- Email: [email protected]
Cascade summing in gamma-ray spectrometry in marinelli-beaker geometries: the third efficiency curve
Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
Cascade summing in gamma-ray spectrometry in marinelli-beaker geometries: the third efficiency curve Menno Blaauw*, Sjoerd J. Gelsema Interfaculty Reactor Institute, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands
Abstract Radionuclides emitting multiple gamma-rays in cascade give rise to summing effects that may be a source of error in the efficiency-curve based interpretation of gamma-ray spectra obtained in highly efficient counting geometries. Correction methods for sources that are small enough for the detector efficiency to be constant over the source volume are well-known. However, in geometries where the detector efficiency is not constant throughout the sample volume, such as Marinelli-beaker geometries, appreciable underestimation of the source activity may still occur if the variation of the efficiencies over the source volume is not accounted for. By introducing a third efficiency curve that accounts for the variation of the detector efficiency over the source volume, we have developed a practical, easy-to-use method that allows for determination of all three efficiency curves from a single, high-resolution gamma-ray spectrum, as well as for accurate correction for cascade summing effects. r 2003 Elsevier Science B.V. All rights reserved. PACS: 07.85.Nc; 29.30.Kv Keywords: Gamma-ray spectrometry; Cascade summing; Coincidence summing; Marinelli beaker; Self-attenuation
1. Introduction It is a well-known fact that photons emitted simultaneously by a single decaying nucleus may be detected as a single photon by a gamma-ray detector. Such cascade summing may result in significant underestimation of the activity of the source if not corrected for. Typically, the fullenergy photopeak efficiency curve is used in conjunction with the total efficiency curve and the decay scheme of the nuclide to perform such corrections [1–4]. However, in geometries where *Corresponding author. Tel.: +31-15-2783528; fax: +31-152783906. E-mail address: [email protected] (M. Blaauw).
the detector efficiency is not constant throughout the sample volume, such as Marinelli-beaker geometries, appreciable under estimation of the source activity may still occur if the variation of the efficiencies over the source volume is not accounted for [5]. Only in 1994, a team of IAEA experts in a meeting on nuclear spectrometry identified ‘‘y important topics for which there are not yet practical, easy-to-use solutions. They relate to the quantification of activities in samples and involve true-coincidence summing and selfattenuation effects’’[6]. For the Marinelli-beaker geometry, solutions are available, three of them as recently as 2001. The oldest solution is laborious but accurate radionuclide-specific calibration, where the
0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-9002(03)01075-1
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
312
detector is calibrated for the radionuclide(s) of interest in the geometry to be used, with a calibration source that matches the unknown source in density and composition and contains the specific radionuclides of interest. Newer methods are efficiency-curve based. Kolotov et al.’s method [7] is based on the mapping of the efficiencies in the space around the detector. Assuming that the introduction of a sample in this space does not affect the peak-tototal efficiency ratio, the detector response for any radionuclide can be obtained by numerical integration of the responses for volume elements that are small enough for the efficiencies to be considered constant within them. This software is available from Canberra. We do not expect it to be applicable to large detectors. Arnold and Sima’s method [8,9] introduces an ‘‘effective total efficiency curve’’ that accounts for self-attenuation as well as for radiation scattered in the sample and reaching the detector afterwards. The corresponding ‘‘GESPECOR’’ software is available from MATEC. The method presented in this paper is based on the introduction of a third efficiency curve that accounts for the variation of the detector efficiencies over the source volume, due to self-attenuation and scattering in the sample, as well as to distance from the detector. The software is incorporated in ORTEC’s GammaVision.
The following is an abbreviated, simplified version of the derivations offered in Ref. [10]. The probability P of detecting a photon in a peak at the energy of interest, under the condition that a specific cascade occurs (so that all transition probabilities are unity) where N photons are emitted and no angular correlations or internal conversion occur, is given by M Y i¼1
eri
/e1 e2 S ¼ /e1 S/e2 S þ rs1 s2
ð2Þ
s2i
is the variance of ei over the source where volume and r is the correlation coefficient. We define the squared-to-linear average ratio l l 2 ¼ /e2 S=/eS2
ð3Þ
and substitute in Eq. (2), obtaining qffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffi /e1 e2 S ¼ /e1 S/e2 Sð1 þ r l12 1 l22 1Þ:
ð4Þ
We now approximate by assuming high correlation (r ¼ 1) between efficiencies as well as similar relative standard deviations, and get
2. Theory
P¼
the index number of the ith photon not contributing, e is the photopeak efficiency, i.e. the probability of the photon depositing all its energy in the active detector volume and et is the total efficiency, i.e. the probability of the photon depositing any amount of energy in the active detector volume. To calculate the total probability of obtaining a count in the peak, Eq. (1) must be summed over all cascades where photons are emitted of which the energies add up to the energy of interest. Eq. (1) and all that follows can easily be generalized to include internal conversion effects. When expanded, Eq. (1) yields products of 1 up to N peak and total efficiencies. If any of these cannot be considered to be constant, P must be averaged over the source volume. We now focus on the volume average of the product of two efficiencies. For any efficiency distribution, we can write
NM Y
ð1 etsj Þ
ð1Þ
j¼1
where M is the number of photons contributing to the peak (MA1; 2; yN), ri is the index number of the ith photon contributing to the photopeak, si is
/e1 e2 S ¼ l1 /e1 Sl2 /e2 S:
ð5Þ
We further approximate by assuming that /e1 e2 yeN S ¼ l1 /e1 Sl2 /e2 SylN /eN S:
ð6Þ
This means that, when we calculate Eq. (1) for a voluminous source, we replace both the full-energy and the total efficiencies e by the products l/oeS in all products of more than one efficiency. Otherwise, we use /oeS unmodified. Calculations can thus be performed with the same speed as for non-voluminous sources, which is why we simplified the derivation relative to [10]. The products l/oet S as a function of energy can be identified as a close relative of Arnold and Sima’s effective total efficiency curve.
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
The normal full-energy peak efficiency curve eðEÞ can be parametrized with four parameters by using Gunnink’s polynomial [11]. The total efficiencies can be related to the peak efficiencies by a linear curve on a log–log scale requiring two more parameters. By Monte Carlo investigations, we established that the squared-to-linear curve lðEÞ can be described with lðEÞ ¼ a0 þ a1 lnðEÞa2
ð7Þ
involving three more parameters. In Eq. (7), a0 depends on the sample-detector geometry exclusively, a1 and a2 on sample density and composition. The three curves together can thus be characterized with only nine parameters. The approximations made will obviously affect the accuracy of the results. However, in our method the parameters defining the efficiency curves are not determined by independent methods, but by fitting all parameters to a single, measured spectrum using the computational method outlined above (this is a logical sequel to the method for well-type detectors presented in Ref. [4]. We, therefore, expect the effects of the approximations to cancel out in part between the calibration and the measurement step.
3. Experimental One calibration source, containing 57Co, 60Co, Y, 109Cd, 113Sn, 137Cs and 139Ce was prepared in a 1 l Marinelli beaker by gravimetrically pipetting an aliquot of a certified liquid calibration Amersham source in 0.5 M hydrochloric acid. A second calibration source was prepared by irradiating a solution of 0.5 g NaBr in 10 ml of 0.5 M hydrochloric acid during 6 min at a thermal neutron flux of 4 1016 m2 s1. Three aliquots were pipetted gravimetrically in small, polyethylene capsules and measured at 25 cm distance from a 18% relative efficiency, calibrated HPGe detector, to determine the specific 82Br activity of the stock solution. From the stock solution, 15 mg was added to 1 l 0.5 M hydrochloric acid in a Marinelli beaker. A mixed-radionuclide test source was prepared independently at the KVI in Groningen, The
88
313
Netherlands, in the frame of a national validation of measurement protocols [5]. The 1 l source was prepared from certified reference sources supplied by PTB in Braunschweig, Germany. The density of this source (1.0125 kg/l) was slightly higher than of the calibration sources (1.000 kg/l) due to high concentrations of KCl. All three sources were measured on the same Ge(Li) detector mentioned earlier. Counting times were chosen to achieve better than 1% precision in all relevant peaks, including the sum peaks of 82Br at 1173 and 1331 keV. The peak areas were determined with in-house software validated with the 1995 IAEA test spectra [12,13]. The nine parameters defining the three efficiency curves were fitted with non-linear least squares methods to the combined lists of peak areas and energies obtained from the calibration sources (the software used was the actual module now incorporated in ORTEC’s GammaVision). To compensate for imprecision in the decay scheme data used, all peak area precisions of better than 1% are set to 1% in the fit. Peaks with area precisions worse than 10% as well as the escape peak of 24Na at 1732 keV were disregarded in the fit. The set of curves obtained was used to interpret the spectrum of the test source. In the interpretation, weighted averages of the activities corresponding to the different peaks of the radionuclides were determined where possible. Only uncertainties due to counting statistics of the peaks in the spectrum of the test sample were propagated. Radionuclide-specific calibration was applied to radionuclides present in both the calibration and the test source and alternative interpretation results obtained.
4. Results The w2r of the fit of the nine parameters to the measured peak areas was 2.2 at 40–9=31 degrees of freedom (w2r was 25 at 40–6 d.f. if the lðEÞ curve was kept at unity in the fit). The ratios of measured and reproduced peak areas are shown in Fig. 1, as well as the associated standardized residuals (a.k.a. as z-scores). Since w2r deviates from unity
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
reproduced over measured ratio
314
Table 1 Interpretation results for the test sample, obtained with the new method, and results obtained with radionuclide-specific calibration in selected cases
1.15 1.10 1.05 1.00 0.95
Amersham source
24 Na ,82 Br source 82
0.90
Br sum peaks
Radionuclide
Known activity
3-curve method
40
18672 137.370.9 116.770.6 126.470.9 129.870.7 144.870.9
18174 13970.7 118.270.5 129.670.5 129.570.5 144.070.7
0.85
z-scores
0
500
1000
1500 2000 Photon energy (keV)
2500
3000
K Co 60 Co 133 Ba 134 Cs 137 Cs 57
4 3 2 1 0 -1 -2 -3 -4 -5 0
500
1000
1500 2000 Photon energy (keV)
2500
3000
peak-to-total ratio
full-energy peak efficiency
Fig. 1. The results of the fit of the nine parameters defining the three efficiency curves to the two measured and combined calibration spectra.
136.171.2 116.770.8
141.071.2
All uncertainties are 1 SD.
significantly, some sources of variation as yet unaccounted for are present. Especially the 1650 keV datapoint shows a discrepancy. This might be due to our neglecting angular correlations or to imperfections in the decay scheme. The efficiency curves themselves are shown in Fig. 2. The interpretation result of the test source is shown in Table 1.
0.010
5. Discussion and conclusions
0.001 0.40 0.30 0.20
0.10 0.09 0.08 1.24
squared-to-linear ratio
Radionuclidespecific
1.22 1.20 1.18 1.16 100
1000 Photon energy (keV)
Fig. 2. The three efficiency curves resulting from the fit to the measured, combined calibration spectra.
Our third efficiency curve is closely related to Arnold and Sima’s, however; the effective curve they propose appears to exclusively address the volume effects on coincidence losses (that are determined by the total efficiencies), whereas our third curve covers the volume effects both on summing in and on summing out. The three efficiency curves as defined in this paper can account for the detector response (sum peaks as well as ‘‘natural’’ peaks) for a voluminous source containing radionuclides emitting one (57Co, 109Cd, 113Sn, 137Cs and 139Ce), two (24Na, 60 Co, 88Y) or multiple photons (82Br, 134Cs) simultaneously. The accuracy of the results obtained with the new method is equal to that of results obtained with radionuclide specific calibration. Since we did not assume a specific sample shape or composition in our derivations, we expect the method to also be accurate for other sampledetector geometries (pillboxes, bottles, etc.), at all sample densities lower than 1 g/ml, as well as at
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
larger sample dimensions and higher sample densities, but have not yet established the ranges of applicability. Since the calibration step in this method requires the measurement of only one mixed calibration source with a suitable composition (e.g. 57Co, 88Y, 109 Cd, 113Sn, 134Cs, 137Cs and 139Ce, 60Co may give deconvolution problems with the 134Cs sum peak at 1174 keV), we feel we have achieved ‘‘a practical, easy-to-use solution’’ for the problem of cascade summing corrections in voluminous sources such as Marinelli beakers. As yet, however, calibration for each sample geometry is still required.
References . [1] K. Debertin, U. Schotzig, Nucl. Instr. and Meth. 158 (1979) 471.
315
[2] L. Moens, J. de Donder, F. Lin Xilei, A. De Corte, A. de Wispelaere, J. Simonits, Hoste, Nucl. Instr. and Meth. 187 (1981) 451. [3] T.M. Semkow, G. Mehmood, P.P. Parekh, M. Virgil, Nucl. Instr. and Meth. A 290 (1990) 437. [4] M. Blaauw, Nucl. Instr. and Meth. A 419 (1998) 146. [5] E.R.v.d. Graaf, R.J. de Meijer, Appl. Radiat. Isot. 57 (2002) 73. [6] IAEA Technical Document 1024, IAEA, Vienna, Austria, 1994. [7] V.P. Kolotov, M.J. Koskelo, J. Radioanal. Nucl. Chem. 233 (1998) 95. [8] O. Sima, D. Arnold, Appl. Radiat. Isot. 53 (2000) 51. [9] D. Arnold, O. Sima, J. Radioanal. Nucl. Chem. 248 (2001) 365. [10] S.J. Gelsema, Ph.D. Thesis, University of Technology, Delft, 2001. [11] R. Gunnink, A.L. Prindle, J. Radioanal. Nucl. Chem. Art. 160 (1992) 330. [12] M. Blaauw, V. Osorio Fernandez, W. Westmeier, Nucl. Instr. and Meth. A 387 (1997) 410. [13] M. Blaauw, Nucl. Instr. and Meth. A 432 (1999) 74.
Cascade summing in gamma-ray spectrometry in marinelli-beaker geometries: the third efficiency curve Menno Blaauw*, Sjoerd J. Gelsema Interfaculty Reactor Institute, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands
Abstract Radionuclides emitting multiple gamma-rays in cascade give rise to summing effects that may be a source of error in the efficiency-curve based interpretation of gamma-ray spectra obtained in highly efficient counting geometries. Correction methods for sources that are small enough for the detector efficiency to be constant over the source volume are well-known. However, in geometries where the detector efficiency is not constant throughout the sample volume, such as Marinelli-beaker geometries, appreciable underestimation of the source activity may still occur if the variation of the efficiencies over the source volume is not accounted for. By introducing a third efficiency curve that accounts for the variation of the detector efficiency over the source volume, we have developed a practical, easy-to-use method that allows for determination of all three efficiency curves from a single, high-resolution gamma-ray spectrum, as well as for accurate correction for cascade summing effects. r 2003 Elsevier Science B.V. All rights reserved. PACS: 07.85.Nc; 29.30.Kv Keywords: Gamma-ray spectrometry; Cascade summing; Coincidence summing; Marinelli beaker; Self-attenuation
1. Introduction It is a well-known fact that photons emitted simultaneously by a single decaying nucleus may be detected as a single photon by a gamma-ray detector. Such cascade summing may result in significant underestimation of the activity of the source if not corrected for. Typically, the fullenergy photopeak efficiency curve is used in conjunction with the total efficiency curve and the decay scheme of the nuclide to perform such corrections [1–4]. However, in geometries where *Corresponding author. Tel.: +31-15-2783528; fax: +31-152783906. E-mail address: [email protected] (M. Blaauw).
the detector efficiency is not constant throughout the sample volume, such as Marinelli-beaker geometries, appreciable under estimation of the source activity may still occur if the variation of the efficiencies over the source volume is not accounted for [5]. Only in 1994, a team of IAEA experts in a meeting on nuclear spectrometry identified ‘‘y important topics for which there are not yet practical, easy-to-use solutions. They relate to the quantification of activities in samples and involve true-coincidence summing and selfattenuation effects’’[6]. For the Marinelli-beaker geometry, solutions are available, three of them as recently as 2001. The oldest solution is laborious but accurate radionuclide-specific calibration, where the
0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-9002(03)01075-1
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
312
detector is calibrated for the radionuclide(s) of interest in the geometry to be used, with a calibration source that matches the unknown source in density and composition and contains the specific radionuclides of interest. Newer methods are efficiency-curve based. Kolotov et al.’s method [7] is based on the mapping of the efficiencies in the space around the detector. Assuming that the introduction of a sample in this space does not affect the peak-tototal efficiency ratio, the detector response for any radionuclide can be obtained by numerical integration of the responses for volume elements that are small enough for the efficiencies to be considered constant within them. This software is available from Canberra. We do not expect it to be applicable to large detectors. Arnold and Sima’s method [8,9] introduces an ‘‘effective total efficiency curve’’ that accounts for self-attenuation as well as for radiation scattered in the sample and reaching the detector afterwards. The corresponding ‘‘GESPECOR’’ software is available from MATEC. The method presented in this paper is based on the introduction of a third efficiency curve that accounts for the variation of the detector efficiencies over the source volume, due to self-attenuation and scattering in the sample, as well as to distance from the detector. The software is incorporated in ORTEC’s GammaVision.
The following is an abbreviated, simplified version of the derivations offered in Ref. [10]. The probability P of detecting a photon in a peak at the energy of interest, under the condition that a specific cascade occurs (so that all transition probabilities are unity) where N photons are emitted and no angular correlations or internal conversion occur, is given by M Y i¼1
eri
/e1 e2 S ¼ /e1 S/e2 S þ rs1 s2
ð2Þ
s2i
is the variance of ei over the source where volume and r is the correlation coefficient. We define the squared-to-linear average ratio l l 2 ¼ /e2 S=/eS2
ð3Þ
and substitute in Eq. (2), obtaining qffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffi /e1 e2 S ¼ /e1 S/e2 Sð1 þ r l12 1 l22 1Þ:
ð4Þ
We now approximate by assuming high correlation (r ¼ 1) between efficiencies as well as similar relative standard deviations, and get
2. Theory
P¼
the index number of the ith photon not contributing, e is the photopeak efficiency, i.e. the probability of the photon depositing all its energy in the active detector volume and et is the total efficiency, i.e. the probability of the photon depositing any amount of energy in the active detector volume. To calculate the total probability of obtaining a count in the peak, Eq. (1) must be summed over all cascades where photons are emitted of which the energies add up to the energy of interest. Eq. (1) and all that follows can easily be generalized to include internal conversion effects. When expanded, Eq. (1) yields products of 1 up to N peak and total efficiencies. If any of these cannot be considered to be constant, P must be averaged over the source volume. We now focus on the volume average of the product of two efficiencies. For any efficiency distribution, we can write
NM Y
ð1 etsj Þ
ð1Þ
j¼1
where M is the number of photons contributing to the peak (MA1; 2; yN), ri is the index number of the ith photon contributing to the photopeak, si is
/e1 e2 S ¼ l1 /e1 Sl2 /e2 S:
ð5Þ
We further approximate by assuming that /e1 e2 yeN S ¼ l1 /e1 Sl2 /e2 SylN /eN S:
ð6Þ
This means that, when we calculate Eq. (1) for a voluminous source, we replace both the full-energy and the total efficiencies e by the products l/oeS in all products of more than one efficiency. Otherwise, we use /oeS unmodified. Calculations can thus be performed with the same speed as for non-voluminous sources, which is why we simplified the derivation relative to [10]. The products l/oet S as a function of energy can be identified as a close relative of Arnold and Sima’s effective total efficiency curve.
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
The normal full-energy peak efficiency curve eðEÞ can be parametrized with four parameters by using Gunnink’s polynomial [11]. The total efficiencies can be related to the peak efficiencies by a linear curve on a log–log scale requiring two more parameters. By Monte Carlo investigations, we established that the squared-to-linear curve lðEÞ can be described with lðEÞ ¼ a0 þ a1 lnðEÞa2
ð7Þ
involving three more parameters. In Eq. (7), a0 depends on the sample-detector geometry exclusively, a1 and a2 on sample density and composition. The three curves together can thus be characterized with only nine parameters. The approximations made will obviously affect the accuracy of the results. However, in our method the parameters defining the efficiency curves are not determined by independent methods, but by fitting all parameters to a single, measured spectrum using the computational method outlined above (this is a logical sequel to the method for well-type detectors presented in Ref. [4]. We, therefore, expect the effects of the approximations to cancel out in part between the calibration and the measurement step.
3. Experimental One calibration source, containing 57Co, 60Co, Y, 109Cd, 113Sn, 137Cs and 139Ce was prepared in a 1 l Marinelli beaker by gravimetrically pipetting an aliquot of a certified liquid calibration Amersham source in 0.5 M hydrochloric acid. A second calibration source was prepared by irradiating a solution of 0.5 g NaBr in 10 ml of 0.5 M hydrochloric acid during 6 min at a thermal neutron flux of 4 1016 m2 s1. Three aliquots were pipetted gravimetrically in small, polyethylene capsules and measured at 25 cm distance from a 18% relative efficiency, calibrated HPGe detector, to determine the specific 82Br activity of the stock solution. From the stock solution, 15 mg was added to 1 l 0.5 M hydrochloric acid in a Marinelli beaker. A mixed-radionuclide test source was prepared independently at the KVI in Groningen, The
88
313
Netherlands, in the frame of a national validation of measurement protocols [5]. The 1 l source was prepared from certified reference sources supplied by PTB in Braunschweig, Germany. The density of this source (1.0125 kg/l) was slightly higher than of the calibration sources (1.000 kg/l) due to high concentrations of KCl. All three sources were measured on the same Ge(Li) detector mentioned earlier. Counting times were chosen to achieve better than 1% precision in all relevant peaks, including the sum peaks of 82Br at 1173 and 1331 keV. The peak areas were determined with in-house software validated with the 1995 IAEA test spectra [12,13]. The nine parameters defining the three efficiency curves were fitted with non-linear least squares methods to the combined lists of peak areas and energies obtained from the calibration sources (the software used was the actual module now incorporated in ORTEC’s GammaVision). To compensate for imprecision in the decay scheme data used, all peak area precisions of better than 1% are set to 1% in the fit. Peaks with area precisions worse than 10% as well as the escape peak of 24Na at 1732 keV were disregarded in the fit. The set of curves obtained was used to interpret the spectrum of the test source. In the interpretation, weighted averages of the activities corresponding to the different peaks of the radionuclides were determined where possible. Only uncertainties due to counting statistics of the peaks in the spectrum of the test sample were propagated. Radionuclide-specific calibration was applied to radionuclides present in both the calibration and the test source and alternative interpretation results obtained.
4. Results The w2r of the fit of the nine parameters to the measured peak areas was 2.2 at 40–9=31 degrees of freedom (w2r was 25 at 40–6 d.f. if the lðEÞ curve was kept at unity in the fit). The ratios of measured and reproduced peak areas are shown in Fig. 1, as well as the associated standardized residuals (a.k.a. as z-scores). Since w2r deviates from unity
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
reproduced over measured ratio
314
Table 1 Interpretation results for the test sample, obtained with the new method, and results obtained with radionuclide-specific calibration in selected cases
1.15 1.10 1.05 1.00 0.95
Amersham source
24 Na ,82 Br source 82
0.90
Br sum peaks
Radionuclide
Known activity
3-curve method
40
18672 137.370.9 116.770.6 126.470.9 129.870.7 144.870.9
18174 13970.7 118.270.5 129.670.5 129.570.5 144.070.7
0.85
z-scores
0
500
1000
1500 2000 Photon energy (keV)
2500
3000
K Co 60 Co 133 Ba 134 Cs 137 Cs 57
4 3 2 1 0 -1 -2 -3 -4 -5 0
500
1000
1500 2000 Photon energy (keV)
2500
3000
peak-to-total ratio
full-energy peak efficiency
Fig. 1. The results of the fit of the nine parameters defining the three efficiency curves to the two measured and combined calibration spectra.
136.171.2 116.770.8
141.071.2
All uncertainties are 1 SD.
significantly, some sources of variation as yet unaccounted for are present. Especially the 1650 keV datapoint shows a discrepancy. This might be due to our neglecting angular correlations or to imperfections in the decay scheme. The efficiency curves themselves are shown in Fig. 2. The interpretation result of the test source is shown in Table 1.
0.010
5. Discussion and conclusions
0.001 0.40 0.30 0.20
0.10 0.09 0.08 1.24
squared-to-linear ratio
Radionuclidespecific
1.22 1.20 1.18 1.16 100
1000 Photon energy (keV)
Fig. 2. The three efficiency curves resulting from the fit to the measured, combined calibration spectra.
Our third efficiency curve is closely related to Arnold and Sima’s, however; the effective curve they propose appears to exclusively address the volume effects on coincidence losses (that are determined by the total efficiencies), whereas our third curve covers the volume effects both on summing in and on summing out. The three efficiency curves as defined in this paper can account for the detector response (sum peaks as well as ‘‘natural’’ peaks) for a voluminous source containing radionuclides emitting one (57Co, 109Cd, 113Sn, 137Cs and 139Ce), two (24Na, 60 Co, 88Y) or multiple photons (82Br, 134Cs) simultaneously. The accuracy of the results obtained with the new method is equal to that of results obtained with radionuclide specific calibration. Since we did not assume a specific sample shape or composition in our derivations, we expect the method to also be accurate for other sampledetector geometries (pillboxes, bottles, etc.), at all sample densities lower than 1 g/ml, as well as at
M. Blaauw, S.J. Gelsema / Nuclear Instruments and Methods in Physics Research A 505 (2003) 311–315
larger sample dimensions and higher sample densities, but have not yet established the ranges of applicability. Since the calibration step in this method requires the measurement of only one mixed calibration source with a suitable composition (e.g. 57Co, 88Y, 109 Cd, 113Sn, 134Cs, 137Cs and 139Ce, 60Co may give deconvolution problems with the 134Cs sum peak at 1174 keV), we feel we have achieved ‘‘a practical, easy-to-use solution’’ for the problem of cascade summing corrections in voluminous sources such as Marinelli beakers. As yet, however, calibration for each sample geometry is still required.
References . [1] K. Debertin, U. Schotzig, Nucl. Instr. and Meth. 158 (1979) 471.
315
[2] L. Moens, J. de Donder, F. Lin Xilei, A. De Corte, A. de Wispelaere, J. Simonits, Hoste, Nucl. Instr. and Meth. 187 (1981) 451. [3] T.M. Semkow, G. Mehmood, P.P. Parekh, M. Virgil, Nucl. Instr. and Meth. A 290 (1990) 437. [4] M. Blaauw, Nucl. Instr. and Meth. A 419 (1998) 146. [5] E.R.v.d. Graaf, R.J. de Meijer, Appl. Radiat. Isot. 57 (2002) 73. [6] IAEA Technical Document 1024, IAEA, Vienna, Austria, 1994. [7] V.P. Kolotov, M.J. Koskelo, J. Radioanal. Nucl. Chem. 233 (1998) 95. [8] O. Sima, D. Arnold, Appl. Radiat. Isot. 53 (2000) 51. [9] D. Arnold, O. Sima, J. Radioanal. Nucl. Chem. 248 (2001) 365. [10] S.J. Gelsema, Ph.D. Thesis, University of Technology, Delft, 2001. [11] R. Gunnink, A.L. Prindle, J. Radioanal. Nucl. Chem. Art. 160 (1992) 330. [12] M. Blaauw, V. Osorio Fernandez, W. Westmeier, Nucl. Instr. and Meth. A 387 (1997) 410. [13] M. Blaauw, Nucl. Instr. and Meth. A 432 (1999) 74.