)3 -— ‘1 + ‘2 7’ ‘2 K2 (A12) where Al = [ 2(1 ~ P)2 - $+ c [ “ j ‘)2+ 2K1(1 - ‘“)+ K2 c 11 c A2 ~ 2K1(1 - ‘)] C2 (A13) (A14) A3 ~ ‘CK:/(C + K2)3 (A15) B2 = (1 - ~)2/c2 (A16) Defining (A17) we have in Eq (A12) the five integrals 1(1,a’,b), 1(2,a’,b), 1(3,a’,b), 1(1,a,b), and 1(2,a,b) Evaluating these integralswith the help of tables,18 I(l,a’,b)= (A18) n m I(2,a’,b)= I(3,a’,b)= ‘ ‘(’’a’b) = & xa’ 3/2 (a,z - b2) ‘ I(2,a,b)= na (A19) 3/2 (a2 - b2) n(2a’2 + b2) 5/2 2(a’2 - b2) (A20) so that r I = 2X (2a’2 + b2)A3 a’A ‘1 ~+ (a12 -b2)3’2 + 2(a12 -b2)”2 + aB ‘1 - = ‘A21) 3/2 (a2 - b2) ● Eq (A21) together with Eq (Al) is our desired solution Using Eqs (A4) to (A6) to evaluate Eqs (A13) to (A16), we get ‘1=v:J2-~+2v1 (A22) :11 A2 (~ “’K ‘3 ‘2 a 42 = - + + K1)2 (A23) (1 - ~) (A24) = (m/v )2 (A25) - b2 = + (K2 - 1)( 112 + COS2a’ ) - 2K~ Cos a ‘ E (A26) I a’ - b ‘1 = K - = - - + - C n = B & , , , (C + K2) + K2 + C+K 1) r =~ “ 1), r I (with KI + J : I(with K K) = A J , 471K2 47tK2 + K) 43 J K1 K @ “ R — — a — — J= Want Spect — — July P SOC 10 ~, No 1, - A — REFERENCES PP 219, 221 ‘“ *AA?.! w - - - - - I Ibwwannos I ... merit for heating viii COMPTON SCATTERINGOF PHOTONS FROM ELECTRONS IN THERMAL (MAXWELLIAN )MOTION: ELECTRON HEATING by Joseph J Devaney ABSTRACT The Compton differentialscatteringof photons from a... ComptonScattering of Photonsfrom Electrons in Thermal (Maxwellian) Motion: Electron Heating JosephJ.Devaney —@ L& & : .)= xm.;”; !m?g’-+, = —=a m ! ‘–“ , Ty ; - , —- :’=l_., , ,., - ,, TABLE OF. .. COMPARISONOF THE WIENKE-LATHROPAPPROXIMATIONTO THE EXACT COMPTON SCATTERINGOF PHOTONS FROM A MAXWELLIAN ELECTRON GAS 14 WIENKE-LATHROPFITTED APPROXIMATIONTO COMPTON SCATTERING 14 VIII THE TOTAL COMPTON