- Email: [email protected]
Precise efficiency calibration of an HPGe detector up to 3.5 MeV, with measurements and Monte Carlo calculations
ARTICLE IN PRESS
Applied Radiation and Isotopes 60 (2004) 173–177
Precise efficiency calibration of an HPGe detector up to 3.5 MeV, with measurements and Monte Carlo calculations R.G. Helmera,*, N. Nicab, J.C. Hardyb, V.E. Iacobb a
Idaho National Engineering and Environmental Laboratory, Idaho Accelerator Center, Idaho State University, P.O. Box 1625, Pocatello, Idaho Falls, ID 83415-2114, USA b Cyclotron Institute, Texas A&M University, College Station, TX, USA
Abstract Previously we used relative and absolute efficiency measurements combined with Monte Carlo calculations to define the efficiency of an HPGe gamma-ray detector with 0.2% accuracy from 50 to 1400 keV. This work has been extended to 4.8 MeV with measurements of relative efficiencies from 24Na, 56Co, and 66Ga sources. The combined results of experiment and calculation yield an efficiency curve up to 3.5 MeV with 0.4% accuracy. Single- and double-escape peak contributions also agree with calculation if positron annihilation-in-flight is incorporated. r 2003 Elsevier Ltd. All rights reserved. Keywords: Ge detector; Precise efficiency; Measurements 20–3500 keV; Monte Carlo
1. Introduction The authors have previously reported (Hardy et al., 2002; Helmer et al., 2003) on the determination of the efficiency curve for a coaxial, 280-cm3 n-type Ge g-ray detector, based on a combination of measured relative efficiencies for nine radionuclides, the measured absolute efficiency for the g rays from 60Co, and efficiencies calculated with CYLTRAN (Halbleib and Mehlhorn, 1986), a Monte Carlo photon and electron transport code. With the measured physical dimensions of the detector, only minor adjustments were required, well within tolerances, to obtain excellent agreement between the measured and calculated efficiency values (Helmer et al., 2003). From these results, we estimated the uncertainty in the final efficiency curve to be 0.2% between 50 and 1400 keV. An extensive description of the methods used to develop this curve is given in Helmer et al. (2003).
*Corresponding author. Tel.: +1-208-526-4157; fax: +1208-526-9267. E-mail address: [email protected] (R.G. Helmer).
Without changing the detector parameters in any way, we have now extended this work to 4.8 MeV by the continued use of both measured relative efficiencies and CYLTRAN-calculated efficiencies.
2. Measured efficiencies Relative efficiencies for the full-energy (FE) peaks have been measured for the energies of the strong g-rays from the decays of 24Na, 56Co, and 66Ga. We produced 24 Na by thermal neutron capture on Na2CO3 at the Texas A&M reactor, the source material being deposited on, and covered by 0.08-mm plastic foil. For 66Ga, we used a 66Zn beam from the Texas A&M K500 Cyclotron to initiate the reaction 1H (66Zn, n) 66Ga, the recoiling 66 Ga nuclei being separated from other reaction products in the MARS recoil spectrometer, and then implanted into 0.08-mm plastic foil. The 56Co source was purchased from Isotopes Products Laboratory in the form of an evaporated metallic salt under 2.77 mm of plastic. The efficiency measurements were made, and the spectra were analyzed, in the same manner as described in Helmer et al. (2003). Since the g rays from these
0969-8043/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2003.11.012
ARTICLE IN PRESS R.G. Helmer et al. / Applied Radiation and Isotopes 60 (2004) 173–177
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Table 1 Decay data used in our relative efficiency determinations. The 24Na data are from IAEA Co-ordinated Research Program (1991); those for 56Co are from the evaluation in Baglin et al. (2002); and for 66Ga are from the measurements and recommendations in Baglin et al. (2002). The g-ray energies are nominal values and the intensities (emission probabilities) are per 100 decays for 24Na, and are relative values for 56Co and 66Ga Nuclide
Eg (keV)
Ig
24
1368 2754 847 977 1038 1175 1238 1360 1771 1811 1964 2015 2035
99.9936(15) 99.846(6)a 100.0(2) 1.425(9) 14.09(5) 2.255(14) 66.3(3) 4.274(15) 15.52(5) 0.643(4) 0.715(6) 3.041(16) 7.79(4)
Na
56
Co
Nuclide
66
Ga
Eg (keV)
Ig
2598 3202 3253 3273 834 1039 1333 2190 2752 3229 3381 4086 4806
17.02(6) 3.22(3) 7.90(6) 1.864(15) 15.93(5) 100.0(3) 3.175(12) 14.42(5) 61.35(23) 4.082(19) 3.960(19) 3.445(18) 5.03(3)
a We have measured Ig ¼ 0:000645 (14) for the 3867-keV g ray from 24Na. We then used this result to revise slightly the Ig value quoted for the 2754-keV g ray in IAEA Co-ordinated Research Program (1991).
nuclides are generally in cascade and the source–detector distance was 15 cm, corrections for coincidence summing were made: for most g rays these corrections were less than 1%; they were never larger than B2%. The relative FE efficiencies deduced from these measurements depend on the relative g-ray emission probabilities used for these nuclides. On the one hand, the value for the two strong 24Na g rays is well known (IAEA Co-ordinated Research Program, 1991). On the other, those for the more complex decays of 56Co and 66 Ga have not been well established in the past because they depended on Ge detector efficiencies, which were not well determined themselves. However, there have been recent measurements and evaluations of these emission probabilities that have improved the situation. The major improvement has come with the use of g rays from neutron-capture reactions to aid in the detector efficiency calibration above 2.75 MeV, the highest energy that can be covered with long-lived radioactive sources. The values we have used for all three sources are given in Table 1 and are from IAEA Co-ordinated Research Program (1991) and Baglin et al., (2002). Only those g rays with emission probabilities known to 1% or better were considered and, among those, we only retained those that were cleanly observed—uncontaminated by escape peaks, for example—with measured areas in our spectra determined with uncertainties of 1% or better. For many of the g rays, our measured spectra also yielded single-escape (SE) and double-escape (DE) peak relative efficiencies. We have found it most useful to express these results as ratios of escape peak areas to that of the corresponding FE peak. These ratios are
independent of the g-ray emission probabilities and of any coincidence summing corrections.
3. Calculated efficiencies In Helmer et al. (2003), we obtained a very accurate efficiency curve between 50 and 1400 keV based on detailed source measurements and CYLTRAN (Halbleib and Mehlhorn, 1986) Monte Carlo calculations. The calculations used the measured physical properties of our detector, with adjustments being made only in the nominal dead-layer thicknesses, which could not be independently measured, and some fine-tuning of the detector radius, well within its measurement tolerance. With these dimensions unchanged, we used the same code to compute the detector efficiencies at the 26 energies (834–4806 keV) tabulated in Table 1 for the three nuclides measured here. These calculations were done in a manner to obtain the desired statistical uncertainty on the SE and DE peaks as well as the FE peaks.
4. Comparison of FE peak efficiencies The measured and calculated FE peak efficiencies are compared in Fig. 1. In this comparison, there are only three free parameters—a single scaling factor for each nuclide. We determined these scaling factors by minimizing the weighted differences between the measured values and the Monte Carlo efficiency values: that is, by means of a least-squares fit. Each of these nuclides has one or more g rays below 1400 keV where the FE
ARTICLE IN PRESS R.G. Helmer et al. / Applied Radiation and Isotopes 60 (2004) 173–177
175
5. Comparison of escape-peak ratios
Fig. 1. Differences between the measured efficiencies (points with uncertainties) and the corresponding Monte Carlo calculated values (line at 0). The differences are calculated as experiment minus calculation, divided by calculation, and are expressed as a percent. The points can be identified with individual sources as follows: 24Na, solid squares; 56Co, open diamonds; 60Co, x’s; 66Ga, open circles; and 88Y, open squares. Note that the point for the 4.8-MeV gamma ray from 66Ga is at 3.8%; only the bottom of its error bar is visible at the upper right of the figure.
efficiency curve has already been determined precisely (Helmer et al., 2003). Therefore, the efficiencies at the higher energies presented here are completely consistent with those at the lower energies reported in Helmer et al. (2003). In general, the largest component in the uncertainties given for these measured efficiencies is from the known emission probabilities (see Table 1). Note also that Fig. 1 includes the results of two measurements from our previous work (Hardy et al., 2002; Helmer et al., 2003), those of 60Co and 88Y. The former provides an absolute efficiency determination that anchors the overall efficiency curve, while the latter yields a useful efficiency ratio that overlaps the energy region of interest in the present work. As indicated in Fig. 1, the agreement between measured and calculated efficiencies below 3500 keV is excellent. Over this energy region, the normalized chisquared for the 24 points from 24Na, 56Co and 66Ga (with 21 degrees of freedom) is 0.70. We suggest that the uncertainty in the deduced efficiencies is 0.4% between 1400 and 3500 keV. Above 3500 keV there are two 66Ga g rays with emission probabilities claimed to better than 1%, both of which deviate significantly from the Monte Carlo calculations. Whether these represent real discrepancies between experimental and theoretical efficiencies, or a systematic error in the accepted emission probabilities can only be decided by further experiments. For now, we forego any estimation of the uncertainty in our calibration curve above 3500 keV.
In contrast to the excellent agreement in Fig. 1, when the measured SE/FE and DE/FE ratios were first compared with the Monte Carlo calculations, they were found to be consistently lower than the calculated ratios by up to 7%. We then learned from an author of the CYLTRAN code (Kensek, 2003) that the program follows each pair-produced positron as it loses energy by scattering, until it reaches thermal energy and annihilates, producing two 511-keV photons, which are then followed until they escape or are absorbed. In that process, no account is taken of positron annihilation-inflight (AIF). Clearly, since the positrons that annihilate in flight produce photons with energies that are significantly different from 511 keV, the SE and DE peak areas calculated by the code need to be reduced by the known probabilities for AIF. Calculated values for these probabilities are given in Browne and Firestone (1986) where they can be seen to depend primarily on the positron energy and secondarily on the material in which the annihilation occurs. Thus, for each particular g ray of energy Eg ; we reduced the SE and DE efficiency values from the Monte Carlo calculation by the AIF probability determined for a positron with energy 12 (Eg 1022) keV. Sample values for the AIF probabilities (Browne and Firestone, 1986) determined in this way are 2.3%, 3.9%, and 5.5% for g-ray energies of 2, 3, and 4 MeV, respectively. As shown in Fig. 2, when the calculated escape ratios are reduced to account for AIF, the agreement with the measured values is excellent. Note that in this comparison, the uncertainties that contribute are only those obtained from the peak area determinations and from the statistics of the Monte Carlo calculations.
Fig. 2. Measured escape ratios (points with uncertainties) compared with CYLTRAN Monte Carlo calculations (curves) that have been corrected for positron annihilation in flight.
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Table 2 Portion of a Monte Carlo calculated spectrum for g rays for 1173 keV showing the regions of the FE peak and the two peaks from Ge K X-ray escape. The energy bins have a width of 0.2 keV and the lower energy of the bin is given. The values for the spectrum are the counts per 100 g rays emitted into a 30 cone; a total of 108 g rays were emitted Energy(keV)
Counts
Energy(keV)
Counts
1173.23 1173.03 1172.83 1172.63
3.2294 (18) 0.000097 (10) 0.000095 (10) 0.000108 (10)
1163.8 1163.6 1163.4(Ka ) 1163.2 1163.0
0.000169 0.000161 0.000926 0.000138 0.000149
This comparison and the one in Fig. 1 do not take into account any possible change in the FE peak areas that could occur when the 511-keV annihilation photons are replaced by those from AIF. It is expected, though, that such a change would be very small and likely negligible in the context of the uncertainties that appear in both figures. The exact magnitude of the effect on the FE peak will need to await a change in the Monte Carlo code.
(13) (13) (30) (12) (12)
Energy (keV)
Counts
1162.7 1162.5 1162.3 (Kb ) 1162.1 1161.9
0.000174 0.000174 0.000359 0.000160 0.000171
(13) (13) (19) (13) (13)
Table 3 Ratio of the area of the sum of the two Ge K X-ray peaks to that of the FE peak as calculated with the CYLTRAN Monte Carlo code Energy(keV)
Ratio 104
Energy(keV)
Ratio 104
100.0 300.0 500.0 700.0 900.0
18.5 (1) 3.01 (9) 2.97 (9) 3.01 (11) 2.87 (10)
1173.2 1368.6 1836.1 2754.0 3866.5
2.99 3.25 3.24 3.48 3.78
(8) (10) (15) (17) (21)
6. Ge K X-ray escape from outer detector surface In the course of our careful analysis of the measured peaks from the 24Na sources, it was determined that a small peak occurs systematically about 10 keV below the FE peaks at both 1368 and 2754 keV. In order to confirm that this small peak was, in fact, due to escape of Ge K Xrays from the detector, the corresponding Monte Carlo calculations were run with a binning structure that showed the Ka and Kb X-ray-escape components. In Table 2, the tally in some of the Monte Carlo bins, each 0.2 keV wide, around the peak regions are given for a gray energy of 1173 keV. This clearly shows the two Ge K X-ray escape peaks. The calculated relative intensities of the sum of these Ge K X-ray escape peaks, (Ka +Kb )/FE, are given in Table 3 for g rays covering a range of energies from 100 to 3867 keV. Evidently the ratio is almost independent of g-ray energy above 300 keV and at those energies is presumably due to the loss of X-rays from all surfaces of the crystal, the cylindrical outer surface, the inner core surface and the back surface. The areas of the Ge X-ray escape peaks are difficult to determine accurately from experiment because they are not well resolved from the much larger FE peak, and are only marginally distinguishable from its low-energy tail, a tail whose shape cannot be independently determined. Nevertheless, it is clear that the observed Ge K X-ray escape ratios at 1368 and 2754 keV are larger than those calculated by at least a factor of 5. Although we cannot explain this difference, we would note that in Helmer et al., (2003) a smaller discrepancy, but in the same sense, was observed for low-energy g rays, where the Ge
X-rays escaped out of the front face of the detector. The difference in both cases may result from physical irregularities at the crystal surfaces or local variations in the charge collection not accounted for by the Monte Carlo calculations, which provide a spectrum of the energy deposited in the detector volume, but not the spectrum of charge actually collected. There may, of course, be some other limitation in the code evident only in this rather subtle effect. 7. Conclusion We have shown that it is possible to produce Monte Carlo calculated efficiencies that are as accurate as the currently possible measurements up to approximately 3.5 MeV and that a combination of Monte Carlo calculations with accurately measured relative efficiencies can generate an efficiency curve that is accurate to about 0.2% from 50 to 1400 keV and to about 0.4% from there to 3500 keV. Acknowledgements We wish to thank Dr. Ronald P. Kensek, Sandia National Laboratory for his consultation on matters related to the CYLTRAN Monte Carlo code. The work at Texas A&M was supported by the US Department of Energy under Grant No. DE-FG03-93ER40773 and by the Robert A. Welch Foundation.
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References Baglin, C.M., Browne, E., Norman, E.B., Moln!ar, G.L., Belgya, T., R!evay, Zs., Szelecs!enyi, F., 2002. 66Ga: standard for high-energy calibration of Ge detectors. Nucl Instrum. Methods A 481, 365–377. Browne, E., Firestone, R.B., 1986, Table of Radioactive Isotopes. Wiley, New York, p. D-6. Halbleib, J.A., Mehlhorn, T.A., 1986. The integrated tiger series (ITS) of coupled electron/photon Monte Carlo transport codes. Nucl. Sci. Eng. 92, 338–339. Hardy, J.C., Iacob, V.E., Sanchez-Vega, M., Effinger, R.T., Lipnik, P., Mayes, V.E., Willis, D.K., Helmer, R.G., 2002.
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Precise efficiency calibration of an HPGe detector: source measurements and Monte Carlo calculations with sub-percent precision. Int. J. Appl. Radiat. Isot. 56, 65–69. Helmer, R.G., Hardy, J.C., Iacob, V.E., Sanchez-Vega, M., Neilson, R.G., Nelson, J., 2003. The use of monte carlo calculations in the determination of a Ge detector efficiency curve. Nucl. Instrum. Methods A 511, 360–381. IAEA Co-ordinated Research Program, 1991. X-ray and gamma-ray standards for detector calibration, IAEATECDOC-619. Kensek, R.P., 2003. private communication.
Applied Radiation and Isotopes 60 (2004) 173–177
Precise efficiency calibration of an HPGe detector up to 3.5 MeV, with measurements and Monte Carlo calculations R.G. Helmera,*, N. Nicab, J.C. Hardyb, V.E. Iacobb a
Idaho National Engineering and Environmental Laboratory, Idaho Accelerator Center, Idaho State University, P.O. Box 1625, Pocatello, Idaho Falls, ID 83415-2114, USA b Cyclotron Institute, Texas A&M University, College Station, TX, USA
Abstract Previously we used relative and absolute efficiency measurements combined with Monte Carlo calculations to define the efficiency of an HPGe gamma-ray detector with 0.2% accuracy from 50 to 1400 keV. This work has been extended to 4.8 MeV with measurements of relative efficiencies from 24Na, 56Co, and 66Ga sources. The combined results of experiment and calculation yield an efficiency curve up to 3.5 MeV with 0.4% accuracy. Single- and double-escape peak contributions also agree with calculation if positron annihilation-in-flight is incorporated. r 2003 Elsevier Ltd. All rights reserved. Keywords: Ge detector; Precise efficiency; Measurements 20–3500 keV; Monte Carlo
1. Introduction The authors have previously reported (Hardy et al., 2002; Helmer et al., 2003) on the determination of the efficiency curve for a coaxial, 280-cm3 n-type Ge g-ray detector, based on a combination of measured relative efficiencies for nine radionuclides, the measured absolute efficiency for the g rays from 60Co, and efficiencies calculated with CYLTRAN (Halbleib and Mehlhorn, 1986), a Monte Carlo photon and electron transport code. With the measured physical dimensions of the detector, only minor adjustments were required, well within tolerances, to obtain excellent agreement between the measured and calculated efficiency values (Helmer et al., 2003). From these results, we estimated the uncertainty in the final efficiency curve to be 0.2% between 50 and 1400 keV. An extensive description of the methods used to develop this curve is given in Helmer et al. (2003).
*Corresponding author. Tel.: +1-208-526-4157; fax: +1208-526-9267. E-mail address: [email protected] (R.G. Helmer).
Without changing the detector parameters in any way, we have now extended this work to 4.8 MeV by the continued use of both measured relative efficiencies and CYLTRAN-calculated efficiencies.
2. Measured efficiencies Relative efficiencies for the full-energy (FE) peaks have been measured for the energies of the strong g-rays from the decays of 24Na, 56Co, and 66Ga. We produced 24 Na by thermal neutron capture on Na2CO3 at the Texas A&M reactor, the source material being deposited on, and covered by 0.08-mm plastic foil. For 66Ga, we used a 66Zn beam from the Texas A&M K500 Cyclotron to initiate the reaction 1H (66Zn, n) 66Ga, the recoiling 66 Ga nuclei being separated from other reaction products in the MARS recoil spectrometer, and then implanted into 0.08-mm plastic foil. The 56Co source was purchased from Isotopes Products Laboratory in the form of an evaporated metallic salt under 2.77 mm of plastic. The efficiency measurements were made, and the spectra were analyzed, in the same manner as described in Helmer et al. (2003). Since the g rays from these
0969-8043/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2003.11.012
ARTICLE IN PRESS R.G. Helmer et al. / Applied Radiation and Isotopes 60 (2004) 173–177
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Table 1 Decay data used in our relative efficiency determinations. The 24Na data are from IAEA Co-ordinated Research Program (1991); those for 56Co are from the evaluation in Baglin et al. (2002); and for 66Ga are from the measurements and recommendations in Baglin et al. (2002). The g-ray energies are nominal values and the intensities (emission probabilities) are per 100 decays for 24Na, and are relative values for 56Co and 66Ga Nuclide
Eg (keV)
Ig
24
1368 2754 847 977 1038 1175 1238 1360 1771 1811 1964 2015 2035
99.9936(15) 99.846(6)a 100.0(2) 1.425(9) 14.09(5) 2.255(14) 66.3(3) 4.274(15) 15.52(5) 0.643(4) 0.715(6) 3.041(16) 7.79(4)
Na
56
Co
Nuclide
66
Ga
Eg (keV)
Ig
2598 3202 3253 3273 834 1039 1333 2190 2752 3229 3381 4086 4806
17.02(6) 3.22(3) 7.90(6) 1.864(15) 15.93(5) 100.0(3) 3.175(12) 14.42(5) 61.35(23) 4.082(19) 3.960(19) 3.445(18) 5.03(3)
a We have measured Ig ¼ 0:000645 (14) for the 3867-keV g ray from 24Na. We then used this result to revise slightly the Ig value quoted for the 2754-keV g ray in IAEA Co-ordinated Research Program (1991).
nuclides are generally in cascade and the source–detector distance was 15 cm, corrections for coincidence summing were made: for most g rays these corrections were less than 1%; they were never larger than B2%. The relative FE efficiencies deduced from these measurements depend on the relative g-ray emission probabilities used for these nuclides. On the one hand, the value for the two strong 24Na g rays is well known (IAEA Co-ordinated Research Program, 1991). On the other, those for the more complex decays of 56Co and 66 Ga have not been well established in the past because they depended on Ge detector efficiencies, which were not well determined themselves. However, there have been recent measurements and evaluations of these emission probabilities that have improved the situation. The major improvement has come with the use of g rays from neutron-capture reactions to aid in the detector efficiency calibration above 2.75 MeV, the highest energy that can be covered with long-lived radioactive sources. The values we have used for all three sources are given in Table 1 and are from IAEA Co-ordinated Research Program (1991) and Baglin et al., (2002). Only those g rays with emission probabilities known to 1% or better were considered and, among those, we only retained those that were cleanly observed—uncontaminated by escape peaks, for example—with measured areas in our spectra determined with uncertainties of 1% or better. For many of the g rays, our measured spectra also yielded single-escape (SE) and double-escape (DE) peak relative efficiencies. We have found it most useful to express these results as ratios of escape peak areas to that of the corresponding FE peak. These ratios are
independent of the g-ray emission probabilities and of any coincidence summing corrections.
3. Calculated efficiencies In Helmer et al. (2003), we obtained a very accurate efficiency curve between 50 and 1400 keV based on detailed source measurements and CYLTRAN (Halbleib and Mehlhorn, 1986) Monte Carlo calculations. The calculations used the measured physical properties of our detector, with adjustments being made only in the nominal dead-layer thicknesses, which could not be independently measured, and some fine-tuning of the detector radius, well within its measurement tolerance. With these dimensions unchanged, we used the same code to compute the detector efficiencies at the 26 energies (834–4806 keV) tabulated in Table 1 for the three nuclides measured here. These calculations were done in a manner to obtain the desired statistical uncertainty on the SE and DE peaks as well as the FE peaks.
4. Comparison of FE peak efficiencies The measured and calculated FE peak efficiencies are compared in Fig. 1. In this comparison, there are only three free parameters—a single scaling factor for each nuclide. We determined these scaling factors by minimizing the weighted differences between the measured values and the Monte Carlo efficiency values: that is, by means of a least-squares fit. Each of these nuclides has one or more g rays below 1400 keV where the FE
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175
5. Comparison of escape-peak ratios
Fig. 1. Differences between the measured efficiencies (points with uncertainties) and the corresponding Monte Carlo calculated values (line at 0). The differences are calculated as experiment minus calculation, divided by calculation, and are expressed as a percent. The points can be identified with individual sources as follows: 24Na, solid squares; 56Co, open diamonds; 60Co, x’s; 66Ga, open circles; and 88Y, open squares. Note that the point for the 4.8-MeV gamma ray from 66Ga is at 3.8%; only the bottom of its error bar is visible at the upper right of the figure.
efficiency curve has already been determined precisely (Helmer et al., 2003). Therefore, the efficiencies at the higher energies presented here are completely consistent with those at the lower energies reported in Helmer et al. (2003). In general, the largest component in the uncertainties given for these measured efficiencies is from the known emission probabilities (see Table 1). Note also that Fig. 1 includes the results of two measurements from our previous work (Hardy et al., 2002; Helmer et al., 2003), those of 60Co and 88Y. The former provides an absolute efficiency determination that anchors the overall efficiency curve, while the latter yields a useful efficiency ratio that overlaps the energy region of interest in the present work. As indicated in Fig. 1, the agreement between measured and calculated efficiencies below 3500 keV is excellent. Over this energy region, the normalized chisquared for the 24 points from 24Na, 56Co and 66Ga (with 21 degrees of freedom) is 0.70. We suggest that the uncertainty in the deduced efficiencies is 0.4% between 1400 and 3500 keV. Above 3500 keV there are two 66Ga g rays with emission probabilities claimed to better than 1%, both of which deviate significantly from the Monte Carlo calculations. Whether these represent real discrepancies between experimental and theoretical efficiencies, or a systematic error in the accepted emission probabilities can only be decided by further experiments. For now, we forego any estimation of the uncertainty in our calibration curve above 3500 keV.
In contrast to the excellent agreement in Fig. 1, when the measured SE/FE and DE/FE ratios were first compared with the Monte Carlo calculations, they were found to be consistently lower than the calculated ratios by up to 7%. We then learned from an author of the CYLTRAN code (Kensek, 2003) that the program follows each pair-produced positron as it loses energy by scattering, until it reaches thermal energy and annihilates, producing two 511-keV photons, which are then followed until they escape or are absorbed. In that process, no account is taken of positron annihilation-inflight (AIF). Clearly, since the positrons that annihilate in flight produce photons with energies that are significantly different from 511 keV, the SE and DE peak areas calculated by the code need to be reduced by the known probabilities for AIF. Calculated values for these probabilities are given in Browne and Firestone (1986) where they can be seen to depend primarily on the positron energy and secondarily on the material in which the annihilation occurs. Thus, for each particular g ray of energy Eg ; we reduced the SE and DE efficiency values from the Monte Carlo calculation by the AIF probability determined for a positron with energy 12 (Eg 1022) keV. Sample values for the AIF probabilities (Browne and Firestone, 1986) determined in this way are 2.3%, 3.9%, and 5.5% for g-ray energies of 2, 3, and 4 MeV, respectively. As shown in Fig. 2, when the calculated escape ratios are reduced to account for AIF, the agreement with the measured values is excellent. Note that in this comparison, the uncertainties that contribute are only those obtained from the peak area determinations and from the statistics of the Monte Carlo calculations.
Fig. 2. Measured escape ratios (points with uncertainties) compared with CYLTRAN Monte Carlo calculations (curves) that have been corrected for positron annihilation in flight.
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Table 2 Portion of a Monte Carlo calculated spectrum for g rays for 1173 keV showing the regions of the FE peak and the two peaks from Ge K X-ray escape. The energy bins have a width of 0.2 keV and the lower energy of the bin is given. The values for the spectrum are the counts per 100 g rays emitted into a 30 cone; a total of 108 g rays were emitted Energy(keV)
Counts
Energy(keV)
Counts
1173.23 1173.03 1172.83 1172.63
3.2294 (18) 0.000097 (10) 0.000095 (10) 0.000108 (10)
1163.8 1163.6 1163.4(Ka ) 1163.2 1163.0
0.000169 0.000161 0.000926 0.000138 0.000149
This comparison and the one in Fig. 1 do not take into account any possible change in the FE peak areas that could occur when the 511-keV annihilation photons are replaced by those from AIF. It is expected, though, that such a change would be very small and likely negligible in the context of the uncertainties that appear in both figures. The exact magnitude of the effect on the FE peak will need to await a change in the Monte Carlo code.
(13) (13) (30) (12) (12)
Energy (keV)
Counts
1162.7 1162.5 1162.3 (Kb ) 1162.1 1161.9
0.000174 0.000174 0.000359 0.000160 0.000171
(13) (13) (19) (13) (13)
Table 3 Ratio of the area of the sum of the two Ge K X-ray peaks to that of the FE peak as calculated with the CYLTRAN Monte Carlo code Energy(keV)
Ratio 104
Energy(keV)
Ratio 104
100.0 300.0 500.0 700.0 900.0
18.5 (1) 3.01 (9) 2.97 (9) 3.01 (11) 2.87 (10)
1173.2 1368.6 1836.1 2754.0 3866.5
2.99 3.25 3.24 3.48 3.78
(8) (10) (15) (17) (21)
6. Ge K X-ray escape from outer detector surface In the course of our careful analysis of the measured peaks from the 24Na sources, it was determined that a small peak occurs systematically about 10 keV below the FE peaks at both 1368 and 2754 keV. In order to confirm that this small peak was, in fact, due to escape of Ge K Xrays from the detector, the corresponding Monte Carlo calculations were run with a binning structure that showed the Ka and Kb X-ray-escape components. In Table 2, the tally in some of the Monte Carlo bins, each 0.2 keV wide, around the peak regions are given for a gray energy of 1173 keV. This clearly shows the two Ge K X-ray escape peaks. The calculated relative intensities of the sum of these Ge K X-ray escape peaks, (Ka +Kb )/FE, are given in Table 3 for g rays covering a range of energies from 100 to 3867 keV. Evidently the ratio is almost independent of g-ray energy above 300 keV and at those energies is presumably due to the loss of X-rays from all surfaces of the crystal, the cylindrical outer surface, the inner core surface and the back surface. The areas of the Ge X-ray escape peaks are difficult to determine accurately from experiment because they are not well resolved from the much larger FE peak, and are only marginally distinguishable from its low-energy tail, a tail whose shape cannot be independently determined. Nevertheless, it is clear that the observed Ge K X-ray escape ratios at 1368 and 2754 keV are larger than those calculated by at least a factor of 5. Although we cannot explain this difference, we would note that in Helmer et al., (2003) a smaller discrepancy, but in the same sense, was observed for low-energy g rays, where the Ge
X-rays escaped out of the front face of the detector. The difference in both cases may result from physical irregularities at the crystal surfaces or local variations in the charge collection not accounted for by the Monte Carlo calculations, which provide a spectrum of the energy deposited in the detector volume, but not the spectrum of charge actually collected. There may, of course, be some other limitation in the code evident only in this rather subtle effect. 7. Conclusion We have shown that it is possible to produce Monte Carlo calculated efficiencies that are as accurate as the currently possible measurements up to approximately 3.5 MeV and that a combination of Monte Carlo calculations with accurately measured relative efficiencies can generate an efficiency curve that is accurate to about 0.2% from 50 to 1400 keV and to about 0.4% from there to 3500 keV. Acknowledgements We wish to thank Dr. Ronald P. Kensek, Sandia National Laboratory for his consultation on matters related to the CYLTRAN Monte Carlo code. The work at Texas A&M was supported by the US Department of Energy under Grant No. DE-FG03-93ER40773 and by the Robert A. Welch Foundation.
ARTICLE IN PRESS R.G. Helmer et al. / Applied Radiation and Isotopes 60 (2004) 173–177
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