Consistency test of coincidence-summing calculation methods for extended sources

Consistency test of coincidence-summing calculation methods for extended sources 1 O. Simaa,b, A. De Vismes Ottc, M. S. Diasd, P. Dryake, L. Ferreuxf, D. Guraub, S. Hurtadog, P. Jodlowskih, 2 K. Karfopoulosi, M. F. Koskinasd, M. Laubensteinj, Y. K. Leek, M. C. Lépyl, A. Lucab, M. O. Menezesd, 3 D. S. Moreirad, J. Nikoličm, V. Peyresn, P. Saganowskiq, M. I. Savvao, R. Semmlerd, J. Solce, T. T. Thanhp, 4 K. Tyminskaq, Z. Tyminskiq, T. Vidmarr, I. Vukanacm, H. Yucels 5 aPhysics Department, University of Bucharest, Bucharest-Magurele, Romania 6 bHoria Hulubei National Institute for R & D in Physics and Nuclear Engineering (IFIN-HH), Bucharest-7 Magurele, Romania 8 cIRSN/LMRE, Orsay, France 9 dNuclear Metrology Laboratory, Nuclear and Energy Research Institute - IPEN-CNEN/SP, São Paulo, Brazil 10 eCzech Metrology Institute, Brno, Czech Republic 11 fIRSN / PSE-ENV / SAME / LMN, Vésinet, France 12 gUniversity of Sevilla, Sevilla, Spain 13 hFaculty of Physics and Applied Computer Science, AGH University of Science and Technology, Cracow, 14 Poland 15 iGreek Atomic Energy Commission, Environmental Radioactivity Monitoring Department, Athens, Greece 16 jLaboratori Nazionali del Gran Sasso, Istituto Nazionale di Fisica Nucleare, Assergi, Italy 17 kDEN/DANS/DM2S/SERMA, CEA-Saclay, France 18 lLaboratoire National Henri Becquerel, CEA Saclay, France 19 mLaboratory for Radiation Measurements, Department of Radiation and Environmental Protection, Vinča 20 Institute of Nuclear Sciences, Belgrade, Serbia 21 nLaboratorio de Metrología de Radiaciones Ionizantes, CIEMAT, Madrid, Spain 22 oINRASTES, National Centre for Scientific Research “Demokritos”, Athens, Greece 23 pUniversity of Science, VNU-HCM, Faculty of Physics & Engineering Physics, Department of Nuclear 24 Physics, Ho Chi Minh City, Viet Nam 25 qRadioisotope Centre, POLATOM, NCBJ, Poland 26 rSCK•CEN, Belgian Nuclear Research Centre, Boeretang 200, 2400 Mol, Belgium 27 sAnkara University, Institute of Nuclear Sciences, Ankara, Turkey 28 29 Abstract 30 An internal consistency test of the calculation of coincidence-summing correction factors FC for volume 31 sources is presented. The test is based on exact equations relating the values of FC calculated for three ideal 32 measurement configurations. The test is applied to a number of 33 sets of FC values sent by 21 teams. Most 33 sets passed the test, but not the results obtained using the quasi-point source approximation; in the latter case 34 the test qualitatively indicated the magnitude of the bias of FC. 35 36 Keywords: Gamma-ray spectrometry; HPGe detector; Coincidence-summing corrections; Self-consistency 37 test 38 1. Introduction 39 Coincidence-summing effects represent one of the factors affecting the calibration of the full energy peak 40 (FEP) efficiency for high efficiency HPGe detectors. These effects are enhanced in present day 41 measurements, due to the desire and possibility to achieve high efficiency measurement conditions. 42 In view of the importance of the coincidence-summing corrections and of the complex procedures applied 43 for evaluating them, the Gamma-Ray Spectrometry Working Group (GSWG) of the International 44 Committee for Radionuclide Metrology (ICRM) carried out a series of actions intended to test the quality of 45 these procedures (Lépy et al., 2010; Lépy et al., 2012; Vidmar et al., 2014; Vidmar et al., 2016). As a 46 continuation, this work presents the results of an action proposed in the meeting of the GSWG organized 47 during the ICRM Conference in Buenos Aires (ICRM GSWG webpage, 2018). It has the specific task of 48 testing the internal self-consistency of the methods applied for the computation of coincidence-summing 49 corrections for volume sources. The proposed test is based on exact relations that should be fulfilled in 50 certain ideal measurement configurations. More precisely, the results obtained using any computation 51 method for one such configuration should be related by exact equations to the results given by the same 52 method for other configurations. Thus, this test does not require experimental data (avoiding the problem of 53 experimental uncertainties and the conformity of the detector model used in the computations with the actual 54 one) or comparisons of a method with other methods (avoiding the debate concerning the selection of a 55 particular reference method). 56 57 2. Theoretical Background 58 From the point of view of activity measurement by gamma-ray spectrometry, the occurrence of coincidence-59 summing has two types of effects: coincidence-losses from existing peaks and coincidence summing-in 60 effects (Debertin and Helmer, 1988; Gilmore, 2008). The latter add counts to existing peaks or produces 61 peaks (pure sum peaks) which cannot be assigned to any photon emitted by the decay of the nuclide. 62 In the absence of coincidence-summing effects, the net count rate in every peak (energy Ei) is proportional 63 with the nuclide activity A and the photon emission probability pi =I  (Ei): 64 𝑁(𝐸)=𝜀(𝐸)∙𝑝∙𝐴 (1) 65 In a given measurement configuration, the FEP efficiency  (Ei) depends only on the energy of the photon. In 66 the presence of coincidence-summing effects, the probability of recording a count in the peak of energy Ei 67 depends also on the probability that other photons (energy Ej) interact with the detector simultaneously. The 68 count rate equation can be written similarly to Eq. 1, by introducing the apparent efficiency  app(Ei): 69 𝑁(𝐸,𝑋)= 𝜀(𝐸,𝑋)∙𝑝(𝑋) ∙ 𝐴(𝑋) = 𝜀(𝐸)∙𝐹(𝐸,𝑋)∙𝑝(𝑋)∙𝐴(𝑋) (2) 70 In this equation FC(Ei,X) represents the coincidence-summing correction factor for the peak of energy Ei of 71 the nuclide X. In the case of a point source and neglecting the angular correlation effects, it can be written as 72 (Sima and Arnold 2000): 73 𝐹(𝐸,𝑋)= 1−𝑝 𝑝∙  𝐸+𝑝 𝑝∙  𝐸∙  (𝐸)− ⋯ +𝑝 𝑝∙𝜀𝐸∙𝜀𝐸 𝜀(𝐸) ⬚ , ⬚ , ⬚  74 −  𝑝 𝑝 𝜀𝐸∙𝜀𝐸∙  (𝐸) 𝜀(𝐸) ⬚ ,, +⋯ 75 (3) 76 The first sum describes the coincidence losses from the peak of energy Ei due to the simultaneous detection 77 of photon Ej (pair coincidences), summed over all photons emitted in cascades with the photon of energy Ei; 78 the second sum describes triple coincidences and so on. The next sums from the equation (which are the 79 only terms contributing in the case of pure sum peaks) describe the summing-in contribution, when relevant; 80 the first term corresponds to complete energy deposition of the photons of energies Ep and Eq, with 81 Ep+Eq=Ei, summed over all possible combinations, whereas the second term takes into account the events 82 when some counts of this type are lost from the peak due to coincidence losses produced by the detection of 83 another photon Er. Note that the magnitude C of the coincidence effects is given by the sum terms from 84 Equation (3), i.e. the term C=1-FC in the case of usual peaks (irrespective to the fact that only coincidence 85 losses or both losses and summing in contribute) and C=FC in the case of pure sum peaks. In Eq. 3,  86 represents the FEP efficiency,  the total efficiency, and pij and pijk, ... are the probabilities of simultaneous 87 emission of the groups of (i,j) and (i,j,k), ... photons. In this equation the angular correlation of cascading 88 photons was neglected. 89 In the case of volume sources, the terms become more involved. For example,  (Ej) from equation (3) 90 should be replaced by the quantity 91  𝐸→  𝐸;𝐸=∫𝜀(𝐸,𝑟)∙  (𝐸,𝑟)𝑑𝑣 ∫𝜀(𝐸,𝑟)∙𝑑𝑣 93 (4) 92 where the integration is carried out over the source volume, dv being the volume element around 𝑟. In the 94 presence of angular correlations, the dependence of the efficiency on the direction of emission of the photon 95 should be introduced, the angular correlation function should be included in the integrand and integrations 96 should be carried out over the directions of emission of the photons (the same procedure is valid also in the 97 case of point sources). The expression in equation (4) which can be viewed as an effective total efficiency 98 (Arnold and Sima, 2001) for photon Ej, depending also on the energy Ei. The terms involving two 99 efficiencies in equation (3) should be replaced by similar integrals of products of three efficiencies, and so 100 on. Note that for the evaluation of FC, the variation of the efficiencies as a function of the emission point 101 within the volume source is needed. Thus, contrary to the case of point sources, the efficiency related 102 quantities required for the computation of FC do not have a directly measurable correspondent. Furthermore, 103 due to the dependence of the effective total efficiency on Ej and Ei, not only on Ej, the number of quantities 104 that should be evaluated is higher. Consequently, it is clear that compared to the case of point sources, the 105 computation of FC for volume sources is more difficult and can take much longer time. The angular 106 correlation effects further complicate the evaluation of FC. 107 In view of the complexity of the evaluation of FC in the case of volume sources, in some codes explicit or 108 hidden approximations are applied. The uncertainties associated with some of these approximations can be 109 partly revealed by the test described below. 110 111 3. Self-consistency test 112 Consider three sources, S1, S2 (identical with S1) and S3, with negligible container walls and filled with air 113 (or vacuum). The dimensions of the sources satisfy R1=R2=R3=R, H2=H1, H3=H1+H2. The sources are 114 measured in the configurations a, b and c presented in Fig.1. Note that configuration c corresponds to 115 mounting the first two sources one upon the other on the detector. Each configuration is placed in vacuum 116 (or in air) and there are no materials in the vicinity of the detector and sources. In these conditions the 117 interactions within the sources, in the container walls or outside the sources and detector, are negligible. 118 Then the properties of the S3 source measured in configuration c should be identical with the properties of 119 the composite source, obtained by placing S2 above S1. 120 For example, the contribution to the count rate of the decays from each of the two parts of the composite 121 source is equal to the contribution to the count rate of the sources 1 and 2 measured independently in 122 geometries a and b. This is true both in the case of the peak count rate (either in the absence or in the 123 presence of coincidence-summing effects) and in the case of the total count rate. Therefore the following 124 relations should hold (ICRM GSWG webpage, 2018): 125 𝜀 =𝜀+𝜀 2 (5),   =  +   2 (6) 127 126 𝐹 =𝐹 ∙𝜀+𝐹 ∙𝜀 𝜀+𝜀 (7) 129 128 𝜀  =1 2𝜀  +𝜀  (8) 130 Denote by  c,  c, FCc and 𝜀  the respective quantities computed directly for source S3, measured in 131 geometry c. If the procedure used to compute these quantities is self-consistent, then their values should be 132 equal, within the range of their uncertainties, with  comp,  comp, FCcomp and 𝜀  . 133 The participants to the test were requested to compute the coincidence-summing correction factors FC with a 134 statistical uncertainty better than 1%, for several peaks of 60Co (1173.23, 1332.49, 2505.75 keV), 134Cs 135 (569.33, 604.72, 795.86, 1400.59 keV), 133Ba (276.40, 356.01 keV) and 152Eu (121.78, 1408.01 keV), for the 136 three configurations a, b, c. The parameters of the sources were R=2cm, H1=H2=2.5 cm, H3=5 cm. A 137 simplified model of an n-type HPGe detector (Vidmar et al., 2014; ICRM GSWG webpage, 2018) with 138 radius RD=3 cm, length LD=6 cm, cylindrical inner hole (RH=0.5 cm, LH=4 cm) and negligible dead layer 139 thickness was assumed. The parameters of the Al endcap were: thickness 1 mm, radius 4 cm, distance from 140 the crystal to the endcap 0.5 cm. 141 The participants were asked to report also the values of the full energy peak efficiency and to provide 142 information on the method applied. The use of nuclide data recommended by the ICRM, based on DDEP 143 (DDEP, 2018), was suggested. 144 145 4. Participants 146 A number of 21 individual or group participants took part in this action. Several participants have sent more 147 than one set of results, obtained using different computation methods. Thus, 1 participant has sent 6 sets, 2 148 participants – 3 sets, 3 participants – 2 sets, resulting in a total number of 33 sets of results being received. 149 The main characteristics of the procedures applied, as well as the codes of the results received, are presented 150 in Table 1. The procedures differ in the method applied for the computations of the efficiency (including 151 volume source effect), of the required decay data parameters and in the way in which the efficiencies and the 152 decay data are combined (Sima, 2012). 153 General purpose programs such as MCNP (Briesmeister, 2000; Pelowitz, 2013; Goorley et al., 2013), 154 PENELOPE (Salvat, 2015) or GEANT 4 (Agostinelli et al., 2003; Allison et al., 2006) have been used by 155 several participants for the computation of coincidence-summing corrections (Thanh et al., 2018; García-156 Toraño et al., 2017; Hurtado et al., 2004). In all the cases the value of FC is obtained by running the code to 157 obtain the peak efficiencies from the simulated spectra both in the presence and in the absence of 158 coincidence emission. MCNP-CP (Berlizov, 2012) and MCNPX, PENELOPE with PENNUC (García-159 Toraño et al., 2017) and GEANT 4 with the Radioactive Decay Module (Hauf et al., 2013) were used, the 160 decay being fully simulated using ENSDF (MCNP-CP and GEANT 4) or DDEP (PENNUC) decay data. 161 Another generalist code, developed primarily for applications in reactor physics, (TRIPOLI-4 webpage), 162 was applied for the computation of FC values for the two gamma photons of 60Co. 163 MCNP and PENELOPE were also used for providing input values of the efficiencies for other software 164 applied for the computation of FC. 165 Thus, MCNP was used in combination with ETNA (Lépy et al., 2004; Lépy et al., 2012) by two participants. 166 The complete version of ETNA was used for the evaluation of the efficiencies dependence on the position of 167 the emission point in the volume source, using the efficiency transfer from a reference geometry, for which 168 the efficiency was computed by MCNP. One participant defined as the reference geometries the 169 configurations a, b and c, which were considered also the target geometries. Using the efficiencies 170 dependence on position, ETNA performs a numerical integration for obtaining FC. 171 In the application of the software developed at CMI (Dryak et al., 2012) (2 participants), the volumes of the 172 sources were decomposed in a number of small domains. One participant evaluated the spectra in the 173 presence/absence of coincidences for each domain and then merged them together. FC was obtained from the 174 ratio of the peak count rates in the combined spectra. The other participant used MCNP to compute  ,  and 175 FC for each domain and then averaged the values. The joint emission probabilities were evaluated using a 176 MATLAB script or numerically. 177 SUMCOR (Dias et al., 2018) coupled with MCNP code was used in 3 sets of results. In the first, the 178 efficiencies were computed for a number of 200 (geometries a and b) or 400 (geometry c) randomly sampled 179 points and a Semkow type procedure (Semkow et al., 1990) was applied to obtain individual FC values, that 180 were finally averaged. In the second set, the same point source efficiencies were used for the evaluation of 181 the LS curve, that was further applied for FC computation via a Blaauw and Gelsema (2003) type formalism. 182 The quasi-point source approximation using the efficiencies computed by MCNP for the complete sources 183 was also applied in SUMCOR. 184 PENELOPE was used for providing input values of the efficiencies for 7 sets of FC results obtained with 185 other software, not with PENNUC. SUMCOR was applied with input from PENELOPE in the same way as 186 with input from MCNP (3 sets of results). The complete version of ETNA for volume sources was also 187 applied with input efficiencies for the reference point source evaluated by PENELOPE; in this case ETNA 188 provided the values of elementary efficiencies for emission points within the volume sources and then 189 performed a numerical integration for obtaining FC. In addition, the efficiencies for the complete volume 190 sources were computed using the transfer method from the point source reference (with efficiencies 191 evaluated by PENELOPE) and the FAST version of ETNA (quasi-point source approximation) was applied 192 for obtaining FC. PENELOPE was also used by 2 participants for the computation of the efficiencies 193 required in the input of the TrueCoinc software (Sudar, 2002). 194 The standard version of GESPECOR (Sima et al., 2001; Sima and Arnold, 1996; Arnold and Sima, 2004; 195 Arnold and Sima, 2006) was applied by 7 participants. In this code the joint emission probabilities of groups 196 of photons, evaluated analytically (Sima and Arnold, 2008), are combined with the corresponding integrals 197 of products of efficiencies, obtained by Monte Carlo simulation. The GESPECOR version including the 198 angular correlations was also applied (1 set). Finally, one set of results obtained with GESPECOR 199 deliberately applied the quasi-point source approximation. 200 2 participants reported results obtained using the EFFTRAN software (Vidmar, 2005; Vidmar and Likar, 201 2005; Vidmar and Korun, 2006; Vidmar et al., 2011). In EFFTRAN the effective solid angle method is used 202 for obtaining the spatial dependence of the efficiency within the volume sources, on the basis of the 203 efficiency transfer from a reference geometry. This spatial dependence is used for the evaluation of the LS 204 curve. Finally a Semkow type procedure (Semkow et al., 1990) for the evaluation of the FC is applied, using 205 the peak and total efficiencies for the complete sources combined with the LS curve as proposed by Blaauw 206 (Blaauw and Gelsema, 2003). 2 participants reported results obtained using the EFFTRAN software 207 (Vidmar, 2005; Vidmar and Likar, 2005; Vidmar and Korun, 2006; Vidmar et al., 2011). In EFFTRAN the 208 effective solid angle method is used for obtaining the total efficiency for the sample as a whole, in 209 combination with an approximate way of accounting for the contribution from the scattering within the 210 sample itself (Vidmar and Likar, 2005). The spatial dependence of the efficiency within the volume sources 211 is treated with the LS-curve formalism introduced by Blaauw and Gelsema (2003), but contrary to his 212 original approach, the LS factors are obtained by pure calculation, rather than based on a measurement 213 (Vidmar and Korun, 2006; Vidmar and Kanisch, 2010). To arrive at the full-energy-peak efficiencies the 214 effective solid angle values are multiplied with a peak-to-total ratio of a point source obtained from a 215 deterministic model based on (Vidmar et al, 2001). Finally, the Andreev recursive formulae (Andreev et al., 216 1972) are applied, using the peak and total efficiencies multiplied by the value read off of the LS curve at a 217 corresponding gamma-ray energy. 218 The ISOCS procedure included in the CANBERRA software was used by one participant. The coincidence 219 summing corrections including up to 3 photons are evaluated using decay data from an internal library (Zhu 220 et al., 2009). In the case of extended sources, the effects are evaluated for points located inside of the 221 volume source and then are averaged (Kolotov et al, 1996). 222 In most of the cases, DDEP data were used (DDEP, 2018). Exceptions were the applications of MCNP-CP, 223 GEANT 4, TrueCoinc and ISOCS. 224 In several calculations, only pair coincidences were included, in others all possible coincidences with  225 photons were considered. There were also differences in the procedures applied to deal with X-Rays. 226 Several participants included the contribution of beta rays and of bremsstrahlung radiation to coincidence 227 summing, but the effect is very small and most results reported did not include these effects. With one 228 exception, angular correlations were neglected. 229 Several participants did not report the uncertainties of all the computed quantities. 230 Concerning the completeness of the results sent, it should be mentioned that not every set received included 231 all the requested results. In fact, only the FC values for 1173 and 1332 keV were reported in every set. From 232 the 33 sets of results received, only 19, respectively 21, included results for the sum peaks with energy 1400 233 keV (134Cs) and 2505 keV (60Co); apparently in the standard versions of MCNP, ETNA, EFFTRAN, ISOCS 234 and TrueCoinc it is not possible to make calculations for these pure sum peaks, but a trick applied by several 235 participants allowed obtaining results for pure sum peaks using these codes (see Table 1). 236 237 5. Results 238 According to the principle of the proposed self-consistency test, the value of the quantity x (FC or the 239 efficiency) computed directly for the geometry c, xc, should be equal to the value of the same quantity 240 computed considering the geometry c as composed from a and b, xcomp. For testing purposes, an index of 241 self-consistency is defined by: 242 𝐼𝑆𝐶 =𝑥−𝑥 𝑥 243 (9) 244 The value of ISC can be interpreted as follows. If the procedure applied for the computation of the quantity x 245 would be absolutely correct in the case of geometries a and b, then the relative bias of the quantity x 246 calculated with that procedure for geometry c is equal to the value of ISC. On the other hand, if also the 247 values xa and xb computed with that procedure are biased, then the value of ISC is no longer equal to the 248 relative bias of xc, the relation between the two depending on the specific case and in fact the relative bias of 249 xc can be larger than ISC. 250 In the following, the values of the self-consistency index for coincidence summing effects, ISC(NP) for 251 usual peaks (x=1-FC) and ISC(SP) for pure sum peaks (x=FC), as well as the index ISC(AE) for the apparent 252 efficiency (x=  app), will be presented. 253 In Figs. 2-5, the distribution of the self-consistency index is displayed for selected peaks of each nuclide 254 considered. Most values of the self-consistency index are around 0, but there are several values around -255 0.15, clearly violating the self-consistency. According to Equations (7) and (9), ISC(NP) and ISC(SP) 256 depend both on the values of FC and on the values of the peak efficiencies. However, the evaluation of the 257 self-consistency index for the peak efficiencies demonstrates that with the exception of the set 30, for which 258 the index is around 0.034 in the energy range from 121 to 1408 keV, in all the other cases the values of the 259 peak efficiencies are self-consistent to a high degree. Thus, if ISC(NP) and ISC(SP) are significantly 260 different from zero, this is due to the method applied for the evaluation of FC, and not of the efficiencies. It 261 should be mentioned that the sets with code numbers 9, 19, 24, 26, 27 and 28 were obtained using the quasi-262 point source approximation, either deliberately, or because the software requires simply the efficiency 263 values for the complete volume source. In Table 2 the results obtained when the quasi-point source 264 approximation was deliberately applied (specific versions of GESPECOR, ETNA and SUMCOR) are 265 compared with the results obtained with the same software using the averaging over the volume of the 266 sources. As expected, the relative difference between the values calculated using the two procedures is 267 highest in the case of the source with larger dimensions (geometry c) and smallest in the case of the source 268 in geometry b (same volume as in geometry a, but a weaker relative dependence of the efficiencies on the 269 position of the emission point due to longer distance to the detector). In the case of the sets of results 270 obtained using the quasi-point source approximation, the self-consistency index is around -0.15, but the bias 271 (with respect to volume integration) of the results for geometry c is around -0.23; thus, the fact that also the 272 results for geometries a and b are biased, results in a bias higher than ISC in geometry c. The conclusion that 273 the true bias of the results in geometry c is larger than ISC is valid, even if the reference values from Table 2 274 might also have a small bias. 275 In a correct approach to the computation of FC the self-consistency index should be zero. If this condition is 276 not met, certainly there should be some approximations and inconsistencies in the computational method. 277 Therefore it is interesting to correlate the computed magnitude of the coincidence-summing effects (1-FC for 278 usual peaks and FC for pure sum peaks) for the case of configuration (a) with the self-consistency index. 279 Several examples are given in the figures 6-8. It can be observed that generally many points belong to one of 280 two groups, the first around ISC=0 (self-consistency fulfilled) and the other around ISC=-0.15 (quasi-point 281 source approximation). In the group around ISC=0, the magnitude of coincidence-summing effects for a 282 given energy is roughly the same, whereas in the group around ISC=-0.15, the values are spread over a 283 wider range. In addition, there are also points that do not belong to any of these two groups, as evident in 284 Fig. 7. In these cases the decay schemes (133Ba and 152Eu) are more complex and X-rays contributions to the 285 coincidence effects are significant. The differences between specific computation codes in the description of 286 the X-rays contributions, as well as the possible differences in the decay data libraries, might be responsible 287 for these points being located apart from the two groups. 288 From the point of view of activity calculation, the most important quantity characterizing the measurement 289 is the apparent full energy peak efficiency  app. Generally, for usual peaks the index ISC(AE) is not higher 290 than 4 to 5%; in most cases it is close to 0. This is due to the fact that  app for normal peaks is equal to the 291 product FC  . Because FC is relatively small with respect to 1, even if the relative bias of 1-FC is about 15%, 292 as in the case of quasi-point source approximation, the effect on FC and  app is much smaller and 293 consequently, the relative bias introduced in the value of  app is also small. Thus FC and  app for usual peaks 294 are weakly sensitive to the approximations applied in the computation. This is in contrast with the case of 295 pure sum peaks, for which the value of ISC(AE) is about the same as ISC(NP), around -0.15, when the 296 quasi-point source approximation is applied in the computation. In Figure 9 the self-consistency index of the 297 apparent efficiencies is represented as a function of energy, but keeping only the results of the sets for which 298 at least one absolute value of ISC(AE) is higher than 0.05. 299 Concerning the uncertainties represented in the figures, they correspond to one standard deviation. The 300 values were computed on the basis of the data reported by the participants using uncertainty propagation 301 formula. In this context, the uncertainties of ISC(NP) and ISC(SP) should be interpreted with some caution, 302 because they were computed assuming that the values of FC and of  are not correlated. This is true in the 303 case of some sets of reported results, but not in all cases; however, as the covariance between FC and  was 304 not provided by the participants, it was always disregarded. 305 The proposed self-consistency test may reveal the existence of some problems in the method for the 306 evaluation of the coincidence-summing effects. However, it may be insensitive to some features of the 307 computation method, even if they imply specific approximations and led to biased results. A trivial example 308 is the following: if all efficiencies are multiplied by the same scale factor, the value of ISC for the peak 309 efficiency remains unchanged. Also, ISC(NP), ISC(SP) and ISC(AE) are insensitive to a scale factor applied 310 to all efficiencies if only pair coincidences contribute to summing effects. An example A more complex 311 example is provided by the angular correlation effects. These effects were included only in one set of results. 312 This set passed the self-consistency test, as well as many sets in which angular correlations were neglected. 313 In Table 3 the magnitude of the effect of angular correlations on FC is presented. Of course, the 314 discrepancies with respect to isotropy are higher in the case of geometry b, corresponding to the smallest 315 solid angle. In the case of pure sum peaks, it is higher than 7% in geometry b, but it is not negligible also in 316 the other geometries. The effect on FC for usual peaks is much lower, but the effect on 1-FC for these peaks 317 is almost the same as in the case of pure sum peaks. The small difference between the effect on 1-FC for 318 1173 and 1332 keV and on FC for 2505 keV, all subjected to the same angular correlations, is due to the 319 integration over a larger solid angle in the first case (higher effective solid angle for any interaction than for 320 complete energy deposition in the detector). 321 322 6. Summary and conclusions 323 A self-consistency test to check some features of several procedures applied for the evaluation of 324 coincidence-summing corrections in the case of extended sources was proposed. The test is based on exact 325 relations between the values of FC and of the peak efficiencies calculated for three ideal measurement 326 conditions. These relations should be valid in the case of correct procedures for the evaluation of FC. The 327 test parameter proposed is the index of self-consistency ISC, equal to 0 if the relations are valid. A value of 328 ISC significantly different from 0 suggests that there are some problems in the computation of the 329 coincidence-summing effects and indicates qualitatively the magnitude of the bias of the results. 330 The main purpose of this GSWG action was to provide to each user of gamma-ray spectrometry a simple 331 and convincing test of the quality of the procedure applied in his laboratory for the computation of 332 coincidence-summing corrections. The test is particularly useful for emphasizing the limitations of the 333 quasi-point source approximation in the calculation of coincidence-summing corrections for extended 334 sources. In the test, vacuum (or air) sources are considered, whereas actual matrices of the sources present 335 non-negligible photon interaction probabilities; in such measurements the expected bias is higher, because 336 efficiency dependence on the position of the emission point should be stronger than in the case of vacuum 337 sources (Arnold and Sima, 2001). 338 As with any test, the hypothesis of correctness of the procedure can be ruled out if ISC significantly differs 339 from 0, but the fact that ISC is close to 0 is not a proof of the correctness of the procedure. Therefore the 340 proposed test should be complemented by other tests, including experimental validation of the results. 341 342 References 343 Agostinelli, S., et al., 2003. GEANT4-a simulation toolkit, Nucl. Instrum. Methods A 506, 250-303. 344 Allison, J., et al., 2006. Geant4 developments and applications, IEEE Trans. Nucl. Sci., 53, 270-278. 345 Andreev, D.S., Erokhina, K.I., Zvonov, V.S., Lemberg, I.Kh., 1972. Consideration of cascade transitions in 346 determining the absolute yield of gamma rays. Instrum. Exp. Technol. 15, 1358 (English translation). 347 Arnold, D., Sima, O., 2001. Total versus effective total efficiency in the computation of coincidence 348 summing corrections in gamma-ray spectrometry of volume sources. J. Radioanal. Nucl. Chem. 248, 365-349 370. 350 Arnold, D., Sima, O., 2004. Application of GESPECOR software for the calculation of coincidence 351 summing effects in special cases. Appl. Radiat. Isot. 60, 167-172. 352 Arnold, D., Sima, O., 2006. Calculation of coincidence summing corrections for X-ray peaks and for sum 353 peaks with X-ray contributions. Appl. Radiat. Isot. 64, 1297-1302. 354 Berlizov, A., 2012. A correlated particle source extension of a general purpose Monte Carlo N-particle 355 transport code, MCNP-CP Upgrade Patch Version 3.2. Institute for Nuclear Research National Academy of 356 Sciences of Ukraine, 2012 357 Blauuw, M., Gelsema, S. J., 2003. Cascade summing in gamma-ray spectrometry in marinelli-beaker 358 geometries: the third efficiency curve. Nucl. Instrum. Methods A 505, 311-315. 359 Briesmeister J.C., (Ed.), 2000. MCNP-A General Monte Carlo N-Particle Transport Code, Version 4C. Los 360 Alamos National Laboratory Report LA-13709-M, Los Alamos. 361 DDEP webpage: http://www.nucleide.org/DDEP_WG/DDEPdata.htm 362 Debertin, K., Helmer, R., 1988. Gamma- and X-Ray spectrometry with semiconductor detectors, North 363 Holland (1988). 364 Dias, M. S. et al., 2018. SUMCOR: Cascade summing correction for volumetric sources applying MCNP6. 365 Appl Radiat Isot 134, 205-211. 366 Dryak, P., et al., 2012. Correction of the TRUE summations in volume sources. Appl Rad Isot 70, 2130-367 2132. 368 García-Toraño, E., et al., 2017. Simulation of decay processes and radiation transport times in radioactivity 369 measurements. Nucl. Instrum. Methods B 396, 43-49. 370 F C(ISOT) calculated assuming isotropy of each radiation, FC(AC) calculated including the angular correlation of F C (ISOT)/ F C (AC) - 1 (1 - F C (ISOT))/(1 - F C (AC)) - 1 Geom etry a Geometry b Geometry c Geometry a Geometry b Geometry c - 0.004 - 0.003 - 0.004 0.032 0.073 0.037 - 0.004 - 0.003 - 0.004 0.032 0.073 0.037 - 0.006 - 0.005 - 0.006 0.032 0.080 0.038 - 0.007 - 0.005 - 0.006 0.036 0.084 0.042 0.037 0.076 0.042 0.036 0.075 0.041 513 514 515 516 Fig. 1. Measurement configurations 517 518 519 520 Fig. 2. Self-consistency index for 60Co peaks. The values of ISC(NP) for the 1173.23 keV and 1332.49 keV peaks, and of ISC(SP) for the 2505.75 keV peak are 521 displayed in function of the code of the participants. The uncertainty bars correspond to 1 σ. 522 523 524 Fig. 3. Self-consistency index for 133Ba peaks of energy 276.40 keV and 356.01 keV. The values of ISC(NP) are displayed in function of the code of the participants. 525 526 Fig. 4. Self-consistency index for several peaks of 134Cs. The values of ISC(NP) for the 569.33 keV and 604.72 keV peaks, and of ISC(SP) for the 1400.59 keV peak are displayed 527 in function of the code of the participants. 528 529 Fig. 5. Self-consistency index for 152Eu peaks of energy 121.78 keV and 1408.01 keV. The values of ISC(NP) are displayed in function of the code of the participants. 530 531 Fig. 6. The distribution of the self-coincidence index as a function of the magnitude C of coincidence summing effects in configuration (a) for the 1173.23 keV peak of 60Co. 532 533 Fig. 7. The distribution of the self-coincidence index as a function of the magnitude C of coincidence summing effects in configuration (a) for the 356.01 keV peak of 133Ba. 534 535 Fig. 8. The distribution of the self-coincidence index as a function of the magnitude C of coincidence summing effects in configuration (a) for the pure sum peak of 60Co at 536 2505.75 keV. 537 538 Fig. 9. The self-consistency index for the apparent peak efficiency for the sets of results in which at least one absolute value of ISC(AE) > 0.05. The uncertainty bars were 539 removed for avoiding a too clumsy figure. 540 541