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A simulation study of the Cosmic Ray Inspection and Passive Tomography (CRIPT) muon spectrometer D. Waller

Defence R&D Canada – Ottawa

Technical Memorandum DRDC Ottawa TM 2010-168

October 2010

A simulation study of the Cosmic RayInspection and Passive Tomography (CRIPT)muon spectrometerD. WallerDefence R&D Canada – Ottawa

Defence R&D Canada – OttawaTechnical MemorandumDRDC OTTAWA TM 2010-168October 2010

Principal Author

Original signed by D. Waller

D. Waller

Approved by

Original signed by J. Tremblay-Lutter

J. Tremblay-LutterHead Capabilities for Asymmetric and Radiological Defence and Simulation Section

Approved for release by

Original signed by C. McMillan

C. McMillanDRDC Ottawa Chief Scientist

c© Her Majesty the Queen in Right of Canada as represented by the Minister of NationalDefence, 2010

c© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de laDéfense nationale, 2010

Abstract

The Cosmic Ray Inspection and Passive Tomography (CRIPT) for Special Nuclear Mate-rial (SNM) Detection project is investigating the effectiveness of muon scattering tomog-raphy for detecting SNM and dense radiation-shielding materials like lead. The goal ofCRIPT is to design, build and test a large-scale prototype system that uses naturally oc-curring cosmic-ray-induced muons to image a volume of interest. Measuring the momentaof muons on an event-by-event basis should reduce the required imaging times signifi-cantly. This paper describes relatively simple simulations of the performance of differentmagnetized and unmagnetized muon spectrometer designs. An unmagnetized spectrometeruses the multiple Coulomb scattering of a muon through known absorbers to estimate themuon’s momentum, p. A magnetized spectrometer measures p using both multiple scat-tering and the magnetic deflection inside magnetized iron absorbers. Using only multiplescattering measurements, the resolution of 1/p is 50% or better. The gain in resolutionobtained by magnetizing the iron absorbers is not very significant (less than 10% relativeimprovement) unless the magnetic field in the iron is very high (≥ 1.0 T). An unmagnetizedspectrometer appears to be the preferred choice for CRIPT as it is simpler and likely morecost-effective to implement.

Résumé

Le projet Inspection et tomographie passive par rayonnements cosmiques (CRIPT) pourles matières nucléaires spéciales (MNS) étudie l’efficacité de la tomographie par diffusionmuonique pour détecter les MNS et les matériaux de blindage denses contre les rayonne-ments, comme le plomb. L’objectif de la méthode CRIPT est de concevoir, construire et tes-ter un système prototype à grande échelle qui utilise les muons produits naturellement parles rayons cosmiques pour imager les volumes d’intérêt. En mesurant la quantité de mouve-ment événement par événement, on devrait réduire grandement le temps de prise d’image.Dans le présent document, nous décrivons des simulations relativement simples de la per-formance de différents types de spectromètre à muons magnétisés et non magnétisés. Unspectromètre non magnétisé utilise la diffusion coulombienne multiple d’un muon à tra-vers des absorbeurs connus, afin d’estimer la quantité de mouvement p du muon. Un spec-tromètre magnétisé mesure p en utilisant à la fois la diffusion multiple et la déviationmagnétique à l’intérieur d’absorbeurs de fer magnétisés. En utilisant seulement des me-sures de diffusion multiple, la résolution de 1/p est de 50% ou plus. Le gain en résolutionobtenu par la magnétisation des absorbeurs de fer n’est pas très important (moins de 10%d’amélioration relative), à moins que le champ magnétique dans le fer ne soit très élevé(> 1,0 T). Un spectromètre non magnétisé semble être le choix approprié pour le projetCRIPT, car il est plus simple et probablement plus rentable à appliquer.

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Executive summary

A simulation study of the Cosmic Ray Inspection andPassive Tomography (CRIPT) muon spectrometer

D. Waller; DRDC OTTAWA TM 2010-168; Defence R&D Canada – Ottawa;October 2010.

Introduction: The Cosmic Ray Inspection and Passive Tomography (CRIPT) for SpecialNuclear Material (SNM) Detection project is a Chemical, Biological, Radiological, Nu-clear, Explosives (CBRNE) Research & Technology Initiative (CRTI) project that is inves-tigating the effectiveness of muon scattering tomography for detecting SNM and the denseshielding that would likely surround a smuggled radiological dispersal device (RDD). Thegoal of the CRIPT project is to design, build and test a large-scale prototype system thatwill be able to image air-cargo containers, and nuclear waste containers using naturallyoccurring cosmic-ray-induced muons. Measuring the momenta of muons on an event-by-event basis should reduce the required imaging times significantly. This paper studies thesimulated performance of different magnetized and unmagnetized muon spectrometer de-signs.

Method: Different CRIPT muon spectrometers designs are studied in Monte Carlo simu-lations written in MATLAB. Multiple Coulomb scattering and the deflection of muons in amagnetic field are the two main physical processes included in the simulations. The multi-ple scattering angular distribution is assumed to be Gaussian; non-Gaussian scattering andenergy loss of the muon are not considered in these simulations. The muon spectrometersestimate the momentum, p, of a muon by measuring the muon deflection angles due tomultiple scattering and magnetic fields (if present). A maximum likelihood technique isused to obtain the estimates of 1/p.

Results: Examination of the simulations show that the output of the Monte Carlo simula-tions for the spectrometers behaves as expected. The effect of varying several spectrometerparameters (number of absorber planes, absorber thickness, hit resolution of muon detec-tors, strength of magnetic field in Fe absorbers) on the 1/p resolution is studied.

Conclusions: Using only multiple scattering measurements (i.e. an unmagnetized spec-trometer), the 1/p resolution is < 50%. The resolution is strongly dependent on the numberof layers of scattering Fe absorbers as the uncertainty of 1/p is statistics limited: the morescattering layers, the more scattering angle measurements that are available to constrainthe muon momentum. The 1/p resolution of high momentum muons (pμ ≥ 5 GeV/c) isalso strongly affected by the hit resolution of the spectrometer’s muon trackers. The Fe

DRDC OTTAWA TM 2010-168 iii

absorber thickness does not have a strong effect on the 1/p resolution for an unmagnetizedspectrometer.

The gain in 1/p resolution obtained by magnetizing the Fe absorbers is not very significant(less than 10% relative improvement) until the magnetic field in the Fe is very high (≥ 1.0T). At lower fields, the deflection from multiple scattering is greater than the magneticdeflection so the sign of the muon’s electric charge is frequently estimated incorrectly,leading to poor 1/p resolution. To achieve better 1/p resolution, it is likely cheaper andeasier to increase the number of absorber-detector layers in an unmagnetized spectrometerthan to magnetize the Fe absorbers to very high fields.

Recommendations: It appears that an unmagnetized spectrometer is likely to be the mostcost-effective solution for measuring muon momenta on an event-by-event basis. The base-line design of two absorber-detector layers provides approximately 40% 1/p resolutionwith 2 mm hit resolution for the muon detectors. As the absorber thickness does not havea large effect on the resolution, it is recommended that thin Fe absorbers (5 cm thick) beused to reduce the weight and cost of the spectrometer.

The 1/p resolution functions obtained from this study should be incorporated into imagereconstruction simulations to assess the full impact of different spectrometer designs. Inparallel with this activity, a more detailed simulation of this study’s baseline, unmagnetizedmuon spectrometer should be performed in GEANT4 to test the validity of the resultsobtained in this study.

iv DRDC OTTAWA TM 2010-168

Sommaire

A simulation study of the Cosmic Ray Inspection andPassive Tomography (CRIPT) muon spectrometer

D. Waller ; DRDC OTTAWA TM 2010-168 ; R & D pour la défense Canada –Ottawa ; octobre 2010.

Introduction : Le projet Inspection et tomographie passive par rayonnements cosmiques(CRIPT) pour les matières nucléaires spéciales (MNS) s’inscrit dans le cadre de l’Initia-tive de recherche et de technologie (IRTC) sur les agents chimiques, biologiques, radiolo-giques, nucléaires et explosifs (CBRNE) qui vise à étudier l’efficacité de la tomographiepar diffusion muonique pour détecter les MNS et le blindage dense qui pourrait entou-rer un dispositif de dispersion radiologique (DDR) passé en contrebande. L’objectif duprojet CRIPT est de concevoir, construire et tester un système prototype à grande échellequi pourra imager des conteneurs de fret aérien et des conteneurs de déchets nucléairesen utilisant les muons naturellement induits par le rayonnement cosmique. En mesurant laquantité de mouvement d’un événement individuel, on devrait réduire grandement le tempsde prise d’image. Dans le présent document, nous décrivons des simulations relativementsimples de la performance de différents types de spectromètre à muons magnétisés et nonmagnétisés.

Méthode : Différents types de spectromètres à muons CRIPT sont étudiés à l’aide desimulations de Monte-Carlo écrites en MATLAB. La diffusion coulombienne multiple etla déviation des muons dans un champ magnétique sont les deux principaux processusphysiques inclus dans les simulations. La distribution angulaire de la diffusion multiple estprésumée être gaussienne ; la diffusion et la perte d’énergie non gaussiennes des muonsne sont pas prises en compte dans ces simulations. Les spectromètres à muons estiment laquantité de mouvement p d’un muon en mesurant les angles de déviation du muon dus àla diffusion multiple et aux champs magnétiques (s’il y en a). Une technique du maximumde vraisemblance est utilisée pour obtenir les estimations de 1/p.

Résultats : L’examen des simulations des spectromètres montre qu’elles donnent les résultatsprévus. L’effet de la variation de plusieurs paramètres du spectromètre (nombre de plansabsorbeurs, épaisseur des absorbeurs, résolution en vénement des détecteurs de muons,intensité du champ magnétique des absorbeurs de Fe) sur la résolution de 1/p est étudié.

Conclusions : En utilisant seulement les mesures de diffusion multiple (c.-à-d. avec unspectromètre magnétisé), la résolution de 1/p est < 50%. La résolution est fortementdépendante du nombre de couches d’absorbeurs de Fe, car l’incertitude de 1/p est sta-tistiquement limitée : plus il y a de couches diffusantes, plus on dispose de mesures des

DRDC OTTAWA TM 2010-168 v

angles de diffusion pour contraindre la quantité de mouvement du muon. La résolution de1/p des muons à des valeurs élevées de quantité de mouvement (pμ > 5 GeV/c) est en outrefortement tributaire de la résolution en vénement des traceurs de muons du spectromètre.L’épaisseur des absorbeurs de Fe n’a pas un effet important sur la résolution de 1/p pourun spectromètre magnétisé.

Le gain en résolution de 1/p obtenu en magnétisant les absorbeurs de Fe n’est pas trèsimportant (moins de 10% d’amélioration relative) à moins que le champ magnétique dansle fer ne soit très élevé (> 1,0 T). Dans des champs de faible intensité, la déviation due à ladiffusion multiple est supérieure à la déviation magnétique de sorte que le signe de la chargeélectrique du muon est souvent estimé incorrectement, ce qui entrane une piètre résolutionde 1/p. Pour obtenir une meilleure résolution de 1/p, il est probable moins cher et plusfacile d’augmenter le nombre de couches d’absorbeurs-détecteurs dans un spectromètrenon magnétisé que de magnétiser les absorbeurs de Fe à des champs très élevés.

Recommandations : Il semble qu’un spectromètre non magnétisé soit la solution la plusrentable pour mesurer la quantité de mouvement des muons au cas par cas. La concep-tion nominale des couches d’absorbeurs-détecteurs fournit une résolution de 1/p d’envi-ron 40% avec une résolution en événement de 2 mm pour les détecteurs de muons. Commel’épaisseur des absorbeurs n’a pas un effet important sur la résolution, il est recommandéque des absorbeurs minces de Fe (5 cm d’épaisseur) soient utilisés pour réduire le poidset le coût de l’appareil. Les fonctions de résolution de 1/p obtenues à partir de cette étudedevraient être incorporées dans les simulations de reconstruction d’image pour valuer l’im-pact des différents types de spectromètres. En parallèle à cette activité, une simulation plusdétaillée du scénario de base de cette étude, soit un spectromètre à muons non magnétisé,devrait être effectuée dans GEANT4 pour tester la validité des résultats obtenus dans laprésente étude.

vi DRDC OTTAWA TM 2010-168

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Cosmic-ray muons and Special Nuclear Material (SNM) detection . . . . 1

1.2 CRIPT muon tracking system . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Muon momentum estimation . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Multiple Coulomb scattering in an unmagnetized spectrometer . . . . . . 7

2.1.1 Maximum likelihood fits for unmagnetized spectrometer . . . . . 8

2.2 Magnetized spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Maximum likelihood fits for magnetized spectrometer . . . . . . 9

3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Horizontal displacements and multiple scattering angles . . . . . . . . . . 11

3.2 Unmagnetized 1/p Fe results . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Magnetized Fe 1/p results . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Assessing the impact of different 1/p f it resolutions . . . . . . . . . . . . 27

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

DRDC OTTAWA TM 2010-168 vii

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

List of Abbreviations and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

viii DRDC OTTAWA TM 2010-168

List of figures

Figure 1: The principle of muon scattering tomography for detecting SNM in acargo container . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Figure 2: A side-view of unmagnetized spectrometer . . . . . . . . . . . . . . . . 6

Figure 3: The estimated and true trajectories of a scattered muon through anunmagnetized spectrometer . . . . . . . . . . . . . . . . . . . . . . . . 7

Figure 4: The probability density functions for deflection angles of positive andnegative muons in a magnetized spectrometer . . . . . . . . . . . . . . . 10

Figure 5: The horizontal displacement of the muon tracks as a function of muonpolar angle in an unmagnetized spectrometer . . . . . . . . . . . . . . . 12

Figure 6: The horizontal displacements and angular deflection of 2.0 GeV/cmuon tracks in a two-layer, unmagnetized, Fe spectrometer . . . . . . . 13

Figure 7: The difference between estimated and true scattering angles assumingperfect muon tracker hit resolution. . . . . . . . . . . . . . . . . . . . . 14

Figure 8: The fitted momentum values for 2 GeV/c muons in an unmagnetizedmuon spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 9: The fitted 1/p values for an unmagnetized muon spectrometer . . . . . . 17

Figure 10: The RMS, mean and resolution of 1/p estimates with an unmagnetizedspectrometer for different numbers of Fe absorbers . . . . . . . . . . . . 18

Figure 11: The RMS, mean and resolution of 1/p estimates with an unmagnetizedspectrometer for different thicknesses of Fe absorbers . . . . . . . . . . 19

Figure 12: The effect of the angular resolution of the spectrometer trackers on theRMS, mean and resolution of 1/p estimates with unmagnetizedspectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 13: The effect of the tracker hit resolution on the RMS, mean andresolution of 1/p estimates with an unmagnetized spectrometer . . . . . 21

Figure 14: The resolution of 1/p f it with an unmagnetized spectrometer fordifferent muon momenta and tracker hit resolutions . . . . . . . . . . . . 22

Figure 15: The fitted 1/p values for a magnetic spectrometer with correct andincorrect muon charge assignments . . . . . . . . . . . . . . . . . . . . 25

DRDC OTTAWA TM 2010-168 ix

Figure 16: The resolution of 1/p f it with a magnetic spectrometer for differentmuon momenta and magnetic field strengths . . . . . . . . . . . . . . . 26

Figure 17: The resolutions of 1/p f it with a magnetized spectrometer for differenttracker hit resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 18: The resolution of 1/p f it with a magnetized spectrometer for differentnumbers of Fe absorber layers . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 19: The resolution of 1/p f it for different thicknesses of Fe absorber layers . . 30

Figure 20: The resolution for 1/p as a function of muon momentum . . . . . . . . . 31

List of tables

Table 1: The results of 1/p f it measurements for an unmagnetized spectrometerwith perfect tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Table 2: The fraction of muon tracks with the correct charge assignment, σx[y] =0 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23



Table 5: The resolution of 1/p with no tracker hit uncertainty (σx[y] = 0 mm) fordifferent magnetic field strengths for a 2-layer spectrometer. . . . . . . . 27

Table 6: The resolution of 1/p (with 2 mm tracker hit resolution) for differentmagnetic field strengths for a 2-layer spectrometer. . . . . . . . . . . . . 27

Table 7: The resolution of 1/p (with 4 mm tracker hit resolution) for differentmagnetic field strengths for a 2-layer spectrometer. . . . . . . . . . . . . 27

x DRDC OTTAWA TM 2010-168

1 Introduction

The Cosmic Ray Inspection and Passive Tomography (CRIPT) for Special Nuclear Mate-rial (SNM) Detection project is a Chemical, Biological, Radiological, Nuclear, Explosives(CBRNE) Research & Technology Initiative (CRTI) project that is investigating the effec-tiveness of muon scattering tomography for detecting SNM and the dense shielding thatwould likely surround a smuggled radiological dispersal device (RDD). The goal of theCRIPT project is to design, build and test a large-scale prototype system that will be ableto image air-cargo containers, and nuclear waste containers.

1.1 Cosmic-ray muons and Special Nuclear Material(SNM) detection

A muon is a short-lived sub-atomic particle (τμ = 2.2× 10−6 s at rest) which is a heavierversion of an electron (mμ = 105 MeV/c2). Muons are produced in the Earth’s upper at-mosphere when energetic galactic (or extra-galactic) cosmic rays (usually protons) interactwith gas molecules, producing charged pions (π±) which promptly decay to like-chargedmuons (μ±) and muon neutrinos (νμ or νμ):

p+ 14N,16 O, etc. → X +π±,π+ → μ+ +νμ, (1)π− → μ− +νμ

Although muons are short-lived when they are at rest, many cosmic-ray-induced muonssurvive until they arrive at the Earth’s surface because they have velocities very close to thespeed of light, and thus are subject to relativistic time dilation 1.

In muon scattering tomography (MST), the trajectories of highly-penetrating cosmic-ray-induced muons are measured before and after they pass through a volume of interest (e.g. ashipping cargo container). Figure 1 shows the principle of MST. The muons undergo mul-tiple Coulomb scattering when they pass through matter. The higher the density and atomicnumber of the material, the larger the resulting deflection angle tends to be. More detailsabout multiple Coulomb scattering are provided in Section 2.1. The average momentum ofthe muons is approximately 3 GeV/c, so the typical scattering angle after passage througha cargo container is on the order of tens of milliradians (mrad) 2. After collecting enoughmuon scattering data, it is possible to build a three-dimensional (3D) image of the vol-ume of interest. The scattering densities (related to the variance of the scattering angles

1. When an object moves with a velocity close to the speed of light, time passes more slowly in its frameof reference.

2. At sea level, the momentum spectrum for cosmic-ray induced muons extends from 0 GeV/c to ex-tremely high momenta, with the peak flux at approximately 0.5 GeV/c and rapidly declining as the momen-tum increases: the flux at 5 GeV/c is an order of magnitude lower than the flux at 0.5 GeV/c.

DRDC OTTAWA TM 2010-168 1

and muon momenta) of all the voxels in a volume are estimated in order to determine thedensity/type of material that is present.

The flux of muons used in MST is naturally occurring, so there is no health risk associatedwith operating a MST system 3. The downside to using naturally occurring muons is thatit is challenging to reduce scanning times to operationally desirable times (one minuteor less). The flux at sea level is approximately 10,000 muons m−2 minute−1. In order tominimize scanning times, all the useful information available from the muons must beextracted. An extremely valuable piece of information for interpreting the muon scatteringmeasurements is the event-by-event muon momentum. In addition to the density and atomicnumber of the scattering material, the muon’s typical scattering angle is determined by itsmomentum: the larger the momentum, the smaller the scattering angle. The importance ofmeasuring the muon momentum has been highlighted by several other MST groups [1, 2,3]. These groups have calculated that the imaging times could be reduced by up to a factorof two by using muon momentum information. This paper describes simulation studies thathave been conducted to determine the performance of different muon spectrometer designsfor CRIPT.

1.2 CRIPT muon tracking systemIn order to determine the scattering angles of muons that pass through a volume of inter-est, the trajectories before and after scattering must be measured very precisely. This isachieved by using two or more layers of detectors both above and below the cargo whichmeasure the 3D location where a muon passes through the detector; this is called a “hit”.Two or more 3D hits define a muon’s trajectory; the more hits that are measured for a muon,the better the angular resolution of the fitted trajectory (called a “track”). Multiple similar(or identical) muon detectors are required in the muon spectrometer.

Two muon detection technologies are considered for CRIPT’s muon “trackers”: gas-filleddrift chambers and scintillator-based detectors. The drift chambers measure the ionizationcreated by the muon as it passes through them; their design is inspired by the muon driftchambers used in the OPAL detector [4]. The scintillator-based detectors measure the scin-tillation light that is produced when a muon passes through them. For these simulationstudies, no assumptions are made about which type of detector is used. The only prop-erty of the trackers that is simulated is the hit resolution. The results of this study willhelp to determine which technology is preferred for detecting/tracking muons in CRIPT’sspectrometer.

3. It may be possible in the future to use a man-made, accelerator-based muon beam for MST, but thistopic is currently beyond the scope of the CRIPT project.

2 DRDC OTTAWA TM 2010-168

Figure 1: The principle of muon scattering tomography for detecting SNM in a cargocontainer. Hits in the muon trackers of each module are indicated by the small red circlesalong the muon’s trajectory.

1.3 Muon momentum estimationIn principle, several different techniques may be used to estimate the momentum of a muonthat passes through CRIPT. These include measurements of any of the following:

1. The muon’s deflection angles due to multiple Coulomb scattering through absorbersof known thickness and radiation length;

2. The muon’s deflection in a magnetic field, either in air or through a magnetizedmaterial;

3. The rate of energy loss per unit length of material traversed (dE/dx); or,

4. The Čerenkov radiation that is (or is not) produced in either ring-imaging or (multi-stage) threshold Čerenkov detectors.

The first technique, measuring the deflection angles due to multiple scattering, has beensuccessfully used by the Soudan-2 and MACRO atmospheric neutrino experiments [5, 6]. Avariant of this technique has also been successfully demonstrated in a liquid time-projection


chamber [7]. Multiple scattering has also previously been proposed as a cost-effective so-lution for muon tomography [8]. It has been implemented by the Italian muon tomographygroup with mixed success [9].

The second technique, using a magnetic field to deflect the muon, could be used in asso-ciation with the first technique. This paper investigates the potential of combining thesetwo techniques by comparing the results from simulated, magnetized and un-magnetizedspectrometers.

The other techniques of muon momentum estimation are not currently being consideredfor the CRIPT project as they are less likely to provide cost-effective measurements of pμ.


2 Method

This section describes the methods employed to estimate muon momentum using both un-magnetized (Section 2.1) and magnetized (Section 2.2) spectrometers. The baseline designconsidered for both unmagnetized and magnetized spectrometers is shown in Figure 2. Itassumes that the spectrometer is located immediately below the lower tracking module ofCRIPT and has a depth of 2 m. Figure 2 shows a spectrometer that is composed of two lay-ers of 10 cm thick Fe absorbers and two muon trackers, but the following studies investigatevarious numbers of absorber-tracker layers and different absorber thicknesses.

Simulations of muons’ passage through the spectrometer were performed in MATLABusing Monte Carlo methods. First, a cosmic ray muon’s direction is determined by samplingthe appropriate distributions. The direction with respect to vertical is assumed to follow acos2 θ distribution where θ is the zenith (or polar) angle of the muon 4; θ = 0o correspondsto vertical (downward-going) muons, while θ = 90o for horizontal muons. Only 18% ofmuons are incident at angles larger than 45o. The muon azimuthal angle distribution, φ, isuniform from 0o to 360o.

After the two angles for the muon are determined, the location of the muon’s entry intothe top layer of the spectrometer is determined randomly. From this point, the trajectory ofthe muon through the spectrometer is calculated. This calculation includes the simulationof multiple Coulomb scattering through the iron absorber layers in the spectrometer, andfor a magnetized spectrometer, the magnetic deflection of the muon as it passes througheach layer of magnetized iron. Muon energy loss was not calculated for these simulations;instead, it has been assumed that the effects of energy loss can be corrected for by moresophisticated momentum estimation algorithms.

After the true trajectory of each muon is calculated, the measured hit locations in the track-ers are determined by randomly shifting each hit by a random number sampled from aGaussian whose width is equal to the hit resolution of the tracker, σx[y]. Once all the mea-sured hit locations are determined, the passage of the muon through the tracker is recon-structed by connecting the measured hit locations in each tracker to the location in thepreceding Fe absorber where the muon was assumed to have scattered. In reality, there isno single location in the absorber where the muon scatters, but since only one positionmeasurement is made after each Fe absorber, some assumption must be made about thetrajectory of the muon: the muon’s trajectory is assumed to extend, unscattered, until theabsorber’s mid-point, at which point the muon is deflected towards its next measured hit.Figure 3 shows an example of an estimated trajectory and compares it to the true (and morecomplicated) trajectory.

4. The zenith angle distribution for cosmic-ray-induced muons has some energy dependence which isignored for this simulation; however, the dependence is not strong so the results of this work should still bevalid.


tracker

tracker

tracker

tracker

Fe slab

Fe slab

z

x

�

xz1

�

xz2

plane-to-plane = 100 cm

10 cm

muon

Figure 2: Side-view of unmagnetized spectrometer in x− z plane. The muon scatteringangles in the x− z plane after each layer of Fe are shown by θxz1 and θxz2. Two additionalscattering angles can be measured in the y− z plane. The green circles in each trackerindicate muon hits. The top two trackers are not part of the spectrometer; they comprise thelower tracking module of CRIPT.


unsc

atte

red

traje

ctor

y

initi

altra

ject

ory

true path

tracker hit

tracker

Fe absorber

estim

ated

traj

ecto

rysc

atte

red

traj

ecto

ry

�true

�estimated

bias = �

estimated -�

true

Figure 3: The estimated and true trajectories of a scattered muon through an unmagnetizedspectrometer. The estimate of the scattered muon trajectory, and hence scattering angle,θestimated, assumes that the muon scatters after traversing half of the thickness of the Feabsorber. The true scattering angle , θtrue, can be larger or smaller than θestimated.

2.1 Multiple Coulomb scattering in an unmagnetizedspectrometer

The unmagnetized spectrometer simulated for these studies uses layers of iron, whosethicknesses vary from 5 cm to 15 cm, to deflect muons via multiple Coulomb scattering.The multiple scattering angles follow a distribution that is approximately Gaussian 5. Themost likely scattering angle is zero degrees and the width of the Gaussian distribution, in

5. For this study, the distribution is assumed to be perfectly Gaussian. Subsequent, higher fidelity, sim-ulations will include the non-Gaussian tails in the scattering distributions. Since less than 2% of the muonsundergo non-Gaussian scattering, the Gaussian assumption is adequate for this study.


mrad, is

θMS =13.6βpc

√LX0

[1+0.038ln(L/X0)] , (2)

where β is the velocity of the muon 6 as a fraction of the speed of light, p is the momentumof the muon in MeV/c, L is the path length of the muon through the material, and X0 isthe radiation length of the material [10]. The radiation length of a material is the meandistance a high energy electron must travel before it has lost all but 1/e of its energy bybremsstrahlung. The radiation length (in cm) is related to the density, ρ (g/cm3), and atomicproperties of a material by

X0 =716.4 A

ρZ(Z +1) ln(287/√

Z)(3)

where Z is the number of protons per nucleus, and A is the number of nucleons per nucleus[10]. The radiation length for higher-Z materials is shorter which leads to larger anglemultiple scattering. For L = 10 cm in Fe (X0 = 1.76 cm), θMS = 17 mrad for a 2 GeV/cmuon.

In addition to simulating the scattering angle of the muon, its horizontal displacement withrespect to its initial trajectory must also be calculated. The multiple scattering angle andhorizontal displacement are strongly correlated. The horizontal displacement, yd , is givenby

yd = g1LθMS/√

12+g2LθMS/2, (4)

where g1 and g2 are independent, unit Gaussian random numbers, θMS is the width ofthe scattering angle distribution (from Equation 2), and L is the path length of the muonthrough the material. The correlation is achieved by setting the multiple scattering angleequal to g2θMS.

2.1.1 Maximum likelihood fits for unmagnetized spectrometer

For a muon that passes through the spectrometer, two independent, orthogonal scatteringangles are measured after each absorber. For simplicity we define these angles in the spec-trometer’s x− z and y− z planes (z points down to the ground). Using the known value ofX0 for Fe and the estimated path length of the muon in the absorber, L, Equation 2 can bere-arranged to estimate 1/p.

An estimate of 1/p is provided by a maximum likelihood fit to the scattering angle data.First, for each layer of absorber that is traversed, the standard deviation of the multiplescattering angle, θMS, is calculated according to Equation 2 using an assumed value of 1/p

6. For most of the cosmic ray induced muons at the Earth’s surface, β is very close to 1: β = 0.994 for pμ= 1 GeV/c and β = 0.9999 for pμ = 10 GeV/c.


(and a corresponding value of β). The measured multiple scattering angles in the x− z andy− z planes, θxz[yz], are then compared to a Gaussian distribution

P(θxz[yz]) =1√2πθ

e−θ2xz[yz]

2θ2 (5)

whose width, θ =√

θ2MS +σ2θxz[yz]

, depends on θMS and the uncertainty in the scatteringangle measurement, σθxz[yz] in the x− z[y− z] plane. The uncertainty of the scattering anglemeasurement depends on the hit resolution of the muon trackers, σx[y], the tracker-absorberseparation, z, and the projection of the zenith angle of the muon in the x− z[y− z] plane,ψxz[yz]:

σθxz[yz] =1

1+ tan2 ψxz[yz]

√2

zσx[y]. (6)

The value of 1/p (and corresponding β) that minimizes the negative log likelihood function

− log(L) = −[

N∑i=1

log(P(θxz))+N∑

i=1

log(P(θyz))

](7)

for the N scattering layers is the best estimate of 1/p: 1/p f it .

2.2 Magnetized spectrometerIn addition to simulating unmagnetized Fe absorbers, magnetized absorbers have been in-vestigated. Muons are electrically charged particles so they are deflected by the magneticfield inside the iron. The resulting radius of curvature for the muon, R (cm), depends on thestrength of the magnetic field, B (T), and its momentum transverse to the direction of themagnetic field, pT (GeV/c):

R =100pT0.3B

. (8)

The resulting deflection angle, θB is approximately equal to L/R for small deflection angles,where L is the length of the muon’s path through the iron absorber. The direction of thedeflection depends on the charge (positive or negative) of the muon and the angle betweenthe muon’s momentum vector and the magnetic field.

2.2.1 Maximum likelihood fits for magnetized spectrometer

The Gaussian functions (Equation 5) that appear in the maximum likelihood function usedto determine 1/p f it for an unmagnetized spectrometer (Equation 7) must be modified for amagnetized spectrometer. Although the width of the Gaussian distribution does not change,


its central value is offset depending on the magnitude of the magnetic angular deflection,θBxz[yz]:

P(θxz[yz]) =1√2πθ

e−(θxz[yz]−θBxz[yz] )

2

2θ2 (9)

Since the sign of θBxz[yz] depends on the sign of the muon’s electric charge, the chargeis an additional free parameter in the fit. Figure 4 illustrates how the probability densityfunctions (PDFs) for the deflection angle differ for positive and negative muon charges.The separation between the PDFs depends on the magnetic field strength and the muon’s(transverse) momentum. In the fitting algorithm, both muon charge hypotheses are testedand the one that yields the smaller value for − log(L) is selected.

deflection angle (mrad)

prob

(q,p

)

q� = -1q� = +1

f(p,B)

0+10 -10

Figure 4: Probability density functions (PDFs) for deflection angles of positive and nega-tive muons in a magnetized spectrometer. The separation between the two PDFs dependson �pμ and �B. The width of the PDFs depends on pμ and the number of radiation lengths ofmaterial traversed. The blue rectangles represent notional deflection angle measurements.In this case the positive charge hypothesis is preferred.


3 Results3.1 Horizontal displacements and multiple scattering

anglesBefore determining the momentum resolutions obtained with different spectrometer de-signs, low-level details of the Monte Carlo simulations were studied to ensure their quality.A number of different distributions, whose properties were easily calculated, were investi-gated.

Figure 5 displays the horizontal displacements of the muons trajectories in the muon track-ers with respect to the expected hit locations had no multiple scattering occurred in the(unmagnetized) Fe absorbers. The scatter plots in this figure do not suggest that there areany problems with the Monte Carlo simulations, as there are no unexpected correlations be-tween the displacements and the muon’s initial zenith (polar) and azimuthal angles. Also,the magnitude of the horizontal displacements, typically a few centimetres per metre ofvertical displacement, is consistent with expectations.

Figure 6 shows histograms of the horizontal displacements of 2 GeV/c muons after passagethrough unmagnetized Fe absorbers as well as a scatter plot of the displacements in x andy, and the distribution of multiple scattering angles. The histograms are consistent withGaussian distributions (as expected) and the scatter plot shows no correlation between thex and y displacements. Finally, the root mean squared (RMS) of the multiple scatteringangle distribution, 21 mrad, is consistent with the expectation for 2 GeV/c muons.

Figure 7 displays comparisons of the true and measured muon scattering angles. Plots (a)and (b) show the strong correlation between the true and measured angles in both the x− zand y− z planes. The non-zero widths of the scattering angle residual histograms (plots(c) and (d)) result from the approximation of the measured scattering angle (as alreadydiscussed in Figure 3). The two lower scatter plots in Figure 7 show that there are nocorrelations between (e) scattering angles in orthogonal planes (x− z and y− z), and (f)scattering angles in the same plane (x− z) but after different absorbers (absorbers 1 and 2).


−5 0 50

0.2

0.4

0.6

0.8

x displacement (cm)

pola

r an

gle

(rad

ians

)

−5 0 50

0.2

0.4

0.6

0.8

y displacement (cm)po

lar

angl

e (r

adia

ns)

−5 0 5−1

−0.5

0

0.5

1

x displacement (cm)

cos(

φ)ta

n(θ)

−5 0 5−1

−0.5

0

0.5

1

y displacement (cm)

sin(

φ)ta

n(θ)

Figure 5: The upper plots show the horizontal displacements due to multiple scatteringof muons in tracking chambers 85 cm below 10 cm thick unmagnetized Fe absorbers. Thedisplacements are shown as a function of muon polar (zenith) angle. The lower scatter plotsshow the displacements as a function of the muon azimuthal angle, φ, and zenith angle, θ.


−6 −4 −2 0 2 4 60

20

40

60

80

100

x−displacement [cm]

entr

ies/

0.4

cm

−6 −4 −2 0 2 4 60

20

40

60

80

100

y−displacement [cm]

entr

ies/

0.4

cm

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

x−displacement [cm]

y−di

spla

cem

ent [

cm]

−0.06 −0.04 −0.02 0 0.02 0.040

50

100

150

200

250

300

δ(θ) [radians]

entr

ies/

0.01

rad

ians

Figure 6: The horizontal (x and y) displacements and angular deflection, δ(θ), of 2.0GeV/c muon tracks in a two-layer, unmagnetized, Fe spectrometer. Each Fe absorber is 10cm thick, and the total height of the spectrometer is 2 m. The muon tracker layers are 85cm below each Fe layer. The RMS of the scattering angle distributions in both the xz andyz planes is 21 mrad.


−0.05 0 0.05−0.05

0

0.05

δθxztrue

[radians]

δθx m

easu

red [

radi

ans]

−0.05 0 0.05−0.05

0

0.05

δθyztrue

[radians]

δθy m

easu

red [

radi

ans]

−2 −1 0 1 20

200

400

600

800

1000

δθxzmeasured

− δθxztrue

[mrad]−2 −1 0 1 2

0

200

400

600

800

1000

δθyzmeasured

− δθyztrue

[mrad]

−0.05 0 0.05

−0.05

0

0.05

θyztrue

[radians]

θ xz t

rue [

radi

ans]

−0.05 0 0.05

−0.05

0

0.05

θxz1true

[radians]

θ xz2

true

[rad

ians

](e)

(c)

(a) (b)

(d)

(f)

Figure 7: The difference between estimated and true scattering angles assuming perfectmuon tracker hit resolution. The upper two plots, (a) and (b), show the strong correlationbetween the estimated and true scattering angles in the xz and yz planes if the muon track-ers have perfect hit resolution. The middle two plots, (c) and (d), are histograms of thedifference between the estimated and true scattering angles. For this spectrometer geom-etry (2 m height, 2 layers of 10 cm thick Fe, 85 cm between each Fe and tracker layer),the RMS of the difference is less than 1.0 mrad. The lower left plot, (e), shows the lack ofcorrelation between the scattering angles in the xz and yz planes. The lower right plot, (f),shows the lack of correlation between the two xz scattering angles (xz1 and xz2) after eachof the two Fe absorbers in the spectrometer.


3.2 Unmagnetized 1/p Fe resultsIt is important to note that in Equation 2, θMS depends linearly on 1/p. Since the measuredscattering angles used to determine 1/p f it in the likelihood fits are Gaussian distributed,the uncertainty of 1/p f it is also Gaussian. However, the uncertainty of the inverse, p f it , isnot Gaussian as the relative uncertainties for 1/p f it are typically large (> 25%). As can beseen in Figure 8, the distribution of p f it for 2 GeV/c muons is clearly non-Gaussian. Forthis reason, all the momentum estimation results presented in this paper are for 1/p f it .

For five samples of mono-energetic muons in an unmagnetized spectrometer, the distri-bution of 1/p f it values from scattering angle measurements is shown in Figure 9. Thespectrometer was assumed to be 2 m high, containing two layers of 10 cm thick Fe, with85 cm between each Fe and tracker layer. The muon tracker hit resolution was assumed tobe perfect. The means, RMSs, resolutions and biases of these 1/p f it results are shown inTable 1. These results show that a 12% correction can be applied to 1/p f it to yield goodagreement with 1/ptrue. The bias that remains after this correction (< 2%) is negligiblecompared to the ∼ 37% resolution of the measurements. It is interesting to note that the1/p f it bias is reduced to only -2% if the true scattering angles are used in the fits insteadof the measured scattering angles.

Table 1: The results of 1/p f it measurements for an unmagnetized spectrometer with per-fect tracking. The difference between 1/ptrue and 1/p f it is fairly constant across this rangeof momenta so a correction can be applied without introducing significant extra uncertainty.The resolution, RMS(1/p f it)/< 1/p f it >, is also quite constant for the muon momenta con-sidered.

ptrue 1/ptrue < 1/p f it > RMS(1/p f it) bias resolution(GeV/c) (c/GeV) (c/GeV) (c/GeV) (%) (%)

0.5 2.0 1.74 0.62 -13 361.0 1.0 0.87 0.32 -13 362.0 0.5 0.44 0.16 -12 375.0 0.2 0.18 0.06 -10 37

10.0 0.1 0.09 0.03 -10 36

Figures 10 to 13 show how the RMS, mean value, and resolution (= RMS/mean value) of1/p f it vary as different properties of the simulated unmagnetized spectrometer are varied.All simulated muons have p = 2.0 GeV/c and the spectrometer depth is 2 m (the spacingbetween planes and trackers changes by up to 15 cm as the thickness or number of Fe platesis varied). Unless varied in the plots, the default parameters are– number of Fe absorbers (plates) = 2,– absorber thickness = 10 cm,– angular resolution = 0 mrad, and– hit resolution = 0 mm.


0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

pfit

[Gev/c]

entr

ies/

0.1

GeV

/c

Figure 8: The fitted momentum values for 2 GeV/c muons in an unmagnetized muonspectrometer. The spectrometer is assumed to be 2 m high, containing two layers of 10 cmthick Fe, 85 cm between each Fe and tracker layer. Note the non-Gaussian shape of thedistribution; see text for the explanation of the shape.

The plot on the left of Figure 10 shows that as the number of Fe absorber plates is in-creased, RMS(1/p f it) decreases significantly. This is expected as the statistical precisionof the 1/p f it estimates improves as the number of measured scattering angles increases(two per plane). The middle plot shows that the bias between the true and fit values of1/p increases significantly as the number of plates is increased beyond two. This occursbecause the separation between the Fe plates and the trackers is reduced as the number ofplates increases. This increases the difference between the estimated scattered muon tra-jectory and true trajectory (see Figure 3). Again, using the true multiple scattering angle in


0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

200

400

600

800

1000

1200

1400

1/pfit

[c/GeV]

entr

ies/

(0.0

2 c/

GeV

)0.5 GeV/c2 GeV/c5 GeV/c10 GeV/c1 GeV/c

0.5 GeV/c

2 GeV/c

5 GeV/c

10 GeV/c

1 GeV/c

pμ = 0.5 GeV/c

pμ = 1 GeV/c

pμ = 2 GeV/c

pμ = 5 GeV/c

pμ = 10 GeV/c

Figure 9: The fitted 1/p values for an unmagnetized muon spectrometer. Five differentmuon momenta are considered.

the likelihood fits eliminates this large bias. The resolution (RMS/mean) of 1/p f it is shownin the plot on the right of Figure 10. As expected, the resolution improves as the number ofscattering angle measurements is increased.

Figure 11 shows that the resolution of the spectrometer does not vary much as the thicknessof the Fe absorber is increased. The width of the scattering angle distribution increases asthe square root of the absorber thickness (see Equation 2); however, since the hit resolution(and angular resolution) is assumed to be perfect, increasing the amount of scattering haslittle effect. The increase in bias on the mean value of 1/p f it , < 1/p f it >, as the thicknessincreases occurs for the same reason described in the previous paragraph: the separationbetween the absorber and tracker decreases.

The angular resolution of the unmagnetized spectrometer scattering angle measurementswas varied to generate the plots in Figure 12. As expected, the 1/p f it resolution is degradedas the angular resolution gets worse. For these spectrometer dimensions, 1.5 mrad angular


resolution is equivalent to 1 mm tracker hit resolution.

The effect on 1/p f it of varying the tracker hit resolution is shown in Figure 13. Between 0mm and 2 mm hit resolution, the effect on 1/p f it is not very large. However, for larger hitresolutions and higher muon momenta, there is a significant effect. Those results are shownin Figure 14 and Section 3.3. Figure 14 shows that as the muon momentum increases from0.5 GeV/c to 4.0 GeV/c, the effect of larger hit resolution becomes apparent. This occursbecause higher momentum muons scatter less so their horizontal displacements are smaller(in particular, relative to the hit resolution).

2 4 60

0.05

0.1

0.15

0.2

0.25

number of Fe plates

RM

S(1

/pfit

) [c

/GeV

]

2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

number of Fe plates

<1/

p fit>

[c/G

eV]

2 4 60

0.1

0.2

0.3

0.4

0.5

number of Fe plates

RM

S/<

1/p f

it>

Figure 10: The RMS, mean and resolution of 1/p estimates for 2.0 GeV/c muons withunmagnetized spectrometer for different numbers of Fe absorbers.


5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

absorber thickness [cm]

RM

S (

1/p)

[c/G

eV]

5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


<1/

p fit>

[c/G

eV]

5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


RM

S/<

1/p f

it>

Figure 11: The RMS, mean and resolution of 1/p estimates with an unmagnetized spec-trometer for 2.0 GeV/c muons with different thicknesses of Fe absorbers. Two layers ofabsorber and perfect tracker hit resolution are assumed.


0 5 100

0.05

0.1

0.15

0.2

0.25

angular resolution [mrad]

RM

S(1

/pfit

) [c

/GeV

]

0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


<1/

p fit>

[c/G

eV]

0 5 100

0.1

0.2

0.3

0.4

0.5

0.6


RM

S(1

/pfit

)/<

1/p f

it>

Figure 12: The effect of the angular resolution of the spectrometer trackers on the RMS,mean and resolution of 1/p estimates with unmagnetized spectrometer for 2.0 GeV/cmuons. Two layers of 10 cm thick Fe absorber are assumed.


0 0.1 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

hit resolution (cm)

RM

S(1

/pfit

) [c

/GeV

]

0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

0.6

hit resolution (cm)

<1/

p fit>

[c/

GeV

]

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

hit resolution (cm)

RM

S(1

/pfit

)/<

1/p f

it>

Figure 13: The effect of the tracker hit resolution on the RMS, mean and resolution of 1/pestimates with an unmagnetized spectrometer. The muon momentum is assumed to be 2.0GeV and the spectrometer consists of two layers of 10 cm thick Fe and two layers of muontrackers, each 85 cm below its corresponding Fe layer.


0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

hit resolution [cm]

RM

S(1

/pfit

)/<

1/p f

it>

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

hit resolution [cm]

RM

S(1

/pfit

)/<

1/p f

it>

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

hit resolution [cm]

RM

S(1

/pfit

)/<

1/p f

it>

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

hit resolution [cm]

RM

S(1

/pfit

)/<

1/p f

it>

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

hit resolution [cm]

RM

S(1

/pfit

)/<

1/p f

it>

p = 0.5 GeV/c p = 1.0 GeV/c p = 2.0 GeV/c

p = 3.0 GeV/c p = 4.0 GeV/c

Figure 14: The resolution of 1/p f it with an unmagnetized spectrometer for different muonmomenta (0.5 GeV/c to 4.0 GeV/c) and tracker hit resolutions. The sensitivity to increasedhit resolution is greater for higher momentum muons.


3.3 Magnetized Fe 1/p resultsInterpreting the data from a magnetized spectrometer requires determining the electriccharge of muons on an event-by-event basis. When the magnetic deflection is large com-pared to the deflection from multiple scattering, it is easy to determine the charge correctly.However, for weaker magnetic fields, this is more difficult. Tables 2, 3, and 4 show thefraction of muons that are assigned the correct charge in the likelihood fits as a function ofmuon momentum and magnetic field strength. The difference amongst the three tables isthe hit resolution that is assumed for the spectrometer’s trackers: 0 mm, 2 mm, or 4 mm.As expected, the fraction of muons with the correct charge assignment increases as themagnetic field strength increases, and as the tracker hit resolution improves. The correctcharge fraction is not very sensitive to the muon momentum unless the hit resolution ispoor (σx[y] = 4 mm). When the hit resolution is 4 mm, the correct charge fraction decreasesas the momentum increases, due to the reduced magnetic deflection.

Table 2: The fraction of muon tracks with the correct charge assignment for muon trackerhit resolution σx[y] = 0 mm. The spectrometer uses two layers of magnetized, 10 cm thickFe absorbers.

momentum magnetic field [T][GeV/c] 0.5 1.0 1.5 2.0

0.5 0.767 ± 0.006 0.931 ± 0.004 0.985 ± 0.002 0.9982 ± 0.00061.0 0.780 ± 0.006 0.939 ± 0.003 0.987 ± 0.002 0.9981 ± 0.00062.0 0.782 ± 0.006 0.942 ± 0.003 0.988 ± 0.002 0.9987 ± 0.00055.0 0.789 ± 0.006 0.944 ± 0.003 0.989 ± 0.002 0.9992 ± 0.000410.0 0.794 ± 0.006 0.945 ± 0.003 0.990 ± 0.001 0.9990 ± 0.0004



0.5 0.769 ± 0.006 0.930 ± 0.003 0.984 ± 0.002 0.998 ± 0.0011.0 0.780 ± 0.006 0.937 ± 0.003 0.987 ± 0.002 0.998 ± 0.0012.0 0.783 ± 0.006 0.936 ± 0.003 0.987 ± 0.002 0.999 ± 0.0015.0 0.774 ± 0.006 0.926 ± 0.003 0.984 ± 0.002 0.998 ± 0.001

10.0 0.742 ± 0.006 0.890 ± 0.003 0.962 ± 0.003 0.992 ± 0.001

Figure 15 shows the effect on 1/p f it of estimating the muon charge incorrectly. For bothplots (a) and (b), all the simulated muons had qμ = +1, so the green (dark) histograms forq f itμ = −1 represent incorrect charge assignments. More muons have the wrong charge in




0.5 0.766 ± 0.006 0.930 ± 0.004 0.982 ± 0.002 0.999 ± 0.0011.0 0.779 ± 0.006 0.935 ± 0.003 0.986 ± 0.002 0.998 ± 0.0012.0 0.774 ± 0.006 0.929 ± 0.003 0.983 ± 0.002 0.998 ± 0.0015.0 0.741 ± 0.006 0.889 ± 0.003 0.962 ± 0.003 0.992 ± 0.001

10.0 0.674 ± 0.007 0.803 ± 0.003 0.894 ± 0.004 0.949 ± 0.003

the upper plot (a) as the hit resolution is worse and the magnetic field strength is lower. Inboth plots, the wrong sign sample has 1/p f it values that are smaller (and farther from thetrue value) than the correct sign sample. Consequently, the larger the wrong sign fraction,the worse the 1/p f it resolution.

For one specific spectrometer design (two 10 cm thick Fe absorbers with two 2 mm hitresolution trackers), the effect of varying the magnetic field strength from 0.0 T to 2.0T is shown in Figure 16. The results are separated by muon momentum to show how the1/p f it resolution differs significantly with momentum. The resolution improves as the fieldstrength increases, but the difference between unmagnetized and 1.0 T Fe is only ∼ 10%for a . This means that very high magnetic fields will be required to make a substantialimprovement in the resolution.

Figures 17 to 19 show the effect on the 1/p f it resolution of varying different spectrometerparameters: the tracker hit resolution, number of absorbers, and absorber thickness. A rangeof muon momenta and magnetic field strengths are considered for each set of spectrometerparameters.

Figure 17 shows that degrading the hit resolution from perfect (0 mm) to more realisticresolutions has a significant effect on the 1/p f it resolutions that are obtained for the highermomentum muons. For example, the resolution of the 10 GeV/c muons degrades from ap-proximately 36% to 70% as the hit resolution is varied from 0 mm to 4 mm. The resolutionfor lower momentum muons is affected much less. Tables 5 to 7 show the results used togenerate Figure 17.

Figure 18 shows that increasing the number of Fe absorbers (and associated trackers) fromone to three significantly improves the 1/p f it resolution for lower momentum muons (0.5≤pμ ≤ 2.0 GeV), but has less of an effect on the higher momentum muons.Figure 19 shows that the 1/p f it resolution of lower momentum muons is not significantlyimproved by increasing the thickness of the Fe absorbers (plates); however, increasing the


thickness results in a modest improvement for the resolution of higher momentum muons.This suggests that if the higher momentum muons are important for detecting and/or imag-ing high-Z materials, then thicker absorbers should be considered if the increased cost iswarranted.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

20

40

60

80

100

120

1/pfit

[c/GeV]

entr

ies

per

bin

[0.0

1 G

eV/c

]

0 0.2 0.4 0.6 0.8 1 1.2 1.40

50

100

150

1/pfit

[c/GeV]

entr

ies

per

bin

[0.0

1 G

eV/c

]

(a)

(b)

pμ = 2 GeV/c

| B| = 1.0 T

σx[y] = 2 mm

number of Fe layers = 2

thicknessFe = 10 cm

qμfit = −1

qμfit = +1

qμfit = +1

pμ = 2 GeV/c

| B| = 0.5 T

σx[y] = 4 mm

number of Fe layers = 2

thicknessFe = 10 cm

qμfit = −1

Figure 15: The fitted 1/p values for a magnetic spectrometer with correct (q f itμ = +1)and incorrect (q f itμ = −1) muon charge assignments. The results for 2 GeV/c muons in aspectrometer with 0.5 T field and 4 mm hit resolution are shown in the upper plot (a);the lower plot, (b), shows the results for a spectrometer with a 1.0 T field and 2 mm hitresolution.


0 0.5 1 1.5 20.2

0.25

0.3

0.35

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B field [T]

RM

S(1

/pfit

)/<

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it>

0.5 GeV/c1 GeV/c2 GeV/c5 GeV/c10 GeV/c

σx[y]

= 2.0 mm

Fe thickness = 10 cmnumber of Fe plates = 2

Figure 16: The resolution of 1/p f it with a magnetic spectrometer for different muon mo-menta and magnetic field strengths. The 1/p f it resolution improves as the strength of themagnetic field increases.


Table 5: The resolution of 1/p (RMS(1/p f it)/< 1/p f it >) with no tracker hit uncertainty(σx[y] = 0 mm) for different magnetic field strengths for a 2-layer spectrometer. Each layeruses a 10 cm thick Fe absorber.

momentum magnetic field [T][GeV/c] 0.0 0.5 1.0 1.5 2.0

0.5 0.358 0.354 0.326 0.287 0.2491.0 0.373 0.367 0.333 0.290 0.2502.0 0.374 0.370 0.336 0.293 0.2525.0 0.375 0.369 0.335 0.292 0.251

10.0 0.374 0.368 0.334 0.291 0.251

Table 6: The resolution of 1/p (RMS(1/p f it)/< 1/p f it >) with 2 mm tracker hit resolutionfor different magnetic field strengths for a 2-layer spectrometer. Each layer uses a 10 cmthick Fe absorber.


0.5 0.359 0.354 0.326 0.287 0.2501.0 0.379 0.371 0.336 0.292 0.2522.0 0.394 0.383 0.347 0.301 0.2595.0 0.476 0.438 0.394 0.342 0.291

10.0 0.610 0.551 0.515 0.471 0.414

Table 7: The resolution of 1/p (RMS(1/p f it)/< 1/p f it >) with 4 mm tracker hit resolutionfor different magnetic field strengths for a 2-layer spectrometer. Each layer uses a 10 cmthick Fe absorber.


0.5 0.362 0.356 0.328 0.289 0.2511.0 0.392 0.381 0.344 0.299 0.2572.0 0.444 0.418 0.376 0.327 0.2785.0 0.608 0.549 0.514 0.469 0.412

10.0 0.724 0.659 0.668 0.674 0.657

3.4 Assessing the impact of different 1/p f itresolutions

The different 1/p f it resolutions that are obtained by varying spectrometer parameters mustbe incorporated into simulations of image reconstruction and SNM detection with CRIPT.


0.2

0.3

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0.7

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S(1

/pfit

)/<

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it>

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B Field [T]

0.5 GeV/c

1.0 GeV/c

2.0 GeV/c

5.0 GeV/c

10.0 GeV/c

(c) σx[y] = 4.0 mm

(a) σx[y] = 0.0 mm

(b) σx[y] = 2.0 mm

Fe plate thickness = 10 cmNumber of Fe plates = 2

(c) σx[y] = 4.0 mm

Figure 17: The resolution of 1/p f it with a magnetized spectrometer for different trackerhit resolutions (0 mm, 2 mm and 4 mm for plots (a), (b) and (c)).

This will determine the effect that different spectrometer designs would have on the perfor-mance of CRIPT. These simulations must study how imaging times, probabilities of SNMdetection, and false positives rates vary for different spectrometer designs.

In order to provide useful momentum resolution information, the 1/p f it resolution func-tions (versus 1/ptrue) were determined for different spectrometer designs. Simple functions(an exponential decay plus a constant) were sufficient to fit the simulated 1/p f it results. Anexample of one these resolution functions is shown in Figure 20.


0.2

0.3

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0.6

0.7

RM

S(1

/pfit

)/<

1/p f

it>

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0.8

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B field [T]

0.5 GeV/c1.0 GeV/c2.0 GeV/c5.0 GeV/c10.0 GeV/c

Fe plate thickness = 10 cmσ

x[y] = 2.0 mm

(b) number of Fe plates = 2

(c) number of Fe plates = 3

(a) number of Fe plates = 1

Figure 18: The resolution of 1/p f it with a magnetized spectrometer for different numbersof Fe absorber layers (1, 2 and 3 layers for plots (a), (b) and (c)).


0.2

0.3

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S(1

/pfit

)/<

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it>

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0.5 GeV/c1.0 GeV/c2.0 GeV/c5.0 GeV/c10.0 GeV/c

(c) Fe plate thickness = 15 cm

(b) Fe plate thickness = 10 cm

(a) Fe plate thickness = 5 cm

σx[y] = 2.0 mmNumber of Fe plates = 2

Figure 19: The resolution of 1/p f it for different thicknesses of Fe absorber layers. (5 cm,10 cm and 15 cm layers for plots (a), (b) and (c)).


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.4

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1/pfit

[c/GeV]

RM

S(1

/pfit

)/<

1/p f

it>

simulation resultsbest fit resolution function

resolution function:

y = Ae−Bx + C

A = 0.52 ± 0.05B = 3.85 ± 0.79C = 0.37 ± 0.02

Figure 20: The resolution for 1/p as a function of muon momentum. The best fit parametervalues are shown assuming the resolution function is an exponential decay (Ae−Bx) plus aconstant (C). The spectrometer is assumed to have two 10 cm thick, unmagnetized Feabsorbers and the muon track hit resolution is 4 mm.


4 Conclusions

The results of the simulation cross-checks from Section 3.1 show that the Monte Carlo sim-ulation of CRIPT’s muon spectrometer is behaving as expected. Both the scattering anglesand horizontal displacements caused by multiple Coulomb scattering in the Fe absorbersare consistent with expectations.

Using only multiple scattering measurements (i.e. an unmagnetized spectrometer), the mo-menta of muons can be estimated in an unbiased manner (after appropriate corrections, seeSection 3.2). Previously, 1/p f it resolution < 50% has been shown to be adequate for muonscattering tomography [11]; this resolution goal is achievable using only multiple scatteringmeasurements. The resolution is strongly dependent on the number of layers of scatteringFe absorbers as the uncertainty of 1/p f it is statistics limited: the more scattering layers, themore scattering angle measurements that are available to constrain the muon momentum.The 1/p f it resolution of high momentum muons (pμ ≥ 5 GeV/c) is also strongly affectedby the hit resolution of the spectrometer’s muon trackers. The resolution for lower momen-tum muons is less affected due to their larger scattering angles. The Fe absorber thicknessdoes not have a strong effect on the 1/p f it resolution for an unmagnetized spectrometer.

Magnetizing the Fe absorbers in the spectrometer provides extra information for estimatingmuon momenta; however, the gain in 1/p f it resolution is not very significant (less than10% relative improvement) unless the magnetic field in the Fe is very high (≥ 1.0 T). Forlower fields, the deflection from multiple scattering is greater than the magnetic deflectionso the sign of the muon’s electric charge is frequently estimated incorrectly (25% wrongsign for B = 0.5 T), leading to poor resolution for 1/p f it . If the 1/p f it resolution needs tobe better than that obtained with the baseline, unmagnetized spectrometer (approximately40% resolution with two layers of 10 cm thick Fe absorbers with 2 mm tracker hit resolutionand a total spectrometer depth of 2 m), it is likely cheaper and easier to increase the numberof absorber-detector layers than to magnetize the Fe absorbers to very high field strengths.

The impact of the different spectrometer designs on the performance of CRIPT will ulti-mately be determined by simulating the image reconstruction performance using a varietyof 1/p functions that represent different spectrometer designs. By comparing the imagingperformance and cost of different designs, the best muon spectrometer design for CRIPTcan be determined.


5 Recommendations

This study has shown how the measurement of 1/p f it depends on a number of muon spec-trometer design parameters. With the caveat that more detailed simulations will be per-formed in the future, it appears that an unmagnetized spectrometer is likely to be the mostcost-effective solution for measuring muon momenta on an event-by-event basis. The base-line design of two absorber-detector layers provides approximately 40% 1/p f it resolutionwith 2 mm hit resolution for the muon detectors. As the absorber thickness does not have alarge effect on the resolution, it is recommended that a thin Fe absorber (5 cm) be used toreduce the weight and cost of the spectrometer.

The 1/p f it resolution functions obtained from this study should be incorporated into imagereconstruction simulations to assess the full impact of different spectrometer designs. Inparallel with this activity, a more detailed simulation of this study’s baseline, unmagnetizedmuon spectrometer should be performed in GEANT4 to test the validity of the resultsobtained in this study.


References

[1] Morris, Chris et al. (2008), Tomographic Imaging with Cosmic Ray Muons, Scienceand Global Security, 16, 37–53.

[2] Pesente, Silvia et al. (2009), First results on material identification and imaging witha large-volume muon tomography prototype, Nucl. Instrum. Meth., A604, 738–746.

[3] Hohlmann, Marcus, Gnanvo, Kondo, Grosso, L., Locke, J.B., Quintero, A., andMitra, Debasis (2009), Design and Construction of a First Prototype MuonTomography System with GEM Detectors for the Detection of Nuclear Contraband,In IEEE Nuclear Science Symposium Conference Record, 2009.

[4] Allison, John et al. (1991), The Diamond shaped cathode pads of the OPAL muonbarrel drift chambers, Nucl. Instrum. Meth., A310, 527–534.

[5] Sanchez, Mayly C. et al. (2003), Observation of atmospheric neutrino oscillations inSoudan 2, Phys. Rev., D68, 113004.

[6] Ambrosio, M. et al. (2002), Muon energy estimate through multiple scattering withthe MACRO detector, Nucl. Instrum. Meth., A492, 376–386.

[7] Ankowski, A. et al. (2006), Measurement of through-going particle momentum bymeans of multiple scattering with the ICARUS T600 TPC, Eur. Phys. J. C, 48(2),667–676.

[8] Green, J.A et al. (2006), Optimizing the Tracking Efficiency for Cosmic Ray MuonTomography, In IEEE Nuclear Science Symposium Conference Record, 2006,pp. 285–288.

[9] Benettoni, M. et al. (2007), Muon radiography with the CMS Muon Barrelchambers, In IEEE Nuclear Science Symposium Conference Record, 2007,pp. 1021–1025.

[10] C. Amsler et al. (Particle Data Group) (2008), Multiple scattering through smallangles, Physics Letters B, 667, 1.

[11] Armitage, John et al. (2009), Cosmic Ray Inspection and Passive Tomography forSNM Detection, In International Conference on Applications of Nuclear Techniques,pp. 24–35.


List of Abbreviations and Symbols3D three-dimensionalA number of nucleons per nucleusB magnetic fieldβ velocity of muon as a fraction of the speed of lightc speed of light in vacuumCBRNE Chemical, Biological, Radiological, Nuclear, ExplosivesCBSA Canadian Border Services AgencyCRIPT Cosmic Ray Inspection and Passive TomographyCRTI Chemical, Biological, Radiological, Nuclear, Explosives Research & Technology InitiativeDRDC Defence Research and Development CanadaFe irongi Gaussian random numberGeV giga-electron-voltL path lengthL likelihoodmrad milliradiansMeV mega-electron-voltMST muon scattering tomographyN nitrogenO oxygenp momentumpμ muon momentumpT transverse momentump f it muon momentum from fitptrue true muon momentumqμ muon chargeR radius of curvatureρ densityRDD radiological dispersal deviceRMS root mean squaredSNM special nuclear materialT teslaθMS width of multiple scattering angle distributionX0 radiation lengthyd horizontal displacementZ number of protons per nucleus


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Defence R&D Canada – Ottawa3701 Carling Avenue, Ottawa, Ontario, CanadaK1A 0Z4

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3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriateabbreviation (S, C or U) in parentheses after the title.)

A simulation study of the Cosmic Ray Inspection and Passive Tomography (CRIPT) muonspectrometer

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Waller, D.

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October 2010

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DRDC OTTAWA TM 2010-168

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( X ) Unlimited distribution( ) Defence departments and defence contractors; further distribution only as approved( ) Defence departments and Canadian defence contractors; further distribution only as approved( ) Government departments and agencies; further distribution only as approved( ) Defence departments; further distribution only as approved( ) Other (please specify):

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The Cosmic Ray Inspection and Passive Tomography (CRIPT) for Special Nuclear Material(SNM) Detection project is investigating the effectiveness of muon scattering tomography fordetecting SNM and dense radiation-shielding materials like lead. The goal of CRIPT is to design,build and test a large-scale prototype system that uses naturally occurring cosmic-ray-inducedmuons to image a volume of interest. Measuring the momenta of muons on an event-by-event ba-sis should reduce the required imaging times significantly. This paper describes relatively simplesimulations of the performance of different magnetized and unmagnetized muon spectrometerdesigns. An unmagnetized spectrometer uses the multiple Coulomb scattering of a muon throughknown absorbers to estimate the muon’s momentum, p. A magnetized spectrometer measuresp using both multiple scattering and the magnetic deflection inside magnetized iron absorbers.Using only multiple scattering measurements, the resolution of 1/p is 50% or better. The gainin resolution obtained by magnetizing the iron absorbers is not very significant (less than 10%relative improvement) unless the magnetic field in the iron is very high (≥ 1.0 T). An unmagne-tized spectrometer appears to be the preferred choice for CRIPT as it is simpler and likely morecost-effective to implement.

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dirty bombradiological dispersal deviceRDDcosmic raymuonmultiple scatteringmagnetic deflectionspecial nuclear materialSNMimprovised nuclear deviceIND

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A simulation study of the Cosmic Ray Inspection and Passive Tomography (CRIPT… · 2012. 8. 3. · sures de diffusion multiple, la resolution de 1´ /p est de 50% ou plus. Le gain