γ-ray tracking in germanium: the backtracking method

Nuclear Instruments and Methods in Physics Research A 477 (2002) 391–396

g-ray tracking in germanium: the backtracking method

J. van der Marel*, B. Cederwall

Physics Department Frescati, Royal Institute of Technology, Frescativ .agen 24, S-10405 Stockholm, Sweden

Abstract

In the framework of a European TMR network project the concept for a g-ray tracking array is being developed for

nuclear physics spectroscopy in the energy range of B10 keV up to several MeV. The tracking array will consist of alarge number of position-sensitive germanium detectors in a spherical geometry around a target. Due to the highsegmentation, a Compton scattered g-ray will deposit energy in several different segments. A method has beendeveloped to reconstruct the tracks of multiple coincident g-rays and to find their initial energies. By starting from the

final point the track can be reconstructed backwards to the origin with the help of the photoelectric and Compton cross-sections and the Compton scatter formula. Every reconstructed track is given a figure of merit, thus allowingsuppression of wrongly reconstructed tracks and g-rays that have scattered out of the detector system. This so-called

backtracking method has been tested on simulated events in a shell-like geometry for germanium and in planargeometries for silicon, germanium and CdTe. r 2002 Elsevier Science B.V. All rights reserved.

PACS: 29.30.k; 07.85

Keywords: Tracking; g-ray spectroscopy

1. Introduction

In nuclear spectroscopy Euroball [1] and Gam-masphere [2] represent the state of the art ingermanium detector arrays. In order to address theeven higher demands on future detector systems, anew concept for g-ray spectroscopy is underdevelopment: g-ray tracking [3]. In g-ray trackingthe interactions of a Compton scattered g-ray arecombined in such a way that the track of the g-rayis reconstructed and the initial energy and first(and other) interaction points can be deduced.

Up to now, two methods for g-ray tracking havebeen reported: the backtracking method [4] and aclustering method [5]. Both methods are developedfor nuclear spectroscopy applications and there-fore are focussed on a detector geometry thatresembles a shell around an emission point in thecenter. Because of the constraints on the energyresolution the detector material of choice isgermanium.

In this paper, the concept of the backtrackingmethod is presented briefly and its performancefor a germanium shell as can be used in nuclearspectroscopy is demonstrated for ‘realistic’ eventsas can occur in experiments. Furthermore theapplication of the method in planar detectors ofdifferent materials is demonstrated. g-ray trackingin planar detectors can be useful in applications

*Corresponding author. Present address: Nederlands Meet-

instituut. P.O. Box 80000, 3508 TA Utrecht, The Netherlands.

Tel.: +31-30-253-9098; fax: +31-30-253-9095.

E-mail address: [email protected] (J. van der Marel).

0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 8 3 7 - X

where precise position-sensitive detection of rela-tively high energetic X-rays or g-rays is importantsuch as in computed tomography.

2. Reconstruction algorithm

The reconstruction method is explained in detailin Ref. [4]. In this paper only a brief overview willbe given.

The backtracking method is based on theobservation that in germanium most final photo-electric interactions after Compton scattering havean energy deposition in a narrow energy bandbetween 100 and 250 keV, independent of theinitial energy (which should be above this range).For silicon and CdTe similar energy bands can befound at 50–150 and 150–300 keV, respectively.

An interaction that deposited an energy in theabove mentioned regions has a rather highprobability (B40%) to be a final point. This is agood starting point (end point of the track) s forthe reconstruction of the g-ray track. Then theclosest interaction can be tested with the help ofthe photoelectric cross-sections to see whether it isa suitable candidate for the last but one point s� 1of the track. Since the reconstruction starts at theend of a track, the energy of the g-ray before theinteraction occurred is known, so that absolutecross-sections can be used. If the result of the test,which we call the figure of merit wi for point i; isabove a certain predefined threshold the recon-struction proceeds to the second but last points� 2: With these two points and using the Comptonscatter formula

cos yc ¼ 1�mec2 1

Es�

1

Es�1

� �ð1Þ

in which yc is the scattering angle,mec

2 ¼ 0:511MeV is the rest mass of the electron,Es�1 is the g-ray energy in MeV before thescattering and Es is the energy (MeV) afterscattering, the direction in which point s� 2 hasto be looked for can be calculated. The point thatis closest to that direction is tested with the help ofthe Compton scatter cross-sections to see whetherit is in a certain predefined range. If this test is alsosatisfactory, the reconstruction can proceed in the

same way. After every step, it is tested whether theprobability that the g-ray comes from the emissionpoint is above a certain threshold. If this is thecase, the track is terminated. Recursions have beenbuilt-in so that if a step results in an unsatisfactorywi another interaction point for the current step orthe previous step is considered. If a trackterminates the total figure of merit wtot iscalculated. The wtot can be used later in theanalysis to reject poorly reconstructed g-rays org-rays that scattered out of the detector system.Interaction points that cannot be attributed to atrack are assumed to be the result of directphotoelectrical absorption of the g-ray and obtaina wtot related to the probability for photelectricabsorption.

3. Backtracking in a spherical detector

The backtracking method has been developedfor nuclear spectroscopy, where many (position-sensitive) detectors are placed in a sphericalgeometry around an emission point. In this typeof application, many coincident g-rays within abroad energy range may have to be detected. Forthe calculation a germanium sphere with an innerradius of 15 cm and an outer radius of 24 cm hasbeen chosen. With the help of the GEANTsimulation package [6] events have been madeconsisting of 25 coincident g-rays with energiesfrom 100 keV up to 2.5MeV with an energyspacing of 100 keV. The g-rays are emitted inrandom directions.

To make the calculation somewhat more realis-tic, the resolving distance dres; which indicates theminimum distance at which two interactions canbe seen as separate interactions, is limited. Ifpoints are closer to each other than dres; the pointsare combined to one point: the new energy is theadded energy of the points and the new position isthe average position of the points, weighted bytheir energies. Also some noise (Gaussian shaped)spos has been added to the position. To the energyof every interaction point, a fixed amount ofelectronic noise is added as well as a variableamount of statistical noise. An energy threshold of5 keV per interaction has been applied.

J. van der Marel, B. Cederwall / Nuclear Instruments and Methods in Physics Research A 477 (2002) 391–396392

In Fig. 1, the reconstruction efficiency e (definedas the number of reconstructed events in aphotopeak divided by the number of events fromthe GEANT simulation in that peak) is plotted asa function of the g-ray energy E: To show theresult of selection on wtot on the efficiency,different curves have been plotted. As can be seenfrom Fig. 1 higher selection criteria result in worseefficiency. However, the peak-to-total ratio (the

ratio between the contents of the photopeaks andthe total number of counts in the spectrum)increases considerably as can be seen in Table 1.

4. Backtracking in a planar detector

The backtracking method can also be applied onplanar detectors of different materials. The size ofthe detectors in these calculations is 20� 20 cmand 2 cm thickness. Also, here it is assumed thatthe g-ray source is at a known position: on aperpendicular line through the center of thedetector at a distance of 5 cm from the detectorsurface. Twenty g-rays are emitted simultaneouslyin random directions of which a limited numberwill hit the detector to simulate the occurrence ofmultiple hits in a detector.

Calculations have been performed using thesame method as in Section 3 to limit the resolvingdistance of the detector system and to add positionnoise. For the different materials, different energieswhich can be considered to be characteristicfor that material have been chosen. For the

0 0.5 1 1.5 2 2.5energy (MeV)

0

0.2

0.4

0.6

0.8

1

reco

nstr

uctio

n ef

ficie

ncy

dres = 1 mm, σpos = 0.2 mmdres = 1 mm, σpos = 0.2 mm, wtot > 0.1dres = 1 mm, σpos = 0.2 mm, wtot > 0.2dres = 5 mm, σpos = 1.0 mmdres = 5 mm, σpos = 1.0 mm, wtot > 0.1dres = 5 mm, σpos = 1.0 mm, wtot > 0.2

Fig. 1. Reconstruction efficiency for several position resolutions and selection criteria for events of 25 coincident g-rays of differentenergy in germanium.

Table 1

Peak-to-total ratios for the reconstructed realistic spectrum for

some different selection criteria and for different resolving

distances dres and position noises spos

dres spos Selection P=T(mm) (mm)

1.0 0.2 All tracks 0.40

1.0 0.2 wtot > 0:1 0.59

1.0 0.2 wtot > 0:2 0.68

5.0 1.0 All tracks 0.30

5.0 1.0 wtot > 0:1 0.45

5.0 1.0 wtot > 0:2 0.53

J. van der Marel, B. Cederwall / Nuclear Instruments and Methods in Physics Research A 477 (2002) 391–396 393

calculation of silicon, g-rays with an energy of200 keV have been used, for germanium 511 keVand for CdTe 1.332MeV. In Fig. 2 the reconstruc-tion efficiencies and the peak-to-total ratios areplotted as a function of the resolving distance. Forcomparison, the peak-to-total ratios as calculatedfrom the GEANT simulations are: 0.126 forsilicon, 0.397 for germanium and 0.416 for CdTe.

A more realistic way to include the limitedposition resolution of a detector is to divide thedetector in cells of fixed dimensions. One can thinkof a stack of pixel detectors or double-sided stripdetectors. The energy of all interactions thatdeposit energy in a certain cell is added and theposition is the center of the cell. No additionalposition noise is added. In these calculations the

0 0.1 0.2 0.3 0.4 0.5resolving distance (cm)

0

0.2

0.4

0.6

0.8

1

reco

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germaniumcadmium telluridesilicon

0 0.1 0.2 0.3 0.4 0.5resolving distance (cm)

0

0.1

0.2

0.3

0.4

0.5

peak

−t0−

tota

l rat

io


Fig. 2. Efficiency and peak-to-total ratio as a function of the resolving distance dres: The position noise spos ¼ dres=5: Reconstructed

tracks with wtot > 0:10 have been selected.

0 0.5 1 1.5 2cell size (mm)

0

0.2

0.4

0.6

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1

reco

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uctio

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ficie

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0 0.5 1 1.5 2cell size (mm)

0

0.1

0.2

0.3

0.4

0.5

peak

−t0−

tota

l rat

io


Fig. 3. Efficiency and peak-to-total ratio as a function of the cell size. The cells are cubic. Reconstructed tracks with wtot > 0:10 have

been selected.


cells are cubic. To compare the results with thosein which the resolving distance was limited, thesame number of coincident g-rays and the samemultiplicities have been used. The results areplotted in Fig. 3 (note the different scale on theX-axis).

The reconstruction efficiency and the peak-to-total ratio depend on the selection of thereconstructed tracks by means of wtot: Thisdependence is shown in Fig. 4. For germanium,the peak-to-total ratio increases for higher selec-tions of wtot; but for CdTe there is a clear optimumat wtot > 0:2: For silicon the peak-to-total ratiohardly depends on selection on wtot: In all cases thereconstruction efficiency decreases with increasingconstraints on wtot; most dramatically for siliconand CdTe.

5. Conclusion

The backtracking method can be successfullyapplied for the reconstruction of Compton scat-tered g-rays in a position-sensitive detector system.The method has been developed for a geometry inwhich many detectors are placed as a shell aroundthe g-ray emission point. Due to the constraints onthe energy resolution, the detector material forapplications in nuclear spectroscopy is germa-

nium. Up to now the method has only been testedon simulated events, due to the absence of a realdetector system. It has also been shown that themethod can be used in planar geometries ofdifferent materials, i.e. germanium, silicon andCdTe.

In both geometries, it becomes clear that for thebest results in the reconstruction the positionresolution should be as good as possible. Thereconstruction efficiency drops considerably, espe-cially at resolving distances above 1mm. As can beexpected, also in the case of cubic detector cells theefficiency drops for large cells. The reconstructionefficiencies for the different materials are similar,for germanium it is best and for cadmium tellurideit is worst. A reason for this is that the energyresolution for the individual interactions forgermanium is best and for CdTe worst, a featurethat is taken into account in the simulations. Dueto these inaccuracies in energy the reconstructionis less precise, which results in a lower figure ofmerit for the tracks. In CdTe this can be clearlyseen by the dramatic decrease of the reconstruc-tion efficiency and by the decrease of the peak-to-total ratio for high selections on wtot: The peak-to-total ratio also decreases for worse positionresolutions. Here, germanium and CdTe are verysimilar, but silicon is much worse. Also in theGEANT simulations this is the case. In silicon

0 0.1 0.2 0.3 0.4 0.5figure of merit

0

0.2

0.4

0.6

0.8

1

reco

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uctio

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ficie

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0 0.1 0.2 0.3 0.4 0.5figure of merit

0

0.2

0.4

0.6

0.8

1

peak

toto

tal r

atio


Fig. 4. Efficiency and peak-to-total ratio as a function of the figure of merit. The detectors consist of cubic cells of 0.3mm.

J. van der Marel, B. Cederwall / Nuclear Instruments and Methods in Physics Research A 477 (2002) 391–396 395

many interactions deposit a very small amount ofenergy and due to the energy threshold of 5 keV,many interactions are excluded from the tracks.For silicon this threshold can probably be lowered.For a selection with wtot > 0:10 the peak-to-totalratio is in the best case as good as from thesimulation. However, by adjusting the selection abetter peak-to-total ratio can be obtained at theprice of a somewhat lower efficiency.

g-ray tracking in planar detectors can be appliedin e.g. computed tomography when relatively highX-ray energies are used, so that the probability ofCompton scattering in the detector system isrelatively high. In silicon and CdTe the peak-to-total ratio cannot be improved dramatically, butespecially when more than one g-ray at a time canreach the detector system, g-ray tracking can bebeneficial to distinguish the different g-rays.

We have outlined the concept of the back-tracking method for g-ray tracking. The perfor-mance of this method has been demonstrated forvarious conditions using Monte-Carlo simulateddata. The results show that the position resolutionis very critical for future detector systems based ong-ray tracking. In the case of silicon and CdTe, amore detailed study is necessary in order to

investigate the feasability to obtain the sameperformance as for germanium. It should also bepointed out that our tracking results for germa-nium still are relatively preliminary and thatfurther improvements are foreseen.

Acknowledgements

This work was supported by the Commissionof the European Communities within theTMR programme under contract No.ERBFMRXCT97-0123.

References

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[3] I.Y. Lee, Nucl. Instr. and Meth. A 422 (1999) 195.

[4] J. van der Marel, B. Cederwall, Backtracking as a way to

reconstruct Compton scattered g-rays, Nucl. Instr. and

Meth. A 437 (1999) 538.

[5] G.J. Schmid, et al., Nucl. Instr. and Meth. A 430 (1999) 69.

[6] GEANT, Detector description and simulation tool, CERN,

Geneva, Switzerland.


γ-ray tracking in germanium: the backtracking method

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