voidpp点点通2004

SAND2004-4782PrintedSeptember2004ITS5TheoryManualDocumentRev.
1.
2September17,2004R.
P.
Kensek,B.
C.
Franke,T.
W.
LaubDraftVersion1.
22ITS5TheoryManualTABLEOFCONTENTS1Scope.
32GeneralLimitations.
33PhotonTransport43.
1InteractionsModeled.
53.
1.
1PhotoelectricEffect53.
1.
1.
1PhotoelectronProduction53.
1.
1.
2Relaxation.
53.
1.
2IncoherentScattering.
63.
1.
3CoherentScattering73.
1.
4PairProduction.
74ElectronTransport.
74.
1TransportMechanics.
74.
1.
1BoundaryCrossingDetails.
84.
2InteractionsModeled.
84.
2.
1CollisionalEnergyLoss.
84.
2.
2ElasticScattering.
84.
2.
3InelasticScattering94.
2.
4Bremsstrahlung.
94.
2.
5Knock-ons94.
2.
6ImpactIonization.
104.
2.
6.
1VacancyRelaxation.
105GeometryTracking106Tallies.
107BiasingOptions.
108SourceDescriptions.
109Statistics.
1010RandomNumberGenerators.
1011References10DraftVersion1.
23ITS5TheoryManual1ScopeThisdocumentdescribesthemodelingofthephysics(andeventuallyfeatures)intheIntegratedTIGERSeries(ITS)codes[Franke04]whichislargelypulledfromvarioussourcesintheopenliterature(especially[Seltzer88],[Seltzer91],[Lorence89],[Halbleib92]),althoughthosesourcesoftendescribetheETRANCodefromwhichthephysicsengineofITSisderived,notnecessarilyidentical.
Thisismeanttobeanevolvingdocument,withmorecoverageanddetailastimegoeson.
Assuch,entiresectionsarestillincomplete.
Presently,thisdocumentcoversthecontinuous-energyITScodeswithmorecompletenessonphotontransport(thoughelectrontransportwillnotbecompletelyignored).
Inparticular,thisdocumentdoesnotcovertheMultigroupcode,MCODES(externallyappliedelectromagneticfields),orhigh-energyphenomena(photonpair-production).
Inthisversion,equationsarelargelylefttothereferencesthoughtheymaybepulledinovertime.
2GeneralLimitationsAsimulationisonlyasgoodastheunderlyingmodel.
Thecontinuous-energyITScodes(withoutexternallyappliedelectromagneticfields)ismodeledafterthelinear,time-independentBoltzmannEquationforphotonsandelectrons(additionalassumptionsonhowtheelectronsarehandledwillbediscussedshortly),whichessentiallysaysparticlesstreambetweencollisionpoints.
Thefrequencyandtypeofcollisionsaregivenintermsofcrosssectionswhichareprobabilitiesperpathlength.
Thephysicsisembodiedinthedetailsofwhathappensatcollisionpoints,whichmayincludehowtheenergyanddirectionoftheincomingparticlechange,whethertheincomingparticleisabsorbed,andhowmanynewparticlesemergeincludingdetailsoftheirenergiesanddirections.
Here,"timeindependence"meansthatneitherthesourcenormaterialpropertiesdependontime.
TheBoltzmannEquationis"linear"whichmeanstheparticleinteractionsdonotmodifythematerialpropertiesandtheparticlesdonotinteractwitheachother.
Thecrosssectionsarebaseduponthoseofelementsofneutralatoms.
Inparticular,molecularandsolid-stateeffectsarenotmodeled.
Theseeffectstendtobeofgreaterimportanceatlowerenergies.
TheAtomicFormFactorassociatedwithphotoncoherentscatteringisparticularlysensitivetothis(e.
g.
see[Poletti02]).
ThesolecasewhereITSaddressesmolecularorsolid-stateeffects(atleastapproximately)istheparametersfortheBethestoppingpowerofelectrons,wherealternatevaluesareusedforlow-Zelementsdependingonwhetheritisusedinacompoundorwhetheritisagasorsolid[Berger88].
Themodeledcrosssectionstendtoberathersmoothfunctionsofenergy,exceptfordiscontinuitiesinphotoncrosssectionsatedgeenergies.
Realcrosssections[Hubbell99]exhibitoscillationsneartheseedgeenergies,sothecodesmaybelessaccurateformono-energeticphotonsnearedgeenergiesofthematerialtheyaretransportingthrough.
Forthecaseofaspectrumofphotons,sucheffectsarebelievedtoquicklywashout.
Similarily,realelectroncrosssectionsmayexhibitlargeoscillationsforsmallangulardeflectionsduetosolid-stateeffects.
ThissuggeststransportcodessuchasITS"arenotintendedorsuitableforapplicationsinthethin-film,pluralscatteringregime,whereinterferenceeffectsdependingonthestructureofthematerialplayanimportantrole"[Berger88].
For3Dcalculations,distancesof1nmforthenon-CADcodes,or10nmfortheCADcodes(includingaCG-onlysimulationusinganexecutablefromtheCADcodes)mustbe"small"comparedtozonethicknessesinthemodeledgeometry.
Intracking,whenaparticleismovedtoaboundary,the3Dcodesdeterminewhichzoneisenterednextbymakingaqueryatapointslightlybeyondtheboundary(basedonthesedistances)inthedirectionoftheparticle.
Generally,electrontransportismoreapproximatethanphotontransport.
Electrontransportistreatedina"ClassI"[Berger63]condensed-historyapproach,wheretheaccumulatedeffectsofscatteringoverapathlengtharetreatedinsteadofindividualinteractions(detailsaregiveninthesectionsonelectrontransport).
Sinceindividualscattersarenotmodeledalongthepathlength,thepathlengthshouldbefairlysmallcomparedtoregionsofinterest(sometimesathirdissufficient[Jensen88],thoughthiswilldependontheDraftVersion1.
24ITS5TheoryManualapplication).
Fromanangular-deflectionperspective,thispathlengthistheITS"substep"(seesection4),whichinprinciple(butnotwiththepresentimplementation)canbereducedtoanarbitrarysize.
Presently,ITSusesunstablerecursionrelationswhichtranslatetoafixed,smallnumberoftermstobesummed[Seltzer88],whichwouldbeinsufficientforconvergenceforverysmallpathlengths.
Fromanenergy-stragglingperspective,thepathlengthisthe"step"size.
Also,thepresentimplementationinITSusesseveralnearest-neighboralgorithms,theeffectsofwhichshouldbebettercharacterized.
3PhotonTransportPhotonsmoveinstraightlinesbetweencollisionsitesandaremodeledassuch.
Thefrequencyofcollisionsisdeterminedbythetotalcrosssectionofall(modeled)photoninteractions.
Adistancetocollisionissampledbasedonanexponentialdistribution.
Inthecontinuous-energymembersoftheITScodes,thissamplingisdonemechanicallyassumingphotoncollisionforcingisineffect.
First,thedistancetoboundaryofthecurrentzoneiscalculatedandthenaturalprobabilityofaninteractionalongthatdistancetoboundaryiscalculated.
Iftheuserhadrequestedphotoncollisionforcinginthatzone,thisnaturalprobabilitywouldbeusedtomodifytheweightofthephotonwhichwouldinteractwiththeuser'sspecifiedprobability.
Bydefault,thenaturalprobabilityisusedasthe"forcing"probabilitywhichismathematicallyequivalenttonoforcingatall.
Thedetailsofwhathappensatacollisionsitearespecifiedbythemodelingofthetypeofcollisionwhichoccurred.
Therelativeprobabilityofanyoneinteractionisgivenbytherelativesizeofthecorrespondingcrosssections.
Whennointeractionoccursinagivenzone,thephotonismovedtotheboundary.
Thenextzoneisdeterminedinslightlydifferentwaysdependingonthedimensionalityofthemembercode:Inthe3Dcodes(ACCEPT),apointischosenacertaindistancebeyondtheboundaryinthedirectionofthephoton.
Thisdistanceis1nmfornon-CADand10nmforCAD(includingtheCG-onlyoptioninCAD).
Thezonecontainingthatpointisidentifiedasthenextzoneforthephoton.
Inthe2.
5Dcodes(CYLTRAN),thenextzoneisdeterminedbycomparingtoeithertheaxialorradialboundariesofasetofcandidatezonesbasedonthesurfaceexitedfromthepreviouszone.
Inthe1Dcodes(TIGER),thenewzoneindexissimplyonegreaterorsmallerthanthepreviouszone,dependingonthedirectionofthephoton.
Afterbeingmoved(toeitheraboundaryorcollisionsite–ifthephotonhasnotbeenabsorbed),theprocessofsamplingdistancetocollisionrepeats(untilthephotonhasescapedtheboundariesofthemodelgeometry).
Nosamplingofdistancetocollisionoccursinvoidregions.
Thepre-processeddataofthecrosssectionsfromNISTforeachinteractionaregivenasvaluesonan80-pointenergygrid(spanningtherangeof1keVto1GeV)foreachelement.
Higher-Z(atomicnumber)elementswillhavemoreenergypoints,astheenergyrangeisbrokenupinto"tables"betweentheedgeenergies(atwhichthephotoelectriccrosssectionshavediscontinuitiesinenergy).
Thesevaluesaremappedontoafinegrid(ofabout3000energypoints–thegridisadaptivewithmorepointsaddednearedge-energydiscontinuites)bylog-logquadraticinterpolation.
(TheXCOM[Berger87]program,uponwhichourmethodsarebased,usesdifferentinterpolationprocedures.
XCOMuseslog-logcubicsplineinterpolationabovetheK-edge,andlog-loglinearinterpolationbelow.
SpotcheckingvaluesobtainedfromourfinegridandXCOM'sinterpolationmethods,thedifferencesaregenerallylessthan1%andoftenmuchlessawayfromedgeenergies.
ThereisevidenceNIST[Seltzer93]maybeexploringuseoflog-logHermitepolynomialinterpolation"toavoidtheoccasionalnumericalexcursionfoundforcubicsplines,"sothebestinterpolationchememaybeamovingtarget.
)ValuesarethenusedintheMonteCarlowithasimplenearest-neighborlookup,wheretheenergyatwhichthecrosssectionisevaluatedisthegeometricmeanoftheendpointsoftheenergybin.
Crosssectionsforcompoundsareobtainedfromweightedsumsoverthecorrespondingcoefficients(i.
e.
fractionsofweightsoftheatomicconstituents)fortheelements.
[Berger87]DraftVersion1.
25ITS5TheoryManual3.
1InteractionsModeled3.
1.
1PhotoelectricEffectThetotalphotoelectricabsorptioncrosssectionsarebasedoncalculationsfromScofield[Scofield73].
However,thesecrosssectionswerenotrenormalizedusingtherelativisticHartree-FockcorrectionfactorssuppliedbyScofieldsincereviewsofSalomanandHubbell[Saloman87]indicatethatagreementwithexperimentisbetterwhentherenormalizationisnotdone.
AlthoughthesecrosssectionswerealsousedinthepubliclyreleasedITS3.
0in1992[Halbleib92],themedicalphysicscommunityapparentlyhasonlyrecently"rediscovered"thebenefitofuseofthese(non-renormalized)crosssections[DeMarco02].
Theaccuracyofthephotoelectriccrosssectionsaregivenin[Hubbell99]as:PhotonenergyrangeSolidGas0.
5-1keV10-20%5%1-5keV5%5%5-100keV2%2%100keV–10MeV1-2%1-2%10MeV–100GeV2-5%2-5%Strictlyspeaking,thistableiscomparingmeasurementstotheEPDLdataset,butagainthesearebasedonthesamenon-renormalizedScofieldcalculations(withaslightlydifferentinterpolationscheme).
Thisassessmentcomesfromasystematiccomparison[Saloman88]tothemassiveNISTdatabasewhere,forexampleinthe5-100keVrange,almosteachelementiscomparedwithmeasurementsfromseveralsources.
Thisisthedominantinteractioncrosssectionforlow-energyphotons.
3.
1.
1.
1PhotoelectronProductionTheprobabilityofhavingaparticularshellionizedistheratioofthephotoelectriccrosssectionevaluatedoneithersideofthediscontinuityattheshelledgeenergy(forphotonenergiesabovethatedgeenergy,butotherwiseassumedtoindependentofthephoton'senergy).
Forthestandardcodes,onlytheK-shellofthehighest-Zelementineachmaterialistreated.
ForthePCODES,allelementsaretreateddowntoanaverage-Mandaverage-Nshell.
Theangleoftheemergingphoto-electron(withrespecttotheparentphoton)isdescribedbytheFischerdistributionatlowerenergiesandSauterformulaathigherenergies(bothcanbefoundin[Lorence89]).
Forthestandardcodes(non-PCODES),theswitchingenergyis50keV[Seltzer88]andthedistributionshavebeenpre-calculatedbyNISTonthirteenenergiesand21equi-probablereduced-anglebins.
Thesamplingproceedsbylinearlyinterpolatingonenergyandanangularterm,(1-βcosθ),whichisatypicalfactorintheBethe-Heitlerformulas.
ForthePCODES,theswitchingenergyisafunctionofatomicnumber[MacCallum73]andthedistributionsaresampledanalyticallybyeitherdirectsamplingfromFischerorarejectiontechniquefortheSauterdistribution.
Theenergyoftheemergingphoto-electronisthedifferencebetweentheenergyofthephotonhavingthephotoelectricinteractionandthebindingenergyoftheatomicshellinvolved.
Forthestandardcodes,thebindingenergiesarethosefrom[Carlson75].
ThebindingenergiesforthePCODESaresettobeconsistentwiththeNISTphotoelectricedgeenergies.
Forshellenergiesbelow1keVforthePCODEs,weusethebindingenergiesof[Carlson75].
Bindingenergiesaregenerallywellknown,andthedifferencesinthesedatasetstendtobelessthan1%.
3.
1.
1.
2RelaxationIfashellhasbeenionizedthroughthephotoelectricevent,theresidualshellbindingenergymustrelaxthroughtheproductionoffluorescencephotonsorAugerelectrons.
Foreachshell,thedatawhichareneededarethefluorescenceefficiencyandrelativeprobabilitiesofwhichothershell(s)maybeinvolvedwhichinturnaffectstheenergiesoftheemittedfluorescencephotonsorAugerelectrons.
DraftVersion1.
26ITS5TheoryManualInthestandardcodes,onlytheK-shellistreated.
A1984fit(overatomicnumberZ)isusedfortheK-shellfluorescenceefficiencies.
See[Hubbell94]fordiscussionandlatestsuggestionoftabulatedvalues.
Efficienciesforhigh-Zmaterials(whichgeneratethemostfluoresence)areratherstabletothe1%level.
Low-ZelementssuchasAlhavechangedbyabout20%overtheyears,andisnowknowntoabout10%[Hubbell94].
Thebranchingratios(whichK-shellfluorescenceorwhichAugerelectrons)aretakenfrom[Bambynek72].
Theassessmentofagreementofthesefluorescencebranchingratiostomeasuredvaluesareaslargeas20%forlow-Zandoftheorderof7%athighZ[Bambynek72].
ThedataforthePCODES(whichneedsimilardatabutformoreshells,includingCoster-KronigtransitionsfortheLshells)comefrom10referencescitedin[Halbleib75].
Atthistime,itisnotclearwhichquantitiycamefromwhichsource.
3.
1.
2IncoherentScatteringThetotalcrosssectionforincoherent(Compton)scatteringistakenfrom[Hubbell75]obtainedfromusingaproductoftheanalyticKlein-Nishinaformulaandnon-relativisticHartree-Fockincoherent-scatteringfunctionsS(Z,x)(wherexisthemomentumtransferandZistheatomicnumber).
ThesamplingoftheangleofthescatteredphotonisperformedbysamplingfromtheKlein-NishinacrosssectionthenrejectingonS(Z,x)forlargeenergies(energieslargerthan0.
003MeVtimesthesquarerootofaneffectiveatomicnumberforthegivenmaterial),otherwisesamplingonS(Z,x)andrejectingonKlein-Nishina.
WeareconsistentinusingthesameS(Z,x)tosamplefromthetotalordifferential(inangle)crosssections.
WhentheKlein-Nishinacrosssectionissampled,Kahn'smethod[Kahn56]isusedforenergieslessthan5MeVwhileKoblinger'smethod[Koblinger75]isusedotherwise.
Bothmethodsuseanalyticexpressionandaremathematicallyequivalent(butnotnecessarilyasefficient)intheirrangeofvalidity.
Koblinger'smethodcannotbeusedbelowabout1.
4MeVsinceoneoftheterms,treatedasaprobability,becomesnegative.
TheswitchingenergiesweredeterminedbyNISTasgoodchoicesforefficientsamplingtimes.
Whenevaluatingtheincoherentscatteringfunctions,cubicsplineinterpolationisused(whichisadditionallylog-logformomentumtransferslargerthan2inverseAngstroms).
S(Z,x)isassumedtobeunityabovemomentumtransfersof100inverseAngstroms.
WhenxissampledfromS(Z,x),cubicsplineinterpolationisusedonthecumulativeprobabilityfunction.
Forcompounds,aneffective(normalizedtounityforlargex)incoherentscatteringfunctionisusedas[Seltzer89]Seff(x)=Σiwi(Zi/Ai)[S(x,Zi)/Zi]/where=Σiwi(Zi/Ai).
Oncetheangleofthescatteredphotonisdetermined,itsenergyandthatoftheemergingelectronaredeterminedbyComptonkinematics.
Hubbell[Hubbell97]assessedtheincoherent-scatteringfunctionapproachtobe5%orbetterinforwardscatteringanglesandforlowandmediumZmaterials.
ForlargeanglesandhighZvaluesthisapproachmaybetoohighbyasmuchas20%.
HealsoreferstoS-Matrixtheorywhichmayprovidemoreaccuratecrosssectionsthantheuseoftheincoherentscatteringfunction,butadmitsmorecomparisonswithexperimentaldataareneeded.
Meanwhile,thereisinterestinasimplertheorybutonewhichcanincorporateComptonDopplerbroadening–whichisthoughttohaveaveryminoreffectonenergydeposition(duetoenergydepositionbeingtheintegralofthebroadenedquantity)butadramaticeffectonforexample,apulseheightdistribution.
EvaluationsbyNIST[Rao04]onthisalternateapproachtomodelingComptonscatteringresultedindifferencesinthetotalcrosssectionsbylessthan1%above50keV,butcouldbeaslargeas10%at5keVforparticularelements.
Theyconclude"Wenote,unfortunately,thatthereisascarcityofabsoluteexperimentaldatafortotalscatteringcrosssections.
"[Rao04](WehaveastudentinternimplementingaformoftheComptonDopplerbroadeningsowecaneventuallymakeourownassessmentsofitsimpactonourapplications.
)DraftVersion1.
27ITS5TheoryManual3.
1.
3CoherentScatteringThetotalcrosssectionsforcoherent(Rayleigh)scatteringistakenfrom[Hubbell79]obtainedfromusingaproductoftheanalyticThompsonformulacombinedwithrelativisticHartree-FockatomicformfactorsF(Z,x)(wherexisthemomentumtransferandZistheatomicnumber).
ThesamplingoftheangleofthescatteredphotonisperformedbysamplingfromtheThompsoncrosssectionthenrejectingonF(Z,x)forsmallenergies(energiessmallerthan0.
002MeVtimesapowerofaneffectiveatomicnumberforthegivenmaterial),otherwisesamplingonF(Z,x)andrejectingontheThompsonformula.
Unlikethoseusedforthetotalcrosssections(usedtodeterminetherelativeprobabilityofhavingaCoherentinteraction),theF(Z,x)usedfortheanglesamplingarethenon-relativisticHartree-Fockatomicformfactorsof[Hubbell75].
Whenevaluatingtheatomicformfactors,cubicsplineinterpolationisused(whichisadditionallylog-logformomentumtransferslargerthan0.
1inverseAngstroms).
F(Z,x)isassumedtobezeroaboveamaterial-dependentmomentumtransfer(calculatedasxforwhichF(Z,x)=1.
E-6).
WhenxissampledfromF(Z,x),cubicsplineinterpolationisusedonthecumulativeprobabilityfunctionwhichisadditionallylogarithmicinmomemtumtransferforsampledprobabilitiesgreaterthan0.
1inverseAngstroms.
Forcompounds,aneffective(normalizedtounityatx=0)atomicformfactorisusedas[Seltzer89]Feff(x)=Σiwi(Zi2/Ai)[F(x,Zi)/Zi2]/where=Σiwi(Zi/Ai).
Itisassumednoenergylossisassociatedwithacoherentscatteringinteraction.
3.
1.
4PairProductionPairproductionisnotyetdiscussed.
4ElectronTransportElectrontransportistreatedina"ClassI"[Berger63]condensed-historyapproach,wheretheaccumulatedeffectsofscatteringoverapathlengtharetreatedinsteadofindividualinteractions.
Pathlengthsaretreatedontwolevels[Seltzer88]inITS.
Forenergy-losssampling,a"step"sizeisdefinedasthedistanceoverwhichaspecifiedaverageenergylossoccurs(theITSdefaultisabout8%sothat,after8stepstheelectron'senergyis,ontheaverage,halved.
).
Thesestepsizesarenaturallymappedtoalogarithmicenergygridonwhichmostoftheelectroncrosssectionsareevaluated.
Thisstepisfurtherdividedintomultiple(from2-15withlargernumbersforhigheratomicnumberofthematerial)substepstohelpcapturethespatialvariation("wiggliness"[Seltzer91])ofrealelectrontracks.
Thespatialdisplacementoverasubstepisdescribedinthefollowingsectionon"transportmechanics".
4.
1TransportMechanicsITSusesthesimplesttransportmechanicsscheme[Berger63]:straight-linemotioninthedirectionoftheunscatteredelectron,withtheaccumulatedangulardeflectionimposedattheendofthesubstep.
Otherapproaches(attemptingtobetterdescribethe"wiggliness"ofthescatteredelectronsoverasubstep)includestraight-linemotionwithexplicitpathlengthcorrections[Seltzer91],singleormultiplehinges[Kawrakow98],orsamplingfromdistributionsgeneratedfromsingle-scatteredcodes(whichdescribeDraftVersion1.
28ITS5TheoryManualwheretheelectronendsup,butnothowitgotthere).
[Ofthose,thesinglerandomly-placedhingehassomeattractiveproperties(approximatelypreservingmoments)whilemaintainingsimplicityforboundarycrossing.
ThismaybeexploredinfutureversionsofITS.
]4.
1.
1BoundaryCrossingDetailsWiththesimplestraight-aheadtransport,thespatiallocationofaboundarycrossingisstraightforward.
However,themachineryofthecondensed-historyalgorithms,whichisbasedonprecalculatingsamplingdistributionsoverapre-determinedpathlength,needtobemodified,sinceboththestepandsubteppathlengthshavebeentruncated.
Theenergyofanelectronwhichhascrossedaboundaryisresampledbasedontruncatedstep-size.
Thisenergylossisappliedtotheelectronattheboundary.
Theenergylossofthetruncatedsubsetissimplythescaledenergyloss(basedonthefullstep)forthatfractionofthesubstep.
Thisintroducesalackofcorrelationoftheenergylossalongthesubstepsalreadytraversed,butattemptstomaximizethepathlengthusedinthesampling(i.
e.
atruncatedstepinsteadofatruncatedsubstep).
Ontheaverage,themeanenergydepositedwillbepreserved,buttheenergystraggling(spread)fortheenergydepositedwillbedifferentfromtheenergystragglingoftheenergylostbytheelectronontheothersideoftheboundary.
Theangulardeflectionsarenowsampledwithaprocedure[Jensen88]whichrepresentsthemultipleangularsacatteringdistributionasacombinationofamodifiedGaussian[Berger88]withasingle-scatteringlarge-angletail.
Ifthepathlengthissufficientlyshort,onlythesinglescatteringangulardeflectionswillbesampled.
Thenumberoflarge-anglesinglescatters(ortotalsinglescatteringwhenthepathlengthistoosmalltohavemultiplescattering)isbasedonaPoissiondistributionoftheaveragenumberofthosescatters.
Thesingle-scatteringdistributions(forthesetruncatedsubsteps)arerepresentedanalyticallyasscreenedRutherforddistributions,evenbelow256keV,usingSeltzer'smodifiedscreeningparameter[Seltzer88]topreservethefirsttransportmomentofthemoreaccuratephase-shiftcalculations.
Thisaccumulatedscatteringisthenimposedattheboundary(i.
e.
theendofthetruncatedsubstep).
Finally,accumulatedscatteringswhichbringtheparticlebackintothepresentzone(i.
e.
,havenotcrossedtheboundary)arerejected.
Thenumberofanysecondariesproducedissimplyscaledbackbytheratioofthetruncatedtofullsubstep.
4.
2InteractionsModeled4.
2.
1CollisionalEnergyLossTheenergylossoverastepissampledfromamodified[Seltzer91]Landau/Blunck-Leisegangdistribution.
TwomodificationsweremadebySeltzer:(1)afinitemaximumenergylosswasdeterminedtopreservethemeanstoppingpower(calculatedfromtheBetheformulaforthetotalstoppingpower),and(2)asemi-empiricalcorrectiontothewidthwhichdramaticallyimprovescomparisonswithexperimentforsmallerstepsizes.
Withinastep,thesampledenergylossispartitionedequallyamongthesubsteps.
4.
2.
2ElasticScatteringTheGoudsmit-Saunderson(GS)distributionisaninfiniteserieswhosecoefficientsinvolvethetransportmoments(angularmoments)ofthesingle-scatteringcrosssection.
ThemethodofevaluationusedinITSwaslargelydevelopedinthelate50'searly60'swhencomputationalpowerwasverylimitedsoapremiumwasplacedonanalyticmanipulations.
Amorerobustmethodhasbeendescribedin[Berger88].
Atenergiesabove256keV,thecrosssectionisrepresentedastheproductofascreenedRutherfordwiththeratiooftheMotttoRutherford.
In[Berger88],Bergergivesatableoftheestimatederrorasabout5%inUat274keV(fallingoffathigherenergiesorlower-Z).
ThescreeningparameterηisessentiallythatofMoliere,butmodifiedbySeltzer[Seltzer88]whichweusetosmooththetransitiontotheuseofphase-shiftcalculationsbelow256keV.
Seltzer'smodificationisdesignedtoapproximatelypreservethefirsttransportmomentofthephase-shiftcrosssection.
TheMotttoRutherfordratioistreatedasnumericaldatafromNIST:evaluatedforeveryelement(Z=1to100)at18energiesand5angles(every45degrees).
(ThedataincludesasimilartabulationofMott-positronstoRutherfordratios.
)ThefunctionisinterpolatedbyDraftVersion1.
29ITS5TheoryManualpiecewisequadratic(Lagrange)interpolationinenergy.
Finally,thiscrosssectionisexpandedinpowersof(1-cosθ+η)wherefivecoefficientsaredeterminedbymatchingexactlyat5angles(every45degrees).
Atenergiesbelow256keV,phase-shiftcalculations[Riley75]havebeenperformedandexpandedina12-parameterfitofasumofdifferentpowersof(1-cosθ+B)combinedwithasumofafewLegendrePolynomials,whereBisoneofthefittingparameters.
Thefittingcoefficientshavebeendeterminedtopreservethefirsttwentytransportmomentsofthephase-shiftcalculations.
(WedohavenumericaltablesofBerger'sextensionofRiley'scrosssectionsto1MeV.
[Berger93])Byrepresentingthesinglescatteringcrosssectioninthisway,thetransport-momentintegrals(todeterminetheGScoefficients)canbedoneanalyticallyusingrecursionrelations,whichunfortunatelyareunstable.
Tomaintainaccuracytheseriesissummedto240terms(withthenexttermrepresentingtheunscatteredprobability)intheforwarddirectionwhenη1.
E-4,theseriesissummedtoexp(1.
794-.
397lnη),butneverlessthan10,inthebackwarddirection[Seltzer88].
Eachcoefficientisevaluatedattwoenergies:atthebeginningandendofthesubstep(whichiscenteredwithinthecorrespondingstep).
Thesameangulardistributionwillthenbeusedthroughoutthestep.
Thoughthismayintroducesomebiasinsituationswhereaboundaryiscrossedbeforethestepcyclecanbecompleted.
Thispermitsthecoefficientsofa2-parameterfit[Spencer55]todescribetheenergydependenceoftheGScoefficients,sotheymaybeintegratedanalyticallyoverenergy(i.
e.
overthesubstep)toaccountfortheenergylossoftheelectronoverthesubstep.
Thefinaldistributionisrepresentedasthedelta-functionplusahistograminangle(withadefaultsizeof33bins,morecloselyspacedatsmallangles).
AllofthesecomputationstakeplacewithinthecrosssectiongeneratingcodeXGEN,sothattheMonteCarloneedonlysamplefromthecumulativeprobabilityofthathistogram.
Thecumulativedistributionisnormalizedtothecomplementofthedelta-functionscattering.
4.
2.
3InelasticScatteringTheangulardeflectionsduetoinelasticscatteringareapproximatedbyincreasingtheelastic-scatteringcrosssectionby(Z+1)/ZwhiletheGScoefficientsaredetermined.
Withthistreatment,thelarge-anglecollisionswillbeuncorrelatedwiththeassociatedlargerenergyloss.
Thisincreasesthecrosssectionatallangles,sotheresultingdistributionisreducedbyZ/(Z+1)atallanglesgreaterthanthekinematicalcutoffangle[Seltzer88].
Analternativeprocedure,incorporatingthecorrectionperhapsalittlemoreconsistently,hasbeensuggestedin[Berger88].
4.
2.
4BremsstrahlungBremsstrahlungproductionissampledfromaPoissondistributionovereachsubstep.
Theenergyofeachphotonissubtractedfromtheparentelectron(hencethetwoarecorrelatedunlikeknock-onproduction).
Thestartingpositionofthephotonissampleduniformlyalongthesubsteppath.
TheangulardistributionofthebremsstrahlungphotonissampledfromBethe-Heitlercrosssections(Bornapproximations)[Koch59]differentialinbothenergyandangleoftheemittedphoton.
Phase-shiftcalculationsfortheseangulardistributionsarealsoavailableatorbelow500keV[Kissel83]andthedifferencesdiscussedin[Seltzer91].
Thedirectionoftheparentelectronistakentobethatatthebeginningorendofthesubstep,whicheveriscloser.
Thiscanleadtomeasurableartifacts[Faddegon93]andasimplesolutionhasbeenproposed[Hughes97]thoughnotyetimplementedinITS.
Thebremsstrahlungcrosssections,differentialinenergy,arediscussedmorefullyin[Seltzer88b].
Forenergiesbelow2MeV,theyincludephase-shiftcalculationsofPratt[Pratt77](withlinearextrapolationtoincludeZ=1andZabove92to100).
Seltzerevaluates"agreementisgenerallywithinthecombinedlimitsofexperimentaluncertaintyandatheoreticaluncertaintyestimatedtobe5to10%.
"[Seltzer88b]4.
2.
5Knock-onsTheproductionofknock-onelectronsissampledaccordingtotheMollercrosssection(whichignoresbindingeffects).
Theangulardistributionoftheknock-onsisbasedonMollerkinematicswiththeDraftVersion1.
210ITS5TheoryManualreferencedirectionoftheparentelectrontakentobethatatthebeginningorendofthesubstep,whicheveriscloser.
Thestartingpositionsoftheseknock-onsaresampleduniformlyoverthesubstep.
Nochangeoftheparentelectron'senergy(ordirection)ismadesincethisisalreadytakenintoaccountthroughusingthetotalstoppingpowerandLandua/Blunck-Leisegangdistribution(andourtreatmentofinelasticscattering).
Hence,theenergylossduetoknock-onsareuncorrelatedfromtheparentelectron.
4.
2.
6ImpactIonization4.
2.
6.
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