32.Passageofparticlesthroughmatter32.PASSAGEOFPARTICLESTHROUGHMATTER.. - PDF document

32.Passageofparticlesthroughmatter32.PASSAGEOFPARTICLESTHROUGHMATTER......................232.1.Notation....................232.2.Electronicenergylossbyheavyparticles......232.2.1.Momentsandcrosssections..........232.2.2.Maximumenergytransferinasinglecollision.....................432.2.3.Stoppingpoweratintermediateener-gies......................532.2.4.Meanexcitationenergy............632.2.5.Densitye®ect................732.2.6.Energylossatlowenergies..........932.2.7.Energeticknock-onelectrons(rays).....1032.2.8.Restrictedenergylossratesforrela-tivisticionizingparticles.............1032.2.9.Fluctuationsinenergyloss..........1132.2.10.Energylossinmixturesandcom-pounds.....................1432.2.11.Ionizationyields..............1432.3.Multiplescatteringthroughsmallangles......1532.4.Photonandelectroninteractionsinmat-ter........................1732.4.1.Collisionenergylossesbye.........1732.4.2.Radiationlength..............1832.4.3.Bremsstrahlungenergylossbye.......1932.4.4.Criticalenergy...............2132.4.5.Energylossbyphotons............2132.4.6.Bremsstrahlungandpairproductionatveryhighenergies...............2532.4.7.Photonuclearandelectronuclearin-teractionsatstillhigherenergies..........2632.5.Electromagneticcascades.............2732.6.Muonenergylossathighenergy.........3032.7.Cherenkovandtransitionradiation........3332.7.1.OpticalCherenkovradiation.........3332.7.2.CoherentradioCherenkovradiation......3432.7.3.Transitionradiation.............35K.A.Oliveetal.(PDG),Chin.Phys.C38,090001(2014)(http://pdg.lbl.gov)August21,201413:18 32.Passageofparticlesthroughmatter32.PASSAGEOFPARTICLESTHROUGHMATTERRevisedSeptember2013byH.Bichsel(UniversityofWashington),D.E.Groom(LBNL),andS.R.Klein(LBNL).Thisreviewcoverstheinteractionsofphotonsandelectricallychargedparticlesinmatter,concentratingonenergiesofinterestforhigh-energyphysicsandastrophysicsandprocessesofinterestforparticledetectors(ionization,Cherenkovradiation,transitionradiation).Muchofthefocusisonparticlesheavierthanelectrons(,p,etc.).Althoughthechargenumberzoftheprojectileisincludedintheequations,onlyz=1isdiscussedindetail.Muonradiativelossesarediscussed,asarephoton/electroninteractionsathightoultrahighenergies.Neutronsarenotdiscussed.ThenotationandimportantnumericalvaluesareshowninTable32.1.32.1.Notation32.2.Electronicenergylossbyheavyparticles[1{33]32.2.1.Momentsandcrosssections:Theelectronicinteractionsoffastchargedparticleswithspeedv=¯coccurinsinglecollisionswithenergylossesW[1],leadingtoionization,atomic,orcollectiveexcitation.Mostfrequentlytheenergylossesaresmall(for90%ofallcollisionstheenergylossesarelessthan100eV).Inthinabsorbersfewcollisionswilltakeplaceandthetotalenergylosswillshowalargevariance[1];alsoseeSec.32.2.9below.Forparticleswithchargezemoremassivethanelectrons(\heavy"particles),scatteringfromfreeelectronsisadequatelydescribedbytheRutherforddi®erentialcrosssection[2],d¾R(W;) dW=2¼r2emec2z2 2(12W=Wmax) W2;(32:1)whereWmaxisthemaximumenergytransferpossibleinasinglecollision.Butinmatterelectronsarenotfree.Wmustbe¯niteanddependsonatomicandbulkstructure.ForelectronsboundinatomsBethe[3]used\BornTheorie"toobtainthedi®erentialcrosssectiond¾B(W;) dW=d¾R(W;¯) dWB(W):(32:2)ElectronicbindingisaccountedforbythecorrectionfactorB(W).ExamplesofB(W)andd¾B=dWcanbeseeninFigs.5and6ofRef.1.Bethe'stheoryextendsonlytosomeenergyabovewhichatomice®ectsarenotimportant.Thefree-electroncrosssection(Eq.(32:1))canbeusedtoextendthecrosssectiontoWmax.AthighenergiesBisfurthermodi¯edbypolarizationofthemedium,andthis\densitye®ect,"discussedinSec.32.2.5,mustalsobeincluded.Lessimportantcorrectionsarediscussedbelow.ThemeannumberofcollisionswithenergylossbetweenWandW+dWoccurringinadistance±xisNe±x(d¾=dW)dW,whered¾(W;)=dWcontainsallcontributions.Itisconvenienttode¯nethemomentsMj()=Ne±xZWjd¾(W;) dWdW;(32:3)August21,201413:18 32.PassageofparticlesthroughmatterTable32.1:Summaryofvariablesusedinthissection.Thekinematicvariablesandhavetheirusualrelativisticmeanings. SymbolDe¯nitionValueor(usual)units ¯nestructureconstante2=4¼²0~c1=137:035999074(44)MincidentparticlemassMeV/c2Eincidentpart.energy°Mc2MeVTkineticenergy,(1)Mc2MeVWenergytransfertoanelectronMeVinasinglecollisionkbremsstrahlungphotonenergyMeVmec2electronmassc20:510998928(11)MeVreclassicalelectronradiuse2=4¼²0mec22:8179403267(27)fmNAAvogadro'snumber6:02214129(27)1023mol1zchargenumberofincidentparticleZatomicnumberofabsorberAatomicmassofabsorbergmol1K4¼NAr2emec20:307075MeVmol1cm2ImeanexcitationenergyeV(Notabene!)(¯°)densitye®ectcorrectiontoionizationenergyloss~!pplasmaenergyp hZ=Ai£28:816eVp 4¼Ner3emec2=®j¡!ingcm3Neelectrondensity(unitsofre)3wjweightfractionofthejthelementinacompoundormixturenj/numberofjthkindofatomsinacompoundormixtureX0radiationlengthgcm2EccriticalenergyforelectronsMeVE¹ccriticalenergyformuonsGeVEsscaleenergyp 4¼=®mec221.2052MeVRMMoliµereradiusgcm2 sothatM0isthemeannumberofcollisionsin±x,M1isthemeanenergylossin±x,(M2M1)2isthevariance,etc.ThenumberofcollisionsisPoisson-distributedwithmeanM0.Neiseithermeasuredinelectrons/g(Ne=NAZ=A)orelectrons/cm3(Ne=NA½Z=A).Theformerisusedthroughoutthischapter,sincequantitiesofinterest(dE=dx,X0,etc.)varysmoothlywithcompositionwhenthereisnodensitydependence.August21,201413:18 32.Passageofparticlesthroughmatter Muon momentum110100Stopping power [MeV cm2/g] Lindhard-Scharff Bethe Radiative Radiativeeffectsreach 1% Without d Radiativelosses bg 0.0010.010.1110100100010 Minimumionization Emc Nuclearlosses m- + Anderson-Ziegler Fig.32.1:Stoppingpower(=h¡dE=dxi)forpositivemuonsincopperasafunctionof¯°=p=Mcovernineordersofmagnitudeinmomentum(12ordersofmagnitudeinkineticenergy).Solidcurvesindicatethetotalstoppingpower.Databelowthebreakat¯°0:1aretakenfromICRU49[4],anddataathigherenergiesarefromRef.5.Verticalbandsindicateboundariesbetweendi®erentapproximationsdiscussedinthetext.Theshortdottedlineslabeled\"illustratethe\Barkase®ect,"thedependenceofstoppingpoweronprojectilechargeatverylowenergies[6].dE=dxintheradiativeregionisnotsimplyafunctionof.32.2.2.Maximumenergytransferinasinglecollision:ForaparticlewithmassM,Wmax=2mec222 1+2°me=M+(me=M)2:(32:4)Inolderreferences[2,8]the\low-energy"approximationWmax=2mec222,validfor2°meM,isoftenimplicit.Forapionincopper,theerrorthusintroducedintodE=dxisgreaterthan6%at100GeV.For2°meM,Wmax=Mc22.Atenergiesoforder100GeV,themaximum4-momentumtransfertotheelectroncanexceed1GeV/c,wherehadronicstructuree®ectssigni¯cantlymodifythecrosssections.ThisproblemhasbeeninvestigatedbyJ.D.Jackson[9],whoconcludedthatforhadrons(butnotforlargenuclei)correctionstodE=dxarenegligiblebelowenergieswhereradiativee®ectsdominate.Whilethecrosssectionforrarehardcollisionsismodi¯ed,theaveragestoppingpower,dominatedbymanysoftercollisions,isalmostunchanged.August21,201413:18 32.Passageofparticlesthroughmatter32.2.3.Stoppingpoweratintermediateenergies:Themeanrateofenergylossbymoderatelyrelativisticchargedheavyparticles,M1=±x,iswell-describedbythe\Betheequation,"dE dx=Kz2Z A1 21 2ln2mec222Wmax I22(¯°) 2:(32:5)Itdescribesthemeanrateofenergylossintheregion0:1¯°1000forintermediate-Zmaterialswithanaccuracyofafew%.Withthesymbolde¯nitionsandvaluesgiveninTable32.1,theunitsareMeVg1cm2.Wmaxisde¯nedinSec.32.2.2.Atthelowerlimittheprojectilevelocitybecomescomparabletoatomicelectron\velocities"(Sec.32.2.6),andattheupperlimitradiativee®ectsbegintobeimportant(Sec.32.6).BothlimitsareZdependent.AminordependenceonMatthehighestenergiesisintroducedthroughWmax,butforallpracticalpurposeshdE=dxiinagivenmaterialisafunctionofalone.Fewconceptsinhigh-energyphysicsareasmisusedashdE=dxi.Themainproblemisthatthemeanisweightedbyveryrareeventswithlargesingle-collisionenergydeposits.Evenwithsamplesofhundredsofeventsadependablevalueforthemeanenergylosscannotbeobtained.Farbetterandmoreeasilymeasuredisthemostprobableenergyloss,discussedinSec.32.2.9.ThemostprobableenergylossinadetectorisconsiderablybelowthemeangivenbytheBetheequation.InaTPC(Sec.33.6.5),themeanof50%{70%ofthesampleswiththesmallestsignalsisoftenusedasanestimator.Althoughitmustbeusedwithcautionsandcaveats,hdE=dxiasdescribedinEq.(32:5)stillformsthebasisofmuchofourunderstandingofenergylossbychargedparticles.Extensivetablesareavailable[4,5,pdg.lbl.gov/AtomicNuclearProperties/].Forheavyprojectiles,likeions,additionaltermsarerequiredtoaccountforhigher-orderphotoncouplingtothetarget,andtoaccountforthe¯nitesizeofthetargetradius.ThesecanchangedE=dxbyafactoroftwoormorefortheheaviestnucleiincertainkinematicregimes[7].Thefunctionascomputedformuonsoncopperisshownasthe\Bethe"regionofFig.32.1.MeanenergylossbehaviorbelowthisregionisdiscussedinSec.32.2.6,andtheradiativee®ectsathighenergyarediscussedinSec.32.6.OnlyintheBetheregionisitafunctionofalone;themassdependenceismorecomplicatedelsewhere.ThestoppingpowerinseveralothermaterialsisshowninFig.32.2.Exceptinhydrogen,particleswiththesamevelocityhavesimilarratesofenergylossindi®erentmaterials,althoughthereisaslowdecreaseintherateofenergylosswithincreasingZ.Thequalitativebehaviordi®erenceathighenergiesbetweenagas(Heinthe¯gure)andtheothermaterialsshowninthe¯gureisduetothedensity-e®ectcorrection,(¯°),discussedinSec.32.2.5.Thestoppingpowerfunctionsarecharacterizedbybroadminimawhosepositiondropsfrom¯°=3:5to3.0asZgoesfrom7to100.ThevaluesofminimumionizationasafunctionofatomicnumberareshowninFig.32.3.Inpracticalcases,mostrelativisticparticles(e.g.,cosmic-raymuons)havemeanenergylossratesclosetotheminimum;theyare\minimum-ionizingparticles,"ormip's.Eq.(32:5)maybeintegratedto¯ndthetotal(orpartial)\continuousslowing-downapproximation"(CSDA)rangeRforaparticlewhichlosesenergyonlythroughionizationandatomicexcitation.SincedE=dxdependsonlyon,R=MisafunctionofE=MorAugust21,201413:18 32.Passageofparticlesthroughmatter 1.010100100010000 1.01010010001.0101001000 p/Mc Muon momentum (GeV/c) H2 liquid He gas C Al Fe 1.010100100010000 Figure32.2:Meanenergylossrateinliquid(bubblechamber)hydrogen,gaseoushelium,carbon,aluminum,iron,tin,andlead.Radiativee®ects,relevantformuonsandpions,arenotincluded.Thesebecomesigni¯cantformuonsinironfor¯°�1000,andatlowermomentaformuonsinhigher-Zabsorbers.SeeFig.32.23.pc=M.Inpractice,rangeisausefulconceptonlyforlow-energyhadrons(RI,whereIisthenuclearinteractionlength),andformuonsbelowafewhundredGeV(abovewhichradiativee®ectsdominate).R=Masafunctionof¯°=p=McisshownforavarietyofmaterialsinFig.32.4.ThemassscalingofdE=dxandrangeisvalidfortheelectroniclossesdescribedbytheBetheequation,butnotforradiativelosses,relevantonlyformuonsandpions.32.2.4.Meanexcitationenergy:\Thedeterminationofthemeanexcitationenergyistheprincipalnon-trivialtaskintheevaluationoftheBethestopping-powerformula"[10].Recommendedvalueshavevariedsubstantiallywithtime.Estimatesbasedonexperimentalstopping-powermeasurementsforprotons,deuterons,andalphaparticlesandonoscillator-strengthdistributionsanddielectric-responsefunctionsweregiveninICRU49[4].SeealsoICRU37[11].Thesevalues,showninFig.32.5,havesincebeenwidelyused.Machine-readableversionscanalsobefound[12].August21,201413:18 32.Passageofparticlesthroughmatter 0.51.01.52.02.5 125102050100 HHeLiBeBCNONeSn Solids H2 gas: 4.10H2 liquid: 3.97 2.35 Ð 0.28 ln(Z) áÐdE/dxñ (MeV gÐ1cm2) Figure32.3:Stoppingpoweratminimumionizationforthechemicalelements.Thestraightlineis¯ttedforZ�6.AsimplefunctionaldependenceonZisnottobeexpected,sinceh¡dE=dxialsodependsonothervariables.32.2.5.Densitye®ect:Astheparticleenergyincreases,itselectric¯eld°attensandextends,sothatthedistant-collisioncontributiontoEq.(32:5)increasesasln¯°.However,realmediabecomepolarized,limitingthe¯eldextensionande®ectivelytruncatingthispartofthelogarithmicrise[2{8,15{16].Atveryhighenergies,±=2!ln(~!p=I)+ln¯°1=2;(32:6)where(¯°)=2isthedensitye®ectcorrectionintroducedinEq.(32:5)and~!pistheplasmaenergyde¯nedinTable32.1.AcomparisonwithEq.(32:5)showsthatjdE=dxjthengrowsasln¯°ratherthanln22,andthatthemeanexcitationenergyIisreplacedbytheplasmaenergy~!p.Theionizationstoppingpowerascalculatedwithandwithoutthedensitye®ectcorrectionisshowninFig.32.1.Sincetheplasmafrequencyscalesasthesquarerootoftheelectrondensity,thecorrectionismuchlargerforaliquidorsolidthanforagas,asisillustratedbytheexamplesinFig.32.2.Thedensitye®ectcorrectionisusuallycomputedusingSternheimer'sparameteriza-tion[15]:(¯°)=8��&#x-4.4;䦁&#x-4.4;䦁:2(ln10)x Cifxx1;2(ln10)x C+a(x1x)kifx0xx1;0ifxx0(nonconductors);0102(xx0)ifxx0(conductors)(32:7)Herex=log10=log10(p=Mc). C(thenegativeoftheCusedinRef.15)isobtainedbyequatingthehigh-energycaseofEq.(32:7)withthelimitgiveninEq.(32:6).Theotherparametersareadjustedtogiveabest¯ttotheresultsofdetailedcalculationsformomentabelowMcexp(x1).Parametersforelementsandnearly200compoundsandmixturesofinterestarepublishedinavarietyofplaces,notablyinRef.16.ArecipeforAugust21,201413:18 32.Passageofparticlesthroughmatter 0.020.5 0.10.50.21.05.02.010.050.020.0Proton momentum (GeV/c)0.050.020.10.50.21.05.02.010.0Muon momentum (GeV/c)bg = p/Mc 1 2 5 10 20 50 100 200 500 1000 2000 5000100002000050000 252525 H2 liquidHe gasPbFeC Figure32.4:Rangeofheavychargedparticlesinliquid(bubblechamber)hydrogen,heliumgas,carbon,iron,andlead.Forexample:ForaK+whosemomentumis700MeV/c,¯°=1:42.ForleadwereadR=M396,andsotherangeis195gcm2(17cm).¯ndingthecoe±cientsfornontabulatedmaterialsisgivenbySternheimerandPeierls[17],andissummarizedinRef.5.Theremainingrelativisticrisecomesfromthe2growthofWmax,whichinturnisdueto(rare)largeenergytransferstoafewelectrons.Whentheseeventsareexcluded,theenergydepositinanabsorbinglayerapproachesaconstantvalue,theFermiplateau(seeSec.32.2.8below).Atevenhigherenergies(e.g.,�332GeVformuonsiniron,andataconsiderablyhigherenergyforprotonsiniron),radiativee®ectsaremoreimportantthanionizationlosses.Theseareespeciallyrelevantforhigh-energymuons,asdiscussedinSec.32.6.August21,201413:18 32.Passageofparticlesthroughmatter 0102030405060708090100 Iadj/Z (eV)Z Barkas & Berger 1964Bichsel 1992 ICRU 37 (1984)\r \r Figure32.5:Meanexcitationenergies(dividedbyZ)asadoptedbytheICRU[11].Thosebasedonexperimentalmeasurementsareshownbysymbolswitherror°ags;theinterpolatedvaluesaresimplyjoined.ThegreypointisforliquidH2;theblackpointat19.2eVisforH2gas.TheopencirclesshowmorerecentdeterminationsbyBichsel[13].Thedash-dottedcurveisfromtheapproximateformulaofBarkas[14]usedinearlyeditionsofthisReview.32.2.6.Energylossatlowenergies:ShellcorrectionsC=ZmustbeincludedinthesquarebracketsofofEq.(32:5)[4,11,13,14]tocorrectforatomicbindinghavingbeenneglectedincalculatingsomeofthecontributionstoEq.(32:5).TheBarkasform[14]wasusedingeneratingFig.32.1.Forcopperitcontributesabout1%at¯°=0:3(kineticenergy6MeVforapion),andthecorrectiondecreasesveryrapidlywithincreasingenergy.Equation32.2,andthereforeEq.(32:5),arebasedona¯rst-orderBornapproximation.Higher-ordercorrections,againimportantonlyatlowerenergies,arenormallyincludedbyaddingthe\Blochcorrection"z2L2()insidethesquarebrackets(Eq.(2.5)in[4]).Anadditional\Barkascorrection"zL1()reducesthestoppingpowerforanegativeparticlebelowthatforapositiveparticlewiththesamemassandvelocity.Ina1956paper,Barkasetal.notedthatnegativepionshadalongerrangethanpositivepions[6].Thee®ecthasbeenmeasuredforanumberofnegative/positiveparticlepairs,includingadetailedstudywithantiprotons[18].Adetaileddiscussionoflow-energycorrectionstotheBetheformulaisgiveninICRU49[4].Whenthecorrectionsareproperlyincluded,theBethetreatmentisaccuratetoabout1%downto0:05,orabout1MeVforprotons.For0:01¯0:05,thereisnosatisfactorytheory.Forprotons,oneusuallyreliesonthephenomenological¯ttingformulaedevelopedbyAndersenandZiegler[4,19].AstabulatedinICRU49[4],thenuclearpluselectronicprotonstoppingpowerincopperis113MeVcm2g1atT=10keV(¯°=0:005),risestoamaximumof210MeVcm2g1atT120keV(¯°=0:016),thenfallsto118MeVcm2g1atT=1MeV(¯°=0:046).August21,201413:18 32.PassageofparticlesthroughmatterAbove0.5{1.0MeVthecorrectedBethetheoryisadequate.Forparticlesmovingmoreslowlythan0:01c(moreorlessthevelocityoftheouteratomicelectrons),Lindhardhasbeenquitesuccessfulindescribingelectronicstoppingpower,whichisproportionalto[20].Finally,wenotethatatevenlowerenergies,e.g.,forprotonsoflessthanseveralhundredeV,non-ionizingnuclearrecoilenergylossdominatesthetotalenergyloss[4,20,21].32.2.7.Energeticknock-onelectrons(rays):ThedistributionofsecondaryelectronswithkineticenergiesTIis[2]d2N dTdx=1 2Kz2Z A1 2F(T) T2(32:8)forITWmax,whereWmaxisgivenbyEq.(32:4).Hereisthevelocityoftheprimaryparticle.ThefactorFisspin-dependent,butisaboutunityforTWmax.Forspin-0particlesF(T)=(12T=Wmax);formsforspins1/2and1arealsogivenbyRossi[2](Sec.2.3,Eqns.7and8).AdditionalformulaearegiveninRef.22.Equation(32:8)isinaccurateforTclosetoI[23].raysofevenmodestenergyarerare.Fora1particle,forexample,onaverageonlyonecollisionwithTe�10keVwilloccuralongapathlengthof90cmofArgas[1].AraywithkineticenergyTeandcorrespondingmomentumpeisproducedatananglegivenbycos=(Te=pe)(pmax=Wmax);(32:9)wherepmaxisthemomentumofanelectronwiththemaximumpossibleenergytransferWmax.32.2.8.Restrictedenergylossratesforrelativisticionizingparticles:FurtherinsightcanbeobtainedbyexaminingthemeanenergydepositbyanionizingparticlewhenenergytransfersarerestrictedtoTWcutWmax.TherestrictedenergylossrateisdE dxTWcut=Kz2Z A1 21 2ln2mec222Wcut I22 21+Wcut Wmax 2:(32:10)ThisformapproachesthenormalBethefunction(Eq.(32:5))asWcut!Wmax.Itcanbeveri¯edthatthedi®erencebetweenEq.(32:5)andEq.(32:10)isequaltoRWmaxWcutT(d2N=dTdx)dT,whered2N=dTdxisgivenbyEq.(32:8).SinceWcutreplacesWmaxintheargumentofthelogarithmictermofEq.(32:5),the¯°termproducingtherelativisticriseintheclose-collisionpartofdE=dxisreplacedbyaconstant,andjdE=dxjTWcutapproachestheconstant\Fermiplateau."(Thedensitye®ectcorrectioneliminatestheexplicit¯°dependenceproducedbythedistant-collisioncontribution.)ThisbehaviorisillustratedinFig.32.6,whererestrictedlossratesfortwoexamplesofWcutareshownincomparisonwiththefullBethedE=dxandtheLandau-Vavilovmostprobableenergyloss(tobediscussedinSec.32.2.9below).August21,201413:18 32.Passageofparticlesthroughmatter /x for:Restricted energy loss for: 0.11.010.0100.01000.0 = 1600 Figure32.6:BethedE=dx,twoexamplesofrestrictedenergyloss,andtheLandaumostprobableenergyperunitthicknessinsilicon.Thechangeof¢p=xwiththicknessxillustratesitsalnx+bdependence.Minimumionization(dE=dxjmin)is1.664MeVg1cm2.Radiativelossesareexcluded.Theincidentparticlesaremuons.\Restrictedenergyloss"iscutatthetotalmeanenergy,notthesingle-collisionenergyaboveWcutItisoflimiteduse.Themostprobableenergyloss,discussedinthenextSection,isfarmoreusefulinsituationswheresingle-particleenergylossisobserved.32.2.9.Fluctuationsinenergyloss:Fordetectorsofmoderatethicknessx(e.g.scintillatorsorLArcells),*theenergylossprobabilitydistributionf(¢;¯°;x)isade-quatelydescribedbythehighly-skewedLandau(orLandau-Vavilov)distribution[24,25].Themostprobableenergylossis[26]yp=ln2mc222 I+ln I+j2(¯°);(32:11)where=(K=2)hZ=Ai(x=¯2)MeVforadetectorwithathicknessxingcm2,andj=0:200[26].zWhiledE=dxisindependentofthickness,¢p=xscalesasalnx+b.Thedensitycorrection(¯°)wasnotincludedinLandau'sorVavilov'swork,butitwaslater *G0:05{0.1,whereGisgivenbyRossi[Ref.2,Eq.2.7(10)].ItisVavilov's[25].Itisproportionaltotheabsorber'sthickness,andassuchparameterizestheconstantsdescribingtheLandaudistribution.ThesearefairlyinsensitivetothicknessforG0:1,thecaseformostdetectors.yPracticalcalculationscanbeexpeditedbyusingthetablesofandfromthetextver-sionsofthemuonenergylosstablestobefoundatpdg.lbl.gov/AtomicNuclearProperties.zRossi[2],Talman[27],andothersgivesomewhatdi®erentvaluesforj.Themostprobablelossisnotsensitivetoitsvalue.August21,201413:18 32.PassageofparticlesthroughmatterincludedbyBichsel[26].Thehigh-energybehaviorof(¯°)(Eq.(32:6))issuchthatp¡!¯°�100ln2mc2 (~!p)2+j:(32:12)ThustheLandau-Vavilovmostprobableenergyloss,liketherestrictedenergyloss,reachesaFermiplateau.TheBethedE=dxandLandau-Vavilov-Bichsel¢p=xinsiliconareshownasafunctionofmuonenergyinFig.32.6.Theenergydepositinthe1600mcaseisroughlythesameasina3mmthickplasticscintillator. ) [MeVElectronic energy loss [MeV]Energy loss [MeV cm/g]150100500.40.50.60.70.81.00.90.81.00.60.40.20.0Landau-VavilovBichsel (Bethe-Fano theory) Dp D fwhm )/(D)/M(¥)10 GeV muon1.7 mm Si 1.21.41.61.82.02.22.4 &#x-137;�.06; Figure32.7:Electronicenergydepositdistributionfora10GeVmuontraversing1.7mmofsilicon,thestoppingpowerequivalentofabout0.3cmofPVCscintillator[1,13,28].TheLandau-Vavilovfunction(dot-dashed)usesaRutherfordcrosssectionwithoutatomicbindingcorrectionsbutwithakineticenergytransferlimitofWmax.ThesolidcurvewascalculatedusingBethe-Fanotheory.M0(¢)andM1(¢)arethecumulative0thmoment(meannumberofcollisions)and1stmoment(meanenergyloss)incrossingthesilicon.(SeeSec.32.2.1.ThefwhmoftheLandau-Vavilovfunctionisabout4fordetectorsofmoderatethickness.¢pisthemostprobableenergyloss,andhidividedbythethicknessistheBethehdE=dxi.Thedistributionfunctionfortheenergydepositbya10GeVmuongoingthroughadetectorofaboutthisthicknessisshowninFig.32.7.Inthiscasethemostprobableenergylossis62%ofthemean(M1(hi)=M1(1)).Foldinginexperimentalresolutiondisplacesthepeakofthedistribution,usuallytowardahighervalue.90%ofthecollisions(M1(hi)=M1(1))contributetoenergydepositsbelowthemean.Itistheveryrarehigh-energy-transfercollisions,extendingtoWmaxatseveralGeV,thatdrivesthemeanintothetailofthedistribution.Thelargeweightoftheserareeventsmakesthemeanofanexperimentaldistributionconsistingofafewhundredeventssubjecttolarge°uctuationsandsensitivetocuts.ThemeanoftheenergylossgivenbytheBetheAugust21,201413:18 32.Passageofparticlesthroughmatter 100200300400500600 0.501.001.502.002.50 640 m (149 mg/cm320 m (74.7 mg/cm160 m (37.4 mg/cm 80 m (18.7 mg/cm 500 MeV pion in silicon Mean energyloss rate w f(D/x) D/x (eV/mm) Dp/x D/x (MeV g-1 cm2) Figure32.8:Stragglingfunctionsinsiliconfor500MeVpions,normalizedtounityatthemostprobablevaluep=x.Thewidthwisthefullwidthathalfmaximum.equation,Eq.(32:5),isthusill-de¯nedexperimentallyandisnotusefulfordescribingenergylossbysingleparticles.\ItrisesaslnbecauseWmaxincreasesasathighenergies.Themostprobableenergylossshouldbeused.Apracticalexample:Formuonstraversing0.25inchesofPVTplasticscintillator,theratioofthemostprobableElossratetothemeanlossrateviatheBetheequationis[0:69;0:57;0:49;0:42;0:38]forT=[0:01;0:1;1;10;100]GeV.Radiativelossesaddlessthan0.5%tothetotalmeanenergydepositat10GeV,butadd7%at100GeV.ThemostprobableElossraterisesslightlybeyondtheminimumionizationenergy,thenisessentiallyconstant.TheLandaudistributionfailstodescribeenergylossinthinabsorberssuchasgasTPCcells[1]andSidetectors[26],asshownclearlyinFig.1ofRef.1foranargon-¯lledTPCcell.AlsoseeTalman[27].While¢p=xmaybecalculatedadequatelywithEq.(32:11),thedistributionsaresigni¯cantlywiderthantheLandauwidthw=4[Ref.26,Fig.15].Examplesfor500MeVpionsincidentonthinsilicondetectorsareshowninFig.32.8.ForverythickabsorbersthedistributionislessskewedbutneverapproachesaGaussian.Themostprobableenergyloss,scaledtothemeanlossatminimumionization,isshowninFig.32.9forseveralsilicondetectorthicknesses. \Itdoes¯ndapplicationindosimetry,whereonlybulkdepositisrelevant.August21,201413:18 32.Passageofparticlesthroughmatter 130.330300101001000 80 m (18.7 mg/cm160 m (37.4 mg/cm = 640 m (149 mg/cm)\r320 m (74.7 mg/cm Figure32.9:Mostprobableenergylossinsilicon,scaledtothemeanlossofaminimumionizingparticle,388eV/m(1.66MeVg1cm2).32.2.10.Energylossinmixturesandcompounds:Amixtureorcompoundcanbethoughtofasmadeupofthinlayersofpureelementsintherightproportion(Braggadditivity).Inthiscase,dE dx=XwjdE dxj;(32:13)wheredE=dxjjisthemeanrateofenergyloss(inMeVgcm2)inthejthelement.Eq.(32:5)canbeinsertedintoEq.(32:13)to¯ndexpressionsforhZ=Ai,hIi,andhi;forexample,hZ=Ai=PwjZj=Aj=PnjZj=PnjAj.However,hIiasde¯nedthiswayisanunderestimate,becauseinacompoundelectronsaremoretightlyboundthaninthefreeelements,andhiascalculatedthiswayhaslittlerelevance,becauseitistheelectrondensitythatmatters.Ifpossible,oneusesthetablesgiveninRefs.16and29,thatincludee®ectiveexcitationenergiesandinterpolationcoe±cientsforcalculatingthedensitye®ectcorrectionforthechemicalelementsandnearly200mixturesandcompounds.Otherwise,usetherecipeforgiveninRef.5and17,andcalculatehIifollowingthediscussioninRef.10.(Notethe\13%"rule!)32.2.11.Ionizationyields:Physicistsfrequentlyrelatetotalenergylosstothenumberofionpairsproducedneartheparticle'strack.Thisrelationbecomescomplicatedforrelativisticparticlesduetothewanderingofenergeticknock-onelectronswhoserangesexceedthedimensionsofthe¯ducialvolume.Foraqualitativeappraisalofthenonlocalityofenergydepositioninvariousmediabysuchmodestlyenergeticknock-onelectrons,seeRef.30.Themeanlocalenergydissipationperlocalionpairproduced,W,whileessentiallyconstantforrelativisticparticles,increasesatslowparticlespeeds[31].Forgases,Wcanbesurprisinglysensitivetotraceamountsofvariouscontaminants[31].Furthermore,ionizationyieldsinpracticalcasesmaybegreatlyin°uencedbysuchfactorsAugust21,201413:18 32.Passageofparticlesthroughmatterassubsequentrecombination[32].32.3.MultiplescatteringthroughsmallanglesAchargedparticletraversingamediumisde°ectedbymanysmall-anglescatters.Mostofthisde°ectionisduetoCoulombscatteringfromnucleiasdescribedbytheRutherfordcrosssection.(However,forhadronicprojectiles,thestronginteractionsalsocontributetomultiplescattering.)Formanysmall-anglescattersthenetscatteringanddisplacementdistributionsareGaussianviathecentrallimittheorem.Lessfrequent\hard"scattersproducenon-Gaussiantails.TheseCoulombscatteringdistributionsarewell-representedbythetheoryofMoliµere[34].AccessiblediscussionsaregivenbyRossi[2]andJackson[33],andexhaustivereviewshavebeenpublishedbyScott[35]andMotzetal.[36].ExperimentalmeasurementshavebeenpublishedbyBichsel[37](lowenergyprotons)andbyShenetal.[38](relativisticpions,kaons,andprotons).*Ifwede¯ne0=rmsplane=1 p 2rmsspace;(32:14)thenitissu±cientformanyapplicationstouseaGaussianapproximationforthecentral98%oftheprojectedangulardistribution,withanrmswidthgivenby[39,40]0=13.6MeV ¯cpzp x=X0h1+0:038ln(x=X0)i:(32:15)Herep,¯c,andzarethemomentum,velocity,andchargenumberoftheincidentparticle,andx=X0isthethicknessofthescatteringmediuminradiationlengths(de¯nedbelow).Thisvalueof0isfroma¯ttoMoliµeredistributionforsinglychargedparticleswith=1forallZ,andisaccurateto11%orbetterfor103x=X0100.Eq.(32:15)describesscatteringfromasinglematerial,whiletheusualprobleminvolvesthemultiplescatteringofaparticletraversingmanydi®erentlayersandmixtures.Sinceitisfroma¯ttoaMoliµeredistribution,itisincorrecttoaddtheindividual0contributionsinquadrature;theresultissystematicallytoosmall.ItismuchmoreaccuratetoapplyEq.(32:15)once,after¯ndingxandX0forthecombinedscatterer.Thenonprojected(space)andprojected(plane)angulardistributionsaregivenapproximatelyby[34]1 2¼µ20exp8���:2space 2209���;d;(32:16)1 p 2¼µ0exp8����:2plane 2209����;dµplane;(32:17)whereisthede°ectionangle.Inthisapproximation,2space(2plane;x+2plane;y),wherethexandyaxesareorthogonaltothedirectionofmotion,andddµplane;xdµplane;y.De°ectionsintoplane;xandplane;yareindependentandidenticallydistributed. *Shenetal.'smeasurementsshowthatBethe'ssimplermethodsofincludingatomicelectrone®ectsagreesbetterwithexperimentthandoesScott'streatment.August21,201413:18 32.Passageofparticlesthroughmatter xsplane yplaneYplaneqplane x/2 Figure32.10:QuantitiesusedtodescribemultipleCoulombscattering.Theparticleisincidentintheplaneofthe¯gure.Fig.32.10showstheseandotherquantitiessometimesusedtodescribemultipleCoulombscattering.Theyarermsplane=1 p 3rmsplane=1 p 30;(32:18)yrmsplane=1 p 3xµrmsplane=1 p 3xµ0;(32:19)srmsplane=1 4p 3xµrmsplane=1 4p 3xµ0:(32:20)Allthequantitativeestimatesinthissectionapplyonlyinthelimitofsmallrmsplaneandintheabsenceoflarge-anglescatters.Therandomvariabless,,y,andinagivenplanearecorrelated.Obviously,yxÃ.Inaddition,yandhavethecorrelationcoe±cientyµ=p 3=20:87.ForMonteCarlogenerationofajoint(yplane;µplane)distribution,orforothercalculations,itmaybemostconvenienttoworkwithindependentGaussianrandomvariables(z1;z2)withmeanzeroandvarianceone,andthensetyplane=z1xµ0(12yµ)1=2=p 3+z2yµxµ0=p 3(32:21)=z1xµ0=p 12+z2xµ0=2;(32:22)plane=z20:(32:23)Notethatthesecondtermforyplaneequalsxµplane=2andrepresentsthedisplacementthatwouldhaveoccurredhadthede°ectionplanealloccurredatthesinglepointx=2.ForheavyionsthemultipleCoulombscatteringhasbeenmeasuredandcomparedwithvarioustheoreticaldistributions[41].August21,201413:18 32.Passageofparticlesthroughmatter32.4.PhotonandelectroninteractionsinmatterAtlowenergieselectronsandpositronsprimarilyloseenergybyionization,althoughotherprocesses(M¿llerscattering,Bhabhascattering,e+annihilation)contribute,asshowninFig.32.11.Whileionizationlossratesriselogarithmicallywithenergy,bremsstrahlunglossesrisenearlylinearly(fractionallossisnearlyindependentofenergy),anddominatesabovethecriticalenergy(Sec.32.4.4below),afewtensofMeVinmostmaterials32.4.1.Collisionenergylossesbye:Stoppingpowerdi®erssomewhatforelectronsandpositrons,andbothdi®erfromstoppingpowerforheavyparticlesbecauseofthekinematics,spin,charge,andtheidentityoftheincidentelectronwiththeelectronsthatitionizes.CompletediscussionsandtablescanbefoundinRefs.10,11,and29.Forelectrons,largeenergytransferstoatomicelectrons(takenasfree)aredescribedbytheM¿llercrosssection.FromEq.(32:4),themaximumenergytransferinasinglecollisionshouldbetheentirekineticenergy,Wmax=mec2(1),butbecausetheparticlesareidentical,themaximumishalfthis,Wmax=2.(TheresultsarethesameifthetransferredenergyisorifthetransferredenergyisWmax.Thestoppingpowerisbyconventioncalculatedforthefasterofthetwoemergingelectrons.)The¯rstmomentoftheM¿llercrosssection[22](dividedbydx)isthestoppingpower:dE dx=1 2KZ A1 2lnmec222fmec2(1)=2g I2+(12)21 2ln2+1 81 2#(32:24)ThelogarithmictermcanbecomparedwiththelogarithmictermintheBetheequation(Eq.(32:2))bysubstitutingWmax=mec2(1)=2.Thetwoformsdi®erbyln2.Electron-positronscatteringisdescribedbythefairlycomplicatedBhabhacrosssection[22].Thereisnoidenticalparticleproblem,soWmax=mec2(1).The¯rstmomentoftheBhabhaequationyieldsdE dx=1 2KZ A1 2lnmec222fmec2(1)g 2I2+2ln22 1223+14 +1+10 (+1)2+4 (+1)3:(32:25)FollowingICRU37[11],thedensitye®ectcorrectionhasbeenaddedtoUehling'sequations[22]inbothcases.Forheavyparticles,shellcorrectionsweredevelopedassumingthattheprojectileisequivalenttoaperturbingpotentialwhosecentermoveswithconstantvelocity.Thisassumptionhasnosoundtheoreticalbasisforelectrons.TheauthorsofICRU37[11]estimatedthepossibleerrorinomittingitbyassumingthecorrectionwastwiceasgreatasforaprotonofthesamevelocity.AtT=10keV,theerrorwasestimatedtobe2%forwater,9%forCu,and21%forAu.AsshowninFig.32.11,stoppingpowersfore,e+,andheavyparticlesarenotdramaticallydi®erent.Insilicon,theminimumvalueforelectronsis1.50MeVcm2/g(atAugust21,201413:18 32.Passageofparticlesthroughmatter=3:3);forpositrons,1.46MeVcm2/g(at=3:7),andformuons,1.66MeVcm2/g(at=3:58).32.4.2.Radiationlength:High-energyelectronspredominantlyloseenergyinmatterbybremsstrahlung,andhigh-energyphotonsbye+epairproduction.ThecharacteristicamountofmattertraversedfortheserelatedinteractionsiscalledtheradiationlengthX0,usuallymeasuredingcm2.Itisboth(a)themeandistanceoverwhichahigh-energyelectronlosesallbut1=eofitsenergybybremsstrahlung,and(b)7 9ofthemeanfreepathforpairproductionbyahigh-energyphoton[42].Itisalsotheappropriatescalelengthfordescribinghigh-energyelectromagneticcascades.X0hasbeencalculatedandtabulatedbyY.S.Tsai[43]:1 X0=4®r2eNA AnZ2Lradf(Z)+ZL0rado:(32:26)ForA=1gmol1,4®r2eNA=A=(716:408gcm2)1.LradandL0radaregiveninTable32.2.Thefunctionf(Z)isanin¯nitesum,butforelementsuptouraniumcanberepresentedto4-placeaccuracybyf(Z)=a2(1+a2)1+0:202060:0369a2+0:0083a40:002a6;(32:27)wherea=®Z[44].Table32.2:Tsai'sLradandL0rad,foruseincalculatingtheradiationlengthinanelementusingEq.(32:26). ElementZLradL0rad H15.316.144He24.795.621Li34.745.805Be44.715.924Others�4ln(184:15Z1=3)ln(1194Z2=3) Theradiationlengthinamixtureorcompoundmaybeapproximatedby1=X0=Xwj=Xj;(32:28)wherewjandXjarethefractionbyweightandtheradiationlengthforthejthelement.August21,201413:18 32.Passageofparticlesthroughmatter Figure32.11:Fractionalenergylossperradiationlengthinleadasafunctionofelectronorpositronenergy.Electron(positron)scatteringisconsideredasionizationwhentheenergylosspercollisionisbelow0.255MeV,andasM¿ller(Bhabha)scatteringwhenitisabove.AdaptedfromFig.3.2fromMesselandCrawford,Electron-PhotonShowerDistributionFunctionTablesforLead,Copper,andAirAbsorbers,PergamonPress,1970.MesselandCrawforduseX0(Pb)=5.82g/cm2,butwehavemodi¯edthe¯gurestore°ectthevaluegivenintheTableofAtomicandNuclearPropertiesofMaterials(X0(Pb)=6.37g/cm2).32.4.3.Bremsstrahlungenergylossbye:Atveryhighenergiesandexceptatthehigh-energytipofthebremsstrahlungspectrum,thecrosssectioncanbeapproximatedinthe\completescreeningcase"as[43]d¾=dk=(1=k)4®r2e(4 34 3y+y2)[Z2(Lradf(Z))+ZL0rad]+1 9(1y)(Z2+Z);(32:29)wherey=k=Eisthefractionoftheelectron'senergytransferredtotheradiatedphoton.Atsmally(the\infraredlimit")thetermonthesecondlinerangesfrom1.7%(lowZ)to2.5%(highZ)ofthetotal.Ifitisignoredandthe¯rstlinesimpli¯edwiththede¯nitionofX0giveninEq.(32:26),wehaved¾ dk=A X0NAk4 34 3y+y2:(32:30)Thiscrosssection(timesk)isshownbythetopcurveinFig.32.12.Thisformulaisaccurateexceptinneary=1,wherescreeningmaybecomeincomplete,andneary=0,wheretheinfrareddivergenceisremovedbytheinterferenceofbremsstrahlungamplitudesfromnearbyscatteringcenters(theLPMe®ect)[45,46]anddielectricsuppression[47,48].Theseandothersuppressione®ectsinbulkmediaarediscussedinSec.32.4.6.August21,201413:18 32.Passageofparticlesthroughmatter 00.40.81.2 00.250.50.751 yd 100 GeV Figure32.12:Thenormalizedbremsstrahlungcrosssectionkd¾LPM=dkinleadversusthefractionalphotonenergyy=k=E.Theverticalaxishasunitsofphotonsperradiationlength.Withdecreasingenergy(E10GeV)thehigh-ycrosssectiondropsandthecurvesbecomeroundedasy!1.CurvesofthisfamilarshapecanbeseeninRossi[2](Figs.2.11.2,3);seealsothereviewbyKoch&Motz[49]. 25102050100200 Copper\r0 = 12.86 g cm-2\rEc = 19.63 MeV\r\rdE/dx ´ X0 (MeV)Electron energy (MeV) ToBremsxactbrestra Figure32.13:Twode¯nitionsofthecriticalenergyEc.Exceptattheseextremes,andstillinthecomplete-screeningapproximation,thenumberofphotonswithenergiesbetweenkminandkmaxemittedbyanelectrontravellingadistancedX0isN=d X0"4 3lnkmax kmin4(kmaxkmin) 3E+k2maxk2min 2E2#:(32:31)August21,201413:18 32.Passageofparticlesthroughmatter 125102050100 5 10 20 50100200400 610 MeV________ Z + 1.24710 MeV________Z + 0.92 Solids HHeLiBeBCNONeSn Figure32.14:Electroncriticalenergyforthechemicalelements,usingRossi'sde¯nition[2].The¯tsshownareforsolidsandliquids(solidline)andgases(dashedline).Thermsdeviationis2.2%forthesolidsand4.0%forthegases.(ComputedwithcodesuppliedbyA.Fass¶o.)32.4.4.Criticalenergy:Anelectronlosesenergybybremsstrahlungataratenearlyproportionaltoitsenergy,whiletheionizationlossratevariesonlylogarithmicallywiththeelectronenergy.ThecriticalenergyEcissometimesde¯nedastheenergyatwhichthetwolossratesareequal[50].Amongalternatede¯nitionsisthatofRossi[2],whode¯nesthecriticalenergyastheenergyatwhichtheionizationlossperradiationlengthisequaltotheelectronenergy.Equivalently,itisthesameasthe¯rstde¯nitionwiththeapproximationjdE=dxjbremsE=X0.Thisformhasbeenfoundtodescribetransverseelectromagneticshowerdevelopmentmoreaccurately(seebelow).Thesede¯nitionsareillustratedinthecaseofcopperinFig.32.13.TheaccuracyofapproximateformsforEchasbeenlimitedbythefailuretodistinguishbetweengasesandsolidorliquids,wherethereisasubstantialdi®erenceinionizationattherelevantenergybecauseofthedensitye®ect.WedistinguishthesetwocasesinFig.32.14.Fitswerealsomadewithfunctionsoftheforma=(Z+b),butwasfoundtobeessentiallyunity.SinceEcalsodependsonA,I,andotherfactors,suchformsareatbestapproximate.ValuesofEcforbothelectronsandpositronsinmorethan300materialscanbefoundatpdg.lbl.gov/AtomicNuclearProperties.32.4.5.Energylossbyphotons:Contributionstothephotoncrosssectioninalightelement(carbon)andaheavyelement(lead)areshowninFig.32.15.Atlowenergiesitisseenthatthephotoelectrice®ectdominates,althoughComptonscattering,Rayleighscattering,andphotonuclearabsorptionalsocontribute.Thephotoelectriccrosssectionischaracterizedbydiscontinuities(absorptionedges)asthresholdsforphotoionizationofvariousatomiclevelsarereached.PhotonattenuationlengthsforavarietyofelementsareshowninFig.32.16,anddatafor30eVk100GeVforallelementsareavailablefromthewebpagesgiveninthecaption.Herekisthephotonenergy.August21,201413:18 32.Passageofparticlesthroughmatter 10 eV1 keV1 MeV1 GeV100 GeV = 82) Cross section (barns/atom)Cross section (barns/atom) - experimental stot Figure32.15:Photontotalcrosssectionsasafunctionofenergyincarbonandlead,showingthecontributionsofdi®erentprocesses[51]:p:e:=Atomicphotoelectrice®ect(electronejection,photonabsorption)Rayleigh=Rayleigh(coherent)scattering{atomneitherionizednorexcitedCompton=Incoherentscattering(Comptonscatteringo®anelectron)nuc=Pairproduction,nuclear¯elde=Pairproduction,electron¯eldg:d::=Photonuclearinteractions,mostnotablytheGiantDipoleResonance[52].Intheseinteractions,thetargetnucleusisbrokenup.Original¯guresthroughthecourtesyofJohnH.Hubbell(NIST).August21,201413:18 32.Passageofparticlesthroughmatter Ð4Ð5Ð610 eV100 eV1 keV10 keV100 keV1 MeV10 MeV100 MeV1 GeV10 GeV100 GeV (g SiC FePbHSn Figure32.16:Thephotonmassattenuationlength(ormeanfreepath)=1=(¹=½)forvariouselementalabsorbersasafunctionofphotonenergy.Themassattenuationcoe±cientis¹=½,whereisthedensity.TheintensityIremainingaftertraversalofthicknesst(inmass/unitarea)isgivenbyI=I0exp(t=¸).Theaccuracyisafewpercent.Forachemicalcompoundormixture,1=¸e®PelementswZ=¸Z,wherewZistheproportionbyweightoftheelementwithatomicnumberZ.TheprocessesresponsibleforattenuationaregiveninFig.32.11.Sincecoherentprocessesareincluded,notalltheseprocessesresultinenergydeposition.Thedatafor30eVE1keVareobtainedfromhttp://www-cxro.lbl.gov/optical constants(courtesyofEricM.Gullikson,LBNL).Thedatafor1keVE100GeVarefromhttp://physics.nist.gov/PhysRefData,throughthecourtesyofJohnH.Hubbell(NIST).August21,201413:18 32.Passageofparticlesthroughmatter Figure32.17:ProbabilityPthataphotoninteractionwillresultinconversiontoane+epair.Exceptforafew-percentcontributionfromphotonuclearabsorptionaround10or20MeV,essentiallyallotherinteractionsinthisenergyrangeresultinComptonscatteringo®anatomicelectron.Foraphotonattenuationlength(Fig.32.16),theprobabilitythatagivenphotonwillproduceanelectronpair(without¯rstComptonscattering)inthicknesstofabsorberisP[1exp(t=¸)]. 00.250.50.751 x = E/k Pair production(X0NA/A) dsLPM/dx1 TeV10 TeV100 TeV1 PeV10 PeV 1 EeV100 PeV Figure32.18:Thenormalizedpairproductioncrosssectiond¾LPM=dy,versusfractionalelectronenergyx=E=k.August21,201413:18 32.PassageofparticlesthroughmatterTheincreasingdominationofpairproductionastheenergyincreasesisshowninFig.32.17.UsingapproximationssimilartothoseusedtoobtainEq.(32:30),Tsai'sformulaforthedi®erentialcrosssection[43]reducestod¾ dx=A X0NA14 3x(1x)(32:32)inthecomplete-screeninglimitvalidathighenergies.Herex=E=kisthefractionalenergytransfertothepair-producedelectron(orpositron),andkistheincidentphotonenergy.Thecrosssectionisverycloselyrelatedtothatforbremsstrahlung,sincetheFeynmandiagramsarevariantsofoneanother.Thecrosssectionisofnecessitysymmetricbetweenxand1x,ascanbeseenbythesolidcurveinFig.32.18.SeethereviewbyMotz,Olsen,&Kochforamoredetailedtreatment[53].Eq.(32:32)maybeintegratedto¯ndthehigh-energylimitforthetotale+epair-productioncrosssection:=7 9(A=X0NA):(32:33)EquationEq.(32:33)isaccuratetowithinafewpercentdowntoenergiesaslowas1GeV,particularlyforhigh-Zmaterials.32.4.6.Bremsstrahlungandpairproductionatveryhighenergies:Atultra-highenergies,Eqns.32.29{32.33willfailbecauseofquantummechanicalinterferencebetweenamplitudesfromdi®erentscatteringcenters.Sincethelongitudinalmomentumtransfertoagivencenterissmall(/k=E(Ek),inthecaseofbremsstrahlung),theinteractionisspreadoveracomparativelylongdistancecalledtheformationlength(/E(Ek)=k)viatheuncertaintyprinciple.Inalternatelanguage,theformationlengthisthedistanceoverwhichthehighlyrelativisticelectronandthephoton\splitapart."Theinterferenceisusuallydestructive.Calculationsofthe\Landau-Pomeranchuk-Migdal"(LPM)e®ectmaybemadesemi-classicallybasedontheaveragemultiplescattering,ormorerigorouslyusingaquantumtransportapproach[45,46].Inamorphousmedia,bremsstrahlungissuppressedifthephotonenergykislessthanE2=(E+ELPM)[46],where*ELPM=(mec2)2®X0 4~c½=(7:7TeV/cm)X0 :(32:34)Sincephysicaldistancesareinvolved,X0=½,incm,appears.Theenergy-weightedbremsstrahlungspectrumforlead,kd¾LPM=dk,isshowninFig.32.12.WithappropriatescalingbyX0=½,othermaterialsbehavesimilarly.Forphotons,pairproductionisreducedforE(kE)�kELPM.Thepair-productioncrosssectionsfordi®erentphotonenergiesareshowninFig.32.18.IfkE,severaladditionalmechanismscanalsoproducesuppression.Whentheformationlengthislong,evenweakfactorscanperturbtheinteraction.Forexample,theemittedphotoncancoherentlyforwardscattero®oftheelectronsinthemedia. *Thisde¯nitiondi®ersfromthatofRef.54byafactoroftwo.ELPMscalesasthe4thpowerofthemassoftheincidentparticle,sothatELPM=(1:41010TeV/cm)X0=½foramuon.August21,201413:18 32.PassageofparticlesthroughmatterBecauseofthis,fork!pE=me104,bremsstrahlungissuppressedbyafactor(kme=!pE)2[48].Magnetic¯eldscanalsosuppressbremsstrahlung.Incrystallinemedia,thesituationismorecomplicated,withcoherentenhancementorsuppressionpossible.Thecrosssectiondependsontheelectronandphotonenergiesandtheanglesbetweentheparticledirectionandthecrystallineaxes[55].32.4.7.Photonuclearandelectronuclearinteractionsatstillhigherenergies:Atstillhigherphotonandelectronenergies,wherethebremsstrahlungandpairproductioncross-sectionsareheavilysuppressedbytheLPMe®ect,photonuclearandelectronuclearinteractionspredominateoverelectromagneticinteractions.Atphotonenergiesaboveabout1020eV,forexample,photonsusuallyinteracthadronically.Theexactcross-overenergydependsonthemodelusedforthephotonuclearinteractions.TheseprocessesareillustratedinFig.32.19.Atstillhigherenergies(&#x-406;&#x.660;1023eV),photonuclearinteractionscanbecomecoherent,withthephotoninteractionspreadovermultiplenuclei.Essentially,thephotoncoherentlyconvertstoa0,inaprocessthatissomewhatsimilartokaonregeneration[56]. [eV]10log101214161820222426 (Interaction Length) [m]10log !1 BH Mig A" A" + Mig Figure32.19:InteractionlengthforaphotoniniceasafunctionofphotonenergyfortheBethe-Heitler(BH),LPM(Mig)andphotonuclear(°A)crosssections[56].TheBethe-Heitlerinteractionlengthis9X0=7,andX0is0.393minice.Similarprocessesoccurforelectrons.AselectronenergiesincreaseandtheLPMe®ectsuppressesbremsstrahlung,electronuclearinteractionsbecomemoreimportant.Atenergiesabove1021eV,theseelectronuclearinteractionsdominateelectronenergyloss[56].August21,201413:18 32.Passageofparticlesthroughmatter32.5.ElectromagneticcascadesWhenahigh-energyelectronorphotonisincidentonathickabsorber,itinitiatesanelectromagneticcascadeaspairproductionandbremsstrahlunggeneratemoreelectronsandphotonswithlowerenergy.Thelongitudinaldevelopmentisgovernedbythehigh-energypartofthecascade,andthereforescalesastheradiationlengthinthematerial.Electronenergieseventuallyfallbelowthecriticalenergy,andthendissipatetheirenergybyionizationandexcitationratherthanbythegenerationofmoreshowerparticles.Indescribingshowerbehavior,itisthereforeconvenienttointroducethescalevariablest=x=X0;y=E=Ec;(32:35)sothatdistanceismeasuredinunitsofradiationlengthandenergyinunitsofcriticalenergy. 0.0000.0250.0500.0750.1000.125020406080100(1/E0)dE/dtt = depth in radiation lengthsNumber crossing plane30 GeV electron\rEnergyPhotons\r´ 1/6.8Electrons 05101520 Figure32.20:AnEGS4simulationofa30GeVelectron-inducedcascadeiniron.Thehistogramshowsfractionalenergydepositionperradiationlength,andthecurveisagamma-function¯ttothedistribution.Circlesindicatethenumberofelectronswithtotalenergygreaterthan1.5MeVcrossingplanesatX0=2intervals(scaleonright)andthesquaresthenumberofphotonswithE1:5MeVcrossingtheplanes(scaleddowntohavesameareaastheelectrondistribution).Longitudinalpro¯lesfromanEGS4[57]simulationofa30GeVelectron-inducedcascadeinironareshowninFig.32.20.Thenumberofparticlescrossingaplane(veryclosetoRossi's¦function[2])issensitivetothecuto®energy,herechosenasatotalenergyof1.5MeVforbothelectronsandphotons.Theelectronnumberfallso®morequicklythanenergydeposition.Thisisbecause,withincreasingdepth,alargerfractionofthecascadeenergyiscarriedbyphotons.Exactlywhatacalorimetermeasuresdependsonthedevice,butitisnotlikelytobeexactlyanyofthepro¯lesshown.Ingascountersitmaybeveryclosetotheelectronnumber,butinglassCherenkovdetectorsandotherdeviceswith\thick"sensitiveregionsitisclosertotheenergydeposition(totaltrackAugust21,201413:18 32.Passageofparticlesthroughmatterlength).Insuchdetectorsthesignalisproportionaltothe\detectable"tracklengthTd,whichisingenerallessthanthetotaltracklengthT.PracticaldevicesaresensitivetoelectronswithenergyabovesomedetectionthresholdEd,andTd=TF(Ed=Ec):AnanalyticformforF(Ed=Ec)obtainedbyRossi[2]isgivenbyFabjaninRef.58;seealsoAmaldi[59].Themeanlongitudinalpro¯leoftheenergydepositioninanelectromagneticcascadeisreasonablywelldescribedbyagammadistribution[60]:dE dt=E0b(bt)a1ebt ¡(a)(32:36)Themaximumtmaxoccursat(a1)=b.Wehavemade¯tstoshowerpro¯lesinelementsrangingfromcarbontouranium,atenergiesfrom1GeVto100GeV.Theenergydepositionpro¯lesarewelldescribedbyEq.(32:36)withtmax=(a1)=b=1:0(lny+Cj);j=e;°;(32:37)whereCe=0:5forelectron-inducedcascadesandC=+0:5forphoton-inducedcascades.TouseEq.(32:36),one¯nds(a1)=bfromEq.(32:37)andEq.(32:35),then¯ndsaeitherbyassumingb0:5orby¯ndingamoreaccuratevaluefromFig.32.21.Theresultsareverysimilarfortheelectronnumberpro¯les,butthereissomedependenceontheatomicnumberofthemedium.AsimilarformfortheelectronnumbermaximumwasobtainedbyRossiinthecontextofhis\ApproximationB,"[2](seeFabjan'sreviewinRef.58),butwithCe=1:0andC=0:5;weregardthisassupersededbytheEGS4result. CarbonAluminumIronUranium0.30.40.50.60.70.8 1010010001 by = E/Ec Figure32.21:Fittedvaluesofthescalefactorbforenergydepositionpro¯lesobtainedwithEGS4foravarietyofelementsforincidentelectronswith1E0100GeV.Valuesobtainedforincidentphotonsareessentiallythesame.The\showerlength"Xs=X0=bislessconvenientlyparameterized,sincebdependsuponbothZandincidentenergy,asshowninFig.32.21.AsacorollaryofthisAugust21,201413:18 32.PassageofparticlesthroughmatterZdependence,thenumberofelectronscrossingaplanenearshowermaximumisunderestimatedusingRossi'sapproximationforcarbonandseriouslyoverestimatedforuranium.Essentiallythesamebvaluesareobtainedforincidentelectronsandphotons.Formanypurposesitissu±cienttotakeb0:5.Thelengthofshowersinitiatedbyultra-highenergyphotonsandelectronsissomewhatgreaterthanatlowerenergiessincethe¯rstor¯rstfewinteractionlengthsareincreasedviathemechanismsdiscussedabove.Thegammafunctiondistributionisvery°atneartheorigin,whiletheEGS4cascade(orarealcascade)increasesmorerapidly.AsaresultEq.(32:36)failsbadlyforaboutthe¯rsttworadiationlengths;itwasnecessarytoexcludethisregioninmaking¯ts.Because°uctuationsareimportant,Eq.(32:36)shouldbeusedonlyinapplicationswhereaveragebehaviorisadequate.Grindhammeretal.havedevelopedfastsimulationalgorithmsinwhichthevarianceandcorrelationofaandbareobtainedby¯ttingEq.(32:36)toindividuallysimulatedcascades,thengeneratingpro¯lesforcascadesusingaandbchosenfromthecorrelateddistributions[61].Thetransversedevelopmentofelectromagneticshowersindi®erentmaterialsscalesfairlyaccuratelywiththeMoliµereradiusRM,givenby[62,63]RM=X0Es=Ec;(32:38)whereEs21MeV(Table32.1),andtheRosside¯nitionofEcisused.InamaterialcontainingaweightfractionwjoftheelementwithcriticalenergyEcjandradiationlengthXj,theMoliµereradiusisgivenby1 RM=1 EsXwjEcj Xj:(32:39)MeasurementsofthelateraldistributioninelectromagneticcascadesareshowninRefs.62and63.Ontheaverage,only10%oftheenergyliesoutsidethecylinderwithradiusRM.About99%iscontainedinsideof3:5RM,butatthisradiusandbeyondcompositione®ectsbecomeimportantandthescalingwithRMfails.Thedistributionsarecharacterizedbyanarrowcore,andbroadenastheshowerdevelops.TheyareoftenrepresentedasthesumoftwoGaussians,andGrindhammer[61]describesthemwiththefunctionf(r)=2rR2 (r2+R2)2;(32:40)whereRisaphenomenologicalfunctionofx=X0andlnE.Athighenoughenergies,theLPMe®ect(Sec.32.4.6)reducesthecrosssectionsforbremsstrahlungandpairproduction,andhencecancausesigni¯cantelongationofelectromagneticcascades[46].August21,201413:18 32.Passageofparticlesthroughmatter32.6.MuonenergylossathighenergyAtsu±cientlyhighenergies,radiativeprocessesbecomemoreimportantthanionizationforallchargedparticles.Formuonsandpionsinmaterialssuchasiron,this\criticalenergy"occursatseveralhundredGeV.(Thereisnosimplescalingwithparticlemass,butforprotonsthe\criticalenergy"ismuch,muchhigher.)Radiativee®ectsdominatetheenergylossofenergeticmuonsfoundincosmicraysorproducedatthenewestaccelerators.Theseprocessesarecharacterizedbysmallcrosssections,hardspectra,largeenergy°uctuations,andtheassociatedgenerationofelectromagneticand(inthecaseofphotonuclearinteractions)hadronicshowers[64{72].Asaconsequence,attheseenergiesthetreatmentofenergylossasauniformandcontinuousprocessisformanypurposesinadequate.Itisconvenienttowritetheaveragerateofmuonenergylossas[73]dE=dx=a(E)+b(E)E:(32:41)Herea(E)istheionizationenergylossgivenbyEq.(32:5),andb(E)isthesumofe+epairproduction,bremsstrahlung,andphotonuclearcontributions.Totheapproximationthattheseslowly-varyingfunctionsareconstant,themeanrangex0ofamuonwithinitialenergyE0isgivenbyx0(1=b)ln(1+E0=E¹c);(32:42)whereE¹c=a=b.Fig.32.22showscontributionstob(E)foriron.Sincea(E)0:002GeVg1cm2,b(E)EdominatestheenergylossaboveseveralhundredGeV,whereb(E)isnearlyconstant.Theratesofenergylossformuonsinhydrogen,uranium,andironareshowninFig.32.23[5]. Muon energy (GeV)0123456789106 b(E) (g-1cm2)Ironbtotalbpairbbremsstrahlungbnuclear102101103104105 Figure32.22:Contributionstothefractionalenergylossbymuonsinironduetoe+epairproduction,bremsstrahlung,andphotonuclearinteractions,asobtainedfromGroometal.[5]exceptforpost-Borncorrectionstothecrosssectionfordirectpairproductionfromatomicelectrons.August21,201413:18 32.Passageofparticlesthroughmatter Figure32.23:Theaverageenergylossofamuoninhydrogen,iron,anduraniumasafunctionofmuonenergy.ContributionstodE=dxinironfromionizationandtheprocessesshowninFig.32.22arealsoshown.The\muoncriticalenergy"E¹ccanbede¯nedmoreexactlyastheenergyatwhichradiativeandionizationlossesareequal,andcanbefoundbysolvingE¹c=a(E¹c)=b(E¹c).Thisde¯nitioncorrespondstothesolid-lineintersectioninFig.32.13,andisdi®erentfromtheRosside¯nitionweusedforelectrons.Itservesthesamefunction:belowE¹cionizationlossesdominate,andaboveE¹cradiativee®ectsdominate.ThedependenceofE¹conatomicnumberZisshowninFig.32.24.Theradiativecrosssectionsareexpressedasfunctionsofthefractionalenergyloss.Thebremsstrahlungcrosssectiongoesroughlyas1=ºovermostoftherange,whileforthepairproductioncasethedistributiongoesas3to2[74].\Hard"lossesarethereforemoreprobableinbremsstrahlung,andinfactenergylossesduetopairproductionmayverynearlybetreatedascontinuous.Thesimulated[72]momentumdistributionofanincident1TeV/cmuonbeamafteritcrosses3mofironisshowninFig.32.25.Themostprobablelossis8GeV,or3.4MeVg1cm2.Thefullwidthathalfmaximumis9GeV/c,or0.9%.Theradiativetailisalmostentirelyduetobremsstrahlung,althoughmostoftheeventsinwhichmorethan10%oftheincidentenergylostexperiencedrelativelyhardphotonuclearinteractions.Thelattercanexceeddetectorresolution[75],necessitatingthereconstructionoflostenergy.TablesinRef.5listthestoppingpoweras9.82MeVg1cm2fora1TeVmuon,sothatthemeanlossshouldbe23GeV(23GeV/c),fora¯nalmomentumof977GeV/c,farbelowthepeak.Thisagreeswiththeindicatedmeancalculatedfromthesimulation.Electromagneticandhadroniccascadesindetectormaterialscanobscuremuontracksindetectorplanesandreducetrackinge±ciency[76].August21,201413:18 32.Passageofparticlesthroughmatter ___________\r (Z + 2.03)0.879___________\r (Z + 1.47)0.838 125102050100 HHeLiBeBCNONeSn SolidsGases Figure32.24:Muoncriticalenergyforthechemicalelements,de¯nedastheenergyatwhichradiativeandionizationenergylossratesareequal[5].Theequalitycomesatahigherenergyforgasesthanforsolidsorliquidswiththesameatomicnumberbecauseofasmallerdensitye®ectreductionoftheionizationlosses.The¯tsshowninthe¯gureexcludehydrogen.Alkalimetalsfall3{4%abovethe¯ttedfunction,whilemostothersolidsarewithin2%ofthefunction.Amongthegasestheworst¯tisforradon(2.7%high). 9509609709809901000 1 TeV muons\ron 3 m Fe Mean\r977 GeV/cMedian\r987 GeV/c dN/dp [1/(GeV/c)] FWHM\r9 GeV/c Figure32.25:Themomentumdistributionof1TeV/cmuonsaftertraversing3mofironascalculatedwiththeMARS15MonteCarlocode[72]byS.I.Striganov[5].August21,201413:18 32.Passageofparticlesthroughmatter32.7.Cherenkovandtransitionradiation[33,77,78]Achargedparticleradiatesifitsvelocityisgreaterthanthelocalphasevelocityoflight(Cherenkovradiation)orifitcrossessuddenlyfromonemediumtoanotherwithdi®erentopticalproperties(transitionradiation).Neitherprocessisimportantforenergyloss,butbothareusedinhigh-energyandcosmic-rayphysicsdetectors. qcgch Cherenkov wavefrontParticle velocity v = bcv = vg Figure32.26:Cherenkovlightemissionandwavefrontangles.Inadispersivemedium,c+=900.32.7.1.OpticalCherenkovradiation:TheanglecofCherenkovradiation,relativetotheparticle'sdirection,foraparticlewithvelocity¯cinamediumwithindexofrefractionniscosc=(1=n¯)ortanc=p 2n21p 2(11=n¯)forsmallc;e:g:ingases.(32:43)Thethresholdvelocitytis1=n,andt=1=(12t)1=2.Therefore,tt=1=(2+2)1=2,where=n1.ValuesofforvariouscommonlyusedgasesaregivenasafunctionofpressureandwavelengthinRef.79.Forvaluesatatmosphericpressure,seeTable6.1.DataforothercommonlyusedmaterialsaregiveninRef.80.PracticalCherenkovradiatormaterialsaredispersive.Let!bethephoton'sfrequency,andletk=2¼=¸beitswavenumber.Thephotonspropageatthegroupvelocityvg=d!=dk=c=[n(!)+!(dn=d!)].Inanon-dispersivemedium,thissimpliestovg=c=n.Inhisclassicalpaper,Tamm[81]showedthatfordispersivemediatheradiationisconcentratedinathinconicalshellwhosevertexisatthemovingcharge,andwhoseopeninghalf-angleisgivenbycot=d d!(!tanc)!0=tanc+2!n(!)dn d!cotc!0;(32:44)where!0isthecentralvalueofthesmallfrequencyrangeunderconsideration.(SeeFig.32.26.)Thisconehasaopeninghalf-angle,and,unlessthemediumisnon-dispersive(dn=d!=0),c+=900.TheCherenkovwavefront`sideslips'alongAugust21,201413:18 32.Passageofparticlesthroughmatterwiththeparticle[82].Thise®ecthastimingimplicationsforringimagingCherenkovcounters[83],butitisprobablyunimportantformostapplications.Thenumberofphotonsproducedperunitpathlengthofaparticlewithchargezeandperunitenergyintervalofthephotonsisd2N dEdx=®z2 ~csin2c=2z2 remec211 2n2(E)370sin2c(E)eV1cm1(z=1);(32:45)or,equivalently,d2N dxd¸=2¼®z2 211 2n2():(32:46)TheindexofrefractionnisafunctionofphotonenergyE=~!,asisthesensitivityofthetransducerusedtodetectthelight.Forpracticaluse,Eq.(32:45)mustbemultipliedbythethetransducerresponsefunctionandintegratedovertheregionforwhich¯n(!)�1.FurtherdetailsaregiveninthediscussionofCherenkovdetectorsintheParticleDetectorssection(Sec.33ofthisReview).Whentwoparticlesareclosetogether(lateralseparation1wavelength),theelectromagnetic¯eldsfromtheparticlesmayaddcoherently,a®ectingtheCherenkovradiation.Becauseoftheiroppositecharges,theradiationfromane+epairatcloseseparationissuppressedcomparedtotwoindependentleptons[84].32.7.2.CoherentradioCherenkovradiation:CoherentCherenkovradiationisproducedbymanychargedparticleswithanon-zeronetchargemovingthroughmatteronanapproximatelycommon\wavefront"|forexample,theelectronsandpositronsinahigh-energyelectromagneticcascade.Thesignalscanbevisibleabovebackgroundsforshowerenergiesaslowas1017eV;seeSec.34.3.3formoredetails.ThephenomenoniscalledtheAskaryane®ect[85].Neartheendofashower,whentypicalparticleenergiesarebelowEc(butstillrelativistic),achargeimbalancedevelops.PhotonscanCompton-scatteratomicelectrons,andpositronscanannihilatewithatomicelectronstocontributeevenmorephotonswhichcaninturnComptonscatter.Theseprocessesresultinaroughly20%excessofelectronsoverpositronsinashower.ThenetnegativechargeleadstocoherentradioCherenkovemission.Theradiationincludesacomponentfromthedeceleratingcharges(asinbremsstrahlung).Becausetheemissioniscoherent,theelectric¯eldstrengthisproportionaltotheshowerenergy,andthesignalpowerincreasesasitssquare.Theelectric¯eldstrengthalsoincreaseslinearlywithfrequency,uptoamaximumfrequencydeterminedbythelateralspreadoftheshower.Thiscuto®occursatabout1GHzinice,andscalesinverselywiththeMoliereradius.Atlowfrequencies,theradiationisroughlyisotropic,but,asthefrequencyrisestowardthecuto®frequency,theradiationbecomesincreasinglypeakedaroundtheCherenkovangle.Theradiationislinearlypolarizedintheplanecontainingtheshoweraxisandthephotondirection.Ameasurementofthesignalpolarizationcanbeusedtohelpdeterminetheshowerdirection.ThecharacteristicsofthisradiationhavebeennicelydemonstratedinaseriesofexperimentsatSLAC[86].AdetaileddiscussionoftheradiationcanbefoundinRef.87.August21,201413:18 32.Passageofparticlesthroughmatter32.7.3.Transitionradiation:Theenergyradiatedwhenaparticlewithchargezecrossestheboundarybetweenvacuumandamediumwithplasmafrequency!pisI=®z2~!p=3;(32:47)where~!p=q 4¼Ner3emec2=®=q (ing/cm3)hZ=Ai£28:81eV:(32:48)Forstyreneandsimilarmaterials,~!p20eV;forairitis0.7eV.ThenumberspectrumdN=d(~!divergeslogarithmicallyatlowenergiesanddecreasesrapidlyfor~!=°~!p�1.Abouthalftheenergyisemittedintherange0:1~!=°~!p1.Inevitableabsorptioninapracticaldetectorremovesthedivergence.Foraparticlewith=103,theradiatedphotonsareinthesoftx-rayrange2to40keV.Thedependenceoftheemittedenergythuscomesfromthehardeningofthespectrumratherthanfromanincreasedquantumyield. 1001000 With absorption200 foils Single interface x-ray energy ), differential yield per interface (keV/keV) Figure32.27:X-rayphotonenergyspectraforaradiatorconsistingof20025mthickfoilsofMylarwith1.5mmspacinginair(solidlines)andforasinglesurface(dashedline).Curvesareshownwithandwithoutabsorption.AdaptedfromRef.88.Thenumberofphotonswithenergy~!�~!0isgivenbytheanswertoproblem13.15August21,201413:18 32.PassageofparticlesthroughmatterinRef.33,N(~!�~!0)=®z2 "ln~!p ~!012+2 12#;(32:49)withincorrectionsoforder(~!0=°~!p)2.Thenumberofphotonsabovea¯xedenergy~!0~!pthusgrowsas(ln)2,butthenumberabovea¯xedfractionof~!p(asintheexampleabove)isconstant.Forexample,for~!�°~!p=10,N=2:519®z2=¼=0:59%z2.Theparticlestays\inphase"withthexrayoveradistancecalledtheformationlength,d(!)=(2c=!)(1=°2+2+!2p=!2)1.Mostoftheradiationisproducedinthisdistance.Hereisthex-rayemissionangle,characteristically1=°.For=1=°theformationlengthhasamaximumatd(°!p=p 2)=°c=p 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