150
22
POSTSCRIPTANDPDFTOOLS
wherez
chargeoftheincidentparticle
NAv
Avogadro’snumber
Z
atomicnumberofthematerial
A
atomicweightofthematerial
density
x
thicknessofthematerial
measures thecontributionofthecollisions withenergytransferclose to
Emax. Foragivenabsorber,tendstowardslargevaluesifxislargeand/orif
issmall. Likewise, tendstowardszero ifxissmalland/orifapproaches
1.
The valueofdistinguishestworegimeswhichoccurinthedescriptionof
ionisation
uctuations:
1. A largenumberofcollisionsinvolvingthelossofall ormostoftheincident
particle energyduring the traversalofanabsorber.
As thetotal energytransferiscomposedofa multitudeofsmall energy
losses,wecanapplythecentrallimittheoremanddescribethe
uctuations
bya Gaussiandistribution. . This s case is applicable to non-relativistic
particlesandisdescribedbythe inequality>10(i.e. . whenthe e mean
energylossintheabsorberisgreaterthanthemaximumenergytransfer
ina single collision).
2. Particlestraversingthincountersandincidentelectronsunderanycondi-
tions.
Therelevant inequalities anddistributions are 0:01 1 << < 10, , Vavilov
distribution, and<0:01, Landaudistribution.
Anadditionalregimeisdenedbythecontributionofthecollisionswithlow
energytransferwhichcanbeestimatedwiththe relation=I0, whereI0 isthe
meanionisationpotentialoftheatom.Landautheoryassumesthatthenumber
ofthese collisionsishigh, , and consequently, , ithasa a restriction =I0 0 1.
InGEANT(seeURLhttp://wwwinfo.cern.ch/asdoc/geant/geantall.html),
thelimitofLandautheoryhasbeensetat=I0=50. Below thislimitspecial
modelstaking into accounttheatomicstructureofthematerialareused. This
isimportantinthinlayersandgaseousmaterials. Figure1 showsthe behaviour
of=I0asa functionofthelayerthicknessforanelectronof100 keVand1GeV
ofkineticenergyinArgon, SiliconandUranium.
Inthefollowing sections, the dierent t theories and modelsforthe energy
loss
uctuationare described. . First, , the e Landau u theory andits limitations
arediscussed,andthen,the VavilovandGaussianstragglingfunctionsandthe
methodsinthethinlayersandgaseousmaterialsarepresented.
2
Landautheory
Foraparticleofmassmxtraversingathicknessofmaterialx,theLandauprob-
abilitydistributionmaybe writtenintermsofthe universal Landaufunction
()as[1]:
f(;x)
=
1
()
2
erehw
elcitraptnedicni ehtfo o charge
vAN
rebmuns’ordagovA
Z
l aireta mehtforebmunci i atom
A
l aireta mehtfothgiewci i atom
ytis den
x
laireta mehtfosse e thickn
otesolcrefsnart ygrene hti i wsnoisilloc ehtfo noitubirtnoc ehtserusaem
fi ro/dnaegralsi xfi seul avegral sdrawotsdnet,rebrosbanevi garoF.xamE
sehcaorppa fi ro/dnalla mssi i xfi orezsdrawotsdnet ,esi i weki L.ll amssi i
1.
fo noitpircsedeht t ni i rucco hci hwsemi ger owt t sehsiugnitsi i dfoeulavehT
:snoitautcu
noitasi ion
tnedicni ehtfotsomrollafossol ehtgnivlovnisnoisillocforebmunegral l A 1.
.rebrosba nafolasrevarteht t gnirudygrene e particle
ygrenella msfo edutitl umafo desopmocsi i refsnart ygrenelatot ehtsA
snoitautcu
ehtebircseddna meroehtti millartnecehtylppanace e losses,w
citsi vitaler-nonot t el l bacil ppa si i esacsihT T .noitubirtsi dnaissua Ga a by
nae meht nehw.e.i( 01 1 ytilauqenie e hty y bdebircsedsid nap p articles>
refsnart ygrene e mumixa a meht t nahtretaerg g si rebrosba ehtni ssol y y energ
.)noisillocelgnisa in
-i dnoc ynarednusnortceletnedicni dnasretnuocni ht t gnisrevartselcitraP P 2.
.sn tio
voli vaV,01 1 1 0:0er r asn oitubirtsiddnase itilauqenitn avelereh T<<
.noitubirtsiduadnaL,10:0 dna,noitud d istrib<
wol hti wsnoisillocehtfonoitubirtnocehtybdenedsi i emi gerl anoitiddanA
ehtsi 0I erehw,0I= noitalereht t hti i wdetamitseebnac hcihwrefsnart t energy
rebmunehttahtsemussayroeht uadnaL. motaehtfolaitnetopnoitasi i noi nae m
.1 0I= noitcirtser a sahti i ,yltneuqesnoc c dna, hgihsi i snoisilloc c eseht t of
,)lmth.llatnaeg/tnaeg/codsa/hc.nrec.ofniwww//:ptthLRUees( TNAEG In
laicepsti mil l siht woleB.05 =0I=tates s neebsahyroeht uadnaLfo ti mil l the
sihT. desueralairetamehtfo erutcurts ci mota ehttnuocca otni gnikatsledo m
ruoivahebehtswohs1 erugi F.slaireta msuoesagdnasreyal ni ht t ni i tnatropmi i is
VeG1dna Vek001fo nortcelenarofssenkcihtreyal ehtfonoitcnufasa0I= of
. muinarUdna nociliS, nogrAni ygrene citenik k of
ygrene ehtrofsledomdnaseiroehttnereideht,snoitces gni wollof f eht t In
snoitati mil sti i dna a yroeht t uadnaLeht,tsri i F . debircsedera a noitautcu
loss
eht dnasnoitcnuf gnilggartsnaissuaGdna a volivaVeht,neht t dna,dessucsid d are
. detneserperaslairetamsuoesagdnasreyal nihteht t ni i sdohte m
2
yroeht uadnaL
-borpuadnaLeht,xlaireta mfossenkcihtagnisrevartx mssamfoelcitrapar r Fo
noitcnuf uadnaLl l asrevinuehtfo smret t ni i nettirwebyamnoitubirtsi i dytili i ab
:]1[sa)(
) f(;x
=
1
)(
2
(a)
(b)
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114
ch-psextra2.tex,v:2.27
2007/08/13
