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A new method for imaging nuclear threats using cosmic ray muons C.L.Morris,JeffreyBacon,KonstantinBorozdin,HaruoMiyadera,JohnPerry,EvanRose,
ScottWatson,andTimWhite
LosAlamosNationalLaboratory,LosAlamos,NM,87545USA.
DerekAberle,J.AndrewGreenandGeorgeG.McDuff
NationalSecurityTechnologies,LosAlamos,NM,87544USA.
ZarijaLukić
LawrenceBerkeleyNationalLaboratory,Berkeley,CA,94720USA.
EdwardC.Milner
SouthernMethodistUniversity,Dallas,TX,75205,USA.
AbstractMuontomographyisatechniquethatusescosmicraymuonstogeneratethree
dimensionalimagesofvolumesusinginformationcontainedintheCoulombscatteringof
themuons.Advantagesofthistechniquearetheabilityofcosmicraystopenetrate
significantoverburdenandtheabsenceofanyadditionaldosedeliveredtosubjectsunder
studyabovethenaturalcosmicrayflux.Disadvantagesincludetherelativelylongexposure
timesandpoorpositionresolutionandcomplexalgorithmsneededforreconstruction.
Herewedemonstrateanewmethodforobtainingimprovedpositionresolutionand
statisticalprecisionforobjectswithsphericalsymmetry.
Introduction Cosmicraymuonscatteringmeasurementsprovideamethodforimagingtheinternal
structureofobjectsusingtheinformationcontainedinthenaturallyoccurringcosmicray
fluxfoundatthesurfaceoftheearth[1‐3].TheuseofthesignalfromCoulombscattering
enablesthreedimensionalimagingofcomplexsceneswithhighsensitivitytoobjects
composedofmaterialswithhighatomiccharge(Z).Thismakesthistechniqueuniquely
suitablefordetectingandmeasuringthepropertiesofatomicexplosivesthatmaybe
inaccessiblewithothertechniques.
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Ifanuclearmaterialhasbeendetecteditisimportanttobeabletomeasuredetailsofits
constructioninordertocorrectlyevaluatethethreat.Thefluxofcosmicraysislowsoit
takesverylongexposurestoproduceimageswithhighresolution.Inthispaperweshow
howonecantakeadvantageofsphericalsymmetrytoimprovethestatisticalprecisionof
muonimaging.
Three Dimensional Imaging Althoughcosmicraysarehighlypenetratingandcanimagethroughconsiderable
overburden,thefluxislimited.Thetimesrequireddetectingthepresenceofquantitiesof
uraniumorplutoniumnecessarytocreateanuclearexplosionareontheorderofminutes,
andthetimesneededtoimagethesedeviceswith~2cmresolutionareontheorderof
hours.
ImagingwithcosmicraysisbasedonmeasuringthemultipleCoulombscatteringofthe
muons.Thedominantpartofthemultiplescatteringpolar‐angulardistributionisGaussian:
2
dN
N  2 0 2

e
,
d 20 2
1)
theFermiapproximation,whereisthepolarangleand0isthemeanmultiplescattering
angle,whichisgivenapproximatelyby:
0 
14.1 MeV
p
l
X0
2)
Themuonmomentumandvelocityarepandrespectively,listhematerialthickness,and
X0istheradiationlengthforthematerial.Thisequationneedstobeconvolvedwiththe
cosmicraymomentumspectruminordertodescribetheangulardistribution.
Thereareseveralalgorithmsforgeneratingtomographicimagesusingtheinputandthe
outputtrajectoriesofthecosmicraysdescribedbySchultzetal.[2]Hereweuseavery
simplemethodwithascenecomposedofvoxels.Foreachvoxelaone‐dimensional
histogramofscatteringanglein1mradstepsfrom0to200mradiscreated.Thehistogram
isincrementedforeverycosmicrayforwhichtheincidentcosmicrayinterceptsthevoxel
andforwhichthedistancebetweentheinputandtheoutputcosmicraysintheplaneofthe
voxelislessthan2cm.Forlargescatteringanglesthisrequirementassociatesmeasured
scatteringeventswithdefinedvoxelsinwhichthemostofthescatteringoccurred.
Thescatteringdistributionforeachvoxelisfittedwithamodelthatusessevenmomentum
groups,[4]pi,toapproximatethemuonspectrumwith,
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2
dN
A  2
 sin( ) i2 e 20 i
d
 0i
 0i 
,
3)
14.1 l
pi X 0
toapproximatethemuonenergydistribution.
Themodelhasbeencalibratedwithdatatakenthroughthreethicknessesoflead,5.08,
10.16and15.24cm.Theamplitudes,Ai,ofeachenergygroup,aswellastheintrinsic
angularresolutionandafixednumberofradiationlengthsduetothedrifttubesandother
structuralelementsofthemuondetectorswerefittedtominimizethelogarithmofthe
likelihoodfunctionassumingthedataweredescribebyaPoisondistribution.Thismodel
doesnotaccountforchangesintheshapeofthemuonspectrumduetostopping.A
maximumlikelihoodfittothesetofleaddataisshowninError!Referencesourcenot
found..Alsoshownisthedecompositionofoneofthedatasetsintoitsmomentumgroups.
Figure1)Multi‐groupfittotheleadcalibrationdata.Ontheleftisaplotofthefitstodifferent
thicknessesoflead,theplotontherightshowsthedecompositionofthefitintoitsmultipleGaussian
components.
Imageswereconstructedbyfittingtheangulardistributionforeachvoxeltoobtainthe
averagenumberofradiationlengthsofmaterialthattheensembleofhistogramentrieshas
traversed.Thisreconstructionalgorithmisscalabletolargedatasets,issimpletocompute,
andprovidesnearoptimaluseofthescatteringinformation.However,itdoesn’toptimally
usethevertexinformation.
Wehaveimagedseveralsphericallysymmetricobjects:nestedsphericalshellsofcopper
andtantalum,thecoppershellalone,andahollowleadball.Theouterradiiwere6.5,4.5,
and10cmandtheinnerradiiwere4.5,1,and2.5cmforthecopper,tantalumandlead
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shellsrespectively.Cartesianslicesthroughthethreedimensionaltomograms,centeredon
theobject,areshowninFigure2.
Figure2)Cartesianslicesthroughthetomographsofthethreeobjects.Theyarepresentedonthe
samepositionandgreyscales.Thegreyscaleislinearbetween0and80radiationlengthsfromblack
towhiterespectively.
One Dimension Imaging Thecenterofeachobject(xc,yc,zc)wasestimatedbyfindingthecentroidofthesignalfrom
theobjectusingtheCartesianslicesshowninFigure2.Thewereusedforaone‐
dimensionalreconstruction.Atrajectorywasdefinedbythelinex(s)where:
x( s)  x0  x s
y ( s)  y 0  y s 4)
z ( s )  z 0  z s
Here(x0,y0,x0)isapointonthelinewithdirectioncosines(x’,y’,x’).Thepointof
minimumdistancebetween(xc,yc,zc)and(x0,y0,x0)isgivenbys=s0:
s0 
( xc  x0 ) x   ( y c  y 0 ) y   ( z c  z 0 ) z 
. x 2  y  2  z  2
5)
r0  ( x( s0 )  xc ) 2  ( y ( s0 )  y c ) 2  ( z ( s0 )  z c ) 2 . 6)
Theradiusofclosestapproachis:
AtwodimensionalhistogramofscatteringangleversusradiusisshowninFigure3.
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Figure3)Left)scatteringanglevs.radiusfortheleadsphericalshell.Thegreyscaleisproportional
tothelogarithmofthenumberofcountsperbin.Ontherightareplotsofcountsvs.scatteringangle
takenalongthelinesshownintheplotontheleft.
Asphericallysymmetricobjectcanbedescribedbyasetofshellsatriwiththickness
dr  ri 1  ri 1  / 2 andofamaterialwithradiationlength,X0i,andweighteddensityvi/X0i.
Theradiationlengthweightedpathlength,Li,asafunctionofricanbeobtainedfromthe
datashowninthe2‐dimensionalhistogrambyusingthemulti‐groupfittingtechnique
describedabove(Equation3)foreachradialbin.ThefittotheleaddatashowninFigure1
hasbeencorrectedby12%toaccountfortheaverage1/cos()increaseinthicknessofthe
leadintheplanargeometryandthenusedhere.
TheLiarerelatedtothesetofradiationlengthweightedvolumedensities,vj,byapath
lengthvectorPij(seeFigure4):
Li  Pij
vj
X0 j
. 7)
8)
Thepathlengthvectoristhelengthaparticleatritraversesthroughshellj:
Pi , j  0
for i  j
2
dr 

2
Pi , j  2  r j    r j for i  j
2

2
. 2
dr 
dr 


2
2
Pi , j  2  r j    r j - 2  r j 1    r j 1 for i  j
2
2


Thiscanbesolvedforthevj/X0jusingtheregularizationtechniquesdescribedinPress.[5]
Thistechniquedampenstheon‐axisnoisethatarisesinconventionalAbelinversions[6]
whichisimportantherebecauseofthepoorstatistics,especiallyatsmallr.
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Figure4)AnIllustrationofthepath‐lengthmatrix,Pi,j.
Theresultsforthethreeobjects,eachwith24hoursofexposure,areshowninFigure5.
Thevj/X0iforeachofthematerialsstudiedhere,copper,tantalum,andlead,arewith10%
ofthetabulatedvalues.[7]Onecaneasilydistinguishthevoidinsideeachoftheshells,even
the1cmradiusvoidatthecenterofthetantalumshell.Ananalysisofthewidthofthe
edgesinFigure5giveapositionresolutionof3mm.
Figure5)Radiationlengthweighteddensityvsradius.Theradiihavebeenmirroredaroundr=0(the
dataatnegativerarethesameasthedataforpositiver).Horizontallinesshowthetabulatedvalueof
/X.[7]
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Itisworthnotingthattheseobjectsaredifficulttostudywithconventionalx‐ray
radiography.Whilethecavitycanbeobservedwithrelativelycrudecollimationtechniques,
quantifyingthecavitydensitytakessophisticatedanti‐scattertechniques.[8]
Conclusion Cosmicrayscatteringdataweretakenonasetofsphericallysymmetricobjects.Thedata
wereanalyzedassumingsphericalsymmetry.Thedatawerestoredinatwodimensional
histogramofscatteringanglevs.radius.Theangulardistributionswerefittedbyasumof
Gaussianswhoseamplitudeswerefixedbyfitstodatatakenonasetofplanarobjects.This
resultedinone‐dimensionalplotsofthicknessesinradiationlengthsforeachoftheobjects.
ThesewereanalyzedwitharegularizedAbelinversiontechniqueyieldingradiation‐length‐
weightedvolumedensities.Theseresultsallowedaquantitativeevaluationofthematerial
compositionoftheobjects.
Acknowledgements WewouldliketoacknowledgehelpfromDaveSchwellenbachandWendiDreesensetting
upthehardwareandsoftwarethathasenabledthesemeasurements.Thisworkwas
supportedinpartbytheUnitedStatesDepartmentofEnergy,theUnitedStates
DepartmentofState,andtheDefenseThreatReductionAgencyoftheUnitedStates
DepartmentofDefense.
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