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The jackknife as an approach for uncertainty assessment in gamma spectrometric measurements of uranium isotope ratios
Nuclear Instruments and Methods in Physics Research B 268 (2010) 2535–2538
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
The jackknife as an approach for uncertainty assessment in gamma spectrometric measurements of uranium isotope ratios H. Ramebäck a,*, A. Vesterlund a, A. Tovedal a, U. Nygren a, L. Wallberg b, E. Holm c, C. Ekberg d, G. Skarnemark d a
Swedish Defence Research Agency, FOI, Division of CBRN Defence and Security, SE-901 82 Umeå, Sweden Swedish Radiation Safety Authority, SE-117 16 Stockholm, Sweden Norwegian Radiation Protection Authority, Postboks 55, 1332 Østerås, Norge d Chalmers University of Technology, Department of Chemical and Biological Engineering, Nuclear Chemistry, SE-412 96 Göteborg, Sweden b c
a r t i c l e
i n f o
Article history: Received 16 March 2010 Received in revised form 7 May 2010 Available online 15 May 2010 Keywords: Uranium Isotope ratio Enrichment Gamma spectrometry Uncertainty
a b s t r a c t The jackknife as an approach for uncertainty estimation in gamma spectrometric uranium isotope ratio measurements was evaluated. Five different materials ranging from depleted uranium (DU) to high enriched uranium (HEU) were measured using gamma spectrometry. High resolution inductively coupled plasma mass spectrometry (ICP-SFMS) was used as a reference method for comparing the results obtained with the gamma spectrometric method. The relative combined uncertainty in the gamma spectrometric measurements of the 238U/235U isotope ratio using the jackknife was about 10–20% (k = 2), which proved to be fit-for-purpose in order to distinguish between different uranium categories. Moreover, the enrichment of 235U in HEU could be measured with an uncertainty of 1–2%. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Measurements of nuclear material are important in many areas, e.g. nuclear safeguards, nuclear forensics (illicit trafficking) and environmental studies. For events involving illicit trafficking of radioactive- or nuclear material an initial basic characterization of the material is important [1,2]. This basic characterization involves, for example, measurement of the isotopic composition and should be performed at an incident place. In the case of uranium such characterization should be able to determine the uranium category, i.e. depleted uranium (DU), natural uranium, low enriched uranium (LEU) or high enriched uranium (HEU). Furthermore, relevant uncertainty estimation is essential in order to compare the measurement results with e.g. references or other measurements. Measurement of the isotopic composition of uranium can be done using different methods, e.g. mass spectrometry [3], alpha spectrometry [4] and gamma spectrometry [5]. Due to the long half life of the uranium isotopes mass spectrometry is by far the most sensitive method and offers also the lowest combined uncertainty. Compared to gamma spectrometry alpha spectrometry is also more sensitive. However, alpha- and mass spectrometry is mainly used for measurements of e.g. swipes sampled from a package containing a radioactive material, as well as in a later stage where a * Corresponding author. Tel.: +46 90 106646; fax: +46 90 106803. E-mail address: [email protected] (H. Ramebäck). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.05.055
more careful examination revealing the origin of the material is to be done. Thus at the incident location the only practical option is often gamma spectroscopy. Measurement of uranium composition using gamma spectrometry can be done using different approaches. The so called enrichment meter is based on the measurement of the 186 keV gamma peak from 235U [6]. Another method is based on the analysis of peaks in the X-ray region around 100 keV [7]. A third approach is based on the measurement of gamma rays in the ‘‘high” energy region from 144 to 1001 keV [8,9]. The first method needs calibration with known materials, which is not needed for the latter two. However, no comprehensive approach for the uncertainty estimation in any of the above mentioned methods has been presented. For example, commercial software for evaluating uranium isotopic composition often relies on counting statistics solely for the uncertainty estimation [10] which often results in too low uncertainty statements since other uncertainty sources which often domintates in relation to counting statistics is not taken into account. The importance of not underestimating, as well as not overestimating, measurement uncertainties is that it may result in wrong conclusions and therefore wrong decisions [11]. Since the measurement of uranium isotopic composition using gamma spectrometry involves the fitting of different parameters in a measurement model, where the isotopic ratios are included, a statistic re-sampling method could be suitable for evaluation of the measurement uncertainties. Other sources that may contribute
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to the combined uncertainty is the counting statistics, the assumed half lives of the uranium isotopes and the uncertainties in the gamma-ray emission probabilities. In this work we explored the jackknife [12,13] as an approach for the evaluation of the uncertainty in the measurement of the uranium isotope ratios using gamma spectrometry in the ‘‘high” energy region. One advantage with the jackknife method is that an inherent sensitivity analysis regarding variance contribution from individual data is possible. Five different uranium materials were measured and ICP-SFMS was used as a reference method.
2.1. Theory The efficiency WEj at a gamma ray energy Ej for a gamma spectrometric measurement can be written as
ð1Þ
where Ci,Ej is the net count rate in a full energy peak at gamma ray energy Ej of a radioactive isotope i, Ii,Ej the photon emission probability at Ej for isotope i, ki the decay constant of the radioactive isotope i, and Ni the number of atoms of isotope i emitting a gamma ray at energy Ej. Furthermore, the isotope ratio of two isotopes i and k can be written as
Ri ¼
Ni Nk
ð2Þ
Here, the isotope k is referred to as the reference isotope, which all other isotopes are related to. (Observe that the isotope ratio of the reference isotope k is unity.) The efficiency WEj can now be written as
WEj ¼
C i;Ej : Ii;Ej ki Ri Nk
ð3Þ
Since Nk is constant for all efficiencies WEj for a specific measurement, the relative efficiency, Wrel,Ej, at gamma ray energy Ej can be written as
Wrel;Ej ¼
C i;Ej : Ii;Ej ki Ri
ð4Þ
For uranium isotope measurements and only considering 234U, U and 238U one will have two isotope ratios, e.g. the ratios of 234 U/235U and 238U/235U (since the isotope ratio of 235U relative 235 U is unity). For enriched uranium 232U is often present in different amounts, which means that daughters from 228Th also can be used for the relative efficiency. In such cases the ratio 228Th/235U also has to be introduced into the measurement model. Moreover, one needs also a function that describes the relative detector efficiency over the gamma ray energy interval of interest. One function with such a property is 235
2
Wrel ðEÞ ¼ ec1 þc2 =E
þc3 ðln EÞ2 þc4 ðln EÞ3 þc5 =E
r2 ¼
m m1 X ðRj R Þ2 m j¼1
ð6Þ
where R* is the result of an isotope ratio when all data points are used, Rj is the result when one data j is omitted in the parameter fit and m is the number of data. The variance can be evaluated by repeating this procedure for all data points j. 2.2. Uranium samples
2. Materials and methods
C WEj ¼ i;Ej Ii;Ej ki Ni
the isotope ratios, can be evaluated by means of the jackknife [12,13]. The variance of an isotope ratio Rj can be estimated using
ð5Þ
which is a development of the function used by Anilkumar et al. [14]. By combining Eqs. (4) and (5) the isotope ratios may be calculated. This can be done by fitting the isotope ratios and the parameters c1–c5 in Eq. (5) by means of finding the minimum in the sum of the squared differences, i.e. the error square sum, of Eqs. (4) and (5). The combined uncertainty of the isotope ratios will consist of the counting statistics in the full energy gamma ray peaks, the uncertainty of the photon emission probabilities, the uncertainty in the half lives of the isotopes, and the uncertainty in the parameter estimation. The uncertainty in the parameter estimation, e.g.
Five uranium samples were measured: two HEU materials, one natural uranium reference material in acid solution (IRMM-184); one DU penetrator, and one sample containing waste of LEU. 2.3. Gamma spectrometric measurements The gamma spectrometric measurements in this study were done using p-type coaxial high-purity germanium (HPGe) detectors (EG&G Ortec, Oak Ridge, TN, USA). The relative efficiency of the detectors were about 50%, and the resolution about 1.8 keV@1332 keV. Standard NIM electronics (Nuclear Instrument Module) were used for pulse processing and high voltage power supply (EG&G Ortec, Oak Ridge, TN, USA). GammaVision (EG&G Ortec, ver. 6.01, Oak Ridge, TN, USA) was used for spectrum acquisitions and peak area evaluations. The following full energy gamma ray peaks were used in the measurements: 235
U: the peaks at 143.8, 163.3, 185.7 and 205.3 keV. U: the peaks from 234mPa at 258.3, 742.8, 766.4 and 1001.0 keV 234 U: the peak at 120.9 keV. 238
In addition to these, peaks from daughters of 228Th were used for the HEU materials. These were 238.6 keV from 212Pb, 727.3, 1078.6 and 1620.5 keV from 212Bi, and 583.2 and 860.5 keV from 208 Tl. All photon emission probabilities and half lives where taken from ENSDF [15]. 2.4. Determination of isotope ratios using ICP-SFMS The materials (all except one HEU material) to be analyzed using ICP-SFMS were dissolved in HNO3 and further diluted in 2% HNO3. The measurements were performed on an Element2 ICP-SFMS instrument (Thermo–Finnigan, Bremen, Germany). The sample introduction equipment consisted of a conical nebulizer and a cyclonic spray chamber (GlassExpansion, Melbourne, Australia). Scanning between the isotopes of interest was performed by changing the acceleration voltage (‘‘E-scan”) and the detection mode was pulse counting. The software controlled dead time correction was inactivated and the raw data were extracted to Excel were corrections for dead time and mass bias were performed manually. Three different uranium reference materials were used in the measurements; IRMM-184 (natural composition); IRMM073/1 (near unity relationship between 233U, 235U and 238U) and IRMM073/7 (233U:235U:238U 0.02:1:1), all obtained from the Institute for Reference Materials and Measurements, Geel, Belgium. IRMM073/7 was used to determine the dead time of the system. IRMM073/1 was used to monitor mass bias and IRMM-184 was used for quality control during the measurements performed in this study. All uncertainties were evaluated in accordance with ISO/GUM [11] using the software GUM Workbench [16].
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In this work uncertainties are presented with a coverage factor k = 2, i.e. representing an approximate 95% confidence interval, if not otherwise stated. 3. Results and discussion Fig. 1 shows the result of the parameter fit for one of the HEU material. As observed in the figure, the jackknife procedure results in different fits, i.e. a set of the measurands RU234/U235, RU238/U235, RTh228/U235, from which the uncertainty can be evaluated according to Eq. (6). For this HEU material the isotope ratios of RU238/U235 and RU234/U235 were 0.082 ± 0.018 and 0.0065 ± 0.0043, respectively. This resulted in an enrichment of 235U of (91.9 ± 1.6%). This material was not available for measurement using ICP-SFMS. However, the measurement result indicates the magnitude of measurement uncertainty that is possible to reach using this method. For the other HEU material measured, the isotope ratios of RU238/U235 and RU234/U235 were 0.0583 ± 0.0090 and 0.0138 ± 0.0026, resulting in a 235U enrichment of (93.28 ± 0.88%). The result from the ICP-SFMS measurement of this material was (92.773 ± 0.020%), i.e. the gamma spectrometric result was in agreement with the more precise mass spectrometric result. Moreover, the isotopic abundance of 238U in this material was (5.44 ± 0.79%), as compared to (6.061 ± 0.017%) from the ICP-SFMS measurement, i.e. no significant difference between the results. Finally, the isotopic abundance of 234U was (1.28 ± 0.23%), as compared to the ICPSFMS result of (1.1654 ± 0.0062%), i.e. also an agreement between the methods. For different gamma spectrometric measurement methods, uncertainties of the same magnitude (1–2%) has been stated [17]. However, these measurement uncertainties were based on repeated measurements rather than on a single one. In this work the measurement uncertainties applying the jackknife were evaluated on a single measurement, which is an advantage when long counting times are needed, e.g. when only small sample sizes are available. The measured abundance of 234U in the two HEU materials were (0.60 ± 0.39%) and (1.28 ± 0.23%), respectively. The, somewhat, high uncertainty in the determination of the isotope abundance of 234U reflects the problem involved in measuring this isotope with gamma spectrometry in so called ‘‘thin” samples [5]. This is due to the rapid change in detector response as a function of energy in this region when measuring thin samples, see Fig. 1. This is also observed in the variance for the jackknife method, Eq. (6), since the dominant contributor to the variance comes from the
143.8 keV gamma peak of 235U, i.e. when this peak is omitted in the jackknife procedure. This peak represents about 80% of the total variance in the HEU materials. The problem is possible to overcome by the use of a thin-window detector, such as one with a Be-endcap, when 234U is to be measured [5]. This will improve the precision in the determination of the 234U abundance using gamma spectrometry since its energy characteristics is different in the low energy region, compared to a detector with e.g. an Al-endcap. Fig. 2 shows the result of the parameter fit from the measurement of the DU penetrator. In the measurement of uranium materials ranging from DU to LEU the peak at 258.3 keV from 234m Pa/238U could not be omitted in the jackknife method, since this will lead to erroneous measurement results. This is due to the properties of the empirical response function, Eq. (5), and the big gap between the energy regions of 235U and 234mPa/238U. However, for HEU containing 232U there often is an extra peak from 212 Pb present at 238.6 keV, which is close to 258.3 keV. In such cases it is possible to omit the 258.3 keV peak in the jackknife procedure as well as omitting the 238.6 keV peak. RU238/U235 from the measurement of the DU, natural uranium, and the LEU are shown in Fig. 3. As can be seen in the figure there are no significant differences between result from the gamma spectrometric measurements and the reference values. The figure also shows that it is possible to distinguish between these uranium categories. Factors that will contribute to the combined uncertainties for the type of measurements presented in this work come from counting statistics, uncertainties in gamma-ray emission probabilities, uncertainties in half lives, and the uncertainties in the relative response, i.e. in the parameter fit. For all the measurement results presented here, uncertainty budgets reveal that the major contribution to the combined uncertainty is due to the uncertainties in the parameter fit, and that the contribution to the total variance from this parameter is >85% for all measurements. No a priori calibration is needed for this kind of measurements since an intrinsic response function is established on each measured sample. Uncertainties due to deviations from a reference geometry is therefore not relevant. However, uncertainties in the response function are still there and are taken care of by the jackknife procedure. Erroneous results may be the consequence from not stating realistic measurement uncertainties. For example, the abundance of 235U in IRMM-184 was (0.6600 ± 0.0006%) when analyzed with PC/FRAM [10], as compared to (0.739 ± 0.062%) in this work. The measurement of another natural uranium material resulted in an
3.50E-05
2500
3.00E-05 2000
Response
Response
2.50E-05 1500
1000
2.00E-05 1.50E-05 1.00E-05
500
5.00E-06 0
0
200
400
600
800
1000
1200
1400
1600
1800
Eg (keV)
0.00E+00 100
300
500
700
900
1100
Eg (keV) Fig. 1. The result of the parameter fit for one of the HEU materials. e: 234U, : 235U, h: 234mPa/238U, and s: daughters from 228Th. The solid lines represent the different fits from the jackknife procedure. Data points represents the result from the mean fit, i.e. when all measurement data were used. Uncertainty bars represent 1s counting statistics.
Fig. 2. The result of the parameter fit of the iDU penetrator. : 235U, h: 234mPa/238U. Data points represents the result from the mean fit, i.e. when all measurement data were used. Uncertainties in the data points are within the size of the symbols. The deviating curve is for the case when 1001 keV is omitted in the parameter fit.
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600
Acknowledgement Gamma spectrometry
RU238/U235
500
The Swedish Radiation Safety Authority, SSM, and the Swedish Armed Forces are gratefully acknowledged for funding this work.
ICP-SFMS
400
References
300 200 100 0
DU
LEU
natU
Uranium sample Fig. 3. RU238/U235 in depleted uranium (DU), low enriched uranium (LEU) and natural uranium (IRMM-184) measured using gamma spectrometry. For IRMM-184 the reference value is the certified one, the other reference values are from ICPSFMS measurements.
abundance of 235U of (0.620 ± 0.002%), using PC/FRAM. In this work the result was (0.780 ± 0.061%), which is in agreement with natural composition. A similar low uncertainty, <0.1%, for a uranium measurement was presented by Nguyen and Zsigrai using the MGA++ code [5]. Without knowledge of the performance of these types of software codes, especially for uranium materials around natural composition, one can easily draw wrong conclusions. 4. Conclusions In this work we have applied the jackknife as a method for evaluating the measurement uncertainty in gamma spectrometric uranium isotope ratio measurements in the energy region 144– 1001 keV. For HEU the uncertainty in the 235U enrichment was around 1–2%, which is in the same range as observed by other [17]. As described above the method developed in this paper results in no significant differences between gamma spectrometric measurements and measurements done using ICP-SFMS, which was used as the reference method. Furthermore, the uncertainty estimations in this work are based on the single measurement rather than on repeated measurements as described by others [17].
[1] Nuclear forensics support, IAEA Nuclear Security Series No. 2, IAEA, Vienna, 2006. [2] Global Initiative to Combat Nuclear Terrorism (GICNT), Statement of principles, http://www.state.gov/t/isn/rls/other/126995.htm. [3] S. Richter, A. Alonso, J. Truyens, H. Kühn, A. Verbruggen, R. Wellum, Evaluating the status of uranium isotope ratio measurements using an inter-laboratory comparison campaign, Int. J. Mass Spectrom. 264 (2007) 184. [4] M.P. Rubio Montero, A. Martín Sánchez, A.M. Carrasco Lourtau, Isotopic uranium and plutonium analysis by alpha-particle spectrometry, Nucl. Instrum. Meth. Phys. Res. B 213 (2004) 429. [5] C.T. Nguyen, J. Zsigrai, Gamma-spectrometric uranium age-dating using intrinsic efficiency calibration, Nucl. Instrum. Meth. Phys. Res. B 243 (2006) 187. [6] H.A. Smith Jr., The measurement of uranium enrichment, In: Passive Nondestructive Assay of Nuclear Materials, NUREG/CR-5550, 1991. [7] S. Abousahl, A. Michiels, M. Bickel, R. Gunnink, J. Verplancke, Applicability and limits of the MGAU code for the determination of the enrichment of uranium samples, Nucl. Instrum. Meth. Phys. Res. A 368 (1996) 443. [8] R.O. Korob, G.A. Blasiyh Nuño, A simple method for the absolute determination of uranium enrichment by high-resolution gamma spectrometry, Appl. Radiat. Isot. 64 (2006) 525. [9] A.N. Berlizov, R. Gunnink, J. Zsigrai, C.T. Nguyen, V.V. Tryshyn, Performance testing of the upgraded uranium isotopics multi-group analysis code MGAU, Nucl. Instrum. Meth. Phys. Res. A 575 (2007) 498. [10] T.E. Sampson, T.A. Kelley, D.T. Vo, Application guide to gamma-ray isotopic analysis using the FRAM software, LA-14018, Los Alamos, 2003. [11] ISO: Guide to the Expression of Uncertainty in Measurement, International Organisation for Standardisation, Geneva, Switzerland, ISBN 92-67-10188-9, 1995. [12] R.G. Miller, The jackknife- a review, Biometrika 61 (1974) 1. [13] G. Meinrath, C. Ekberg, A. Landgren, J.O. Liljenzin, Assessment of uncertainty in parameter evaluation and prediction, Talanta 51 (2000) 231. [14] S. Anilkumar, A.K. Deepa, K. Narayani, A.K. Rekha, P.V. Achuthan, G. Krishnamachari, D.N. Sharma, Estimation of 235U in some depleted uranium samples by high resolution gamma-ray spectrometry using 185 keV and 1001 keV gamma energies of 235U and 234mPa, J. Radioanal. Nucl. Chem. 274 (2007) 161. [15] Evaluated Nuclear Structure Data File (ENSDF), National Nuclear Data Center (NNDC), Brookhaven National Laboratory, USA, 2009, www.nndc.bnl.gov/ ensdf. [16] GUM Workbench, Metrodata GmbH, D-79639 Grenzach-Wyhlen, Germany, http://www.metrodata.de/. [17] J. Morel, C. Hill, M. Bickel, A. Alonso-Munoz, S. Napier, B. Thaurel, Results from the international evaluation exercise for uranium enrichment measurements, Appl. Radiat. Isot. 52 (2000) 509.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
The jackknife as an approach for uncertainty assessment in gamma spectrometric measurements of uranium isotope ratios H. Ramebäck a,*, A. Vesterlund a, A. Tovedal a, U. Nygren a, L. Wallberg b, E. Holm c, C. Ekberg d, G. Skarnemark d a
Swedish Defence Research Agency, FOI, Division of CBRN Defence and Security, SE-901 82 Umeå, Sweden Swedish Radiation Safety Authority, SE-117 16 Stockholm, Sweden Norwegian Radiation Protection Authority, Postboks 55, 1332 Østerås, Norge d Chalmers University of Technology, Department of Chemical and Biological Engineering, Nuclear Chemistry, SE-412 96 Göteborg, Sweden b c
a r t i c l e
i n f o
Article history: Received 16 March 2010 Received in revised form 7 May 2010 Available online 15 May 2010 Keywords: Uranium Isotope ratio Enrichment Gamma spectrometry Uncertainty
a b s t r a c t The jackknife as an approach for uncertainty estimation in gamma spectrometric uranium isotope ratio measurements was evaluated. Five different materials ranging from depleted uranium (DU) to high enriched uranium (HEU) were measured using gamma spectrometry. High resolution inductively coupled plasma mass spectrometry (ICP-SFMS) was used as a reference method for comparing the results obtained with the gamma spectrometric method. The relative combined uncertainty in the gamma spectrometric measurements of the 238U/235U isotope ratio using the jackknife was about 10–20% (k = 2), which proved to be fit-for-purpose in order to distinguish between different uranium categories. Moreover, the enrichment of 235U in HEU could be measured with an uncertainty of 1–2%. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Measurements of nuclear material are important in many areas, e.g. nuclear safeguards, nuclear forensics (illicit trafficking) and environmental studies. For events involving illicit trafficking of radioactive- or nuclear material an initial basic characterization of the material is important [1,2]. This basic characterization involves, for example, measurement of the isotopic composition and should be performed at an incident place. In the case of uranium such characterization should be able to determine the uranium category, i.e. depleted uranium (DU), natural uranium, low enriched uranium (LEU) or high enriched uranium (HEU). Furthermore, relevant uncertainty estimation is essential in order to compare the measurement results with e.g. references or other measurements. Measurement of the isotopic composition of uranium can be done using different methods, e.g. mass spectrometry [3], alpha spectrometry [4] and gamma spectrometry [5]. Due to the long half life of the uranium isotopes mass spectrometry is by far the most sensitive method and offers also the lowest combined uncertainty. Compared to gamma spectrometry alpha spectrometry is also more sensitive. However, alpha- and mass spectrometry is mainly used for measurements of e.g. swipes sampled from a package containing a radioactive material, as well as in a later stage where a * Corresponding author. Tel.: +46 90 106646; fax: +46 90 106803. E-mail address: [email protected] (H. Ramebäck). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.05.055
more careful examination revealing the origin of the material is to be done. Thus at the incident location the only practical option is often gamma spectroscopy. Measurement of uranium composition using gamma spectrometry can be done using different approaches. The so called enrichment meter is based on the measurement of the 186 keV gamma peak from 235U [6]. Another method is based on the analysis of peaks in the X-ray region around 100 keV [7]. A third approach is based on the measurement of gamma rays in the ‘‘high” energy region from 144 to 1001 keV [8,9]. The first method needs calibration with known materials, which is not needed for the latter two. However, no comprehensive approach for the uncertainty estimation in any of the above mentioned methods has been presented. For example, commercial software for evaluating uranium isotopic composition often relies on counting statistics solely for the uncertainty estimation [10] which often results in too low uncertainty statements since other uncertainty sources which often domintates in relation to counting statistics is not taken into account. The importance of not underestimating, as well as not overestimating, measurement uncertainties is that it may result in wrong conclusions and therefore wrong decisions [11]. Since the measurement of uranium isotopic composition using gamma spectrometry involves the fitting of different parameters in a measurement model, where the isotopic ratios are included, a statistic re-sampling method could be suitable for evaluation of the measurement uncertainties. Other sources that may contribute
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to the combined uncertainty is the counting statistics, the assumed half lives of the uranium isotopes and the uncertainties in the gamma-ray emission probabilities. In this work we explored the jackknife [12,13] as an approach for the evaluation of the uncertainty in the measurement of the uranium isotope ratios using gamma spectrometry in the ‘‘high” energy region. One advantage with the jackknife method is that an inherent sensitivity analysis regarding variance contribution from individual data is possible. Five different uranium materials were measured and ICP-SFMS was used as a reference method.
2.1. Theory The efficiency WEj at a gamma ray energy Ej for a gamma spectrometric measurement can be written as
ð1Þ
where Ci,Ej is the net count rate in a full energy peak at gamma ray energy Ej of a radioactive isotope i, Ii,Ej the photon emission probability at Ej for isotope i, ki the decay constant of the radioactive isotope i, and Ni the number of atoms of isotope i emitting a gamma ray at energy Ej. Furthermore, the isotope ratio of two isotopes i and k can be written as
Ri ¼
Ni Nk
ð2Þ
Here, the isotope k is referred to as the reference isotope, which all other isotopes are related to. (Observe that the isotope ratio of the reference isotope k is unity.) The efficiency WEj can now be written as
WEj ¼
C i;Ej : Ii;Ej ki Ri Nk
ð3Þ
Since Nk is constant for all efficiencies WEj for a specific measurement, the relative efficiency, Wrel,Ej, at gamma ray energy Ej can be written as
Wrel;Ej ¼
C i;Ej : Ii;Ej ki Ri
ð4Þ
For uranium isotope measurements and only considering 234U, U and 238U one will have two isotope ratios, e.g. the ratios of 234 U/235U and 238U/235U (since the isotope ratio of 235U relative 235 U is unity). For enriched uranium 232U is often present in different amounts, which means that daughters from 228Th also can be used for the relative efficiency. In such cases the ratio 228Th/235U also has to be introduced into the measurement model. Moreover, one needs also a function that describes the relative detector efficiency over the gamma ray energy interval of interest. One function with such a property is 235
2
Wrel ðEÞ ¼ ec1 þc2 =E
þc3 ðln EÞ2 þc4 ðln EÞ3 þc5 =E
r2 ¼
m m1 X ðRj R Þ2 m j¼1
ð6Þ
where R* is the result of an isotope ratio when all data points are used, Rj is the result when one data j is omitted in the parameter fit and m is the number of data. The variance can be evaluated by repeating this procedure for all data points j. 2.2. Uranium samples
2. Materials and methods
C WEj ¼ i;Ej Ii;Ej ki Ni
the isotope ratios, can be evaluated by means of the jackknife [12,13]. The variance of an isotope ratio Rj can be estimated using
ð5Þ
which is a development of the function used by Anilkumar et al. [14]. By combining Eqs. (4) and (5) the isotope ratios may be calculated. This can be done by fitting the isotope ratios and the parameters c1–c5 in Eq. (5) by means of finding the minimum in the sum of the squared differences, i.e. the error square sum, of Eqs. (4) and (5). The combined uncertainty of the isotope ratios will consist of the counting statistics in the full energy gamma ray peaks, the uncertainty of the photon emission probabilities, the uncertainty in the half lives of the isotopes, and the uncertainty in the parameter estimation. The uncertainty in the parameter estimation, e.g.
Five uranium samples were measured: two HEU materials, one natural uranium reference material in acid solution (IRMM-184); one DU penetrator, and one sample containing waste of LEU. 2.3. Gamma spectrometric measurements The gamma spectrometric measurements in this study were done using p-type coaxial high-purity germanium (HPGe) detectors (EG&G Ortec, Oak Ridge, TN, USA). The relative efficiency of the detectors were about 50%, and the resolution about 1.8 keV@1332 keV. Standard NIM electronics (Nuclear Instrument Module) were used for pulse processing and high voltage power supply (EG&G Ortec, Oak Ridge, TN, USA). GammaVision (EG&G Ortec, ver. 6.01, Oak Ridge, TN, USA) was used for spectrum acquisitions and peak area evaluations. The following full energy gamma ray peaks were used in the measurements: 235
U: the peaks at 143.8, 163.3, 185.7 and 205.3 keV. U: the peaks from 234mPa at 258.3, 742.8, 766.4 and 1001.0 keV 234 U: the peak at 120.9 keV. 238
In addition to these, peaks from daughters of 228Th were used for the HEU materials. These were 238.6 keV from 212Pb, 727.3, 1078.6 and 1620.5 keV from 212Bi, and 583.2 and 860.5 keV from 208 Tl. All photon emission probabilities and half lives where taken from ENSDF [15]. 2.4. Determination of isotope ratios using ICP-SFMS The materials (all except one HEU material) to be analyzed using ICP-SFMS were dissolved in HNO3 and further diluted in 2% HNO3. The measurements were performed on an Element2 ICP-SFMS instrument (Thermo–Finnigan, Bremen, Germany). The sample introduction equipment consisted of a conical nebulizer and a cyclonic spray chamber (GlassExpansion, Melbourne, Australia). Scanning between the isotopes of interest was performed by changing the acceleration voltage (‘‘E-scan”) and the detection mode was pulse counting. The software controlled dead time correction was inactivated and the raw data were extracted to Excel were corrections for dead time and mass bias were performed manually. Three different uranium reference materials were used in the measurements; IRMM-184 (natural composition); IRMM073/1 (near unity relationship between 233U, 235U and 238U) and IRMM073/7 (233U:235U:238U 0.02:1:1), all obtained from the Institute for Reference Materials and Measurements, Geel, Belgium. IRMM073/7 was used to determine the dead time of the system. IRMM073/1 was used to monitor mass bias and IRMM-184 was used for quality control during the measurements performed in this study. All uncertainties were evaluated in accordance with ISO/GUM [11] using the software GUM Workbench [16].
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In this work uncertainties are presented with a coverage factor k = 2, i.e. representing an approximate 95% confidence interval, if not otherwise stated. 3. Results and discussion Fig. 1 shows the result of the parameter fit for one of the HEU material. As observed in the figure, the jackknife procedure results in different fits, i.e. a set of the measurands RU234/U235, RU238/U235, RTh228/U235, from which the uncertainty can be evaluated according to Eq. (6). For this HEU material the isotope ratios of RU238/U235 and RU234/U235 were 0.082 ± 0.018 and 0.0065 ± 0.0043, respectively. This resulted in an enrichment of 235U of (91.9 ± 1.6%). This material was not available for measurement using ICP-SFMS. However, the measurement result indicates the magnitude of measurement uncertainty that is possible to reach using this method. For the other HEU material measured, the isotope ratios of RU238/U235 and RU234/U235 were 0.0583 ± 0.0090 and 0.0138 ± 0.0026, resulting in a 235U enrichment of (93.28 ± 0.88%). The result from the ICP-SFMS measurement of this material was (92.773 ± 0.020%), i.e. the gamma spectrometric result was in agreement with the more precise mass spectrometric result. Moreover, the isotopic abundance of 238U in this material was (5.44 ± 0.79%), as compared to (6.061 ± 0.017%) from the ICP-SFMS measurement, i.e. no significant difference between the results. Finally, the isotopic abundance of 234U was (1.28 ± 0.23%), as compared to the ICPSFMS result of (1.1654 ± 0.0062%), i.e. also an agreement between the methods. For different gamma spectrometric measurement methods, uncertainties of the same magnitude (1–2%) has been stated [17]. However, these measurement uncertainties were based on repeated measurements rather than on a single one. In this work the measurement uncertainties applying the jackknife were evaluated on a single measurement, which is an advantage when long counting times are needed, e.g. when only small sample sizes are available. The measured abundance of 234U in the two HEU materials were (0.60 ± 0.39%) and (1.28 ± 0.23%), respectively. The, somewhat, high uncertainty in the determination of the isotope abundance of 234U reflects the problem involved in measuring this isotope with gamma spectrometry in so called ‘‘thin” samples [5]. This is due to the rapid change in detector response as a function of energy in this region when measuring thin samples, see Fig. 1. This is also observed in the variance for the jackknife method, Eq. (6), since the dominant contributor to the variance comes from the
143.8 keV gamma peak of 235U, i.e. when this peak is omitted in the jackknife procedure. This peak represents about 80% of the total variance in the HEU materials. The problem is possible to overcome by the use of a thin-window detector, such as one with a Be-endcap, when 234U is to be measured [5]. This will improve the precision in the determination of the 234U abundance using gamma spectrometry since its energy characteristics is different in the low energy region, compared to a detector with e.g. an Al-endcap. Fig. 2 shows the result of the parameter fit from the measurement of the DU penetrator. In the measurement of uranium materials ranging from DU to LEU the peak at 258.3 keV from 234m Pa/238U could not be omitted in the jackknife method, since this will lead to erroneous measurement results. This is due to the properties of the empirical response function, Eq. (5), and the big gap between the energy regions of 235U and 234mPa/238U. However, for HEU containing 232U there often is an extra peak from 212 Pb present at 238.6 keV, which is close to 258.3 keV. In such cases it is possible to omit the 258.3 keV peak in the jackknife procedure as well as omitting the 238.6 keV peak. RU238/U235 from the measurement of the DU, natural uranium, and the LEU are shown in Fig. 3. As can be seen in the figure there are no significant differences between result from the gamma spectrometric measurements and the reference values. The figure also shows that it is possible to distinguish between these uranium categories. Factors that will contribute to the combined uncertainties for the type of measurements presented in this work come from counting statistics, uncertainties in gamma-ray emission probabilities, uncertainties in half lives, and the uncertainties in the relative response, i.e. in the parameter fit. For all the measurement results presented here, uncertainty budgets reveal that the major contribution to the combined uncertainty is due to the uncertainties in the parameter fit, and that the contribution to the total variance from this parameter is >85% for all measurements. No a priori calibration is needed for this kind of measurements since an intrinsic response function is established on each measured sample. Uncertainties due to deviations from a reference geometry is therefore not relevant. However, uncertainties in the response function are still there and are taken care of by the jackknife procedure. Erroneous results may be the consequence from not stating realistic measurement uncertainties. For example, the abundance of 235U in IRMM-184 was (0.6600 ± 0.0006%) when analyzed with PC/FRAM [10], as compared to (0.739 ± 0.062%) in this work. The measurement of another natural uranium material resulted in an
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Fig. 2. The result of the parameter fit of the iDU penetrator. : 235U, h: 234mPa/238U. Data points represents the result from the mean fit, i.e. when all measurement data were used. Uncertainties in the data points are within the size of the symbols. The deviating curve is for the case when 1001 keV is omitted in the parameter fit.
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The Swedish Radiation Safety Authority, SSM, and the Swedish Armed Forces are gratefully acknowledged for funding this work.
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Uranium sample Fig. 3. RU238/U235 in depleted uranium (DU), low enriched uranium (LEU) and natural uranium (IRMM-184) measured using gamma spectrometry. For IRMM-184 the reference value is the certified one, the other reference values are from ICPSFMS measurements.
abundance of 235U of (0.620 ± 0.002%), using PC/FRAM. In this work the result was (0.780 ± 0.061%), which is in agreement with natural composition. A similar low uncertainty, <0.1%, for a uranium measurement was presented by Nguyen and Zsigrai using the MGA++ code [5]. Without knowledge of the performance of these types of software codes, especially for uranium materials around natural composition, one can easily draw wrong conclusions. 4. Conclusions In this work we have applied the jackknife as a method for evaluating the measurement uncertainty in gamma spectrometric uranium isotope ratio measurements in the energy region 144– 1001 keV. For HEU the uncertainty in the 235U enrichment was around 1–2%, which is in the same range as observed by other [17]. As described above the method developed in this paper results in no significant differences between gamma spectrometric measurements and measurements done using ICP-SFMS, which was used as the reference method. Furthermore, the uncertainty estimations in this work are based on the single measurement rather than on repeated measurements as described by others [17].
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