High purity germanium (HPGe) detectors are ubiquitous in nuclear physics experiments and are also used in numerous low radioactive background detectors. The effect of the position of 60Co and 137Cs point sources on the shape of spectra were studied with Monte Carlo and HPGe detector measurements. We briefly con firm previous work on the position dependence of relative heights of peaks. Spectra taken with the radiation sources placed at locations around the detector were then compared using the Kolmogorov-Smirnov (K-S) goodness-of- t test. We demonstrate that with this method the Compton continuum spectral shape has good sensitivity to the radial location of a point-source, but poor angular resolution. We conclude with a study of the position reconstruction accuracy as a function of the number of counts from the source

Andre, K.

De, R.

Evans, L.T.

Henning, R.

Morgan, E.D.

arXiv

Carolina Digital Repository

Reconstruction of a Radiation Point Source’s Radial Location UsingGoodness-of-Fit Test on Spectra Obtained from an HPGe DetectorL.T. Evans a,b, K. Andre a,b, R. De a, R. Henning a,b,∗, E.D. Morgan a,baTriangle University Nuclear Laboratories, Durham, NC, USAbUniversity of North Carolina, Chapel Hill, NC, USAAbstractHigh purity germanium (HPGe) detectors are ubiquitous in nuclear physics experiments and are also used in numerouslow radioactive background detectors. The effect of the position of 60Co and 137Cs point sources on the shape ofspectra were studied with Monte Carlo and HPGe detector measurements. We briefly confirm previous work on theposition dependence of relative heights of peaks. Spectra taken with the radiation sources placed at locations aroundthe detector were then compared using the Kolmogorov-Smirnov (K-S) goodness-of-fit test. We demonstrate thatwith this method the Compton continuum spectral shape has good sensitivity to the radial location of a point-source,but poor angular resolution. We conclude with a study of the position reconstruction accuracy as a function of thenumber of counts from the source.Key words: High purity germanium (HPGe) detectors, spatial reconstruction, low background, Kolmogorov-Smirnov (K-S)goodness-of-fit test1. IntroductionHigh Purity Germanium (HPGe) detectors areused for γ-ray detection and spectroscopy in nuclearphysics experiments, including low radioactive back-ground detector systems, such as the proposed Ma-jorana experiment [1]. Others have shown and ex-plained how a detected spectrum shape can vary no-ticeably based on the position of a γ-emitting radia-tion source relative to the detector [2,3,4,5]. We showthat the Kolmogorov-Smirnov (K-S) test is sensitiveto these changes as well. We also show that the ra-dial position of nearby γ-emitting radiation sources∗ Corresponding AuthorEmail addresses: ltevans@unc.edu (L.T. Evans),kali@unc.edu (K. Andre), risingstar49@aol.com (R. De),rhenning@physics.unc.edu (R. Henning),edmorgan@email.unc.edu (E.D. Morgan).can be accurately reconstructed using the K-S testto compare spectra taken at different locations.2. ExperimentFor our measurements, we used an EG&G OrtecHPGe detector model P40621A with a germaniumcrystal of length 93 mm and a diameter of 90 mm.It has an active volume of 582 cm3. The detector’scryostat has a 1.0 mm aluminum endcap.We obtained the γ-ray energy spectra of a 60Copoint source at different positions around the detec-tor. A flat piece of acrylic was taped to the detectorcryostat can to hold the source at a set location. Wemoved the source along a grid with a 2 mm spacingthat was marked on the acrylic sheet, as shown inFig. 1. The source was started 5 mm from the de-tector endcap, and data points were taken until thesource was approximately 35 mm away from the de-Preprint submitted to Elsevier 17 July 2021arXiv:0902.4370v2  [nucl-ex]  16 Aug 2009Fig. 1. Spectra were taken by moving the 60Co or 137Cs (inMonte Carlo simulation) source along a grid of resolution2 mm (not to scale). The grid was marked on a flat acrylicsheet that was mounted to the detector cryostat (the graybox).tector. Each data point was taken for 45 seconds,which yielded an average of 80 000 counts per spec-trum. We also varied the exposure time in 5-secondintervals from 5 to 45 seconds for a location 5 mmfrom the outer casing of the detector. This data wasused to quantify statistical effects on the determina-tion of the position of the 60Co source. An MCA ac-quired an energy spectrum from the HPGe detectorduring each run. As shown in Fig. 2, the major peaksidentified on the spectra were the two 60Co peaks, at1.173 MeV and 1.332 MeV, a background 40K peakat 1.460 MeV, a 60Co sum peak at 2.505 MeV, anda background 208Tl peak at 2.614 MeV. 208Tl is adaughter isotope of naturally-occurring 232Th.Analysis programs were coded in C++ and inROOT [6] to compare a spectrum acquired at anarbitrary location (called a test spectrum) withthe spectra taken at all locations (called the refer-ence spectra). Specifically, it applied a well-knowngoodness-of-fit test, the Kolmogorov-Smirnov (K-S)test, between a test spectrum and all the referencespectra. The K-S test finds the maximum distancebetween two cumulative distribution functions, inour case the cumulants of the energy spectra. Thismaximum distance is inversely correlated to thelikelihood that the two distributions came from thesame distribution, and is converted into a proba-bility using the Kolmogorov distribution [7]. Thisversion of the K-S test is ideally used on unbinnedFig. 2. A typical spectrum from the physical HPGe detector(top) and from the Monte Carlo simulation without back-ground (middle) and with background (bottom). Note thetwo 60Co lines at 1.17 MeV and 1.33 MeV in all spectra,a 40K peak at 1.46 MeV for data with background, a 60Cosum peak at 2.51 MeV in all spectra, and a 208Tl peak at2.61 MeV for the spectra with background. The real datashows a saturation effect past channel 8000.data, but with binned data the output of the testis still a reliable measure of how similar the twodistributions are if the bin-size is smaller than theenergy resolution, as is the case here. We chose2Fig. 3. Cumulative distribution functions for two physical60Co spectra taken at different locations (top) and two sim-ulated 137Cs spectra taken at the same locations (bottom).The 137Cs spectrum shown here was simulated without back-ground. The dotted vertical lines show the bins with thelargest difference between the two cumulative distributionfunctions. In both instances the most sensitivity to differ-ences in the distribution was in the cumulant of the lowenergy half of the Compton continuum. The diagram in thebottom right of the right plot shows the locations of the twopositions being compared in both plots. The K-S test valuebetween these two points was equal to zero within machineprecision in both cases.the K-S test as it is sensitive to subtle differencesin the shape of spectra, conceptually simple, andcomputationally fast. We found that the logarithmof the K-S test value was a better indicator thanthe actual value of how correlated two spectra were.Typical cumulative distribution functions used inthe study are shown in Fig. 3. Interestingly, theseresults indicate that the middle of the Comptoncontinuum provides the K-S test with the mostsensitivity to position-dependent differences in thespectral-shapes. It is less sensitive to the peak-to-Compton ratio, for example. This may also be aFig. 4. A simulated 137Cs spectrum with backgrounds. Notethe 0.662 MeV peak from the 137Cs, as well as the back-ground 208Tl peak at 0.583 MeV. The same amount of back-ground as Fig. 2 was used here.manifestation of the K-S test’s propensity to beless sensitive to the tails of distributions. Regard-less, it still indicates that there is position-sensitiveinformation contained in the spectral shape of thelow-energy half of the Compton continuum.A Monte Carlo simulation in GEANT4 [8] wasused to simulate this experiment as well. The simula-tion geometries were created to match the actual ex-periment as closely as possible, and 60Co and 137Cssources were simulated around the HPGe detectorat the same locations the 60Co radiation source wasmeasured around the physical detector. Typical sim-ulation spectra are shown in Figs. 2 and 4. A Gaus-sian energy smearing was applied to include the ef-fect of finite detector energy resolution. 40K and208Tl backgrounds were also simulated. The inten-sity of the background in the simulations was variedto study the effects, as described later.3. Results3.1. Peak AnalysisWe performed a quick analysis to verify ouranalysis against earlier work concerning the peak-to-compton ratio. Fig. 5 shows the change in the1.33 MeV peak area and the sum peak area rel-ative to the total area as a function of positionaround the detector for physical 60Co measure-ments. Obviously, there is little variation in the rel-ative 1.33 MeV peak area, but a clear trend is seenwith the sum peak. When the radiation source wasplaced at the center of the front of the detector and3Fig. 5. Change in the 60Co 1.33 MeV relative peak height(top), the relative 60Co sum peak height (middle), and the137Cs 662 keV relative peak height (bottom) as a functionof source position. Relative peak intensity refers to the peakarea divided by (the total area minus the peak area). Thesource was moved in incremental steps across the face of thedetector and then along the side of the detector, maintain-ing a distance of 30 cm away from the endcap. Each stepcorresponds to an increment in the horizontal location indexon the graphs’ x-axes. The sharp turns in the middle of thebottom plots show where the source was turned from thefront of the detector to the side. We notice that little vari-ation exists in the 60Co 1.33 MeV relative peak height, butthere is great variation in the relative sum peak heights, aswell as the 137Cs 662 keV relative peak height. Errors shownare statistical.halfway down the side of the detector, the relativesum peak area was roughly five times greater thanwhen the radiation source was placed behind thedetector. This is due to the fact that the solid anglesubtended by the detector is greatest at the centerof the front of the detector and the center of theside, as previously described in [9]. This differencewas clearly visible without conducting statisticaltests. The 1.17 MeV peak area differed with roughlythe same distribution as the 1.33 MeV peak, with adifference in amplitude because the two peaks de-pend on the solid angle subtended by the detectorin the same manner [4].Fig. 5 shows the change in the 662 keV peak arearelative to the total area as a function of positionaround the detector for simulated 137Cs data. Unlikethe 1.17 MeV and 1.33 MeV 60Co peaks, a cleartrend is visible.3.2. Statistical Analysis with theKolmogorov-Smirnov TestFig. 6 shows the contour plots of the K-S test val-ues for four different test spectra compared to thereference spectra for physical and simulated 60Codata. The resulting regions with high K-S values ap-pear like rings on the plots, corresponding to ap-proximately spherical shells in three dimensions. Asthe figure shows, the areas of high K-S test valuesdo not necessarily point out the angular position ofthe source. Fig. 6 also indicates good agreement be-tween data and Monte Carlo and gives us confidencein the simulated 137Cs results. We note there is ainteresting ’double-arch’ structure in the bottom-most right plot of Fig. 6. This is likely due to pickinga test point that lies between two reference spec-tra points. As shown in Fig. 7, a similar plot wasmade comparing the K-S test values for histogramsfilled up to (but not including) the 1.17 MeV peakin the 60Co spectrum. These plots were barely dis-tinguishable from those when all peaks and overflowbins were included, showing that although the dif-ference in the sum peak area relative to position wasquite noticeable, the Compton continuum also hassignificant influence on the differences between thespectra. Similar results were seen with the simulated137Cs data. This is consistent with our earlier obser-vations of the dominant effect of the low-energy halfof the Compton continuum on the K-S value.Using test spectra from 20 randomly-selectedpoints, we found that the radial position of the4Fig. 6. Contour maps of K-S test values between the test spectrum and reference spectra for the actual measurements (leftcolumn) and the Monte Carlo simulation (right column) for a 60Co source. The test source position is shown with a blackdot. The white areas outside of contours represent areas where the K-S test value between that point and the black dot is lessthan 10−55. The detector is located at the gray area in the lower left of the diagram. Each graph has been smoothed, andeach contour in this plot represents a difference in the logarithm of the K-S test value of 9.5Fig. 7. Contour maps of the measured values of the K-Stest value between the spectrum at each position and thetest source spectrum for the actual measurements using theentire 60Co spectrum (left) and bins up to, but excluding, the1.17 MeV 60Co peak (right). Although significant changeswere seen in the sum peak height, the contour plots arebasically indistinguishable from one another, indicating thatthe Compton continuum has significant information aboutthe source position as well.radiation source could be found to within 5 mm95% of the time, yet the angular position could befound to this resolution only 20% of the time. Wefound that higher statistics resulted in lower K-Stest values between the test spectrum and referencespectra for reference spectra that were further apartradially. This is shown in Fig. 8. For 60Co, we foundthe resolution to be well-described as a power-lawthat is proportional to a −0.53 ± 0.02 power of de-tector counts, indicating the statistical dependenceof the resolution. For 137Cs, the resolution is betterdescribed by a power-law that is proportional to a−0.60± 0.02 power of detector counts. As shown inFig. 9, with an increase in background, the resolu-tion is still described by a power law with the samepower of detector counts, but the absolute resolu-tion increases as the number of background eventsincreases, as expected. Nor surprisingly, this indi-Fig. 8. Spatial resolution as a function of total detectorcounts for detector measurement 60Co (top), Monte Carlo60Co (middle) and 137Cs (bottom) data without background.We defined radial resolution as the width of the region wherethe logarithm of the K-S test value was greater than -5.The error was defined as the region where the logarithm ofthe K-S test value was greater than -10. The best fit line(solid line) and the upper and lower bound fits (dotted lines)are shown. For 60Co, in both detector measurements andMonte Carlo simulations the resolution is proportional tothe −0.53± 0.02 power of detector counts, and the constantof proportionality is 3.3 ± 0.5 m for detector measurementsand 3.3 ± 0.4 m for Monte Carlo. For 137Cs, the resolutionis proportional to the −0.60±0.02 power of detector counts,and the constant of proportionality is 3.9± 0.4 m6Fig. 9. Best fit lines with data points for resolution of spa-tial reconstruction as a function of total detector countsfor differing amounts of background for Monte Carlo 60Co(top) and 137Cs (bottom) data. In the 60Co plot, the bot-tom line (with square data points) has no background (as inFig. 8), the middle line (with triangular data points) has aratio of 40K peak area to 1.17 MeV 60Co peak area of 0.25,and the top (with circular data points) has ten times morebackground counts the middle one. In the 137Cs plot theseproportions are the same, using the 0.583 MeV 208Tl as thebackground peak instead of 40K, and the 0.662 MeV 137Cspeak instead of the 1.17 MeV 60Co peak. For the highestbackground rate, the fit was extrapolated from higher detec-tor counts, since our simulation was not sensitive to resolu-tions greater than 30 mm. With an increase in background,the proportionality to detector counts in the power law doesnot change, but the constant of proportionality increases.cates that one can still improve the sensitivity of theradial resolution in the case of large backgroundsby collecting more statistics.When the test spectrum is a spectrum that is thesuperposition of spectra from two distinct sources atdifferent positions, the rings of Fig. 6 widen, with theamount it widens determined by how far apart thetwo sources are. These rings are in between wherethe rings for each individual source would be, withthe exact location determined by the difference inFig. 10. Contour map of the measured values of the K-Stest value between the spectrum at each position and thesuperposition of two test source spectra for simulated MonteCarlo data. The location of the two sources of equal strengthare shown. Note that adding an extra source causes thecontours to expand.intensity of the two sources. This is shown in Fig. 10.4. ConclusionsWe were able to accurately reconstruct a bright,nearby radiation point source’s radial position us-ing an array of reference spectra and the K-S test.We were also able to accurately reproduce theseresults using Monte Carlo simulations, indicatingthat this method can be used by comparing detec-tor spectrum to sets of simulated spectra. The K-Stest appears to be most sensitive to spectral shapeeffect in the Compton-continuum. However, usingour method we were not able to gain good angu-lar position information about a radiation source.Future work will determine the limits of this tech-niques using unbinned data and low statistics. Lowbackground HPGe detector arrays, such as Majo-rana, can potentially use this technique along withrelative rates in detectors to determine the locationof unwanted hot-spots in the detector array thatwere inadvertently introduced during construction.This technique is also applicable to other gamma-rayspectrometers, such as NaI and cryogenic bolome-ter detectors. We also want to point out one noteof caution with using the K-S test in the presenceof sharp peaks. The K-S test would indicate incom-patible distributions when comparing spectra withpeaks that are slightly offset from each other, ie.when there exist slight binning misalignments, cal-ibration offsets, or simulation-vs.-data errors. Caremust be exercised the ensure the proper calibration7and alignment of the spectra’s histogram bins.5. AcknowledgementsWe would like to thank the LENA group and Prof.A.E. Champagne for the use of the LENA HPGedetector. This work is supported under in part byDOE NP Grant # DE-FG02-97ER41041 and theState of North Carolina.References[1] C.E. Aalseth et al [Majorana Collaboration] Nucl. Phys.B (Proc. Suppl.) 138 (2005) 224.[2] H.I. Bak, Y.D. Bae, M.S. Kim, and H.D. Choi, Nucl.Instr. and Meth. A 366 (1995) 332-339.[3] R.G. Helmer, R.J. Gehrke, and R.C. Greenwood, Nucl.Instr. and Meth. A 123 (1975) 51-59.[4] G.F. Knoll, in Radiation Detection and Measurement(John Wiley & Sons, 2000).[5] P.C. Lichtenberger and I.K. MacKenzie, Nucl. Instr. andMeth. 116 (1974) 177-179.[6] R. Brun and F. Rademakers, Nucl. Instr. and Meth. A389 (1997) 81-86. See also http://root.cern.ch/.[7] G.K. Kanji in 100 Statistical Tests, 3rd edition (SAGEPublications Ltd., 2006)[8] S. Agostinelli et al. [GEANT 4 Collaboration], Nucl.Instr. and Meth. A 506 (2003) 250.[9] M.J. Prata, Radiat. Phys. and Chem. 67 (2003) 599-603.8

Reconstruction of a radiation point source’s radial location using goodness-of-fit test on spectra obtained from an HPGe detector

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Reconstruction of a radiation point source’s radial location using goodness-of-fit test on spectra obtained from an HPGe detector

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