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where-(t4ht=
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The_vqalueofdistinguishestwo_regimeswhichoGccurinthedescriptionofionisationUUuctuations:&t4ht=

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    AsthetotalenergytransferiscompGosedofamultitudeofsmallenergylosses,HwecanapplythecentrallimittheoremanddescribGetheuctuationsby$`aGaussiandistribution.Thiscaseisapplicabletonon-relativisticparticlesandisdescribGedbytheinequalityƈ>Ɖ10(i.e.whenthemeanenergyglossintheabsorbGerisgreaterthanthemaximumenergytransferinUUasinglecollision). t4ht=

    2. t4ht=
  2. t4ht= t4ht=Particlesutraversingthincountersandincidentelectronsunderanyconditions.t4ht=t4ht=

    TheRrrelevqantinequalitiesanddistributionsare0:01l<l<10.,RrV*avilovdistribution,UUand<0:01.,Landaudistribution.t4ht=

t4ht=t4ht=

t4ht=t4ht=

AnJ~additionalregimeisdenedbythecontributionofthecollisionswithlowenergy,}transferwhichcanbGeestimatedwiththerelationu=I t4ht=0|q t4ht=si,whereI t4ht=0|q t4ht=qҺisthemeanionisation0|q t4ht= m1.InBt emtt10GEANT*(seeURLi?http://wwwinfo.cern.ch/asdoc/geant/geantall.html),Q thelimitofLandau theory(NhasbGeensetatu=I t4ht=0|q t4ht= m=50.Belowthislimitspecialmodelstakingintoaccountthe;atomicstructureofthematerialareused.ThisisimpGortantinthinlayersandgaseous=materials.Figuret4ht=1 t4ht=showsthebGehaviourofu=I t4ht=0|q t4ht=Easafunctionofthelayerthickness=foranelectronof100keV=and1GeVofkineticenergyinArgon,SiliconandUUUranium.1si?t4ht= t4ht= vt4ht=

47i?t4ht@-t4ht=t4ht=PICt4ht>tex4ht.tmpt4ht;8t4ht|t4ht++latexexa2x.gifAt4ht+PSfile=
Figuret4ht@160x1 t4ht=Thedvqariableu=I t4ht=0|q t4ht= BcanbGeusedtomeasurethevalidityrangeoftheLandautheory*.ItdepGendsonthetypeandenergyoftheparticle,Z0,AandtheionisationUUpGotentialofthematerialandthelayerthickness.t4ht=
5-i?2t4ht=
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InF2UUt4ht= t4ht=t4ht= t4ht=Landautheory t4ht=t4ht=t4ht=

F*orlOaparticleofmassm t4ht=x t4ht=u:traversinglOathicknessofmaterial`x.,theLandauprobability?distributionmaybGewrittenintermsoftheuniversalLandaufunction()UUas[t4ht=1 t4ht=]:*t4ht=

t4ht= t4ht=/V#t4ht=*t4ht=1t4ht= ʨ2t4ht=(0t4ht=C t4ht= t4ht=
f(;`x) ʨ t4ht== t4ht=<$33133wfe (֍ff(). t4ht=
* t4ht=
whereUU*t4ht=
t4ht= t4ht=,#t4ht=*t4ht=1t4ht=2t4ht=X0t4ht=9[ t4ht=R*t4ht=1t4ht=2t4ht=X0t4ht=`*t4ht=1t4ht=2t4ht=X0t4ht=9[ t4ht=*t4ht=1t4ht=2t4ht=X0t4ht=9[ t4ht=*t4ht=1t4ht=2t4ht=X0t4ht=9[ t4ht=*t4ht=1t4ht=2t4ht=X0t4ht=9[ t4ht= t4ht=
() t4ht== t4ht=<$133wfe (֍2[icu cmex10Z t4ht=c8i1t4ht=c+i1 t4ht=expkt4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa3x.gif(6uln u+u)8o>t4ht+t4ht!t4ht|t4ht;9!t4ht=dut4ht@160xt4ht@160x n8t4ht@160x%Tt4ht@160xpt4ht@160xU{[c0 t4ht=
 t4ht== t4ht=<$3388t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa4x.gifǺ t4ht+t4ht!t4ht|t4ht;9t4ht=33wfeW (֍\8 9 O!cmsy70R  t4ht=2|q t4ht=9[ t4ht=
90 t4ht== t4ht=0:4227847:::=18 Wh t4ht=
9 t4ht== t4ht=0:577215:::(Euler'sUUconstant)tB t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa5x.gif t4ht+t4ht!t4ht|t4ht;9t4ht= t4ht== t4ht=averageUUenergylossTd t4ht=
 t4ht== t4ht=actualUUenergylossNa t4ht=
t4ht=
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TheUULandauformalismmakestworestrictiveassumptions:&t4ht=

    t4ht=
  1. t4ht= t4ht=Thetypicalenergylossissmallcomparedtothemaximumenergylossinasinglecollision.ThisrestrictionisremovedintheV*avilovtheory(seesectionUUt4ht=3 t4ht=).8i? t4ht=
    2. t4ht=
  2. t4ht= t4ht=ThetypicalenergylossintheabsorbGershouldbelargecomparedtothebinding4-energyofthemosttightlybGoundelectron.F*orgaseousdetectors,typical3energylossesareafewkeVwhichiscomparabletothebindingenergies>-oftheinnerelectrons.Insuchcasesamoresophisticatedapproachwhichaccountsforatomicenergylevels[t4ht=4 t4ht=]isnecessarytoaccuratelysimulatedatadistributions.InGEANT.,aparameterisedmoGdelbyL.Urbt4ht@+áanisUUused(seesectiont4ht=5 t4ht=).t4ht=
t4ht=t4ht=

t4ht=t4ht=

In) addition,theaverage) vqalueoftheLandaudistributionisinnite.SummingtheLandauZuctuationobtainedtotheaverageZenergyfromthedE=dxZtables,Zweobtainavqaluewhichislargerthantheonecomingfromthetable.TheprobabilityQtosamplealargevqalueissmall,soittakesalargenumbGerofsteps(extractions)DfortheaverageDuctuationtobGesignicantlylargerthanzero.ThisintroGducespYadependenceoftheenergylossonthestepsizewhichcanaectcalculations.t4ht=t4ht=

ASsolutiontothishasbGeentointroducealimitonthevqalueofthevariablesampledAbytheLandaudistributioninordertokeeptheaverageuctuationto0.TheUUvqalueobtainedfromtheVGLANDOroutineis: t4ht=

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@In%0borderUUforthistohaveUUaverage0,wemustimpGosethat: t4ht=
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'at4ht=t4ht=

This?isrealisedintroGducinga t4ht=maxH t4ht=H(t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa8x.gif\rj Vt4ht+t4ht!t4ht|t4ht;9t4ht=V)suchthatifonlyvqaluesof t4ht=maxH t4ht={areaccepted,UUtheaverageUUvqalueofthedistributionist4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa9x.gif\rj Vt4ht+t4ht!t4ht|t4ht;9t4ht= *.t4ht=t4ht=

A7parametric7ttotheuniversalLandaudistributionhasbGeenperformed,withfollowingUUresult: t4ht=

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*̍onlyvqaluessmallerthan t4ht=maxH t4ht=دareaccepted,otherwisethedistributionisresampled. Myi?t4ht=

3UUt4ht= t4ht=t4ht= t4ht=V*avilovtheory t4ht=

t4ht=t4ht=

V*avilov[t4ht=5 t4ht=]g:derivedamoreaccuratestragglingdistributionbyintroGducingthekinematic'limitonthemaximumtransferableenergyinasinglecollision,ratherthanusingUUEmax=1.Nowwecanwrite[t4ht=2 t4ht=]:*t4ht=

t4ht= t4ht=/V#t4ht=*t4ht=1t4ht=!n42t4ht=)5P0t4ht=j] t4ht= t4ht=
ft4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa11x.gif(6;`s)uNt4ht+t4ht!t4ht|t4ht;9!t4ht=!n4 t4ht== t4ht=<$33133wfe (֍ff t4ht=vM t4ht=Nt4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa12x.gifb�?vM;; �2b.t4ht+t4ht!t4ht|t4ht;9!t4ht=@ t4ht=
* t4ht=
whereUU*t4ht=
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 t4ht=vM t4ht=Nt4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa13x.gifb�?vM;; �2b.t4ht+t4ht!t4ht|t4ht;9!t4ht=9r t4ht== t4ht=<$133wfe (֍2[icZ t4ht=c8i1t4ht=c+i1 t4ht=t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa14x.gif(6s)!t4ht+t4ht!t4ht|t4ht;9!t4ht=e t4ht=s s t4ht= tdst4ht@160xt4ht@160x n8t4ht@160x%Tt4ht@160xpt4ht@160xU{[c0G t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa15x.gif(6s)!t4ht+t4ht!t4ht|t4ht;9!t4ht= t4ht== t4ht=expt4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa16x.gifbT(18+ �2 �9)b9*t4ht+t4ht!t4ht|t4ht;9!t4ht=t4ht@160x lexp/t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa17x.gif[A (s)}]Lt4ht+t4ht!t4ht|t4ht;9!t4ht=; t4ht=
[t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa18x.gif(6s)!t4ht+t4ht!t4ht|t4ht;9!t4ht=. t4ht== t4ht=sln 8+(s+  t4ht=2|q t4ht=)t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa19x.gif[Aln (s=)+E1|q(s=)]VDt4ht+t4ht!t4ht|t4ht;9!t4ht=e t4ht=s=M t4ht=M;Ή t4ht=
t4ht=
andUU*t4ht=
t4ht= t4ht=,#t4ht=*t4ht=1t4ht=O2t4ht=!0t4ht= t4ht=*t4ht=1t4ht=O2t4ht=!0t4ht= t4ht= t4ht=
E t4ht=1|q t4ht=(zp)O t4ht== t4ht=cZ  t4ht=zpt4ht=1 t4ht=t t4ht=1 r t4ht= se t4ht=t W t4ht= Xdtt4ht@160xt4ht@160x<(theUUexpGonentialintegral)o t4ht=
 t4ht=vM t4ht= t4ht== t4ht=t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa20x.gif^<$$8�N 8$w��fe�W (֍T`u t4ht=
t4ht=
t4ht=t4ht=

TheuV*avilovparametersaresimplyrelatedtotheLandauparameterby t4ht=Ls t4ht=Ì= t4ht=vM t4ht=N=ln 7.-lItcanbGeshownthatas!0.,thedistributionofthevqariable t4ht=Ls t4ht=approaches%LthatofLandau.F*or0:01%Lthetwodistributionsarealreadypracticallyidentical.* ContrarytowhatmanytextbGooksreport,theV*avilovdistribution(t emti10do})eslnotapproximate7theLandaudistributionforsmall.,butratherthedistributionof t4ht=Ls t4ht=dened^abGovetendstothedistributionofthetrue^޺fromtheLandaudensityfunction.RThustheroutineT,GVAVIVsamplesthevqariable t4ht=Ls t4ht= O4ratherthan t4ht=vM t4ht=N.F*or810theV*avilovdistributiontendstoaGaussiandistribution(seenextsection). ]i?t4ht=

4UUt4ht= t4ht=t4ht= t4ht=GaussianTheory t4ht=

t4ht=t4ht=

V*arious0conictingformshave0bGeenproposedforGaussianstragglingfunctions,butmost,oftheseappGeartohave,littletheoreticalorexperimentalbasis.However,ithasbGeen%shown[t4ht=3 t4ht=]thatfor10%theV*avilovdistributioncanbGereplacedbyaGaussianofUUtheform:*t4ht=

t4ht= t4ht=.H#t4ht=*t4ht=1t4ht=52t4ht=50t4ht=5 t4ht= t4ht=
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0Lҡ t4ht=
thusUUimplying*t4ht=
t4ht= t4ht=*#t4ht=*t4ht=1t4ht=2t4ht=t0t4ht=Y t4ht=H*t4ht=1t4ht=2t4ht=t0t4ht=Y t4ht= t4ht=
mean t4ht== t4ht=t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa23x.gif t4ht+t4ht!t4ht|t4ht;9t4ht= t4ht=
[ t4ht=2|q t4ht=  t4ht== t4ht=<$33u t4ht=^2|q t4ht=33wfe R9 (֍ǡ (18  t4ht=2|q t4ht==2)=uEmaxH(18  t4ht=2|q t4ht==2) t4ht=
t4ht=
t4ht=

5UUt4ht= t4ht=t4ht= t4ht=Urbt4ht@+áanmoGdel t4ht=

t4ht=t4ht=

ThemethoGdforcomputingrestrictedenergylosseswith`-rayproductionabGoveYgiventhresholdenergyinGEANTLisaMonteCarlomethoGdthatcanbGeVusedforthinlayers.VItisfastanditcanbeusedforanythicknessofamedium.ApproachingthelimitofthevqalidityofLandau'stheory*,thelosswdistributionapproachessmoGothlytheLandauformasshowninFiguret4ht=2 t4ht=. zQi?t4ht= t4ht= vt4ht=

Gi?t4ht@-t4ht=t4ht=PICt4ht>tex4ht.tmpt4ht;8t4ht|t4ht++latexexa24x.gifAt4ht+PSfile=
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=i?2t4ht=
t4ht=t4ht=

It-isassumedthattheatomshave-onlytwo-energylevelswithbindingenergyE t4ht=1|q t4ht=and(0E t4ht=2|q t4ht=.TheparticleatominteractionwillthenbGeanexcitationwithenergylossE t4ht=1|q t4ht=orU%E t4ht=2|q t4ht=>,oranionisationwithanenergylossdistributedaccordingtoafunctiong[ۺ(E)1=E t4ht=^2|q t4ht=:UU3t4ht=
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t4ht=		(1)t4ht=
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TheUUmacroscopiccross-sectionforexcitations(i=1;2)is3t4ht=
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t4ht=		(2)t4ht=
,̡andUUthemacroscopiccross-sectionforionisationis3t4ht=
 t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa27x.gif0c ̀֍3C=C<$){EmaxPw��fe�f\ #dI�(Emaxx+8I)ln )(T33E��!t�������emr5max '+I33މ��fe�@ |Iv)lόrY��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(3)t4ht=
i? Emaxois&the(XGEANT&cutfor`-proGduction,orthemaximumenergytransferminusmeanionisationCbenergy*,ifitissmallerthanthiscut-ovqalue.Thefollowingnotationisused:(t4ht=

d t4ht=5t4ht=t4ht=t4ht=+t4ht= t4ht=߿ t4ht= t4ht=t4ht=t4ht=+t4ht= t4ht=߿ t4ht= t4ht=t4ht=t4ht=+t4ht= t4ht=߿ t4ht= t4ht=t4ht=t4ht=+t4ht= t4ht=߿ t4ht= t4ht=t4ht= t4ht=
r;C t4ht=
t4ht=
parametersofthemoGdel t4ht=
t4ht=
E t4ht=iTK t4ht= ( t4ht=
t4ht=
atomicenergylevelsr t4ht=
t4ht=
I % t4ht=
t4ht=
meanionisationenergyS t4ht=
t4ht=
f t4ht=iTK t4ht=j t4ht=
t4ht=
oscillatorstrengths t4ht=
t4ht=
 t4ht=
t4ht=t4ht=

TheTmoGdelhastheparametersf t4ht=iTK t4ht=,E t4ht=iTK t4ht=,C andrGt4ht@160x:(0r61).TTheoscillatorstrengthsvf t4ht=iTK t4ht=andtheatomiclevelenergiesE t4ht=iTK t4ht=^shouldsatisfytheconstraints*t4ht=

t4ht= t4ht=*t4ht=*t4ht=1t4ht=Q2t4ht=YS0t4ht=[Xt4ht= t4ht=*t4ht=1t4ht=Q2t4ht=YS0t4ht=[Xt4ht= t4ht= t4ht=
f t4ht=1|q t4ht=+8f t4ht=2|q t4ht=!# t4ht== t4ht=1 t4ht=(4)t4ht= t4ht= t4ht=
f t4ht=1|q t4ht=:lnRE t4ht=1|q t4ht=H޺+8f t4ht=2|q t4ht=:lnE t4ht=2|q t4ht=Q t4ht== t4ht=ln )IFb t4ht=(5)t4ht= t4ht= t4ht=
t4ht=
The,6parameterCScanbGedenedwiththehelpofthemeanenergylossdE=dx,7in,6thefollowing/`way:ThenumbGersofcollisions(n t4ht=iTK t4ht=`,i=1,2fortheexcitationand3fortheionisation)=VfollowthePoissondistributionwithameannumbGerhn t4ht=iTK t4ht=TLi.Inastep̀яxtheUUmeannumbGerUUofcollisionsis3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa28x.gif�hniTKi=̀֍ìяxY��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(6)t4ht=
TheAmeanenergylossdE=dxAinAastepisthesumoftheexcitationandionisationcontributionsUU3t4ht=
 t4ht=
Ɵi?t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa29x.gif <$3dE3w��fe� )Ɵ (֍dxAxєx=\t4ht= t4ht=t4ht=
t4ht=		(7)t4ht=
! F*rom1this,usingtheequations(t4ht=2 t4ht=),(t4ht=3 t4ht=),(t4ht=4 t4ht=)and(t4ht=5 t4ht=),onecandenetheparameterC3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa30x.gif�C~5=<$KdEKw��fe� )Ɵ (֍dxY��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(8)t4ht=
t4ht=t4ht=

TheUUfollowingvqalueshavebGeenchoseninmGEANTfortheotherparameters: t4ht=

t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa31x.gif#P/PHkf2C=^dG0*Cif2Z~42G2=qZ*Cif2Z~4>2�{)�{f1C=18�f2^HkE2C=10Z�^23eV�{)�{E1C=^ UI V&��fe�Ehb~$Y&Y �������emmi5fZ2w2 ^H獑%#|1#-x��W �RPfZ1lHkr6=0:4Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht=
+@WithN9these(vqaluestheatomiclevelE t4ht=2|q t4ht=8correspGondsapproximatelytheK-shellenergyoftheatoms3andZf t4ht=2|q t4ht=̺thenumbGer3ofK-shellelectrons.rzistheonlyvqariablewhichcanbetuned=freely*.ItdeterminestherelativecontributionofionisationandexcitationtotheUUenergyloss.t4ht=t4ht=

TheCenergylossiscomputedwiththeassumptionthatthesteplength(ortherelativei;energyloss)issmall,andinconsequencethecross-sectioncanbGeconsidered9;constantalongthepathlength.Theenergylossduetotheexcitationis3t4ht=
 t4ht=
i?t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa32x.gif ��\ẁяEec=n1|qE1Q+8n2E2Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(9)t4ht=
 where,cn t4ht=1|q t4ht=պandn t4ht=2|q t4ht=aresampledfromPoissondistributionasdiscussedabGove.ThelossdueJtotheionisationcanbGegeneratedfromthedistributiong[ۺ(E)bytheinversetransformationUUmethoGd:*t4ht=

t4ht= t4ht=,t4ht=*t4ht=1t4ht=5Ys2t4ht== 0t4ht=uXt4ht= t4ht=5*t4ht=1t4ht=5Ys2t4ht== 0t4ht=d(Xt4ht= t4ht=v*t4ht=1t4ht=5Ys2t4ht== 0t4ht=d(Xt4ht= t4ht= t4ht=
u=Fc(E)* t4ht== t4ht=cZ  t4ht=It4ht=E t4ht=g[ۺ(x)dx6X t4ht= t4ht=
EZ=Fc t4ht=1 r t4ht= (u)5Ys t4ht== t4ht=<$ЗI33wfe6i #d18uTPHEmaxމfe@Emax '+I8g t4ht=(10)t4ht= t4ht= t4ht=
t4ht= t4ht= t4ht=(11)t4ht= t4ht= t4ht=
* t4ht=
where6PuisauniformrandomnumbGer6PbetweenFc(I)=0andFc(EmaxCm+I)=1.ThecontributionUUfromtheionisationswillbGe3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa33x.gif,ud̀яEic=Fbn�3 X 򝍑ojg=1<$.vIw��fe�: #d18�ujT E��maxiߟމ��fe�@E��max '+IOY��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(12)t4ht=
" :whereTn t4ht=3|q t4ht= isthenumbGerTofionisation(sampledfromPoissondistribution).TheenergyUUlossinastepwillthenbGèєEZ=̀ѓE t4ht=eKK t4ht=+8̀ѓE t4ht=iTK t4ht=غ.9uU t4ht=

5.1UUt4ht= t4ht=t4ht= t4ht=F*astsimulationforn t4ht=3|q t4ht=C16 t4ht=

t4ht=t4ht=

If@thenumbGer@ofionisationn t4ht=3|q t4ht=&isbiggerthan16,afastersamplingmethodcanbeused.OUThepGossibleenergylossintervqalisdividedintwoparts:oneinwhichthenumbGermofcollisionsislargeandthesamplingcanbedonefromaGaussiandistribution0qandtheotherinwhichtheenergylossissampledforeachcollision.Letus&calltheformerintervqal[}I; {I]&theintervqalA,andthelatter[} {I;EmaxH]theintervqalB.A KNliesbGetweenA1andEmaxH=I.AAcollisionwithalossintheintervqalAhappGenswithUUtheprobability3t4ht=
hi? t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa34x.gif PhPc( {)=cZi  I@URI;7g[ۺ(UXE)dEZ=<$K(Emaxx+8I�)( B[�1)Kw��fe�OG (֍EmaxH Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(13)t4ht=
 The[ meanenergylossandthestandarddeviationforthistypGeofcollisionare3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa35x.gif O(ih̀љE�( {)i=<$ |1Kw��fe�T (֍Pc( )zcZi&{ I@5g[ۺ(UXE)dEZ=<$KI #ln L KwfeB (֍ B[81Yt4ht+t4ht!t4ht|t4ht;9t4ht="class="mathdisplay"t4ht=>t4ht= t4ht=t4ht=
t4ht=		(14)t4ht=
andUU3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa36x.gif 7J[ϟ2@( {)=<$ |1Kw��fe�T (֍Pc( )zcZi&{ I@t4ht= t4ht=t4ht=
t4ht=		(15)t4ht=
If3thecollisionnumbGer3ishigha,,weassumethatthenumbGerofthetypGeA3collisionscan<bGecalculatedfromaGaussiandistributionwiththefollowingmeanvqalueandstandardUUdeviation:*t4ht=

t4ht= t4ht=*t4ht=*t4ht=1t4ht=KG2t4ht=c0t4ht=YXt4ht= t4ht=*t4ht=1t4ht=KG2t4ht=c0t4ht=YXt4ht= t4ht= t4ht=
hn t4ht=A t4ht=iKG t4ht== t4ht=n t4ht=3|q t4ht=|rPc( {) ^ t4ht=(16)t4ht= t4ht= t4ht=
[ t4ht=At4ht=2 t4ht= t4ht== t4ht=n t4ht=3|q t4ht=|rPc( {)(18P( {))O t4ht=(17)t4ht= t4ht= t4ht=
ɲi? t4ht=
ItpcisfurtherassumedthattheenergylossinthesecollisionshasaGaussian distributionUUwith*t4ht=
t4ht= t4ht=*t4ht=*t4ht=1t4ht=2t4ht=&0t4ht=GXt4ht= t4ht=*t4ht=1t4ht=2t4ht=&0t4ht=GXt4ht= t4ht= t4ht=
h̀яE t4ht=A t4ht=iX t4ht== t4ht=n t4ht=A t4ht=h̀яE( {)i2) t4ht=(18)t4ht= t4ht= t4ht=
[ t4ht=E;At4ht=2 t4ht= t4ht== t4ht=n t4ht=A t4ht=[ t4ht=2|q t4ht=@( {)%I| t4ht=(19)t4ht= t4ht= t4ht=
t4ht=
TheenergylossofthesecollisioncanthenbGesampledfromtheGaussiandistribution.t4ht=t4ht=

The)/collisionswheretheenergylossisintheintervqalB)$aresampleddirectlyfrom3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa37x.gifg�̀яEB %=Fbn�3�nmA j`X 򝍑 i=1<$>2 {I]Lw��fe�IFß #d18�uiT~E��max '+I�J� I~މ��fe�,ļE��max '+I 򝎒Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(20)t4ht=
"TheUUtotalenergylossisthesumofthesetwoUUtypGesofcollisions:3t4ht=
 t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa38x.gif�)̀єEZ=̀ѓEAp+8̀ѓEBY��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(21)t4ht=
t4ht=t4ht=

The/Vapproximationofequations((t4ht=16 t4ht=),(t4ht=17 t4ht=),(t4ht=18 t4ht=)and(t4ht=19 t4ht=)canbGeusedunderthefollowingUUconditions:*t4ht=

t4ht= t4ht=*t4ht=*t4ht=1t4ht=F_2t4ht=M}0t4ht=FkXt4ht= t4ht=*t4ht=1t4ht=F_2t4ht=M}0t4ht=FkXt4ht= t4ht=בi?*t4ht=1t4ht=F_2t4ht=M}0t4ht=FkXt4ht= t4ht= t4ht=
hn t4ht=A t4ht=i8c[ t4ht=A t4ht=3 t4ht= t4ht=0 t4ht=(22)t4ht= t4ht= t4ht=
hn t4ht=A t4ht=i8+c[ t4ht=A t4ht=3 t4ht= t4ht=n t4ht=3|q t4ht= } t4ht=(23)t4ht= t4ht= t4ht=
h̀яE t4ht=A t4ht=i8c[ t4ht=Eb~;A$ t4ht=F_ t4ht= t4ht=0 t4ht=(24)t4ht= t4ht= t4ht=
* t4ht=
whereAc4.F*romtheequations(t4ht=13 t4ht=),(t4ht=16 t4ht=)and(t4ht=18 t4ht=)andfromtheconditions(t4ht=22 t4ht=)andUU(t4ht=23 t4ht=)thefollowinglimitscanbGederived:3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa39x.gif,̍` min=<$K(n3Q+8c^2|q)(Emaxx+I�)Kw��fe�W%ܟ (֍Lrn3|q(Emaxx+8I�)+cr2Iaኸ  max=<$K(n3Q+8c^2|q)(Emaxx+I�)Kw��fe�W%ܟ (֍Lrcr2|q(Emaxx+8I�)+n3IY��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(25)t4ht=
',̡This,conditionsgivesalowerlimittonumbGeroftheionisationsn t4ht=3|q t4ht=forwhichthefastsamplingUUcanbGedone:3t4ht=
 t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa40x.gif�gn3 0c2Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(26)t4ht=
AsGintheconditions(t4ht=22 t4ht=),(t4ht=23 t4ht=)and(t4ht=24 t4ht=)thevqalueofcisasminimum4,onegetsn t4ht=3|q t4ht= ظ 16.InordertospGeedthesimulation,themaximumvqalueisusedfor {.t4ht=t4ht=

The^SnumbGerofcollisionswithenergylossintheintervqalB^P(thenumbGerofinteractionsTNwhichhastobGesimulateddirectly)increasesslowlywiththetotalnumbGer6ofcollisionsn t4ht=3|q t4ht=|r.ThemaximumnumbGerofthesecollisionscanbeestimatedasUU3t4ht=
 t4ht=
4i? t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa41x.gifUnBW>;max=n3Q�8nA;min n3|q(hnAi�A)Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(27)t4ht=
F*romlthepreviousexpressionsforhn t4ht=A t4ht=iand[ t4ht=A t4ht= Konecanderivethecondition3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa42x.gif,̍u^nB p=0nBW>;max=<$2n3|qc^2Kw��fe�0 (֍n3Q+8cr2Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(28)t4ht=
,̡TheUUfollowingvqaluesareobtainedwithc=4:#\t4ht=

d t4ht=nt4ht=t4ht=t4ht=+t4ht= t4ht=UV t4ht=H@ t4ht=V: t4ht=w t4ht= t4ht=^t4ht=t4ht=+t4ht= t4ht=UV t4ht=H@ t4ht=V: t4ht=w t4ht= t4ht=t4ht=t4ht=+t4ht= t4ht=UV t4ht=H@ t4ht=V: t4ht=w t4ht= t4ht=t4ht=t4ht=+t4ht= t4ht=UV t4ht=H@ t4ht=V: t4ht=w t4ht= t4ht=t4ht=t4ht=+t4ht= t4ht=UV t4ht=H@ t4ht=V: t4ht=w t4ht= t4ht=t4ht= t4ht=
n t4ht=3|q t4ht=h t4ht=
t4ht=
n t4ht=BW>;max9 t4ht=9 t4ht=
t4ht=
t4ht=
t4ht=
n t4ht=3|q t4ht=|r t4ht=
t4ht=
n t4ht=BW>;max9 t4ht=9 t4ht=
t4ht=
16 t4ht=
t4ht=
16: t4ht=
t4ht=
t4ht=
t4ht=
200 t4ht=
t4ht=
sy29.63 t4ht=
t4ht=
20 t4ht=
t4ht=
17.78sx t4ht=
t4ht=
t4ht=
t4ht=
500 t4ht=
t4ht=
sy31.01 t4ht=
t4ht=
50 t4ht=
t4ht=
24.24sx t4ht=
t4ht=
t4ht=
t4ht=
1000 t4ht=
t4ht=
sy31.50 t4ht=
t4ht=
100 t4ht=
t4ht=
27.59sx t4ht=
t4ht=
t4ht=
t4ht=
1 t4ht=
t4ht=
sy32.00 t4ht=
t4ht=
 t4ht=
uU t4ht=

5.2UUt4ht= t4ht=t4ht= t4ht=SpGecialsamplingforlowerUUpartofthespectrum t4ht=

t4ht=t4ht=

If$thesteplengthisverysmall(5mmingases,2-3minsolids)themoGdelgives0Xenergylossforsomeevents.XT*oavoidXthis,theprobabilityof0energylossiscomputedUU3t4ht=
 t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa43x.giffnPc(̀љEZ=0)=e�(hn�1i+hn�2i+hn�3i)Y��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(29)t4ht=
Ifqtheprobabilityisbiggerthan0.01aspGecialsamplingisdone,takingintoaccountVthefactthatinthesecasestheprojectileinteractsonlywiththeouterelectronsboftheatom.AnenergylevelE t4ht=0|q t4ht=ͺ=10eVbischosentocorrespGondtothetouterelectrons.ThemeannumbGertofcollisionscanbecalculatedfrom3t4ht=
i? t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa44x.gif��hni=<$ig1Kw��fe� 8 (֍E0<$>dE>w��fe� )Ɵ (֍dx ѓxY��t4ht+t4ht!t4ht|t4ht;9t4ht=t4ht= t4ht=t4ht=
t4ht=		(30)t4ht=
The9numbGerofcollisionsn9issampledfromPoissondistribution.InthecaseofthethinMnlayers,allthecollisionsareconsideredasionisationsandtheenergylossiscomputedUUas3t4ht=
T t4ht=
t4ht@-t4ht=t4ht;8t4ht!t4ht|t4ht++latexexa45x.gift7̀яEZ=>n X 򝍑wi=1<$-E0w��fe�?lş #d18�T `E��maxlމ��fe�t4ht= t4ht=t4ht=
t4ht=		(31)t4ht=
v.!t4ht=

t4ht= t4ht=t4ht= t4ht=References t4ht=

8"t4ht=
t4ht=

[1]8t4ht@160xt4ht@160xt4ht@160xt4ht= t4ht=L.Landau.OnktheEnergyLossofF*astParticlesbyIonisation.Originally8published$inJ.Phys..,8:201,1944. &ReprintedinD.terHaar,Editor,8L.D.L})andau,Collectedpapers ,UUpage417.PergamonPress,Oxford,1965.8 t4ht=

t4ht=

[2]8t4ht@160xt4ht@160xt4ht@160xt4ht= t4ht=B.Schorr.ProgramsfortheLandauandtheV*avilovdistributionsand8theUUcorrespGondingrandomnumbers.nComp.Phys.Comm..,UU7:216,1974.8 t4ht=

t4ht=

[3]8t4ht@160xt4ht@160xt4ht@160xt4ht= t4ht=S.M.SeltzermeandM.J.Berger. jEnergylossstragglingofprotonsand8mesons.InaStudiesainPenetr})ationofChargedParticlesinMatter'@,aNuclear8ScienceUUSeriest4ht@160x39,Nat.AcademyofSciences,W*ashingtonDC,1964.8 t4ht=

t4ht=

[4]8t4ht@160xt4ht@160xt4ht@160xt4ht= t4ht=R.T*alman.{Onmthestatisticsofparticleidenticationusingionization.8Nucl.Inst.Meth..,UU159:189,1979.Vi?8 t4ht=

t4ht=

[5]8t4ht@160xt4ht@160xt4ht@160xt4ht= t4ht=P*.V.Vavilov.5Ionisationlossesofhighenergyheavyparticles.Soviet8PhysicsJETPٺ,UU5:749,1957. t4ht=

t4ht=
t4ht= t4ht=t4ht= ;yY(t emti10$Y&Y emmi5#Y&Y emmi7"Y&Y emmi10!temr5 temr7t emtt10t emr12tGGemr17t emr10 !", cmsy10 O!cmsy7u cmex10z