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

THE EXACT COMPTON SCATTERINGOF A PHOTON FROM A RELATIVISTICMAXWELL ELECTRON DISTRIBUTION .... COMPARISONSOF THE MEAN SCATTERED PHOTON ENERGY AND THE MEAN HEATING AS GIVEN BY THE EXACT, T

s, h Affiitive Actios@@ @porturdty Empbyes

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viewsandoptions of authorsexpressedhereindo not necessarily stateor reflectthoseof the United

StatssCovernmcnt or anyagencythereof.

LA-1OO65-MS UC-34

Issued: April 1984

Compton Scattering of Photonsfrom Electrons

in Thermal (Maxwellian) Motion:Electron Heating

THE EXACT COMPTON SCATTERINGOF A PHOTON FROM A RELATIVISTIC

MAXWELL ELECTRON DISTRIBUTION

NUMERICAL VERIFICATIONOF THE EXACT COMPTON SCATTERING FROM A

MAXWELL DISTRIBUTION 0.0.00 ● 0.0 0 0.0000THE WIENKE-LATHROPISOTROPICAPPROXIMATIONFOR THE COMPTON

SCATTERINGOF A PHOTON FROM A RELATIVISTICMAXWELL ELECTRON

DISTRIBUTION 000 0 ● 00 0.●.0 00 0.0

NUMERICAL VERIFICATIONOF THE WIENKE-LATHROPISOTROPIC

APPROXIMATIONOF COMPTON SCATTERING

COMPARISONOF THE WIENKE-LATHROPAPPROXIMATIONTO THE EXACT

COMPTON SCATTERINGOF PHOTONS FROM A MAXWELLIAN ELECTRON GAS

WIENKE-LATHROPFITTED APPROXIMATIONTO COMPTON SCATTERING

THE TOTAL COMPTON SCATTERING CROSS SECTION AT TEMPERATURET

APPLICATION TO MONTE CARLO (OR OTHER) CODES: THE EXACT

EQUATIONS●00 0 0000 ● 0 0● 0.0 0 0.0●

APPLICATIONTO MONTE CARLO (OR OTHER) CODES: THE WIENKE-LATHROP

ISOTROPICAPPROXIMATION 0 0● ●0 APPLICATION TO MONTE CARLO (OR OTHER) CODES: THE WIENKE-LATHROP

FITTED APPROXIMATION●.0 0 ●.0 00.●.00000.0●0.000 0.0000.0 THE MEAN SCATTERED PHOTON ENERGY, HEATING: THE EXACT THEORY

THE MEAN SCATTERED PHOTON ENERGY, HEATING: THE WIENKE-LATHROP

17

18

2626

29

31

NTENTs (coNT) TABLE OF CO

xv COMPARISONSOF THE MEAN SCATTERED PHOTON ENERGY AND THE MEAN

HEATING AS GIVEN BY THE EXACT, THE ISOTROPIC APPROXIMATION,AND

THE FITTED APPROXIMATIONTHEORIES

A ScatteredPhoton Energy

B Heating

XVI RECOMMENDATIONS

ACKNOWLEDGMENTS

APPENDIX A $-INTEGRATIONOF THE EXACT COMPTON DIFFERENTIAL CROSS SECTION

APPENDIX B &INTEGRATION OF THE ISOTROPICAPPROXIMATIONCOMPTON DIFFERENTIALCROSS SECTION 0 00●0 00 0 00.●.0

REFERENCES

PAGE

31 31 35 35 39

40

43 44

I ,

vi

EXECUTIVE SUMMARY

The Compton differentialscattering of photons from a.relativisticMaxwellDistributionof electrons is reviewed and the theory and numerical values veri-fied for application to particle transport codes We checked the Wienke exactcovariant theory, the Wienke-Lathropisotropic approximation,and the Wienlce-Lathrop fitted approximation Derivation of the approximationsfrom the exacttheory are repeated The IUein-Nishinalimiting form of the equationswasverified Numerical calculations,primarily of limiting cases, were made aswere comparisonsboth with Wienke’s calculationsand among the various theo-ries An approximate (Cooper and Cummings), simple, accurate, total cross -section as a function of photon energy

Azimuthal integrationof the exact and

but rejected for practical use because

large quantities and are algebraically

and electron temperatureis presented.isotropic cross sections is performedthe results are small differencesofcumbersome

The isotropic approximationis good for photons below 1 MeV and tures below 100 keV The fitted approximationversion discussed here is gener-ally less accurate but does not require integration,replacing the same with atable or with graphs We recommend that the ordinary Klein-Nishinaformula beused 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 great-

er accuracies,higher temperatures,or better specific detail and no tempera-ture or photon energy limits, the exact theory is recommended However, theexact theory effectivelyrequires four multiple integrationsso that within itsaccuracy and temperatureand energy limits the Wienke-Lathropisotropic approx-imation is a simpler solution and is thereby recommendedas such

tempera-The mean energy of a photon scattered from a Maxwell distributedelectrongas is calculatedby four methods: exact; the Wienke-Lathropisotropic andone-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 lation is preferred if it can be efficiently coded for practical use Theisotropic approximationis a suitable compromise between simplicity and accu-racy, but at the extreme end of the parameter range (T = 100 kev incident

calcu-vii

photon energy v’ = 1 keV, scatteringangle 0 = 180°) the error is as high as-28% For mid-range values like 10 to 25 keV, the errors are generally a per-cent 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 thephoton, i.e., the heating, is only adequately given by the exact expression forall parameters in the ranges 1 < T < 100 keV and 1 < v’ < 1000 keV.—— —— For lowdepositionsthe 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 isotropic approximationdoes well (0.13%error for v’ = 1000 keV, 0 = 180°, T = 10 keV, v = 790.7 keV; and 2.7% errorfor v’ = 1000 keV, Cl= 90°, v = 583.5 keV, T = 100 keV) The regular T= OCompton 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, 0 = 180°,

T = 1 keV, v = 795.9 keV) The fitted approximationis without merit for

heating

viii

COMPTON SCATTERINGOF PHOTONS FROM ELECTRONS INTHERMAL (MAXWELLIAN) MOTION: ELECTRON HEATING

byJoseph J Devaney

ABSTRACTThe Compton differentialscatteringof photons from arelativisticMaxwell distributionof electrons is

reviewed The exact theory and the approximate theories

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

elec-tron temperaturesof 10 keV if errors of less than 1.6% in

the total cross section and of about 5% or less in the

dif-ferential cross section can be tolerated Otherwise, for

photons below 1 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

This report critically reviews the exact Compton differentialscattering

of a photon from an electron distributed according to a relativisticMaxwellvelocity distribution We base our study on the form derived by Wienke usingfield theoretic methods.1-7 (ParticularlyEq (1) of Ref 1, whose deriva-tion is presented in Ref 2.) Wienke was the first known to this writer topoint out the simplicityand power of deriving the Compton effect for movingtargets by the coordinate covariant (i.e., invariant in form) techniques ofmodern field theory His derivation is equivalent to,’4but replaced, earliermethods8 which involved the tedious and obscure making of a Lorentz

transformationto the rest frame of the target electron, applying the Nishina Formula, and making a Lorentz transformationback to the laboratoryframe.

Klein-We also criticallyreview the Wienke-Lathropisotropic approximation

to the exact formula which selectivelysubstituteselectron averages into theexact formula so obviating integrationover the electron momenta and colati-tude 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 ments for the isotropicexpression We verify in detail numerical comparisonsbetween the two theories at selected electron temperaturesand initial photonenergies We rewrite the formulas in a form suitable for application,espe-cially for the Los Alamos National Laboratory Monte Carlo neutron-photoncode,MCNP.1O

plausibilityargu-As a further approximation,Wienke and Lathrop have reduced the tropic approximationto a one- or two-parameterfitted approximation,9 which

iso-we also review As always, the choice betiso-ween the methods is complexityversusaccuracy and limitationsof parameter ranges

We include a simple, accurate estimate of the total Compton cross tion We give the mean scattered photon energy and the mean heating of theelectron gas by the photon scattering Both quantitiesare given as a function

sec-of the photon scatteringangle, (3;the electron temperature;and the incidentphoton energy, v’

Because much of this report is devoted to derivationand verification,

we recommend that a user-orientedreader turn first to the recommendationsofSection XVI, then for differentialcross sections, Sections IX to XI as

desired, which give applicationstogether with reference to Figs 2 through 8,which show the accuracy of the cross-sectionapproximations For scatteredphoton energies and heating, refer first to Section xv for comparisonsanderrors, particularlyFig 10 and Tables 11 and III, then as desired SectionsXIII to XIV Refer to the Table of Contents for further guidance

2

II THE EKACZ (XMPTON SCATTERINGOF A PNOTON FROMA REMTIVISTIC MAXWELL

ELECTRONDISTRIBUTION

We will follow the notation and largely the method of Wienkel>9 andfirst choose the “natural” system of units in which h = c = 1 and kT ~ T inkeV Let the incoming photon and electron energies be v’ and s’, and theoutgoing be v and c, with correspondingelectron momenta ~’ and ~, and photonmomenta ~’ and ~, respectively The angle between $’ and ~ shall be (3;q isoriented relative to some fixed laboratory direction by azimuthal angle Oq.The angle between ~’ and ~’ shall be a’ and that between ~ and $’ shall be a.The azimuthal angle between the spherical triangle sides ~’ and <’$’ shall be

4 Thus on the surface of a sphere with origin of the vectors ~,~’, and $’ atthe center of the sphere and intersectionslabelled on the surface thereof, wehave the spherical triangle shown in Fig 1

Fig 1 Angular relations

The law of cosines applied to the triangle of Fig 1 gives

In terms of energy, e, and (vector)momentum, ~, the four-vectormomentum is

(2)

Its square is

Now all four vectors are required to transform (Lorentz transformation)alike,

in particularas the line element

so as to keep (ds)2 invariant Thus P2 is invariant

In the rest frame ~ = o, s ~ m, so that generally for any four-momentum

where we have used Eqs (6) and (5):

P2 = P’222= m , Q = Q’2 = O

Expanding the four-momentumproducts into energy and

by, for example,

and then using Fig

+9p ;1 =plq?

1 to determineCos a’,

where p’ and q’ are, of course the

get finally for Eq (8)

three-momentumproducts

(9)

that, for example,

magnitude of the 3-momenta~’ and ~f, we

(lo)

&’v’ - p’v’ cosa’ = c’v - p’v cos a +vv’ - vv’ cos 9 ,

where we have used q’2 = v’2, q2 = V2 for the zero rest mass photons

The Compton collision,of course, also conserves 3-momentum so that

If now we square Eq (12) and use the relation

& = &‘L+VL+V’L + 2p’v’ cos a’ - 2p’v cos a - 2vv’ cos 0 (14)

The Compton cross section for scattering a photon into a direction solid

angle do, and into an energy interval dv, from a relativisticMaxwell electron

distribution,f(~’), is given by Wienkel to be

I

I

dcs r’

[

()

20

)

+ Etv’-ptv’ cos a’ + c’v-p’v cos a

m is the electron rest energy (or mass, c = 1),

and

r E e2/mc2 is the classical electron radius

We integrate over final photon energy in order to remove the

~-function However, the 6-function is not in the form 6(v - Vo), where V

is constant because Eq (14) shows that e = e(v) We must first use the

identity

u-1

x

from which, using Eq (14), then differentiating,then substituting

C+v = e’+v’, and then Eq (11),

E(+)

for some constant outgoing photon energy, Vo

Substitutingin Eq (15), integratingover v, and replacing V by v(i.e., v is now the outgoing photon energy), we get

Formulas (15) and (16) have been checked by independentcalculationby

C Zemach, TheoreticalDivision, Los Alamos National Laboratory.* In the

form of Eqs (19) and (16) the formulas are the same as those of Pauli8 andGinzburg and Syrovat-Skiillprovided one corrects for the electron motion.The number of events per unit time are equal to the flux times the density ofelectrons times the cross section times the volume For an electron of

velocity ~, the number of events per unit time is increased by the factor

T << 4 MeV, more accurately, for T < 400 keV

*Information from C Zemach, November 1982

We generally prefer to describe electrons by their kinetic energy, K’,rather than momentum ~’, so using &’2 = p’2 + m2 and c’ = m + K’ we have

K2Sm ‘1 (&’ - p’ cos a) = K - Gcos a ●

Substitutinginto the energy equation ((11)j we get simply

The expressions (32), (33), (34), (25) (or (23) and (24))> and (1) stitute the exact Compton scattering cross section of a photon of energy v’from a relativisticMaxwell distributionof electrons, f(Kt), into the solidangle d!2= sin f3dO d$q The o-integrationcan be carried out analytically,see Appendix A, but we found the result both too cumbersomeand too inaccuratefor practical use The latter was caused by small differencesof very large,but exact, quantities that our calculator could not handle.

con-In the limit T + O, Eq (34) or Eq (19) should reduce to the Nishina Formula (i.e., Compton for zero velocity electrons) So it does, as wenow show Observe that the relativisticMaxwellian is normalized such that

6(X # o) = o

SubstitutingEq (36) in Eq (19), using Eq (33) and Eqs (29), (30), and(31), (i.e.,K = 1 = K~ = K2), we get the usual Compton energy equation fromEqo (32)

III- NUMER.ICALVERIFICATIONOF THE EWX COMPTONSCATTERINGFROM A MAXWELL

DISTRIBUTION

Integrationover $ in Eq (34) is possible analyticallybut is both

complicatedand leads to small differences in large quantities (see Appendix

A) Accordingly,we have used Simpson’s rule to integrate over $, a’, and K’

to provide dcs/dOvs 0 By symmetry the cross section is independentof $q

We used angular intervals of 22.50° and variable electron kinetic energy

inter-vals appropriateto yielding an error of about 1% or less Our numerical

calculationsagree with Wienke and Lathrop9 to about 1% or better except at

T = 1 keV, v’ = 1000 keV, and 13= 30°, where the agreementwas only 3.5%

because of our using a coarse K’ interval We found and checked with Wienke

that his latest communication(Ref 9, February 15, 1983) has an erroneous

exact curve in Fig 6 (T = 100 keV, v’ = 1 keV); however, we agree with

ear-lier Wienke-Lathropexact calculationsfor these parameters The parameter

sets for which we have numerically checked the Wienke-Lathropexact angular

distributionsare given in Table I

TABLE IP~~R%TS~~I~~E~~~~-~OP~~

FOR du/deWERE VERIFIED

INCIDENTTEMPERATURE PHOTONENERGY

(As noted, agreement is within the error of our Simpson’s rule

approximationerror, i.e., = 1% except, 1, 1OO(),30°: = 3.5%.)

10

Iv. THE wlENKE-LATHROPTSOTROPICAPPROXIMATIONPOR THE COMPTONSCATTERINGOF

A PHOTONFROM A REUTIVISTIC MAXWELLELECTRONDISTRIBUTION

The exact Compton formula (34) requires three integrationsto providethe differentialscattering cross section dci/dQ It requires four integrations

to provide the total scattering cross section (the fifth integrationover @q

is trivial, yielding 2X because of symmetry) Accordingly,an approximateform

of Eq (34) without such integrationscould be quite useful when the errors ofthe approximationcan be tolerated We derive the Wienke-Lathropisotropicapproximationby a plausibilityargument This approximationremoves the inte-gration over p’ and a’ In place of averaging the covariant Compton expressionover a Maxwell spectrum,key parameters are averaged in that expression,aninexact but reasonableapproach leading to a simpler expression

Because the relativisticMaxwell distributionis isotropic,it is clearthat

where the average, < >, is the Maxwell average over K’ and at The first

approximationthen is to replace cos a’ by its average

which he shows and we verify and expand to be

SubstitutingEq (42) into Eq (31) yields

(i.e., except for $) now applies solely to cos a’, where it gives zero, and to

<p’>2, where it yields Eqs (45) and (46), we perform that averaging in

Eq (19) with K taken to be Eq (33) to obtain the isotropic approximationforthe differentialcross section

12

where r ~ e2/mc2is the classical electron radius, K2 is given by Eq (47)s

K by Eq (46), v = cos i3,and v/v’ is given by Eq (48)

Simpson’s rule For angles, only 8 intervals gave sufficientaccuracy for ourpurposes

The approximation(49) also reduces in the limit T + o to the Nishina formula12>10for unpolarized light, as it should In Eq (46) set-ting T

Klein-now no

= O givesK = 1, which in Eq (47) then gives K2 = 1, i.e.,K2 is

longer a function of $, and thus yields from Eq (48)

v NUMERICALVERIFICATIONOF TRE WIENKE-LK1’RROPISOTROPICAPPROXIMATIONOF

COMPTONSCATTERING

We have numericallyverified Eq (51) against the tions of Wienke and Lathrop.9 We find all points in agreement to within ourerror in reading the curves of Wienke and Lathrop and possibly also includingthe difference in our use of more terms for K, Eq (46) We verified the for-mula (51) for f3= 30°, 60°, 90”, 120°, and 150° for all parameter sets Inaddition, for T = 100 keV, v’ = 1 keV, the differentialcross section was veri-fied at Cl= 80°, 100°, 110°, and 115° because of the peaked behavior of theapproximationnear 100° (see Fig 5)

independentcalcula-VI COMPARISONOF TEE WIENKE-LATHROPAPPROXIMATIONTO TED?EXACT (XXWTON

SCAT1’ERINGOF PHOTONSFROM A MAXWELLIANELECTRONGAS

In Figs 2 through 6 we compare the isotropic approximationversus theexact differentialComp~on scattering cross sections,do/dCl T refers to theelectron temperature,and the photon energy is the initial photon energy v’.The isotropicapproximationcurves are dashed and are taken from Eq (51)

(integratedover $) The solid curves are the exact curves from Eqs (34) and(50) after numerical integrationover K’, $, and a’ The Wienke-Lathrop

approximationis a good one except for T large and T >> v’ However, the totalCompton cross section, o, is very well representedby the Wienke-Lathrop

approximation Figs 7 and 8 show the Compton cross sections integratedoverthe photon scatteringangle, (3 Again, dashed is the isotropic approximation,solid the exact It is evident that the two total cross sections, isotropicand exact, are nearly indistinguishableup to 100-keV temperature Figures 2,

3, 6, 7, and 8 are reproduced,by permission,from Ref 9 (February 15, 1983).Figures 4 and 5 also contain the Wienke-Lathropfitted approximationcurves(dot-dash),which we discuss next

In the event that a poorer approximation(the present version - it could

be made better) than the isotropic is satisfactory,Wienke and Lathrop9 offer

a “fitted approximation”that avoids even the ~-integrationof their isotropicapproximation However, an adjunct plot or plots or tables of the parameters

<K1> and <K2> versus T and v’ are required Moreover the isotropic imation can be integratedanalytically,but perhaps uselessly (see Appendix B)

approx-14

In Eq (34) the functionsK1 and K2 are replaced by the fitting eters<K1> and <K2> and K is taken to be the average (46) independentof K’

param-so that integrationover the Maxwell distributionis trivially performed:

and we are left with

or graphed for computationaluse Wienke and Lathrop9 have carried out such

a fitting to the total cross section It is not the most general, however,because one parameter, <K2>, is fixed, <K2> = K, thus reducing the problem

to a one-parameterfit They also simplify Eq (56) by omitting the K-factorsfrom the last

(;) =

two terms Their one parameter fit is then of the form

<KlhV’(I - ~) + Kmand

with <Kl> given by themg in Fig 9 (reproducedby permission)so as to

match the total cross section (checkedby us at T = 100 keV and 1 kev, andv’ = 1 keV) K is given by Eq (46) Equations (58) and (59) are substituted

in Eq (55) to get the differentialscattering cross section da/dC) As

alleged, this procedure does eliminate integrations,but requires use of atable or graph, Fig 9 We have calculated the differentialcross section bythis one-parameterfitted cross section for T = 1 keV, v’ = 1 kev and found itindistinguishablefrom the isotropic for these parameters However, for

T = 25keV= v’, Fig 3, and T = 100 keV, v’ = 1 keV, Fig 4, the differencesare appreciable We label the one-parameterapproximationin Figs 4 and 5 as

“approximate(fitted).” Especially from Fig 4 do we conclude that the tropic approximationis superior at least to the one-parameterfitted approxi-mation of Eqs (58), (59), and (55) Of course, one may be able to improve thefits by the use of additionalparameters such as <K2>. But again such usemeans additional

iso-VIII TEE TOTAL

1000 keV The accuracy is better than 1%, except at v = 300 keV, T = 150 kev,where an error of 1.9% appears At higher temperaturesup to 200 keV the error

is 3.7% or less We numericallyverified for T = 10 and 100 keV and for v’ = 1

to 1000 keV that the difference between this fit and both the Devaney and

Wienke-Lathropexact total cross sections was less than could be discerned onthe graphs, Figs 7 and 8 (i.e., < l%)

For unpolarizedlight the total Compton cross section for electrons

at rest is given by:12>14

For electrons in thermal Maxwellian motion we correct Eqs (60) or (61)

by the factor (in brackets)13

Note that the use of the Sampson equations,15 also suggested

earlier,16 is limited to T and v’ << 100 keV and they are less accurate

(< 4%)

Ix APPLICATIONm Mom Quuo (0R0Tm3R) CODES: m mm EQUATIONS

The exact differentialcross section for the Compton scattering of aphoton of energy v’ from a Maxwell gas of electrons at a temperatureT is given

by Eqs (l), (34), (33), (32), and (25) (or (23) and (24)) The scattering is

to photon energy, v, and is in a direction solid angle d!J= sin 9 df3d$q and

17

is symmetric in $q The total scattering cross section (i.e., integrated

over all five variables) is given by Eqs (60) or (61) and (62) All ties possible are in energy units

quanti-The equations give the cross section, or with the total cross sectionthe relative probability,of a particularevent For example, to determine aparticular scattering (v’ + v, e,$q) by a particularelectron (K’,a’,$)nointegrationof Eq (34) is required However, usually a user is interested(because of @q symmetry)only in da/de, the differentialscatteringcross

section of the photon into an angle, (3,and to energy, v Thus the other fourvariables must be integratedover The succeedingsections give approximatemethods for such integrations

“The simplest applicationof Monte Carlo is the evaluationof gralsO”17 In fact,highly ~lti-dimensional integrals are likely to be effi-ciently solved by Monte Carlo methods.lo “Every Monte Carlo computationthatleads to quantitativeresults may be regarded as estimatingthe value of a mul-tiple integral.m18 fius, the above equations are amenable to formal solution

inte-by Monte Carlo methods The equivalentMonte Carlo particle odslo may also be employed At first inspectionone might imagine samplingfor electron kinetic energy, K’, from the Maxwell distribution,Eq (25) (or(23)plus (24)), and for the isotropic spherical directionsa’ and $, thenapplying these to Eq (33) and (34) as well as to (32) to determine probabili-ties The actual detailed applicationsof Monte Carlo to the exact equationsare beyond the scope of this work

transportmeth-x. IuwLIamoN m mm umo (OR OTEER)mDE5: THE umntE-LATNROP

Iso-TROPICAPPROXIMATION

If the accuracy of Wienke-Lathropisotropic approximationis tory, see for example, Figs 1 to 7 for a comparisonof it with the exact scat-tering cross section, then considerablesimplificationin the formulas can beachieved Note from the figures that for temperatureswell below 100 keV, say

satisfac-25 keV or lower, the isotropic cross section does very well Note further thatthe total cross section is extremely well representedby the approximation,asFigs 7 and 8 show Thus, if a problem is insensitiveto angular distribu-tions, indeed if fore and aft symmetry only is required, then the isotropicapproximationis good even up to 100-keV temperature Also note that Figs 2,

3, 5, and 6 show extreme behavior so that Fig 5, the worst, describes a rare18

u : 0.09

T= I keV PHOTONENERGY= 1000 keV