A method for the determination of counting efficiencies in gamma-spectrometric measurements with HPGe detectors

Nuclear Instruments and Methods in Physics Research A 382 (1996) 495-502 NUCLEAR INSTRUMENTS 6METnanS IN PHYSICS RESEAncH Section A method for the determination of counting efficiencies in y-spectrometric measurements with HPGe detectors J.P. Bolivar”‘“, R. Garcia-Tenoriob, M. Garcia-Le6n” “Dept. Fisica Aplicada, E.P.S. La Rdbida, Universidad de Huelva, 2/8/9-Pales, Huelva, Spain hDept. Fisica Aplicada, ETS Arquitectura, Avda, Reina Mercedes sin, 4lOl2-Sevilla, Spain L Facultad de Fisica, Universidad de Sevilla. Apdo 106-5, 41080-Sevilla, Spain Received 2 January 1996; revised form received I I June 1996 Abstract In this paper a general method for y-ray efficiency calibration is presented. The method takes into account the differences of densities and counting geometry between the real sample and the calibration sample. It is based on the y-transmission method and gives the correction factorf as a function of Ey, the density and counting geometry. Altough developed for soil samples, its underlying working philosophy is useful for any sample whose geometry can be adequately reproduced. 1. Introduction Generally speaking, y-ray spectrometry provides a fast, multielemental and non-destructive method of radioactivity measurements which is widely used for studying the presence of radionuclides in nature. Although less sensitive than traditional radiochemical methods, all the above mentioned features make such technique superior to them in tnany cases [l-4]. In y-spectrometry the formula for the calculation of the activity concentration, A, of a given radionuclide in a sample of mass (or volume) M is ,‘+-fJPy EM where n is the net count rate under the full-energy peak corresponding to the photon of energy E, emitted by the radionuclide of interest with an emission probability P., and M is the mass (or volume) of the sample. The factor l is the full-energy peak efficiency corresponding to E,. The accurate determination of E is the key problem when y-spectrometry is used for radionuclide measurements. It is, at the same time, the most difficult problem to be successfully solved. As it is known, given a counting geometry, E depends on E,, the density and the composition of the sample. Therefore, by obtaining l (Ey) for a calibration sample, with the same density, A and similar composition as the samples of unknown activities, the problem, is solved, since such function E(E,) can be *Corresponding author. confidently used in Eq. (I ) to obtain A. This is a very common practice in laboratories engaged in environmental radioactivity determinations, since, in principle, it is an easy task. However, it is very difficult to obtain calibration samples with identical p as the real samples. Furthermore, very frequently it is impossible to prepare calibration samples from real samples since not always one has as much an amount of sample as would be needed [5]. As an alternative, many laboratories when, for instance, measuring soil samples from a given geographical area use one of the soil samples as calibration sample. Nevertheless, this practice, although simple, gives rise to systematic errors in A. Indeed, the obtained e(EY) cannot be rigorously applied to all the samples since their densities are not identical [6-g]. Thus, each natural sample has its own l (EY), which should be found for each of it. This is not a realistic approach to the problem of l (Ey) measurement because it is time consuming and many times impossible to accomplish. In our tackling with the measurement of y-emitters in soil samples from the Southwest of Spain we have dealt with samples of clearly different p. It has induced us to develop a method for E determination which helps to overcome the problems described above. Thus, we present in this paper a general method of E determination for each real sample from the function e(E,) obtained for a calibration sample. The method takes into account the differences of density, and even of counting geometry, between the real and the calibration samples, because the soil samples studied have similar compositions [9]. It is based on the transmission method developed by Cutshall et al. [IO] and modified by Joshi Ill]. and allow 0168-9002/96/$15.00 Copyright 01996 Elsevier Science B.V. All rights reserved PII SO168-9002(96)00790-S 496 J.P. Bolivar et al. I Nucl. Instr. and Meth. in Phys. Res. A 382 (1996) 495-502 the direct determination of the full-energy peak efficiencies with relative uncertainties lower than 10%. After presenting the detector and samples used in this work in Section 2 we describe the method in Section 3. Our results are validated in Section 4 and some conclusions close the paper in Section 5. 2. Experimental The calibration procedure was developed for a conventional HPGe coaxial detector, 1.88 keV resolution and 14% relative efficiency linked to standard electronics and to a PC-based 4k multichannel analyzer. Measurements of low levels of radioactivity were possible by surrounding the detector with 10 cm thick lead shielding, internally lined with 2 mm of copper. A cylindrical container, 6.50 cm in diameter, was always used for sample counting, and placed on the top of the detector. The soil selected for calibration was dried overnight at 105°C and subsequently powdered and homogenized. The other samples were prepared in an identical way. A standard “‘Eu solution was used to determine efficiencies at calibration counting conditions. “‘Eu yemissions of low or poorly known emission probability [ 121 were rejected and so only ten y-lines, covering an energy range from 1201500 keV, were chosen to perform the efficiency calibration process. This is the energy range interesting for us and measurable by our conventional detector. Thus, the soil calibration sample (apparent density p, = 1.48 g cm--‘) was spiked with a well known “‘Eu activity. To assure perfect mixing of the spike in the soil matrix, several homogeneity tests were carried out measuring aliquots at the same conditions. As soon as homogeneity was confirmed we proceeded to the calibration process. 3. The method The set of full-energy peak efficiencies, e,, obtained for the calibration soil cannot be used in Eq. (1) to obtain A in a real soil sample. Instead, l , (set of efficiencies for the real sample) is needed which, according to Ref. [IO], for a given counting geometry can be described as I =fc (2) where f is the transmission factor which accounts for the self-absorption relative differences between the real sample and the calibration sample. In fact, f is defined as f=S/C (3) with S and C being respectively the net count rates under the full-energy peak of interest produced by the same activity in the real and in the calibration soil samples. o,,L-- -1, / _I 40 60 120 160 200 240 260 ~66 @I Fig. 1. Full-energy peak efficiency vs sample mass for different energies in the calibration sample. When the counting geometry is cylindrical, as in our case, f can be determined experimentally by performing some photon transmission experiments [lO,ll]. Thus, if S,, and C, are respectively the net count rates under the fullenergy peak of interest obtained by counting a y-emitting point source placed at the top of the real and calibration samples, f becomes 1 - S”/C,, ] _ e-c/l,-r,lL f= ln(C,,/S,) = (K - &L,)L ps and ~4 are the linear attenuation coefficients for the real and calibration soil sample at the energy of interest, L is the cylinder height (or sample height). It is clear that for a given sample height, L, f depends on the real sample density, p,, and the photon energy, E,. So, f should be determined for each p, and E, one is interested in Ref. [ 131. For multi-elemental determinations this practice is time consuming. It is worthwhile, therefore, to find the functional dependence off on p,, EY and L. 100 ENERGY ( keV ) 1ooo Fig. 2. Logarithm of parameter (I of Eq. (5) vs y-emission energy. J.P. Bolivar et al. I Nucl. lnstr. and Meth. in Phys. Res. A 382 (1996) 495-502 491 This will be tried in what follows. However, firstly we will determine the dependence of l , on E, and on L. l J%] = 624Ey -09”yexp(-4.87 X 10m’Vc) (x’, = 0.92) 3. I. l , Determination where E, is expressed in keV and Vc in cm’. ec was determined for each “‘Eu -y-energy selected and for a wide range of L values, i.e. sample mass values, since the diameter of the cylinder is constant. In Fig. 1 we plot l ,(%) against calibration soil sample mass, M,, for some EY values. For each energy it is found that where a and j3 are parameters which depend on E,, and p, and VL are the density and sample volume. In deriving Eq. (8), the corrections due to true coincidence summing effects, related to cascade gamma emissions of lZEu were considered. These corrections were measured experimentally for the different geometries applying the method of Quintana and Fernandez [ 161, and are low (generally less than 10%. especially for the high volumes) as can be expected from the low relative efficiency of our coaxial detector (14%). We have also observed, that if these corrections are not considered, their effects are smoothed in the fittings, producing errors in the estimation of Ed, that range from 5% to 10%. The meanings of a and /3 are clear. a is the full-energy peak efficiency for L = 0, that is, the efficiency function for a disk source of the same diameter as that of the cylinder at the position of the cylinder bottom. On the other hand, p or /?p, gives the relative variation of eL per unit of added mass or volume. 3.2. f Determination In Fig. 2 and Fig. 3 we can see the dependence of a and /l on Ey. According to the above commented meaning of a and, as it has been previously found by other authors [ 14_ 151. a can be adequately described as a potential function of E,. This is confirmed by us, and we obtain The functional dependence off on Ey, p, and L has been determined by performing transmission experiments through soil samples with apparent densities ranging from 0.6 to 1.7 gem-‘. To achieve it, gamma emitting point sources of 22hRa, “‘Cs and “‘Co were used and five heights (L = 1, 2, 3, 4 and 5 cm) explored. In this way a good range of p,, L and EY was studied by using the counting geometry previously described. lna[%] = 6.44 - 0.9091n(E1 IE,) (xi = 1.80) (6) where E,, = 1 keV As it can be seen in Fig. 4 and Fig. 5 the variation of,f with p, for a given EY and L can be described by It is seen, on the other hand, that p is almost independent of E,, at least for EY > 300 keV, and slightly depends on the energy for smaller Ey. Thus, the average value of p in the 300-1500 keV energy interval is (/3)p, = (4.871tO.10) X IO-’ [cm-‘] Finally, E,. becomes (71 where a and b are two parameters which depend on E, and L as it can be deduced from the data presented in Table 1. There we give the results of the fit according to Eq. (9). It is easy to find the meanings of a and b. Indeed, p, ENERGY (k&J 1000 Fig. 3. Parameter p of Eq. (5) vs -y-emission energy (8) I - 352 keV 11 0.6 / _-._A_ ~___-1._.__J__~ 1 0 0.3 0.6 0.9 1.2 1.5 1.6 DENSITY ( 9 cm 3 ) Fig. 4. Correction factor, f. vs apparent density of soil samples for E, = 352 keV and L = 5.0 cm. 498 J.P. Bolivar et al. I Nucl. Instr. and Meth. in Phys. Res. A 382 (1996) 495-502 * 662 keV 0.6 L-1I 0 0.3 0.6 0.9 1.2 1.5 1.0 DENSITY (gem”) Fig. 5. Correction factor, J vs apparent density of soil samples for E, = 662 keV and L = 5.0 cm. Table 1 Experimental values obtained for the parameters a and b (g- ’ cm2) of Eq. (9), together with the reduced chi-squared xi, obtained applying the least-square fitting method, for every height and energy of interest. * = Value obtained for the parameter is not significant at 95% level Energy [keV] L = 4.0 cm L = 5.0 cm a b xi a b xi 186.2 242.0 295.2 351.9 609.3 661.7 1120. 1175. 1333. 1764. Energy [keV] 1.653 0.327 0.453 1.692 0.377 1.042 1.480 0.265 1.238 1.610 0.367 1.580 1.406 0.235 0.772 1.509 0.285 0.983 1.335 0.205 1.092 1.444 0.259 2.196 1.261 0.162 2.143 1.328 0.201 3.983 1.248 0.158 3.423 1.328 0.200 2.900 1.210 0.157 0.829 1.245 0.165 0.763 1.181 0.121 0.687 1.242 0.153 0.269 1.142 0.103 0.954 I.244 0.145 0.467 1.155 0.096 0.323 1.174 0.119 0.605 L = 2.0 cm L = 3.0 cm a b xi a b x:, and /_L~ of Eq. (4) can be transformed into rl, and rl, being respectively the mass attenuation coefficients for the real and calibration samples. Thus we obtain 1 _ e-tn,P,-a,P,)L 1 -e-’ f= =- (rl,P, - Il,P,)L x (10) Obviously, x k (n,p, - 71,p,)L. For n << 1 we obtain 186.2 242.0 295.2 351.9 609.3 661.7 1120. 1175. 1333. 1764 f = 1 _ ix ^- e-xl2 = ea,P,L’~e-n&/~ = ae-hPs (11) Energy [keV] 1.187 0.145 1.220 0.136 1.172 0.111 1.143 0.101 1.109 0.075 1.112 0.082 1.081 0.067 1.099 0.062 1.073 0.051 I .063 0.052 L = I .O cm a b 0.738 I.386 0.229 0.260 0.644 1.347 0.211 0.798 0.518 1.259 0.169 0.580 1.409 1.229 0.153 1.090 0.558 1.173 0.114 1.015 3.45 1 1.169 0.117 4.510 0.647 1.150 0.113 0.739 0.73 1 1.135 0.09 1 0.863 0.917 1.139 0.084 I.455 0.540 1.149 0.08 1 1.056 XR where a = exp(nCpCL/2) and b = qSL/2. The value of f from Eq. (11) is approximately equal to that from Eq. (IO), their differences being lower than 2% for E, > 122 keV. On the other hand, a = e”‘, being a’ = 7,9,L/2, can be approximated for our calibration sample by 1 + a’, and we obtain f =( 1 + yL)e-wP2 = (, + a,)e-hps (12) 186.2 1.136 0.088 1.136 242.0 1.133 0.089 1.071 295.2 1.067 0.052 0.626 351.9 1.049 0.044 I .332 609.3 1.048 0.032 0.609 661.7 1.050 0.039 2.387 1120 1.065* 0.038* 0.614 1175 1.026* 0.027* 1.614 1333 I .037* 0.027* 0.862 1764. 1.009* 0.023 * 0.838 where the differences between a = e”’ and 1 + a’ are less than 6% for E, > 300 keV. Obviously a’ and b depend on E, as the mass attenuation coefficients do. For E, > 100 keV the relationship between n and E, is potential [ 17,181. And this is what we observe in Fig. 6 for L = 5 cm. The same dependence is found for the other L values studied (L = 1, 2, 3 and 4 cm). On the other hand, also from their definitions, a’ and b depend linearly on L. With this in mind the following functional forms for a’ and b can be guessed a’ = C’E,d’, b = cE;” (13) c’ and c being two parameters which depend linearly on L, E, in keV, while d’ and d are two constant quantities proper of the calibration process. The data of Table 2 reveal that Eq. ( 13) was a good choice for fitting a’ and b (derived from data in Table 1) to E, for different L. On the other hand, in Fig. 7 and Fig. 8 it is shown how c’ and c increase linearly with increasing L values and how d’ and d are basically constants for our L range. J.P. Bolivar et al. I Nucl. Instr. and Meth. in Phy. Res. A 382 (1996) 495-502 499 ENERGY (keV) v Fig. 6. Parameters a’ = a - 1 and b (Eq. (9)) vs energy for the gamma emissions studied with L = 5.0 cm. Since L is proportional to V, that is proportional to M,/p,, c’ and c can be simply expressed as functions of the type kM,Ip,. By fitting the data in Fig. 7 and Fig. 8 to a linear function, it is found that c = (0.0450.28) + (0.0243-‘0.0028): s c’ = - (1.6k1.2) + (0.074’0.012): (14) l where M is expressed in grams and p, in g cmm3. On the other hand d’ = 0.543?0.015 and d = Table 2 Values obtained for tire parameters c’, d’, c, d (Fq (13)) and the reduced chi-squareds ,&. obtained applying the least-square fitting method, for each height considered. * = Value obtained for the parameter is not significant at 95% level V [cm’] L [cm] c’ d’ Xf 33.1 1 .o 67.4 2.0 101 3.0 135 4.0 168 5.0 V [cm’] L [cm1 33.7 I .o 67.4 2.0 101 3.0 135 4.0 168 5.0 2.0*2.0* 3.3kl.l 4.62 1.5 10.923.2 11.1+1.5 c ]g-‘cm’] 1.48?1.0* 1.55 ?0.35* 2.40?0.50 3.25kO.90 4.2210.50 0.57?0.16 2.24 0.52~0.06 1 SKI8 0.50+0.05 1.858 0.58iO.05 1.113 0.54+0.02 1.127 d X: 0.57-co. 11 0.835 0.46?0.04 1.150 0.46?0.04 1.010 0.46ZO.04 1.605 0.47+0.02 1.251 L._~..___ L_ _I- ..-i._ .i_Jo 0 1 2 4 5 6 SAMPLE HEIGHT (cm) Fig. 7. Parameters c’ and d’ of Eq. (I 3) vs sample height (L) 0.485~0.021 on the average. Thus. taking the average value of k for c and c‘ (15) and, consequently f= 1 + 0.061 7E~U.i4’~ exp(-0.0243E;” “‘M,) \ (16) which is the functional dependence off on ET, p, and M (or L) we were looking for. oGL--_ -I--1. _~~I. _L 0 1 2 3 4 5 SAMPLE HEIGHT (cm) ‘C “d Fig. 8. Parameters c and d of Eq. (13) vs sample height (L) 500 Table 3 J.P. Bolivar et al. f Nucl. Instr. and Meth. in Phys. Res. A 382 (1996) 495-502 Average specific activities determined in an intercomparison exercise, and in our laboratory, measuring a marine sediment prepared by CIEMAT Radionuclide Energy [keV] Specific activity of our laboratory [Bqlkg] Average specific activity of the intercomparison [Bq/kg] 295.2 79.424.3 74.028.0 351.9 77.5*4.0 74.92 10.2 609.3 66.2k3.4 69528.4 1764.5 77.1 -c4.4 74.729.7 300 42.2t4.0 43.5k7.4 911 36.922.2 39.524.2 661.7 12.1+0.7 12.9?1.3 1460.8 1413270 14801170 The uncertainty off is expected not to exceed 6% even in unfavourable cases and may be as low as 1% for energies higher than 300 keV and density differences up to 0.5 gcme9. By substituting Eq. (16) in Eq. (2) we have the general efficiency calibration equation for our measurement technique. 4. Validation tests The validity of Eq. (2) and Eq. (16) has been tested by performing some tests. We present here the results obtained for a few of them. In Table 3 we present the activities measured for a marine sediment distributed by the Spanish CIEMAT (Centro de Investigaciones Energeticas, Medio Ambientales y Tecnologicas) among a good number of Spanish environmental radioactivity laboratories. The sediment was counted in our geometry and had an apparent density of 1.66 g cmm3. By applying Eq. (16) and Eq. (2) we obtained the results presented in column 3 of Table 3. In the column 4 of the same Table the average specific activities calculated in the intercomparison exercise are given. It is obvious that our results agree very well with them within the quoted la uncertainties. The results of Table 3 provide an additional validation test. As it should be expected we found, within the uncertainty limits, the same specific activity for those radionuclides which are expected to be in secular equilibrium, e.g. for the pair *‘4Pb-2’4Bi. Since their activities were calculated by using different E,, this simple exercise served us to see the validity of Eq. (16) within a wide energy range. In Table 4 we compare the 228Th activities obtained in different sediments by using our y-spectrometry method and the traditional a-spectrometry method. “‘Th was determined both by measuring the 583 keV y-emission from ‘08T1 and the 911 keV y-emission from 228Ac(228Ra). This exercise was done for several soils with different densities, p,, and the counting performed under our cylindrical geometry with L = 5.0 cm. The results show a general noticeable agreement between a-spectrometric and both y-spectrometric determinations. This confirms the goodness of our approach. The exceptions are samples SOT5 and SOT7 for which the agreement is not so obvious. For these cases, however, a-spectrometry determinations are compatible with 228A~ measurements within 2~ The reason for such disagreement is not clear. Table 4 Results of measurements done in sediment samples from South of Spain. *?‘h and ***Ra specific activities were determined by y-ray soectrometrv and also zz*Th bv a-smctrometrv. L _ . Samnle Density lg cm-‘1 2*sTh(208Tl) (583 keV) 22*Th a-spectrometty *2*Ra(22*Ac) (911 keV) SOT1 1.57 20.3 ? 1.4 SOT2 0.98 64.123.6 SOT3 1.69 21.621.5 SOT4 1.09 39.322.8 SOTS 0.96 38.2Z2.5 SOT6 0.94 64.323.6 SOT7 0.89 49.623.1 SOT8 1.00 44.822.8 SDRl 0.76 61.823.6 SDR2 0.93 41.822.5 SDR3 0.90 36.8-tZ.4 SDR4 1.02 37.622.2 19.0+1.1 64.623.1 19.321.3 40.922.4 52.0t2.8 65.8k3.7 61.953.1 43.Ok2.3 62.lr3.1 34.922.6 34.22 1.9 34.825.7 19.9+1.7 58.123.8 20.7+ 1.7 46.723.6 40.2%3.2 58.124.0 50.6k3.8 43.2k3.2 60.9k4.3 43.022.8 39.823.0 40.1 k2.8 J.P. Bolivar et al. I Nucl. Instr. and Meth. in Phys. Res. A 382 (1996) 49-f-502 Table 5 Fittings obtained for Eq. (17). applying the least-square method, for every energy considered 501 Nuclide Energy [keV] P, WI N, [cps/30 gl p/3 [lo-‘cm-‘] 2ZhRa + 21” 186.2 3.51 +.57.2 2.32O-tO.024 6.3420.12 “‘Pb 295.2 18.2 I .346+0.033 5.79ZO.29 “‘Pb 351.9 35.1 1.177?0.014 5.74+0.14 “JBi 609.3 44.6 0.600+0.008 5.4410.08 ?14Bi 1120.3 14.7 0.3311-0.005 5.02?0.18 “‘Bi 1764.5 15.1 0.236?0.009 4.99t0.40 Not obvious analytical errors in the a-spectrometric determinations cannot be excluded [ 191. Nevertheless, the goodness of our method is apparent if one observes the whole set of results. Finally. we have performed an additional exercise to check the validity of Eq. (5) and the assumption of p being a constant. For that, a powdered phosphate rock sample (p = 1.54 g cm ‘) was used. This sample contained high radioactivity concentrations of U [?J] of about 1200 Bqlkg, and daughters. It allowed a more precise determination of the parameter p. Different amounts of the sample (M and, consequently, L or V) were measured in our cylindrical geometry. According to Eq. (1). E is proportional to N = nI(MPY), where n is the net count rate under full-energy peak of interest in counts/s. Consequently, if Eq. (5) is correct it should be N= N,,e w (17) This function was fitted for each energy considered in our exercise to the experimental values obtained for the difference sample amounts. The N,, and p/J values are 10, 0.1 -_-_ 1-l I up 100 t.ooO ENERGY (keV) IO ? E ; ; n & I Fig. 9. Plot of the parameter N0 and p (Eq. (17)) vs y-ray energy for the gamma emissions studied. given in Table 5. The fitting was very satisfactory with values of xi close to unity. And, as it was expected, N,, depends on E, as a does (see Fig. 9). However, the parameter p (or pp) shows a slight dependence on EY as it is observed in Fig. 9. This is demonstrating that, as expected, the effect of self-absorp tion on p is detectable when the accuracy of tneasurements is improved. However, the maximum difference between pp obtained from Eq. (7) and the average of p/3 from Table 5 (Fig. 9) is only around 6%. This means that the use of Eq. (7) is essentially correct for our purposes. 5. Conclusions A general method for y-ray efficiency calibration of Ge coaxial detectors is presented. It is based on the transmission method and, in fact, is a generalization of such procedure. The transmission factor, f. is described as a single function of E,, sample density and geometry. Although the method has been developed for soil samples, its underlying philosophy makes it useful for any sample for which the measuring geometry can be adequately reproduced. Acknowledgments This work has been partially supported by ENRESA, the U.E. project F13-P-CT92-0035 and the project of Junta de Andalucia “Estudio de 10s flujos de radioniclidos en la producci6n de pasta de papel y su aplicaci6n en la optimizaci6n del proceso industrial”. References Ill I21 131 I41 151 F. El-Daoushy and R. Garcia-Tenorio, Nucl. Instr. and Meth. A 356 (1995) 376. P.G. Appleby, P.J. Nolan, F. Oldfield, N. Richardson and S.R. Higgit, Sci. Tot. Environ. 69 (1988) 157. M.E. Kitto, Int. J. Appl. Instr. Part A 42 (1991) 835. E. Cincu, Nucl. Instr. and Meth. A 312 (1992) 226. R.S. Seymour, M.S. Andreaco and .I. Pierce, J. Radioanal. Nucl. Chem. Art. 123 (1988) 529. 502 J.P. Bolivar et al. I Nucl. Instr. and Meth. in Phys. Res. A 382 (1996) 49.5-502 [6] G. Harbottle, Radioact. and Radiochem. 4 (1993) 20. [7] A.S. Murray, R. Marten, A. Johmston and P. Martini, J. Radioanal. Nucl. Chem., Art. 115 (1987) 263. [S] M.E. Kitto, J. Radioanal. Nucl. Chem. Lett. 145 (1990) 175. [9] J.E. 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