Inversion formula: Single elastic scattering cross-section from multiple scattering distribution...
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Volume 60A, number 3 PHYSICS LETTERS 21 February 1977 INVERSION FORMULA: SINGLE ELASTIC SCAT1~ERING CROSS-SECTION FROM MULTIPLE SCATTERING DISTRIBUTION FUNCTION S.D. SHARMA and S. MUKHERJEE Department of Physics, Himachal Pradesh University, Simla-1 71001, India Received 28 October 1976 An expression giving the single elastic scattering differential cross-section in terms of multiple scattering distribu- tion function is derived, under the usual simplifying assumptions. The theory of multiple scattering (ms) of fast, struc- solid angle d&~. Then tureless charged particles or ions in matter has been studied in great detail [1—4]. The scattering is de- P1(O) d~2 = (o(O)/a) d ~2 . (1) scribed theoretically by a distribution function F(O, ci’), We write which gives the probability of a total deflection in the direction (0, P).The •det~mlhati01~ of the function F P (0) d~ = _L ~ (2! + 1 )h 1P (cos 0) dfl, (2) mvolves calculations in two steps. The first step is to 1 1=0 calculate the cross-section for scattering from one target atom. Then a statistical consideration for corn- with h1 = (P1(cos 0)>. Since the collisions have axial pounding successive deflections leads to the ms dis- symmetry, it can be shown that the average value of tribution function F(0, i/i). It has been the practice so any Legendre polynomial after n independent collisions far to express F(O, ~‘)in terms of the single scattering is equal to the nth power of the average value after one cross-section cr(O). It will, however, be useful to have collision. Hence, the probability that after n collisions an expression giving single scattering cross-section in the particle will be scattered through angle 0 into the terms of F(O, di), since experimentally one measures solid angle dfl is the multiple scattering data. Recently, Sigmund and Winterbon [5] gave such an inversion formula. To de- p (0) d~2 = ~ ~ (21 + l)P1(cos 0) h’~ d~2. (3) nve the formula, they solved the relevant transport 4~ i=o I equation in the small angle approximation by the Fourier transform technique~ As pointed out by the The ms distribution function is now defined as authors, the results depend on the analytic properties ,, of the approximated function F(0). It is, however, F(O) sinO dO = ~ P(n) ~~(0)) sin 0 dO , (4) possible to derive an exact inversion formula, which reduces to the results of Sigmund and Winterbon in the where P(n) is the probability that the particle makes small angle limit. The derivation makes use of the n collisions. If the target foil contains N scattering method of Goudsrnit and Saunderson [2]. We make centres per unit volume and has a thickness t, the the usual assumptions, viz. (i) the elastic scattering has average number of collisions is given by axial symmetry, and (ii) that all the particles traverse / ,, — 5 equal paths in matter, which may be taken to be equal ‘~~‘av — Ntc, to the thickness of the target foil. Our results will be which is usually a large number. We may, therefore, valid for any energy if the static potential picture makes assume a Poisson distribution for P(n), viz. sense for that energy, and if the experimentalists can .~—‘ ~ ~ ~fl 16 isolate within acceptable accuracy, the elastic events. ~fl) — ~l in. e ‘~‘a’~~ Let p1 (0) be the probability that a particle in a Combining the relations (3), (4) and (6), we get single collision is scattered through an angle 0 into a 195
Transcript of Inversion formula: Single elastic scattering cross-section from multiple scattering distribution...
Volume60A, number3 PHYSICSLETTERS 21 February1977
INVERSION FORMULA: SINGLE ELASTIC SCAT1~ERINGCROSS-SECTIONFROM MULTIPLE SCATTERING DISTRIBUTION FUNCTION
S.D. SHARMA andS. MUKHERJEEDepartmentof Physics,HimachalPradeshUniversity,Simla-171001,India
Received28 October1976
An expressiongiving thesingleelasticscatteringdifferentialcross-sectionin termsof multiplescatteringdistribu-
tion functionis derived, undertheusualsimplifyingassumptions.
Thetheoryof multiple scattering(ms)of fast,struc- solid angled&~.Thenturelesschargedparticlesor ionsin matterhasbeenstudiedin greatdetail [1—4]. The scatteringis de- P1(O) d~2= (o(O)/a)d ~2. (1)scribedtheoreticallyby a distribution functionF(O, ci’), We writewhich givestheprobability of a totaldeflectionin thedirection(0, P).The•det~mlhati01~of the functionF P (0)d~= _L ~ (2! + 1)h1P (cos0)dfl, (2)mvolvescalculationsin two steps.The first stepis to 1 1=0
calculatethe cross-sectionfor scatteringfrom onetargetatom.Thena statisticalconsiderationfor corn- with h1 = (P1(cos0)>. Since thecollisionshaveaxialpoundingsuccessivedeflectionsleadsto the msdis- symmetry,it canbeshownthatthe averagevalueoftribution functionF(0, i/i). It hasbeenthe practiceso anyLegendrepolynomial aftern independentcollisionsfar to expressF(O, ~‘)in termsof the singlescattering is equalto thenthpowerof the averagevalueafteronecross-sectioncr(O). It will, however,beusefulto have collision. Hence,theprobabilitythatafter n collisionsan expressiongivingsinglescatteringcross-sectionin theparticlewill bescatteredthroughangle0 into thetermsof F(O, di), sinceexperimentallyonemeasures solid angledfl isthemultiple scatteringdata.Recently,SigmundandWinterbon [5] gavesuchan inversionformula. To de- p (0)d~2= ~ ~ (21+ l)P1(cos0) h’~d~2. (3)nvethe formula,theysolvedthe relevanttransport 4~i=o Iequationin thesmall angleapproximationby theFouriertransformtechnique~As pointedout by the The msdistribution functionis now definedasauthors,theresultsdependon theanalyticproperties ,,
of the approximatedfunctionF(0). It is,however, F(O)sinO dO = ~ P(n)~~(0)) sin 0 dO , (4)possibleto derivean exactinversionformula, whichreducesto the resultsof SigmundandWinterbonin the whereP(n) isthe probabilitythat the particlemakessmallanglelimit. The derivationmakesuseof the n collisions.If thetargetfoil containsNscatteringmethodof GoudsrnitandSaunderson[2]. Wemake centresperunit volumeandhasa thicknesst, thetheusualassumptions,viz. (i) the elasticscatteringhas averagenumberof collisionsis givenbyaxial symmetry,and(ii) that all the particlestraverse / ,, — 5equalpathsin matter,which maybetakento be equal ‘~~‘av— Ntc,to thethicknessof thetargetfoil. Our resultswill be which is usuallya largenumber.We may,therefore,valid for anyenergyif thestaticpotentialpicturemakes assumea Poissondistribution for P(n),viz.sensefor thatenergy,andif theexperimentalistscan .~—‘ ~ ~ ~fl 16isolatewithin acceptableaccuracy,the elasticevents. ~fl) — ~l in. e ‘~‘a’~~
Letp1 (0) betheprobabilitythat a particlein a Combiningtherelations(3), (4) and(6), we getsinglecollision is scatteredthroughan angle0 into a
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Volume60A, number3 PHYSICSLETTERS 21 February1977
h1 = 1 + in [21r fF(o)P1(cos0) sin0 dO] Heref(ri) = 2~J0(~O)lnS(~), (12)av
and hence with
a(O)d&2 r—~--~(cos0—l)d&2 S(~)~�~2irfF(0)Jo(nO)OdO, (13)2ir
andthe leadingcorrectiontermis givenbyd~2
+ ~—~- ~ (21+ 1 )P1(cos0) ~ in f F(0)0 dO.The ms distribution function is usuallya modified gaussiandistribution. Thisdeparturefroma gaussiandistribution,in fact,carriesthe information
x in [21rfF(O)P1(cos0)sin o dO] . (8) aboutthe natureof the singlescatteringcross-section.Foran estimationof error,we mayneglectthe modi-
The relation(8) is exact.The resultsof Sigmundand fication andassumethatF(O) e~02.This indicates
Winterboncanbe obtainedfrom (8) for 0 ~ 0, if one that theerror will be of order— ~ in c~.makesthesmall angleapproximation
The authorsarethankfulto the DepartmentofP
1(cos0) J0((l+ +)0) = J0(~0), (9) Atomic Energy,Governmentof India, for financialandalsoreplacesthesummation~1(2l+l)by the inte- support.gralf~’2ndn.Thisgives
OdO ~in [2irf~io)Jo(no)odo] References
[11 W.T. Scott,Rev. Mod. Phys.35 (1963)231,andreferences
It is possibleto makean estimateof the error madein giventherein.writing (10). While theapproximationmadeby the sub- For morerecentdevelopments,see:CJ. Joachin,Quan-
tumCollision Theory(North-Holland,Amsterdam,1975).stitution (9) is a valid one for 0 small for any1, the [2] S. GoudsmitandJ.L. Saunderson,Phys.Rev. 57 (1940)accuracyof the replacementof thesummationover1 24; 58 (1940)36.
by anintegral over i~canbejudgedby recallingEuler’s [31GMoii~re,Z. Naturforsch2a (1947)133; 3a (1948)78.summationformula, [41S. Mukherjee,Phys.Rev.162 (1967)254; 167 (1968)323.
[5] P. SigmundandK.BWinterbon,Nucl. Instr.Meth. 119(1974)541.
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