- Email: [email protected]
A hybrid method to compute accurate efficiencies for volume samples in γ-ray spectrometry
ARTICLE IN PRESS
Applied Radiation and Isotopes 60 (2004) 227–232
A hybrid method to compute accurate efficiencies for volume samples in g-ray spectrometry M. Garc!ıa-Talavera*, V. Pen˜a LIBRA, University of Valladolid, Centro I+D, Campus Miguel Delibes, Valladolid 47011, Spain
Abstract In recent years, Monte Carlo (MC) methods have been increasingly applied to cope with variability in photopeak efficiencies due to matrix effects. But to obtain proper results only by numerical simulation, especially at low energies, sample bulk density and chemical composition must be well characterized. In this paper, we propose a method that combines both experimental measurements and MC simulations, being applicable to matrices of unknown composition. A transmission measurement of a 210Pb point source through the sample allows one to compute accurately its photopeak efficiencies at energies above 46.5 keV. The method is validated for several inorganic and organic matrices measured in Petri dishes geometry. r 2003 Elsevier Ltd. All rights reserved. Keywords: g-ray spectrometry; Monte Carlo; Efficiency; Self-absorption
1. Introduction Self-absorption effects in volume samples are the main drawback to perform accurate analyses by g-ray spectrometry at low energies. Differences in the physico-chemical characteristics of the calibration samples and the sample under study may introduce errors in the activity results. Nevertheless, due to the large variability among environmental matrices, it is not feasible to obtain experimentally an efficiency calibration curve for each of them. Instead, many authors have developed semi-empirical or computational methods to calculate efficiency values or selfabsorption corrections for any given sample. The pioneer method of Cutshall et al. (1983) was devised to compute photopeak efficiencies at 46.5 keV for volume samples of unknown composition. It is based on the transmission measurement of a collimated beam of 210Pb photons through the sample, allowing one to compute its linear attenuation coefficient ðml Þ: Its main shortcoming is that, due to simplifications in the underlying theoretical model, it provides accurate results only for cylindrical geometries whose thicknesses are comprised in a particular range (Miller, 1987). Several *Corresponding author. Tel.: +34-98-31-83-807. E-mail address: [email protected] (M. Garc!ıa-Talavera).
modifications have been proposed to overcome this limitation and extend it to different geometries (Joshi, 1989; Galloway, 1991), but the assumption that detector and source dimensions are small compared to their distance remains implicit. Monte Carlo (MC) simulations constitute a newer approach to the self-absorption problem (Sima and Arnold, 1996; P!erez-Moreno et al., 2002). These techniques are increasingly applied to gamma spectrometry, particularly to efficiency calculations, since they can reproduce any detector–source configuration without simplification and minimize the experimental work. To obtain accurate efficiencies by means of an MC code, it is essential to implement appropriately the bulk density and the elemental composition of the samples. In particular, neglecting chemical composition effects might lead to substantial errors below 100 keV, even among soil or sediment samples (see Sima and Dovlete, 1997; Garc!ıa-Talavera et al., 2000). Nevertheless, the exact chemical compositions of the matrices under study are usually unknown and costly additional analyses by techniques as ICP-AEA or XRF are required in order to assess them. Alternatively, one can use a collimated beam of energy Eg to measure ml ðEg Þ; which is then input in the MC code to compute the photopeak efficiency at that energy (Overwater et al., 1993).
0969-8043/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2003.11.022
ARTICLE IN PRESS 228
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Inspired by the latter approach, we propose a hybrid method, combining experimental measurements and MC simulations, which presents the following advantages: First, it employs a parameterization of the efficiency as a function of ml in order to avoid performing a new MC calculation for each sample under analysis. Second, if the sample chemical composition is unknown, it is able to provide its ml value at any energy above 46.5 keV, based on a single transmission measurement of a 210Pb source through the sample. To test the accuracy and precision of this method, we apply it to calculate the photopeak efficiencies for several inorganic and organic matrices measured in Petri dishes. In the light of these results, we discuss the convenience of routinely using the hybrid method, compared to approaches that simply assume a standard chemical composition for the matrix.
2. Experimental procedure Experimental measurements were carried out with a Canberra n-type HPGe detector, with 117 cm3 active volume, 25% relative efficiency and resolution of 1.1 keV at 122 keV and 2.0 keV at 1.33 MeV. The associated electronic components are a preamplifier model Canberra 2008 and a DSA-2000 Spectrum Analyzer by Canberra. The recorded spectra were analyzed using Genie-PC software. To test the validity of the MC simulation, several point sources ðPÞ were employed. They were prepared by pipetting a well-known volume (o1% uncertainty) of a standard solution on hydrophobic filters. We used a multi-gamma cocktail DAMRI 9ML01-ELMH05/4 containing 241Am, 109Cd, 57Co, 51Cr, 139Ce, 113Sn, 85Sr, 137 Cs, 88Y and 60Co; and a 210Pb solution provided by CIEMAT. Besides, we employed a Petri dish, of 9-cm diameter and 1.8-cm height, which will be referred to as geometry V : It was filled with distilled water and spiked with known amounts of both standard solutions. For the self-absorption study, several types of matrices, specifically soil, coal, milk powder and dried leaves, were measured in Petri dishes. Two soil samples of different compositions were dried at 40 C for 24 h in a muffle oven and sieved with a 2-mm grid. An anthracite sample was dried in the same conditions and afterwards milled. Grapevine leaves were enclosed in a dryer containing silica gel for several days and milled to obtain a homogeneous powder. A commercial milk powder was employed, requiring no additional processing. For all these volume samples, we determined experimentally their efficiencies at 46.5, 59.5 and 88.0 keV, using photon emissions from 210Pb, 241Am and 109Cd, respectively. To do so, we pipetted known amounts of the two standard solutions mentioned above onto the samples and subsequently dried them over 24 h
at 40 C. The samples were carefully mixed prior to measurement to ensure that the radionuclides were homogeneously distributed in the source. The experimental efficiency at energy Eg for matrix k was computed as ekV ðEg Þ ¼
Ng C g ; pn ðEg ÞAn
ð1Þ
where Ng is the net count rate under the corresponding full-energy peak, An the known radionuclide activity, and pn ðEg Þ the emission probability. The net count rate was corrected by radionuclide decay, if necessary. Cg ; denoting the coincidence summing correction factor, equals 1 for the three considered g emissions. To minimize systematic errors due to improper source homogenization or positioning, two replicates of each sample were prepared and measured. The final photopeak efficiencies were obtained as a weighted average of the efficiency values derived from each pair of samples.
3. Monte Carlo simulation We have applied the MC code GEANT 3.21 to simulate the response of the HPGe detector. To test the accuracy of the simulation, several point sources, with photon emissions covering the energy range 46.5–1332 keV, were measured in various positions; the experimental efficiencies were compared to the ones obtained by MC. To obtain a better agreement among them, some detector parameters, as the crystal inner dead layer, were fitted. After this process, we verified the accuracy of the simulation for geometry V : We measured a Petri dish containing spiked water and compared the MC efficiencies to the experimental ones, derived from 210Pb, 241Am, 57Co, 109Cd, 51Cr and 137Cs photons. All the values agreed within the statistical uncertainties, which were on average 2%.
4. Method development We have devised a method to calculate efficiencies for arbitrary volume samples consisting of three separate stages. In the first one, the efficiency at the energies of interest is parameterized as a function of ml : The functional form adopted for e is the same, but the parameters might vary with energy. When analyzing a matrix of known attenuation coefficient, such function can be directly used to calculate eðEg Þ: For samples of unknown composition, the value of ml ðEg Þ has first to be determined experimentally: the two next stages of the method are designed to accomplish it for energies above 46.5 keV. Next, we present in detail an application of the method to our HPGe detector and a Petri dish
ARTICLE IN PRESS M. Garc!ıa-Talavera, V. Pen˜a / Applied Radiation and Isotopes 60 (2004) 227–232
measuring geometry ðV Þ: Through all the process, special attention is given to the uncertainty calculation.
229
0.09
MC methods can be used to obtain an analytical function relating the photopeak efficiency of the sample at energy Eg ; to its linear attenuation coefficient, ml ðEg Þ: In particular, we have applied the GEANT code (v. 3.21) to calculate such function at 46.5, 60.0 and 88.0 keV. We simulated nk calibration matrices of varying Z-values and bulk densities, which cover the usual range of ml found in environmental samples—selected chemical compositions consist in C, SiO2, CaO and Fe2O3, with densities ranging from 0.5 to 2.0 g/cm3. Their photopeak efficiencies at the energies of interest were calculated by means of the MC code. In Fig. 1, we have plotted the efficiency, eV ðEg Þ; as a function of ml ðEg Þ; for g ¼ 46:5; 60:0 and 88.0 keV. Every data set was fitted by weighted least squares using the Levenberg–Marquardt (L–M) algorithm to the following function:
0.06
0.03 0.0
2.0
4.0
0.0
0.6
1.2
0.00
0.25
0.50
ε 60
0.09
0.06
ð2Þ
where a1 represents the photopeak efficiency for an ideal vacuum source measured also in geometry V ; and a2 is the average length traveled within the source by those photons recorded in the full-energy peak. To simplify the fitting, we defined parameter a01 as a1 =a2 : At every energy, the L–M algorithm provides as output the values of a01 and a2 as well as its covariance matrix ðVij Þ: Using such matrix, we can calculate the uncertainties in the efficiencies, se ; from an application of the propagation of errors formula: 2 2 qe qe s2e ¼ V þ V22 11 qa01 qa2 qe qe V12 : þ2 ð3Þ qa01 qa2 The resulting uncertainties in eV at 46.5, 60.0 and 88.0 keV are below 2% in the considered ml range. For matrices of known chemical composition, as aqueous samples, eV and se can be obtained by substituting in Eqs. (2) and (3), respectively, the value of ml : This, in turn, is calculated as the product of the bulk density, which is easily determined in the laboratory, and the mass attenuation coefficient ðmÞ; whose values are tabulated for all the elements from Z=1 to 92. If matrix sample composition is unknown, one could estimate a value of ml from that of similar materials. In
0.03
0.09
ε 88
1 exp½a2 ml ðEg Þ a2 ml ðEg Þ 1 exp a2 ml ðEg Þ 0 ; ¼ a1 ml ðEg Þ
eV ðEg Þ ¼ a1
ε 46
4.1. Efficiency parameterization
0.06
0.03
µ l (cm-1 ) Fig. 1. Variation of the photopeak efficiency ðeg Þ with the linear attenuation coefficient ðml Þ for a Petri dish measuring geometry, at 46.5, 60.0 and 88.0 keV.
ARTICLE IN PRESS M. Garc!ıa-Talavera, V. Pen˜a / Applied Radiation and Isotopes 60 (2004) 227–232
230 20 18
46.5 keV
16 14
u (%) (%
12 10 8
60.0 keV
6 4
88.0 keV
2 0
Air
C
SiO2
CaCO3
Fe2O3
Fig. 2. Relative uncertainty ðum Þ in the efficiency due to a 30% uncertainty in ml ; at 46.5, 60.0 and 88.0 keV. Several representative matrices with bulk density 1 g/cm3 are plotted as reference in the horizontal axis. C is the major component in organic matrices and SiO2, CaCO3 and Fe2O3 are the most abundant constituents of mineral-origin samples.
this case, the error ðsm Þ in the estimate should be accounted for in se by adding the following term to Eq. (3): s2e;m ðqe=qml Þ2 s2m : In Fig. 2, we evaluate this term at 46.5, 60 and 100 keV assuming a relative error in ml of 30%—errors usually exceed that percentage when assigning to the unknown sample the ml (46.5) value of a similar material. From the figure, we can infer that an accurate determination of the efficiency requires a more precise estimate of ml ; especially for mineral samples. 4.2. Transmission experiment When the exact composition of the sample under analysis is unknown, an experimental attenuation measurement can be performed to compute ml ðEg Þ: A 210 Pb source is adequate for this purpose, since environmental analysis by g-spectrometry rarely involve photopeaks at energies lower than 46.5 keV. For a collimated source of 210Pb, the linear attenuation coefficient of the sample can be calculated using the following equation: ml ðEg Þ ¼
0 ln ðN46 =N46 Þ ; t
ð4Þ
0 being the count rate (s1) in the 46.5-keV photopeak N46 due to the source without attenuation, N46 the count rate with attenuation and t the thickness (cm) of the sample. We have modified the classical transmission experiment by measuring a punctual source ðPÞ placed directly
Fig. 3. Schematic drawing of the measuring setup used for the transmission measurements.
over the sample, without collimating (see Fig. 3). Because this measuring configuration maximizes the solid angle detector–source, counting times can be significantly reduced. On the other hand, Eq. (4) cannot be applied because some of the photons impinging the crystal travel distances longer than t within the sample. We used MC simulations to relate ml ðE46 Þ to the 0 transmission factor, TðE46 Þ; defined as N46 =N46 : We reproduced in the code the setup shown in Fig. 3 and calculated TðE46 Þ for the nk calibration matrices. Taking the natural logarithm of TðE46 Þ; the data can be fitted to a straight line through origin by a weighted least-squares analysis (see Fig. 4). Thus, for this measuring setup, we can apply an equation analogous to (4) where t is substituted by the fitted slope of the line, namely tav ¼ 1:85ð2Þ cm. To determine ml ðE46 Þ for a given sample using the previous equation, one has to obtain experimentally TðE46 ): N46 is measured from the transmission of the 46.5-keV photons through the Petri dish containing the 0 matrix of interest and N46 is determined in identical way but using an empty dish. Special care should be taken when placing the 210Pb source, since counting rates are very sensitive to shifts in its position. To calculate the uncertainty in ml ðE46 Þ; the uncertainties in TðE46 Þ and tav should be taken into account. If we set counting times in order to have relative uncertainties in TðE46 Þ lower than 2%, relative uncertainties in ml
ARTICLE IN PRESS M. Garc!ıa-Talavera, V. Pen˜a / Applied Radiation and Isotopes 60 (2004) 227–232
231
To quantify the deviations in the values of m estimated by this procedure, we have performed numerical simulations. We considered the chemical compositions of soil organic matter (C: 56.4%; H: 5.5%; N: 4.1%; S: 1.1%; O: 32.9%), quartz (SiO2) and pure limestone (CaCO3). For each composition i; we calculated mi ðE46 Þ and generated a set of random numbers following a Gaussian probability distribution of mean mi ðE46 Þ and standard deviation 10%. Every number represents an experimental measurement of mi ðE46 Þ; For each of them, we found Zeq and, according to it, estimated mi ðE60 Þ and mi ðE88 Þ: For the three matrices, the means of the resulting data sets agree with the true mi values and the standard deviations at 60.0 and 88.0 keV are lower than 6% and 2%, respectively. Nevertheless, this approach might lead to substantial deviations for matrices with a high content of hydrogen. The reason is that for all elements abundant in environmental matrices the electron density is nearly constant, except for H, which has a higher one and consequently a higher Compton interaction probability. Yet, concentrations of hydrogen above 5–6% are only found in unprocessed vegetal or animal samples, due to its high proportion of water. If the sample is dried or ashed prior to measuring, as is usually the case, most of the H disappears. Only for organic samples treated by wet digestion, this approach is not valid.
would be up to 10% for samples presenting practically no self-absorption and would rapidly decrease to about 1% as ml increases. 4.3. Estimation of m values at higher energies By measuring the transmission of 46.5-keV photons through the sample, we can compute ml ðE46 Þ and, thus, mðE46 Þ: Based on the latter value, it is possible to assign an effective chemical composition to the sample matrix, consisting of a Z-equivalent material. To do so, we built a table containing the values of mðE46:5 Þ for all elements from Z ¼ 1 to 92, based on data from Hubble and Seltzer (1996). For every sample, we select the element, Zeq ; whose tabulated value agrees exactly with the measured one. In case this does not occur, a combination of two elements Z0 and Z0 þ 1 should be taken ðZeq ¼ oZ0 þ ð1 oÞðZ0 þ 1Þ; 0ooo1Þ: The values of m at higher energies are then obtained upon the effective chemical composition assigned to the sample. 6.0
ln(T)
4.0
5. Method validation 2.0
To test the accuracy and precision of the method, we have applied it to compute the photopeak efficiencies at 46.5, 59.5 and 88.0 keV, for soil, coal, milk powder and leaves samples. Values of TðE46 Þ were obtained experimentally from the corresponding transmission measurements. These were used to estimate the mass attenuation coefficients at 46.5, 59.5 and 88.0 keV and, subsequently, the corresponding eV : The calculated efficiencies were compared to the ones found experimentally. As can be seen in Table 1, all the values agree within the statistical uncertainties.
0.0 0.00
1.00
2.00
3.00
µl ( cm-1) Fig. 4. Variation of the transmission factor T with ml at 46.5 keV for a Petri dish measuring geometry.
Table 1 Comparison of efficiencies calculated by the hybrid method ðecg Þ and experimental efficiencies ðeexp g Þ for samples of different bulk densities and chemical compositions Matrix
ec46
eexp 46
ec60
eexp 60
ec88
eexp 88
Soil (silty) Soil (clayey) Coal Milk powder Leaves
0.061(1) 0.067(2) 0.073(2) 0.075(2) 0.076(3)
0.059(2) 0.068(2) 0.073(1) 0.072(2) 0.077(1)
0.070(1) 0.072(1) 0.075(2) 0.076(2) 0.076(2)
0.069(2) 0.070(2) 0.077(1) 0.076(1) 0.075(1)
0.073(1) 0.074(1) 0.075(1) 0.075(1) 0.073(1)
0.074(1) 0.072(1) 0.077(1) 0.075(1) 0.075(1)
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Table 2 Differences ðdg Þ between the efficiencies obtained by assuming a standard chemical composition for the samples and the efficiencies calculated by the hybrid method Matrix
d46 (%)
d60 (%)
d88 (%)
Soil (silty) Soil (clayey) Coal Milk powder Leaves
4.9(2) 11.8(6) 2.3(2) 2.8(2) 1.7(2)
2.8(1) 6.7(3) 1.6(2) 1.1(1) 0.9(1)
0.6(1) 2.0(2) 1.0(1) 0.9(1) 1.0(1)
Furthermore, for this set of samples, we examined the effect in eV of determining m by a transmission experiment instead of assuming an approximate value of m; based on a standard chemical composition. Particularly, we considered a composition of C for the organic samples and of SiO2 for the soil samples. In Table 2 we observe that differences at 46.5 keV are practically negligible for the organic matrices, while for the soils they are up to 12%. At 88.0 keV all differences are below 2%.
6. Conclusions We have devised and validated a method to compute accurate efficiency values for volume samples. We use MC simulations to find a function relating the photopeak efficiency ðeÞ at the energy of interest to the linear attenuation coefficient ðml Þ of the matrix. Application of such function is simple enough if sample chemical composition is known. For matrices of unknown composition, one could estimate an approximate value of ml from that of similar materials. In practice, this procedure is suitable for dehydrated organic samples and, generally, for any type of matrix above 100 keV. Below that energy, a more accurate knowledge on ml is required. To achieve it, we propose a procedure involving an experimental attenuation measurement of the sample at 46.5 keV.
Acknowledgements The authors acknowledge financial support from Iberdrola S.A. and Junta de Castilla y Leon.
References Cutshall, N.H., Larsen, I.L., Olsen, C.R., 1983. Direct analysis of 214Pb in sediment samples: self-absorption corrections. Nucl. Instrum. Methods 206, 309–312. Galloway, R.B., 1991. Correction for sample self-absorption in activity determination by gamma spectrometry. Nucl. Instrum. Methods A 300, 367–373. Garc!ıa-Talavera, M., Neder, H., Quintana, B., Daza, M.J., 2000. Towards a proper modelling of detector and source characteristics in Monte Carlo simulations. Appl. Radiat. Isot. 52, 777–784. Hubble, J.H., Seltzer, S.M., 1996. Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients. http://physics.nist.gov/PhysRefData/XrayMassCoef/ cover.html. Joshi, S., 1989. Determination of 241Am in sediments by direct counting of low-energy photons. Intern. J. Radiat. Appl. Inst., Part A 40, 691–699. Miller, K.M., 1987. Self-absorption corrections for gamma ray spectral measurements of 210Pb in environmental samples. Nucl. Instrum. Methods A 258, 281–285. Overwater, R.M.W., Bode, P., de Goeij, J.M., 1993. Gammaray spectroscopy of voluminous sources. Corrections for source geometry and self-attenuation. Nucl. Instrum. Methods A 324, 209–218. P!erez-Moreno, J., San Miguel, E.J.P., Bol!ıvar, J.L., 2002. A comprehensive calibration method of Ge detectors for lowlevel g-spectrometry measurements. Nucl. Instrum. Methods A 491, 152–162. Sima, O., Arnold, D., 1996. Self-attenuation and coincidence summing corrections calculated by Monte Carlo simulations for gamma-spectrometric measurements with well-type germanium detectors. Appl. Radiat. Isot. 47, 889–893. Sima, O., Dovlete, C., 1997. Matrix effects in the activity measurement of environmental samples. Appl. Radiat. Isot. 48, 59–69.
Applied Radiation and Isotopes 60 (2004) 227–232
A hybrid method to compute accurate efficiencies for volume samples in g-ray spectrometry M. Garc!ıa-Talavera*, V. Pen˜a LIBRA, University of Valladolid, Centro I+D, Campus Miguel Delibes, Valladolid 47011, Spain
Abstract In recent years, Monte Carlo (MC) methods have been increasingly applied to cope with variability in photopeak efficiencies due to matrix effects. But to obtain proper results only by numerical simulation, especially at low energies, sample bulk density and chemical composition must be well characterized. In this paper, we propose a method that combines both experimental measurements and MC simulations, being applicable to matrices of unknown composition. A transmission measurement of a 210Pb point source through the sample allows one to compute accurately its photopeak efficiencies at energies above 46.5 keV. The method is validated for several inorganic and organic matrices measured in Petri dishes geometry. r 2003 Elsevier Ltd. All rights reserved. Keywords: g-ray spectrometry; Monte Carlo; Efficiency; Self-absorption
1. Introduction Self-absorption effects in volume samples are the main drawback to perform accurate analyses by g-ray spectrometry at low energies. Differences in the physico-chemical characteristics of the calibration samples and the sample under study may introduce errors in the activity results. Nevertheless, due to the large variability among environmental matrices, it is not feasible to obtain experimentally an efficiency calibration curve for each of them. Instead, many authors have developed semi-empirical or computational methods to calculate efficiency values or selfabsorption corrections for any given sample. The pioneer method of Cutshall et al. (1983) was devised to compute photopeak efficiencies at 46.5 keV for volume samples of unknown composition. It is based on the transmission measurement of a collimated beam of 210Pb photons through the sample, allowing one to compute its linear attenuation coefficient ðml Þ: Its main shortcoming is that, due to simplifications in the underlying theoretical model, it provides accurate results only for cylindrical geometries whose thicknesses are comprised in a particular range (Miller, 1987). Several *Corresponding author. Tel.: +34-98-31-83-807. E-mail address: [email protected] (M. Garc!ıa-Talavera).
modifications have been proposed to overcome this limitation and extend it to different geometries (Joshi, 1989; Galloway, 1991), but the assumption that detector and source dimensions are small compared to their distance remains implicit. Monte Carlo (MC) simulations constitute a newer approach to the self-absorption problem (Sima and Arnold, 1996; P!erez-Moreno et al., 2002). These techniques are increasingly applied to gamma spectrometry, particularly to efficiency calculations, since they can reproduce any detector–source configuration without simplification and minimize the experimental work. To obtain accurate efficiencies by means of an MC code, it is essential to implement appropriately the bulk density and the elemental composition of the samples. In particular, neglecting chemical composition effects might lead to substantial errors below 100 keV, even among soil or sediment samples (see Sima and Dovlete, 1997; Garc!ıa-Talavera et al., 2000). Nevertheless, the exact chemical compositions of the matrices under study are usually unknown and costly additional analyses by techniques as ICP-AEA or XRF are required in order to assess them. Alternatively, one can use a collimated beam of energy Eg to measure ml ðEg Þ; which is then input in the MC code to compute the photopeak efficiency at that energy (Overwater et al., 1993).
0969-8043/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2003.11.022
ARTICLE IN PRESS 228
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Inspired by the latter approach, we propose a hybrid method, combining experimental measurements and MC simulations, which presents the following advantages: First, it employs a parameterization of the efficiency as a function of ml in order to avoid performing a new MC calculation for each sample under analysis. Second, if the sample chemical composition is unknown, it is able to provide its ml value at any energy above 46.5 keV, based on a single transmission measurement of a 210Pb source through the sample. To test the accuracy and precision of this method, we apply it to calculate the photopeak efficiencies for several inorganic and organic matrices measured in Petri dishes. In the light of these results, we discuss the convenience of routinely using the hybrid method, compared to approaches that simply assume a standard chemical composition for the matrix.
2. Experimental procedure Experimental measurements were carried out with a Canberra n-type HPGe detector, with 117 cm3 active volume, 25% relative efficiency and resolution of 1.1 keV at 122 keV and 2.0 keV at 1.33 MeV. The associated electronic components are a preamplifier model Canberra 2008 and a DSA-2000 Spectrum Analyzer by Canberra. The recorded spectra were analyzed using Genie-PC software. To test the validity of the MC simulation, several point sources ðPÞ were employed. They were prepared by pipetting a well-known volume (o1% uncertainty) of a standard solution on hydrophobic filters. We used a multi-gamma cocktail DAMRI 9ML01-ELMH05/4 containing 241Am, 109Cd, 57Co, 51Cr, 139Ce, 113Sn, 85Sr, 137 Cs, 88Y and 60Co; and a 210Pb solution provided by CIEMAT. Besides, we employed a Petri dish, of 9-cm diameter and 1.8-cm height, which will be referred to as geometry V : It was filled with distilled water and spiked with known amounts of both standard solutions. For the self-absorption study, several types of matrices, specifically soil, coal, milk powder and dried leaves, were measured in Petri dishes. Two soil samples of different compositions were dried at 40 C for 24 h in a muffle oven and sieved with a 2-mm grid. An anthracite sample was dried in the same conditions and afterwards milled. Grapevine leaves were enclosed in a dryer containing silica gel for several days and milled to obtain a homogeneous powder. A commercial milk powder was employed, requiring no additional processing. For all these volume samples, we determined experimentally their efficiencies at 46.5, 59.5 and 88.0 keV, using photon emissions from 210Pb, 241Am and 109Cd, respectively. To do so, we pipetted known amounts of the two standard solutions mentioned above onto the samples and subsequently dried them over 24 h
at 40 C. The samples were carefully mixed prior to measurement to ensure that the radionuclides were homogeneously distributed in the source. The experimental efficiency at energy Eg for matrix k was computed as ekV ðEg Þ ¼
Ng C g ; pn ðEg ÞAn
ð1Þ
where Ng is the net count rate under the corresponding full-energy peak, An the known radionuclide activity, and pn ðEg Þ the emission probability. The net count rate was corrected by radionuclide decay, if necessary. Cg ; denoting the coincidence summing correction factor, equals 1 for the three considered g emissions. To minimize systematic errors due to improper source homogenization or positioning, two replicates of each sample were prepared and measured. The final photopeak efficiencies were obtained as a weighted average of the efficiency values derived from each pair of samples.
3. Monte Carlo simulation We have applied the MC code GEANT 3.21 to simulate the response of the HPGe detector. To test the accuracy of the simulation, several point sources, with photon emissions covering the energy range 46.5–1332 keV, were measured in various positions; the experimental efficiencies were compared to the ones obtained by MC. To obtain a better agreement among them, some detector parameters, as the crystal inner dead layer, were fitted. After this process, we verified the accuracy of the simulation for geometry V : We measured a Petri dish containing spiked water and compared the MC efficiencies to the experimental ones, derived from 210Pb, 241Am, 57Co, 109Cd, 51Cr and 137Cs photons. All the values agreed within the statistical uncertainties, which were on average 2%.
4. Method development We have devised a method to calculate efficiencies for arbitrary volume samples consisting of three separate stages. In the first one, the efficiency at the energies of interest is parameterized as a function of ml : The functional form adopted for e is the same, but the parameters might vary with energy. When analyzing a matrix of known attenuation coefficient, such function can be directly used to calculate eðEg Þ: For samples of unknown composition, the value of ml ðEg Þ has first to be determined experimentally: the two next stages of the method are designed to accomplish it for energies above 46.5 keV. Next, we present in detail an application of the method to our HPGe detector and a Petri dish
ARTICLE IN PRESS M. Garc!ıa-Talavera, V. Pen˜a / Applied Radiation and Isotopes 60 (2004) 227–232
measuring geometry ðV Þ: Through all the process, special attention is given to the uncertainty calculation.
229
0.09
MC methods can be used to obtain an analytical function relating the photopeak efficiency of the sample at energy Eg ; to its linear attenuation coefficient, ml ðEg Þ: In particular, we have applied the GEANT code (v. 3.21) to calculate such function at 46.5, 60.0 and 88.0 keV. We simulated nk calibration matrices of varying Z-values and bulk densities, which cover the usual range of ml found in environmental samples—selected chemical compositions consist in C, SiO2, CaO and Fe2O3, with densities ranging from 0.5 to 2.0 g/cm3. Their photopeak efficiencies at the energies of interest were calculated by means of the MC code. In Fig. 1, we have plotted the efficiency, eV ðEg Þ; as a function of ml ðEg Þ; for g ¼ 46:5; 60:0 and 88.0 keV. Every data set was fitted by weighted least squares using the Levenberg–Marquardt (L–M) algorithm to the following function:
0.06
0.03 0.0
2.0
4.0
0.0
0.6
1.2
0.00
0.25
0.50
ε 60
0.09
0.06
ð2Þ
where a1 represents the photopeak efficiency for an ideal vacuum source measured also in geometry V ; and a2 is the average length traveled within the source by those photons recorded in the full-energy peak. To simplify the fitting, we defined parameter a01 as a1 =a2 : At every energy, the L–M algorithm provides as output the values of a01 and a2 as well as its covariance matrix ðVij Þ: Using such matrix, we can calculate the uncertainties in the efficiencies, se ; from an application of the propagation of errors formula: 2 2 qe qe s2e ¼ V þ V22 11 qa01 qa2 qe qe V12 : þ2 ð3Þ qa01 qa2 The resulting uncertainties in eV at 46.5, 60.0 and 88.0 keV are below 2% in the considered ml range. For matrices of known chemical composition, as aqueous samples, eV and se can be obtained by substituting in Eqs. (2) and (3), respectively, the value of ml : This, in turn, is calculated as the product of the bulk density, which is easily determined in the laboratory, and the mass attenuation coefficient ðmÞ; whose values are tabulated for all the elements from Z=1 to 92. If matrix sample composition is unknown, one could estimate a value of ml from that of similar materials. In
0.03
0.09
ε 88
1 exp½a2 ml ðEg Þ a2 ml ðEg Þ 1 exp a2 ml ðEg Þ 0 ; ¼ a1 ml ðEg Þ
eV ðEg Þ ¼ a1
ε 46
4.1. Efficiency parameterization
0.06
0.03
µ l (cm-1 ) Fig. 1. Variation of the photopeak efficiency ðeg Þ with the linear attenuation coefficient ðml Þ for a Petri dish measuring geometry, at 46.5, 60.0 and 88.0 keV.
ARTICLE IN PRESS M. Garc!ıa-Talavera, V. Pen˜a / Applied Radiation and Isotopes 60 (2004) 227–232
230 20 18
46.5 keV
16 14
u (%) (%
12 10 8
60.0 keV
6 4
88.0 keV
2 0
Air
C
SiO2
CaCO3
Fe2O3
Fig. 2. Relative uncertainty ðum Þ in the efficiency due to a 30% uncertainty in ml ; at 46.5, 60.0 and 88.0 keV. Several representative matrices with bulk density 1 g/cm3 are plotted as reference in the horizontal axis. C is the major component in organic matrices and SiO2, CaCO3 and Fe2O3 are the most abundant constituents of mineral-origin samples.
this case, the error ðsm Þ in the estimate should be accounted for in se by adding the following term to Eq. (3): s2e;m ðqe=qml Þ2 s2m : In Fig. 2, we evaluate this term at 46.5, 60 and 100 keV assuming a relative error in ml of 30%—errors usually exceed that percentage when assigning to the unknown sample the ml (46.5) value of a similar material. From the figure, we can infer that an accurate determination of the efficiency requires a more precise estimate of ml ; especially for mineral samples. 4.2. Transmission experiment When the exact composition of the sample under analysis is unknown, an experimental attenuation measurement can be performed to compute ml ðEg Þ: A 210 Pb source is adequate for this purpose, since environmental analysis by g-spectrometry rarely involve photopeaks at energies lower than 46.5 keV. For a collimated source of 210Pb, the linear attenuation coefficient of the sample can be calculated using the following equation: ml ðEg Þ ¼
0 ln ðN46 =N46 Þ ; t
ð4Þ
0 being the count rate (s1) in the 46.5-keV photopeak N46 due to the source without attenuation, N46 the count rate with attenuation and t the thickness (cm) of the sample. We have modified the classical transmission experiment by measuring a punctual source ðPÞ placed directly
Fig. 3. Schematic drawing of the measuring setup used for the transmission measurements.
over the sample, without collimating (see Fig. 3). Because this measuring configuration maximizes the solid angle detector–source, counting times can be significantly reduced. On the other hand, Eq. (4) cannot be applied because some of the photons impinging the crystal travel distances longer than t within the sample. We used MC simulations to relate ml ðE46 Þ to the 0 transmission factor, TðE46 Þ; defined as N46 =N46 : We reproduced in the code the setup shown in Fig. 3 and calculated TðE46 Þ for the nk calibration matrices. Taking the natural logarithm of TðE46 Þ; the data can be fitted to a straight line through origin by a weighted least-squares analysis (see Fig. 4). Thus, for this measuring setup, we can apply an equation analogous to (4) where t is substituted by the fitted slope of the line, namely tav ¼ 1:85ð2Þ cm. To determine ml ðE46 Þ for a given sample using the previous equation, one has to obtain experimentally TðE46 ): N46 is measured from the transmission of the 46.5-keV photons through the Petri dish containing the 0 matrix of interest and N46 is determined in identical way but using an empty dish. Special care should be taken when placing the 210Pb source, since counting rates are very sensitive to shifts in its position. To calculate the uncertainty in ml ðE46 Þ; the uncertainties in TðE46 Þ and tav should be taken into account. If we set counting times in order to have relative uncertainties in TðE46 Þ lower than 2%, relative uncertainties in ml
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To quantify the deviations in the values of m estimated by this procedure, we have performed numerical simulations. We considered the chemical compositions of soil organic matter (C: 56.4%; H: 5.5%; N: 4.1%; S: 1.1%; O: 32.9%), quartz (SiO2) and pure limestone (CaCO3). For each composition i; we calculated mi ðE46 Þ and generated a set of random numbers following a Gaussian probability distribution of mean mi ðE46 Þ and standard deviation 10%. Every number represents an experimental measurement of mi ðE46 Þ; For each of them, we found Zeq and, according to it, estimated mi ðE60 Þ and mi ðE88 Þ: For the three matrices, the means of the resulting data sets agree with the true mi values and the standard deviations at 60.0 and 88.0 keV are lower than 6% and 2%, respectively. Nevertheless, this approach might lead to substantial deviations for matrices with a high content of hydrogen. The reason is that for all elements abundant in environmental matrices the electron density is nearly constant, except for H, which has a higher one and consequently a higher Compton interaction probability. Yet, concentrations of hydrogen above 5–6% are only found in unprocessed vegetal or animal samples, due to its high proportion of water. If the sample is dried or ashed prior to measuring, as is usually the case, most of the H disappears. Only for organic samples treated by wet digestion, this approach is not valid.
would be up to 10% for samples presenting practically no self-absorption and would rapidly decrease to about 1% as ml increases. 4.3. Estimation of m values at higher energies By measuring the transmission of 46.5-keV photons through the sample, we can compute ml ðE46 Þ and, thus, mðE46 Þ: Based on the latter value, it is possible to assign an effective chemical composition to the sample matrix, consisting of a Z-equivalent material. To do so, we built a table containing the values of mðE46:5 Þ for all elements from Z ¼ 1 to 92, based on data from Hubble and Seltzer (1996). For every sample, we select the element, Zeq ; whose tabulated value agrees exactly with the measured one. In case this does not occur, a combination of two elements Z0 and Z0 þ 1 should be taken ðZeq ¼ oZ0 þ ð1 oÞðZ0 þ 1Þ; 0ooo1Þ: The values of m at higher energies are then obtained upon the effective chemical composition assigned to the sample. 6.0
ln(T)
4.0
5. Method validation 2.0
To test the accuracy and precision of the method, we have applied it to compute the photopeak efficiencies at 46.5, 59.5 and 88.0 keV, for soil, coal, milk powder and leaves samples. Values of TðE46 Þ were obtained experimentally from the corresponding transmission measurements. These were used to estimate the mass attenuation coefficients at 46.5, 59.5 and 88.0 keV and, subsequently, the corresponding eV : The calculated efficiencies were compared to the ones found experimentally. As can be seen in Table 1, all the values agree within the statistical uncertainties.
0.0 0.00
1.00
2.00
3.00
µl ( cm-1) Fig. 4. Variation of the transmission factor T with ml at 46.5 keV for a Petri dish measuring geometry.
Table 1 Comparison of efficiencies calculated by the hybrid method ðecg Þ and experimental efficiencies ðeexp g Þ for samples of different bulk densities and chemical compositions Matrix
ec46
eexp 46
ec60
eexp 60
ec88
eexp 88
Soil (silty) Soil (clayey) Coal Milk powder Leaves
0.061(1) 0.067(2) 0.073(2) 0.075(2) 0.076(3)
0.059(2) 0.068(2) 0.073(1) 0.072(2) 0.077(1)
0.070(1) 0.072(1) 0.075(2) 0.076(2) 0.076(2)
0.069(2) 0.070(2) 0.077(1) 0.076(1) 0.075(1)
0.073(1) 0.074(1) 0.075(1) 0.075(1) 0.073(1)
0.074(1) 0.072(1) 0.077(1) 0.075(1) 0.075(1)
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Table 2 Differences ðdg Þ between the efficiencies obtained by assuming a standard chemical composition for the samples and the efficiencies calculated by the hybrid method Matrix
d46 (%)
d60 (%)
d88 (%)
Soil (silty) Soil (clayey) Coal Milk powder Leaves
4.9(2) 11.8(6) 2.3(2) 2.8(2) 1.7(2)
2.8(1) 6.7(3) 1.6(2) 1.1(1) 0.9(1)
0.6(1) 2.0(2) 1.0(1) 0.9(1) 1.0(1)
Furthermore, for this set of samples, we examined the effect in eV of determining m by a transmission experiment instead of assuming an approximate value of m; based on a standard chemical composition. Particularly, we considered a composition of C for the organic samples and of SiO2 for the soil samples. In Table 2 we observe that differences at 46.5 keV are practically negligible for the organic matrices, while for the soils they are up to 12%. At 88.0 keV all differences are below 2%.
6. Conclusions We have devised and validated a method to compute accurate efficiency values for volume samples. We use MC simulations to find a function relating the photopeak efficiency ðeÞ at the energy of interest to the linear attenuation coefficient ðml Þ of the matrix. Application of such function is simple enough if sample chemical composition is known. For matrices of unknown composition, one could estimate an approximate value of ml from that of similar materials. In practice, this procedure is suitable for dehydrated organic samples and, generally, for any type of matrix above 100 keV. Below that energy, a more accurate knowledge on ml is required. To achieve it, we propose a procedure involving an experimental attenuation measurement of the sample at 46.5 keV.
Acknowledgements The authors acknowledge financial support from Iberdrola S.A. and Junta de Castilla y Leon.
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