Solid angle subtended by a cylindrical detector at a point source in terms of elliptic inte

arXiv:math-ph/0211061v2 2 Feb 2003Solidanglesubtendedbyacylindricaldetector

atapointsourceintermsofellipticintegrals

M.J.Prata1

InstitutoTecnol´ogicoeNuclear(ITN),Sacav´em,Portugal

1Introduction

Theknowledgeofthesolidangle(Ω)subtendedbyaright,finite,circular

cylinderatapointisotropicsourceisrequiredinnumerousproblemsinnu-

clearandradiationphysics.Generally,Ωcanbeexpressedassumoftwo

components:thatsubtendedbythecylindricalsurface(Ωcyl)andtheotherby

eitheroftheendcircles(Ωcirc).Throughtheyearsthiscalculationhasbeen

addressedbyvariousauthorsusingdifferentmethods.Withouttheworryofbeingexhaustivewegivesomeexamplesofsuchworks.Masket(1957)out-linedageneralprocedurebasedonStokestheoremtoreducethedoublein-tegralΩ=󰀃󰀃sinθdθdϕtoacontourintegralinasinglevariable(θorϕ).

ThemethodwasusedtoexpressΩcircandΩcylassingleintegralswhichwere

numericallyintegrated.Extensivetablesdescribingtheseresultsbothforthe

discandtheentirecylinderwerereportedinaseparatework(Masketetal,

1956).Anapproximationtothesolidangledefinedbytwoparallelplanesur-

faceswasdescribedbyGillespie(1970)andappliedinthecasesoftwoequal

rectanglesandtwoequalcirclesinaface-to-facegeometry.Withthismethod

eachsurfaceissubdividedintosmallfiniteareasandthetwodoubleintegrals

arethenreplacedbyadoublesummation.ThecalculationofΩcircwastreated

byGardnerandVerghese(1971)byreplacingthediscwitharegularn-side

polygonofequalarea,forwhichananalyticalexpressionwasgiven.Inasim-

ilarway,Ωcylwasapproximatedbytheanalyticalexpressionforann-side

regularpolyhedralsurface(Vergheseetal,1972).Greenetal(1974)usedthe

MonteCarlomethodtocalculatethecylindersolidanglefortwoheight-to-

radiusratios(1:1,2:1),consideringdistancesfromsourcetocylindercenter

upto12cylinderradiiandangularpositionsofthesourcerangingfrom0oto

90ofromthecylinderaxis.

AnanalyticalexpressionforΩcircintermsofellipticintegralsduetoPhilip

A.Macklin(Macklin,1957)appearsincludedasafootnoteinMasket(1957).

InthepresentworkweshowthatalsoΩcylcanbereducedtoellipticintegrals

andgive,withoutderivation,expressionsforΩcircwhichcanbededucedin

aakinwayandaredifferentfromthatduetoP.A.Macklin.Thesolidan-

gleofthewholecylindercanthenbeexpressedintermsofellipticintegrals

whichareratherwellknownfunctions(e.g.Milne-Thomson,1964)forwhich

computationalgorithmsandtablesarereadilyavailable.

Insteadofblindnumericalintegration,onecanturntothevarietyofnumeric

methodsalreadyexistent,whichenablethefastcalculationofthesolidan-

gleforthewholerangeofparameters.Forinstance,thecompleteintegrals

ofthefirstandsecondkindscanbecomputedusing(i)thepolynomialap-

proximationsduetoHastings(1955)andincludedinMilne-Thomson(1964,

eqs.17.3.33to17.3.36);(ii)theprocessofthearithmetic-geometricmean

(Milne-Thomson,1964,17.6)or(iii)theinfiniteseries(Milne-Thomson,1964,

17.3.11,17.3.12)whichcanbeusedincombinationwithLanden’stransfor-

mationwhenthemodularangleisclosetoπ/2.

Sincethesolidanglecanbedecomposedintoellipticintegrals,anypossibility

offindinggeneralanalyticalexpressionsintermsofelementaryfunctionsis

henceforwardprecluded.Ontheotherhand,thecalculationhasbeenput

underthesoundroofofthesubjectofellipticintegralsandfunctions.

Inarecentwork(Prata,2002)wedescribetheanalyticalcalculationofthe

2solidanglesubtendedbyacylinderatapointcosinesource.Combiningthese

resultswiththosepresentedhere,thecaseofanaxiallysymmetricpointsource

withanangulardistributiongivenbyfk(Ω)=1+aΩ·kcanbetreated

analiticallywhenthesourceaxis(k)isorthogonaltothatofthecylinder.

2SolidAngleCalculation

Thesolidanglesubtendedbyagivensurfaceatapointisotropicsourcecan

bedefinedas

Ωsurf=1

Fig.2.NotationforΩcirc

2.1CalculationofΩcyl0

Fig.3.QuantitiesusedtocalculateΩcyl0

Fromfigs.1and3,itfollowsthat

Ωcyl0(L,r,d)=1/(4π)ϕmax󰀃

ϕminθmax󰀃

θminsinθdθdϕ=1/(2π)ϕo󰀃

0(cosθmin−cosθmax)dϕ

where

ϕmax=−ϕmin=ϕo≡arcsin(r/d),(2)

cosθmin=L/󰀉

r2−(dsinϕ)2.(3)

4Thus,

Ωcyl0=L/(2π)ϕo󰀁

0󰀂L2+ρ2−(ϕ)󰀄−1/2dϕ.(4)

Thenwechangetheintegrationvariabletoγ−representedinfig.3andgiven

by

γ−=π/2−φ−/2(5)

where

φ−/2=arctan[sin(ϕ)ρ−/(r+d−cos(ϕ)ρ−)]

andρ−(ϕ)isobtainedfromeq.3.

Eq.4isrewrittenas

Ωcyl0=L/(2π)π/2󰀁

γo1

L2+ρ2−(γ−)󰀅d2−r2

(d+r)2−4drsin2γ−(7)

and

γo=(π/2+ϕo)/2.(8)

Introducing

m=4rd/(L2+(d+r)2),(9)

n=4rd/(d+r)2,(10)

thereresultsfromeq.7:

ρ2−(γ−)=(d+r)2(1−nsin2γ−),

5󰀉L2+(d+r)2󰀉

L2+(d+r)2=󰀉

1−n;d≥r.

Substitutingintherhsofeq.6yields

Ωcyl0=1/(2π)󰀉√1−n

1−m/n{√