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1
Simulation of Energy LossStraggling
MariaPhysicist
May28, 2006
Contents
1 Introduction
1
2 Landau u theory
2
2.1 Restrictions s . . . .. . . . . .. . . . . . .. . . . . .. . . . . ..
3
3 Vavilov v theory
4
4 Gaussian n Theory
5
5 Urbanmodel
5
5.1 Fast t simulationfor n3 16 6 .. . . . . . .. . . . . .. . . . . ..
7
5.2 Specialsamplingfor r lowerpartofthespectrum . . .. . . . . ..
9
1
Introduction
Duetothestatistical natureof ionisationenergy loss, large
uctuations can
occur intheamount of energydepositedby y aparticletraversinganabsorber
element. Continuous processes suchas multiplescatteringandenergyloss play
a relevant t roleinthelongitudinal and d lateral development t of electromagnetic
andhadronicshowers, andinthecaseofsampling calorimeters themeasured
resolutioncanbesignicantlyaectedbysuch
uctuations intheir activelayers.
The descriptionof f ionisation
uctuations is characterisedbythesignicance
parameter , whichis proportional totheratioof mean n energy loss s to o the
maximumallowedenergytransfer inasinglecollision n withanatomicelectron
=
Emax
Emax is themaximum transferableenergyinasinglecollisionwith h anatomic
electron.
Emax =
2me2
2
1+2
me=mx +(me=mx)2
;
where
= E=mx, E is energyandmx themass oftheincident particle, 2 =
1 1=
2 andme is s theelectronmass. comes fromtheRutherfordscattering
crosssectionandis denedas:
=
2z2e4NAvZx
me2 c2A
= 153:4
z2
2
Z
A
x keV;
1
2
where
z
chargeof theincident particle
NAv Avogadro’s number
Z
atomicnumber of thematerial
A
atomicweight of thematerial
density
x
thickness of thematerial
measures thecontribution n ofthecollisions s withenergytransfer closeto
Emax. For agivenabsorber,tendstowards largevalues if xis largeand/or if
is small.Likewise,tends towards zeroif xissmalland/orif approaches
1.
Thevalueof distinguishestworegimes whichoccur inthedescriptionof
ionisation
uctuations:
1.A largenumber ofcollisionsinvolvingthelossofallor mostoftheincident
particleenergyduringthetraversal ofanabsorber.
As thetotal energytransfer is composedof amultitudeof smallenergy
losses,wecanapplythecentrallimit theorem anddescribethe
uctuations
byaGaussiandistribution. This s caseis applicableto o non-relativistic
particles andis describedbytheinequality >10(i.e. whenthemean
energyloss intheabsorber is greater thanthemaximumenergytransfer
inasinglecollision).
2.Particles traversingthincountersandincident electrons underanycondi-
tions.
The relevant t inequalities anddistributions are0:01< < 10,Vavilov
distribution,and <0:01, Landaudistribution.
Anadditionalregimeis denedbythecontributionofthecollisionswithlow
energytransfer whichcanbeestimatedwiththerelation n =I0, whereI0 is s the
meanionisationpotentialof theatom. Landautheoryassumesthatthenumber
of thesecollisions is high, and consequently,it has arestriction =I0 1.
InGEANT(seeURLhttp://wwwinfo.cern.ch/asdoc/geant/geantall.html),
thelimitofLandautheoryhasbeenset at =I
0
= 50. Below thislimit special
modelstakingintoaccounttheatomicstructureof thematerialareused.This
is important inthinlayers andgaseous materials. Figure1shows thebehaviour
of=I0 as afunctionof thelayerthickness for anelectronof100keVand1GeV
ofkineticenergyinArgon,SiliconandUranium.
Inthe followingsections,thedierent t theories andmodels for theenergy
loss
uctuation n aredescribed. First, the Landau theoryandits s limitations
arediscussed,andthen,theVavilovandGaussianstragglingfunctions andthe
methods inthethinlayers andgaseous materials arepresented.
2 Landautheory
For aparticleofmass mx traversingathickness ofmaterial x,theLandauprob-
abilitydistributionmaybewritteninterms of theuniversal Landau u function
() as[1]:
f(;x) =
1
()
2
3
10 -2
10 -1
1
10
10 2
0 .01
0 .1
1
10
1 00
x/I
0
< 50
Argon
Silicon
Uranium
1 GeV
100 keV
1 GeV
100 keV
Ste p , [c m]
x /I
0
Figure1: Thevariable=I0 canbeusedtomeasurethevalidityrangeof the
Landautheory.It depends onthetypeandenergyof theparticle, Z,Aandthe
ionisationpotentialofthematerialandthelayer thickness.
where
() =
1
2i
Z
c+i1
c i1
exp(ulnu+u)du
c0
=
0 2 ln
Emax
0 =
0:422784:::= 1
=
0:577215:::(Euler’s constant)
=
averageenergyloss
=
actual energyloss
2.1
Restrictions
TheLandauformalism makes tworestrictiveassumptions:
1. Thetypicalenergyloss issmall comparedtothemaximumenergyloss in
asinglecollision. ThisrestrictionisremovedintheVavilov theory(see
section 3).
2. Thetypicalenergyloss intheabsorbershouldbelargecomparedtothe
bindingenergyofthemost tightlyboundelectron. For gaseousdetectors,
typicalenergylosses areafew keVwhichiscomparabletothebindingen-
ergiesof theinner electrons. In suchcases amoresophisticatedapproach
whichaccounts for atomicenergylevels[4] is necessarytoaccuratelysim-
ulatedatadistributions. InGEANT,aparameterisedmodelbyL.Urban is
used(seesection5).
3
4
Inaddition,theaveragevalueof theLandaudistributionis innite. . Sum-
ming theLandau
uctuation obtainedto theaverageenergyfrom m thedE=dx
tables,weobtainavaluewhichis larger thantheonecomingfrom thetable.
The probabilitytosamplealarge valueis s small, soittakes alargenumber
ofsteps (extractions) for theaverage
uctuationtobesignicantlylarger than
zero. Thisintroduces adependenceof theenergylossonthestepsizewhich
canaect calculations.
A solutiontothis has beentointroducealimit onthevalueof thevariable
sampledbytheLandaudistributioninordertokeeptheaverage
uctuationto
0.Thevalueobtainedfrom theGLANDOroutineis:
dE=dx= = = (
0+2+ + ln
Emax
)
Inorder for this tohaveaverage0,wemust imposethat:
=
0 2 ln
E
max
Thisis realised d introducinga
max
()suchthat ifonlyvaluesof
max
areaccepted, theaveragevalueofthedistributionis
.
A parametrict totheuniversalLandaudistributionhas beenperformed,
withfollowingresult:
max =0:60715+1:1934
+ (0:67794+0:052382
)exp(0:94753+0:74442
)
onlyvaluessmaller thanmax areaccepted, otherwisethedistributionis resam-
pled.
3
Vavilov theory
Vavilov[5] derivedamoreaccuratestragglingdistributionbyintroducing the
kinematiclimit onthemaximum transferableenergyinasinglecollision, rather
thanusingEmax = 1. Nowwecanwrite[2]:
f(;s) =
1
v
v;; 2
where
v
v;;2
=
1
2i
Z
c+i1
c i1
(s)esds
c0
(s) = = exp
(1+2
)
exp[ (s)];
(s) = = sln+(s+2)[ln(s=) ) +E1(s=)] e s=;
and
E1(z) =
Z1
z
t 1etdt
(theexponentialintegral)
v =
0 2
4
5
TheVavilovparameters aresimplyrelatedtotheLandau u parameter by
L= v= ln. . It can beshown that as !0, the e distribution of f the
variable
L
approaches that ofLandau. . For r 0:01thetwodistributions
arealreadypracticallyidentical. Contrarytowhat manytextbooks report, the
Vavilovdistributiondoes not approximatetheLandaudistributionfor small,
but rather thedistributionof Ldenedabovetendstothedistributionof the
truefrom theLandaudensity y function. . Thus s theroutine e GVAVIVsamples
thevariableLratherthanv. For 10theVavilovdistribution tends toa
Gaussiandistribution (seenext t section).
4 Gaussian n Theory
Various con
ictingforms havebeenproposedfor Gaussianstragglingfunctions,
but most oftheseappeartohavelittletheoretical or experimental basis. How-
ever, it has been shown[3] that for 10 theVavilov distribution can be
replacedbyaGaussianof theform:
f(;s)
1
q
2
(1 2=2)
exp
( )2
2
2(1 2=2)
thus implying
mean = =
2 =
2
(1 2=2) ) =Emax (1 2=2)
5 Urbanmodel
Themethodforcomputingrestrictedenergylosses with-rayproductionabove
giventhreshold energyin GEANTis s a a MonteCarlomethodthat t canbeused
for thin n layers. It t is fast anditcanbeusedfor any y thickness s of amedium.
Approachingthelimitofthevalidityof Landau’s theory,theloss distribution
approaches smoothlytheLandau u form m as showninFigure2.
Itis assumedthat theatoms haveonlytwoenergylevels withbindingenergy
E1 andE2. Theparticle{atominteractionwillthenbeanexcitationwithenergy
loss E
1
or E
2
,oranionisation withan energyloss s distributedaccordingtoa
functiong(E) 1=E2:
g(E) =
(Emax +I)I
Emax
1
E2
(1)
Themacroscopiccross-sectionforexcitations(i =1; 2)is
i =C
fi
Ei
ln(2m2
2=Ei) 2
ln(2m2
2=I) 2
(1 r)
(2)
andthemacroscopiccross-sectionfor ionisation n is
3
=C
Emax
I(Emax+ I)ln(Emax+I
I
)
r
(3)
5
6
La nda u
40
20
10
5
1
0. 5
dE/d x [ Ge V/cm] ‘
Counts
0
1 00
2 00
3 00
4 00
5 00
6 00
7 00
8 00
9 00
0
0. 01
0. 02
0. 03
0.0 4
0.0 5
0 .0 6
0 .0 7
0 .0 8
0 .09
0 .1
x 10 -4
Figure2: Energylossdistributionfor a3GeV electroninArgonas given by
standard GEANT.Thewidthofthelayers s is givenincentimeters.
Emax is theGEANTcut for -production,or themaximum energytransfer minus
meanionisationenergy,if it is smaller thanthis cut-ovalue. Thefollowing
notationis used:
r;C parameters s ofthemodel
E
i
atomicenergylevels
I
meanionisationenergy
fi
oscillator strengths
Themodelhas theparameters fi,Ei, Candr(0r 1). Theoscillator
strengths fi andtheatomiclevelenergiesEi shouldsatisfytheconstraints
f1 +f2 = 1
(4)
f
1
lnE
1
+ f
2
lnE
2
= lnI
(5)
Theparameter C canbedenedwiththehelpof themeanenergylossdE=dx
in thefollowingway: Thenumbersofcollisions s (ni, i =1,2for theexcitation
and3 for r theionisation)followthePoissondistributionwithameannumber
hnii.Inastepxthemeannumberofcollisions is
hnii =ix
(6)
ThemeanenergylossdE=dxinastepis thesum oftheexcitationandionisation
contributions
dE
dx
x=
"
1E1+2E2 +3
ZE
max
+I
I
E g(E)dE
#
x
(7)
From this,usingtheequations(2),(3), (4)and(5),onecandenetheparameter
C
C =
dE
dx
(8)
6
7
Thefollowingvalues havebeenchoseninGEANTfor theother parameters:
f2 =
0
ifZ 2
2=Z ifZ Z >2
)
f1 =1 f2
E2 = 10Z2eV
)
E1=
I
Ef22
1
f
1
r= 0:4
With thesevalues s theatomiclevel E2 corresponds approximatelytheK-shell
energyof theatoms and Zf2 thenumber of K-shell electrons. r r is s the e only
variablewhichcanbetunedfreely. Itdetermines therelativecontributionof
ionisationandexcitationtotheenergyloss.
Theenergyloss is computedwiththeassumptionthat thestep p length(or
therelativeenergyloss)issmall,and|inconsequence|thecross-sectioncanbe
consideredconstantalongthepathlength. Theenergyloss duetotheexcitation
is
Ee =n1E1+ n2E2
(9)
wheren
1
andn
2
aresampledfromPoissondistributionasdiscussedabove. The
loss duetotheionisationcanbegeneratedfrom thedistributiong(E) by y the
inversetransformationmethod:
u= F(E) ) =
Z
E
I
g(x)dx
E= F 1(u) ) =
I
1 u Emax
Em ax+I
(10)
(11)
whereuis auniform randomnumber betweenF(I) =0andF(Emax+ I) =1.
Thecontributionfromtheionisations will be
E
i
=
n
3
X
j=1
I
1 u
j
E
m ax
Emax+I
(12)
wheren
3
is thenumber ofionisation(sampledfrom Poissondistribution). The
energylossinastepwill thenbeE =Ee +Ei.
5.1
Fast simulation n forn3 16
If thenumber ofionisationn3is bigger than16,afaster samplingmethodcan
beused. Thepossibleenergyloss intervalisdividedin twoparts: oneinwhich
thenumber of collisions is largeandthesamplingcanbedonefrom aGaussian
distributionandtheother inwhichtheenergyloss issampledfor eachcollision.
Let us calltheformer interval [I;I] theinterval A,andthelatter[I;E
max
]
theinterval B. lies between1 andEmax=I. . A A collisionwith aloss in the
intervalA happens withtheprobability
P()=
ZI
I
g(E)dE=
(E
max
+I)( 1)
Emax
(13)
Themeanenergyloss andthestandarddeviationfor this typeof collisionare
hE()i =
1
P()
Z
I
I
Eg(E)dE=
Iln
1
(14)
7
8
and
2() =
1
P()
Z
I
I
E2g(E) dE= I2
1
ln2
( 1)2
(15)
If the e collisionnumber r is high, weassumethat thenumber of the e typeA
collisions can n becalculated from aGaussiandistribution withthefollowing
meanvalueand standarddeviation:
hnAi =
n3P()
(16)
2
A
=
n
3
P()(1 P())
(17)
It is further assumedthat theenergy y loss s inthesecollisions has a a Gaussian
distributionwith
hEAi = nAhE()i
(18)
2
E;A
= n
A
2()
(19)
Theenergyloss ofthesecollisioncanthen n besampledfrom m theGaussiandis-
tribution.
Thecollisions wheretheenergylossis intheinterval Baresampleddirectly
from
E
B
=
n
3
n
A
X
i=1
I
1 u
i
E
max
+I I
Emax+I
(20)
Thetotalenergyloss isthesumofthesetwotypes of collisions:
E =EA+EB
(21)
Theapproximationofequations (16),(17),(18) and(19)canbeusedunder
thefollowingconditions:
hn
A
i c
A
0
(22)
hnAi+ cA
n3
(23)
hE
A
i c
E;A
0
(24)
wherec 4. Fromtheequations(13),(16) and(18) and d fromtheconditions
(22) and(23) thefollowinglimits canbederived:
min =
(n
3
+c2)(E
max
+I)
n3(Emax+ I)+ c2I
max=
(n
3
+c2 )(E
max
+ I)
c2 (Emax+ I)+ n3I
(25)
This conditions gives alower limit tonumberoftheionisationsn
3
forwhichthe
fast samplingcanbedone:
n3 c2
(26)
As intheconditions (22),(23)and(24)thevalueofcis as minimum4, onegets
n
3
16. Inorder tospeedthesimulation, themaximum valueis used d for r .
Thenumber of collisions withenergyloss in n theinterval B (thenumber
ofinteractions which h has s tobesimulated d directly) ) increases slowlywith h the
totalnumber of collisions n3. Themaximum numberofthesecollisionscan n be
estimatedas
n
B;max
=n
3
n
A;min
n
3
(hn
A
i
A
)
(27)
8
9
From thepreviousexpressions forhn
A
i and
A
onecanderivethecondition
n
B
n
B;max
=
2n3c2
n3 +c2
(28)
Thefollowingvalues areobtainedwith h c= = 4:
n3
nB;max
n3 nB;max
16
16
200
29.63
20
17.78
500
31.01
50
24.24
1000
31.50
100 27.59
1
32.00
5.2
Specialsamplingforlower part ofthe spectrum
Ifthesteplengthisverysmall( 5mmingases, 2-3m insolids) themodel
gives 0energyloss for someevents. Toavoidthis,theprobability y of0energy
loss is computed
P(E= 0)= e e (hn1i+hn2i+hn3i)
(29)
Iftheprobabilityis bigger than n 0.01aspecial samplingis s done, takinginto
accountthefact thatinthesecases theprojectileinteracts onlywiththeouter
electrons oftheatom. An n energylevel E0= = 10eV ischosentocorrespondto
theouter electrons. Themeannumberofcollisions canbecalculatedfrom
hni =
1
E0
dE
dx
x
(30)
Thenumberofcollisions nissampledfrom Poissondistribution. . In n thecase
ofthethinlayers, all thecollisions areconsideredas ionisations andtheenergy
loss is computedas
E =
nX
i=1
E
0
1
Emax
Emax+E0
ui
(31)
9
Figure22.6:Ninelogicalpagesononeoutputpage
pages.
22.4.1.3 psbook:rearrangepagesinaPostScriptfileintosignatures
psbook[-q] [-ssignature] [infile] [outfile]
psbook
takesthepagesinaPostScriptdocumentandcreatesanewfileinwhichthepagesarerearranged
as“signatures”,thesectionsinaprintedbook.Inprintingabook,anumberofpagesareprintedona
largesheetofpaper that is thenfolded and bound tomakethefinalvolume.Inmakinga booklet by
hand,itishelpfultoarrangethepagessothatwhenfoldedtheycomeoutintherightorder.eoptions
ch-psextra2.tex,v:2.27
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