A semi-empirical approach to determine gamma activities ( Bq kg - 1 ) in environmental cylindrical samples

Radiation Measurements 43 (2008) 77 – 84 www.elsevier.com/locate/radmeas

A semi-empirical approach to determine gamma activities (Bq kg−1) in environmental cylindrical samples Daniel Francisco Palacios a,∗ , Juan A. Alfonso b , Haydn Barros a , John J. LaBrecque b , Karla Pérez b , Marian R. Lossada c a Universidad Simón Bolívar (USB), Apartado 89000, Caracas, Venezuela b Instituto Venezolano de Investigaciones Científicas (IVIC), Apartado 21827, Caracas, Venezuela c Dirección de Investigación y Postgrado, Universidad Marítima del Caribe (UMC), Caracas, Venezuela

Received 1 May 2007; accepted 2 November 2007

Abstract A semi-empirical method to determine radionuclide concentrations in large environmental samples without the use of reference material and avoiding the typical complexity of Monte-Carlo codes is proposed. The calculation of full-energy peak efficiencies was carried out from a relative efficiency curve (obtained from the gamma spectra data), and the geometric (simulated by Monte-Carlo), absorption, sample and intrinsic efficiencies for energies between 130 and 3000 keV. The absorption and sample efficiencies were determined from the mass absorption coefficients, whereas the intrinsic efficiency was approximated by an empirical function. The deviations between calculated and experimental efficiencies for a reference material in most cases are less than 10%. Radionuclide activities in marine sediment samples calculated by the proposed method and by the experimental comparative method were not significantly different. This new method can be used for routine environmental monitoring when uncertainties up to 10% are acceptable. © 2007 Elsevier Ltd. All rights reserved. Keywords: Full-energy peak efficiency; Gamma-ray spectroscopy; Monte-Carlo calculations; Marine sediments

1. Introduction The specific activity of a radionuclide can be calculated using the following equation: Ai =

(ni − nb ) Ni = , (Ei )Yi mt (Ei )Yi mt

(1)

where ni is the total counts recorded in the considered peak at energy Ei during the effective measurement time t, nb is the background counts under that peak, Yi is the photon emission probability, while m is the sample mass and (Ei ) is the fullenergy peak efficiency (FEPE) at the considered energy Ei ; the parameters Ni , m, t and Yi are easily obtained. There are two methods to determine the FEPE values: the experimental relative or comparative method (CM), using reference materials with representative geometry of measurement, and the ∗ Corresponding author. Tel./fax: +58 212 9063590.

E-mail address: [email protected] (D.F. Palacios). 1350-4487/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.radmeas.2007.11.043

modelization method which simulates the real conditions including the detector. The experimental method has been widely used, especially for routine monitoring of large environmental samples. When this method is employed, the measurement geometries of standards and unknown samples need to be similar, as well as their compositions and densities. Reference materials are only available for a limited number of matrices and can be very expensive. The modelization method (Brun et al., 1987; GEANT, 1994; Briesmeister, 2000; Sima et al., 2001; Oishi et al., 2003) can be used for the most difficult cases (complex geometry, high activities, etc.), but until recently the codes were very difficult to use which has limited the employment of this technique. Analog Monte-Carlo calculations need long computing times as they require a large number of simulated particles (or histories) for an accurate calculation, which can constitute a limiting factor for routine measurements. Exceptions on this matter are the GESPECOR software (Sima et al., 2001, 2004) and the variance reduction technique implemented in the GEANT4 code

78

D.F. Palacios et al. / Radiation Measurements 43 (2008) 77 – 84

(Hurtado et al., 2004). Another drawback of Monte-Carlo calculations of the FEPE is the sensitivity of the computed values to various input data which are sometimes not accurately known (e.g. detector data and source matrix) and/or uncertainties associated to the incomplete charge collection in the crystal. If the density and composition of the samples to be measured are not considered, uncertainties in the determination of the activities can be generated. In order to compensate the matrix effects as thoroughly as possible, it is necessary to use both the experimental and Monte-Carlo simulation methods (Sima and Dovlete, 1997; Briesmeister, 2000; Sima et al., 2001; Oishi et al., 2003). Thus, it would be useful to develop an easier modelization method for gamma spectrometry, which could eliminate the need of reference materials, taking into account at the same time the matrix and geometry of each sample. 2. Materials and methods 2.1. The gamma-ray spectrometric measurements A coaxial Ge detector, PGT p-type (model IGC 2020) with an aluminum window (0.51 mm thick), 111 cm3 active volume, 5.6 cm diameter, 4.9 cm length, 0.5 cm detector-window distance and 0.1 cm dead layer thickness was employed to make the measurements. The detector had a relative efficiency of 25.5% and nominal FWHMs of 0.90 keV at 122.07 keV and 1.92 keV at 1332 keV. The detector was connected to conventional electronic components and the data were collected by a personal computer with a nucleus analog–digital converter/ interface card. To take into account the variable sample size, we used a cylindrical geometry, the samples being enclosed in polyethylene containers with 8.4 cm inner diameter and sample height varying from 1.0 to 10.0 cm. Reference material IAEA-RGTh1 was used to construct the efficiency (FEPE) curves as a function of sample heights. This material was produced by dilution of a thorium ore OKA-2 (2.89% Th, 219 g U g−1 ) with floated silica powder and shows radioactive equilibrium. The experimental efficiency curves were determined by the gamma emissions of 208 Tl, 212 Pb, 212 Bi and 228Ac in the energy range from 129.05 to 2614.6 keV. The measurement time was only 1800 s, considering the world-wide average activity concentration of 232 Th in the natural environment to be about 40.60 Bq kg−1 (Eisenbud, 1987) and a commonly employed counting time for soil and sediment samples around 24–48 h. To consider the possible effects of composition and apparent density, surface marine sediment samples from the central coast of Venezuela were also analyzed and compared by both the semi-empirical method (SEM) and a CM. They were measured in the same type of container as the reference material but sample height was kept constant at 8.0 cm. The counting time was 75,000 s. For measurements, the containers were placed directly on detectorwindow cap. The following reference materials from the International Atomic Agency’s Analytical Quality Control Services were used as the comparators: Soil-6 for 137 Cs, IAEA-RGK1 for 40 K, IAEA-RGU-1 for 214 Bi (226 Ra) and IAEA-RGTh-1 for 208 Tl.

2.2. Detector FEPE Since only full-energy events need to be considered, the detector efficiency at energy Ei can be expressed as the product of five factors: (Ei ) = f (Ei )geom absp (Ei )s (Ei )int (Ei ).

(2)

In the natural decay chain spectrum, there are peaks originating from gamma transitions in cascade. This introduces coincidence-summing effects in the measurements making it necessary to correct the experimental efficiencies obtained, especially when the source-to-detector distance is short, such as in the case of Petri dishes and Marinelli beakers, or when well or high-efficiency detectors are used. Since we used a normal p-type Ge detector, with a relative low efficiency, which reduces also the cascade summing with X- and low-energy  rays, as well as the measurement geometry also had a low efficiency, it can be expected that the coincidence-summing effect would not be significant. Moreover, in similar counting configuration using a semi-planar HPGe, El-Daoushy and García-Tenorio (1995) showed that true coincidence-summing effects due to non-single gamma emitters are negligible considering their experimental uncertainties (3–5%). Pérez-Moreno et al. (2002) also used the same measurement geometry with a 153 cm3 active volume and 38% relative efficiency XtRa detector without considering the coincidence-summing effects. Even in measurements with Marinelli vessels, the cascade-summing correction factors for 208 Tl and 212 Pb were in most cases below 5% (Alfassi, 2000). Therefore, the correction factor for coincidence summing, f (Ei ), can be assumed to be 1. The geometric efficiency, geom , is the fraction of emitted photons that are intercepted by the detector. This factor is essentially independent of photon energy. The absorption efficiency, absp (Ei ), takes into account the effects of intervening materials that absorb some of the incoming radiation before it interacts with the detector. It has the mathematical form absp (Ei ) = e−{[(Ei )x]dh +[(Ei )x]c } ,

(3)

where (Ei ),  and x are the mass absorption coefficient, density and thickness of the detector housing (dh) and container (c). The sample efficiency, s (Ei ), is the reciprocal of the sample self-absorption correction. In a slab of thickness x, the sample efficiency is s (Ei ) =

1 − e−[(Ei )x]s . [(Ei )x]s

(4)

It should be mentioned that both Eqs. (3) and (4) are approximations. They are exact only in the case that a photon travels perpendicular to the absorption layer and detector end cap. The intrinsic efficiency, int (Ei ), is the probability that a gamma ray that enters the detector will interact and give a pulse in the full-energy peak. Empirically, it can be approximated by

D.F. Palacios et al. / Radiation Measurements 43 (2008) 77 – 84

a power law of the following form (Ridikas and Feray, 2003):  −a2 Ei int (Ei ) = a1 with E0 = 1 keV. (5) E0 Thus, the FEPE for an extended source can be expressed as   1 − e−[(Ei )x]s −{((Ei )x)dh +((Ei )x)c } (Ei ) = geom [e ] [(Ei )x]s     Ei −a2 × a1 . (6) E0 When the radioactivity of environmental samples is measured, there are many characteristic gamma peaks which are emitted from the same mother radionuclide from the 238 U and 232 Th series. According to Eq. (1) for two gamma peaks at energies Ei and Ej from the same radionuclide or from radionuclides in secular equilibrium, then: (7)

It can be rearranged as (Ei ) Ni Yj = = C(Ei , Ej ). N j Yi (Ej )

(8)

The relative efficiency curve [(C(Ei , Ej ) vs. Ei ] can be determined from the measured spectrum of each sample, where the peak areas Ni from interference-free gamma rays from a single radionuclide or from several, if they have the same activity, are used. A high-energy (Ek ) intense gamma ray from one of the radionuclides is selected as a reference gamma ray. Its activity (peak area corrected from branching ratio) becomes a reference activity. Actually, once the sample in the container is measured, the measured data N and nuclear data Y are fixed, so that the ratio (Ni /Nk )Yk /Yi can be plotted as a smooth curve. The gamma-ray absorption or attenuation by the sample and materials between the sample and detector can be directly determined, even if the geometrical shapes of sample and the container were not known in advance. From Eq. (8), the FEPEs at energies Ei can be calculated as (Ei ) = (Ek )C(Ei , Ek ).

then be expressed as a1 = 0.11/1332−a2 . From Eqs. (6) and (9) and substituting a1 , it follows that 9.091C(Ei , Ek )   −[(Ei )x]s 1 − e geom [e−{((Ei )x)dh +((Ei )x)c } ] [(Ei )x]s = (Ek )

−1



Ei 1332

.

(10)

1

(9)

That is, the parameters of the curve fitted to the FEPE data will be the same as those of the relative efficiency curve multiplied by a constant [(Ek )]. Although for a point source detected by a simple cylindrical detector, the overall efficiency is commonly determined by two factors, the intrinsic (int ) and geometric (geom ) efficiencies (Ranger, 1999), the attenuation effect in the detector housing (absorption efficiency) should also be considered ( = int geom absp ). In this case, the geometric and theoretical absolute efficiencies of an HPGe detector at 1332 keV at 25 cm can be determined from detector parameters as specified by the manufacturer and the absorption efficiency from the mass attenuation coefficient of 1332 keV gamma rays in Al. The coefficient, a1 , in the expression for the intrinsic efficiency can

−a2

Geometric efficiency for an extended source can be easily determined by the Monte-Carlo technique. First, the geometric efficiency was determined for a point source located at different distances on the detector axis, then the results were compared with the analytical expression (/4). In order to simulate an isotropic gamma-ray emission from a radioactive source, the zenith () and azimuthal ( ) angles employed were 0   and 0  2. By choosing two independent random values,

1 and 2 , with uniform probability between 0 and 1, the initial direction of the emission of a gamma ray from an isotropic source can be generated by imposing that  = arccos(1 − 2 1 ) and = 2 2 . After generating the initial direction of the emission of a gamma ray, coordinates of the intersection point with the detector plane and the distance (da ) to detector axis are calculated. If da rd (detector radius) and  > /2, it is considered that the photon impacted on the top of the detector’s surface and the event is counted. The geometric efficiency is calculated as the number of photons incident on the detector divided by the number emitted by the radiation source. The above sequence is repeated for a predefined number of histories. The detector geometric efficiencies for point sources calculated by the analytical expression and by Monte-Carlo technique are shown in Fig. 1. As can be seen, both results are overlapped indicating that the algorithm developed for Monte-Carlo simulation is adequate. To calculate the geometric efficiency for an extended source (cylindrical), besides the traveling direction, a special subroutine of the program was assigned to the random selection of the gamma-ray emitting point in the sample (z = hs 3 , r = rc 4 ,  = 2 5 , x = r cos , y = r sin , where 3 , 4 and 5 are

Analytical Geometric efficiency

Ni (Ej )Yj = 1. Nj (Ei )Yi

79

Monte Carlo

0.1

0.01

0.001 0

5

10 15 20 Source-detector distance (cm)

25

30

Fig. 1. Comparison of a Monte-Carlo geometric efficiency simulation with the theoretical results (/4) for a point source placed at different source–detector distances.

D.F. Palacios et al. / Radiation Measurements 43 (2008) 77 – 84

min mix

z

100 h = 1 cm h = 2 cm

r

P(x,y,z)

h = 3 cm

xy plane

C (Ei, Ek)

80

10

h = 4 cm h = 5 cm h = 6 cm h = 7 cm

1

h = 8 cm h = 9 cm h = 10 cm

0.1 100 longitudinal section of detector

1d

1000 Energy (keV)

10000

Fig. 3. Dependence of C(Ei , Ek ) on energy for different sample thicknesses (h); Ek = 2614.5 keV from 208 Tl was chosen to be the reference-energy line.

3. Results and discussion rd y

traverse section of detector

3.1. The simulated and experimental FEPE using the IAEA-RGTh-1 reference material

min mix P(x,y,z)

x

Fig. 2. Representation of the critical angles for the gamma-ray incidence on the detector body (ld —detector length).

random values with uniform probability between 0 and 1, hs is the sample thickness and rc is the inner container radius). If r > rd , then the probability that a gamma-ray impacts on the detector lateral side has also to be considered. To do this, another subroutine calculates the critical angles min , max , min and max , as shown in Fig. 2. If the generated angles  and lay between min and max and between min and max , respectively, then the gamma-ray impacts on the detector and the event is counted. The above sequence is repeated for a predefined number of times (by selecting randomly new points within the source). Once C(Ei , Ej ) is determined from the experimental data by Eq. (8), if geom , mass absorption coefficients, densities and thicknesses of container, detector housing and sample are known, then the left component of Eq. (10) can be calculated for each energy Ei . Then, (Ek ) can be determined by a fit to a power equation, so that the FEPEs at energies Ei can be obtained from Eq. (9) and the radionuclide activities from Eq. (1). In this way, we can obtain an efficiency curve for each sample that contains information on the radiation self-absorption within the sample, without the necessity of using reference materials to obtain the efficiency calibration curve. On the other hand, the only radiation property needed for the modelization method is the traveling direction considered in geom , which is relatively simple and can be quickly calculated.

In the IAEA-RGTh-1 reference material there are many characteristic gamma rays which are emitted from 232 Th series and all of them have the same activity. The most intense gamma rays of 228Ac, 212 Pb, 212 Bi and 208 Tl were used. The 2614 keV gamma ray from 208 Tl was selected as the reference gamma ray. The activities computed from the other gamma rays of 208 Tl, and from the gamma rays of the other radionuclides, were related to the reference activity and plotted vs. energy for different sample thicknesses (Fig. 3). The experimental results were fitted to power equations of the type C(Ei , Ek )=a1 (Ei /E0 )−a2 with E0 = 1 keV. The regression coefficients were quite similar, indicating their weak dependence upon sample thicknesses. It is shown in Fig. 3 that it is possible to consider a general function C(Ei , Ek ) for any sample thickness using the average values of regression coefficients (represented by the continuous line). The uncertainties at one sigma level (1 ) for a1 and a2 were 14% and 3%, respectively. In addition to the previous geometric considerations that allow to suppose no significant coincidence-summing effects in our case, it can be assumed that if any count is lost from one peak area due to the summing effect of another in coincidence, then it is likely that this process enriches the number of counts in over-crossing line which belongs to the same nuclide. Since the relative efficiency curve was derived using the least-squares best fit technique for various spectral data corresponding to different nuclides, some of them emitting gamma rays that suffer little or no coincidence summing, the coincidence effects are averaged by this routine procedure. Of course, the above discussion is strictly true if one analyzes all of the contributing gamma lines from each nuclide. Furthermore, for radionuclides emitting photons at several energies, the activity concentration will be calculated from the weighted average of the values obtained from each line. The change of the reference line does not affect the slope of the fitted curve as shown in Fig. 4, which is as well the

D.F. Palacios et al. / Radiation Measurements 43 (2008) 77 – 84

100

0.1 Ek = 2614 KeV, h = 2 cm

h = 2 cm

Ek = 727 KeV, h = 2 cm

h = 4 cm

Ek = 2614 KeV, h = 10 cm

h = 6 cm

Efficiency

C (Ei, Ek)

10

81

Ek = 727 KeV, h = 10 cm

h = 8 cm

0.01

h = 10 cm

1

0.1 100

0.001 100

1000 Energy (keV)

10000

Fig. 4. Dependence of C(Ei , Ek ) on gamma energy for different reference-energy lines and sample thicknesses.

slope of the FEPE curve. The elimination of the gamma line with higher probability of coincidence summing (2614 keV of 208 Tl) (Hossain et al., 2002) causes a very small change in the slope of the curves, but more so for thinner samples, where the coincidence-summing effect would be larger (De Corte et al., 1994). The use of the reference line Ek = 2614 keV makes the dependencies of the relative efficiencies for different sample thicknesses to be described by a same potential function with an error smaller than 10%. This effect can be explained by the weak sample self-absorption for this high-energy line. The dependencies of mass attenuation coefficients with gamma energy for the reference material (RGTh-1), detectorwindow material (Al) and polyethylene (C2 H4 )n from the container material were determined from Berger et al. (2005). The chemical composition of RGTh-1 was estimated to be SiO2 = 99.71% and Th = 0.29%; the other analytes were not significant. Values from XCOM database were fitted to power functions and the regression parameters were used in Eq. (10). Although manufacturers specify the detector’s technical parameters, they are not always accurate enough; thus, a supplementary study is required in order to investigate them. Crystal radius and length, its position inside the end cap, the thickness of the germanium dead layer must be known as accurately as possible to attempt to simulate it by Monte-Carlo methods (Sima et al., 2004). The trial and error method was used to characterize the parameters of the HPGe detector. The experimental and simulated FEPEs were compared and detector parameters were adjusted for optimal correspondence between results. To compensate for the differences between the simulated and experimental results, the active volume of Ge detector resulted to be 1 mm thinner and detector-window distance 0.2 cm longer than the manufacturer specifications. Such a disagreement is not unprecedented. It has been reported that theoretical results based on manufacturer’s specifications of detector size could not adequately predict experimental results (Clouvas et al., 1998; Schoenfeld et al., 2002). In general, the choice of detector parameters to be fixed might sometimes require some trial and error; an alternative procedure to change the parameters (detector-window distance and active volume of detec-

1000 Energy (keV)

10000

Fig. 5. Comparison between efficiencies determined from experimental results (points) and from the developed method (continuous line).

tor) is to consider the radiation absorption in the detector dead layer. The corresponding absorption term for the dead layer is exp −(dl ), where  is the total macroscopic cross-section in germanium and dl the dead layer thickness. The results obtained by the SEM and experimental values of efficiency for different sample thicknesses are shown in Fig. 5. It can be seen that there is a good agreement between both sets of results. The simulated and experimental efficiencies for all the geometries agree with the typical errors of the individual data points (∼10%). The results are better for thicker samples where the possible effects of coincidence summing are minimized. In fact, the probability of coincidence measurements diminishes when the attenuation in the source is larger (Daza et al., 2001). Of course, if close geometry is needed to increase the efficiency or more accurate determinations are required, a cascade correction algorithm or extensive programs and libraries as KORSUM, GESPECOR or ETNA could be used. In general, it can be concluded that the deviations between the computed results and the measured efficiencies are mostly within 10%. These results are comparable to those reported by Vidmar et al. (2001), who used another semi-empirical formula for the FEPE of the HPGe detectors, but their study was limited only to point sources. In the calculations of geometric efficiencies the probability of impacts on detector lateral side was considered; however, we did not take into account the additional attenuation of the crystal holder inside the detector end cap because its dimensions and material compositions were not available from the manufacturer specifications. Nevertheless, our results indicate that the modifications introduced to detector parameters seem to compensate for this effect. Anyway, due to the relation between the sample and detector crystal dimensions, it is expected that most of the radiation emitted by the samples hit the top of the detector face. 3.2. Calculation of radionuclide activities in marine sediment samples The relative efficiency curve was obtained from data, as before from the same radionuclides and their respective gamma rays in the RGTh-1 reference material. All the relative

82

D.F. Palacios et al. / Radiation Measurements 43 (2008) 77 – 84

10

100 CI-LC

Ek = 338 KeV Ac-228 Ek = 351 KeV Pb-214

P9-LC Y10-LC

1

0.1 100

1000 Energy (keV)

C (Ei, Ek)

P7-LC

10

Fig. 6. Dependence of C(Ei , Ek ) on energy for different sediment samples; Ek = 2614.5 keV from 208 Tl was chosen to be the reference-energy line.

efficiency curves, as shown in Fig. 6, can be described by one general power function (continuous line) for any sample using the average values of regression coefficients. The uncertainties at one sigma level (1 ) for the parameters a1 and a2 were 15% and 3%, respectively. This means that compositions and densities of the analyzed samples are very similar. Radionuclide concentrations were calculated from Eq. (1) using the simulated FEPEs, obtained from Eq. (9), but employing both the average relative efficiency curve and that generated for each sample. It is not surprising that the average values of the parameters of the relative efficiency curves for the reference material and the sediment samples were statistically indistinguishable, since in both cases the measurement geometries were identical and the matrix differences were not significant for the studied energy range. According to the experimental and the simulated coincidencesumming correction factors for the gamma rays of the 238 U series reported by Laborie et al. (2002), the correction factors for the 295.2 and 351.9 keV gamma rays from 214 Pb and for the 1764.5 keV gamma ray from 214 Bi were very close to unity in a well geometry, where it is known that coincidence-summing effects can be very high. With more reason, it is expected that in the adopted cylindrical geometry it is not necessary to correct those lines. The comparison of the relative efficiencies for a typical marine sediment sample, obtained from radionuclides of the 232 Th series (using as reference line the 338.3 keV of 228Ac) and from the radionuclides of the 238 U series (using as reference line the 351.9 keV of 214 Pb), is shown in Fig. 7. Those reference energies were chosen by their proximity, since it was demonstrated that for the same set of energies the relative efficiency curves for different reference energies are parallel to each other in a log–log scale. It can be seen that the relative efficiencies corresponding to gamma-ray energies of 214 Bi and 214 Pb are well described by the relative efficiency curve fitted to the spectral data of radionuclides of the 232 Th series, assuming an error of 10%. Again, this demonstrates that the coincidence-summing effects are not significant. As can be seen in Fig. 8, for E > 150 keV the mass attenuation coefficients for different elements are quite similar. Typical soils and sediments are principally formed by Si (∼15%), Al (∼8%), O (∼62%), Fe (∼5%), Ca (∼2%), K (∼2%), F (∼3%),

1

0.1 100

10000

1000 Energy (keV)

10000

Fig. 7. Dependence of C(Ei , Ek ) on energy for the O1-LC sediment sample. Close gamma-ray energies of radionuclides from the 232 Th and 238 U series were taken as reference-energy lines.

1.00 Si Ca K

 (cm2/g)

C (Ei, Ek)

OI-LC

O

0.10

A1 Fe F TSS

0.01 100

1000 Energy (keV)

10000

Fig. 8. Mass attenuation coefficients vs. gamma energy for the most common elements in environmental samples and for a typical soil or sediment (TSS) matrix.

in a percentage depending on the geological environment. The dependence of mass attenuation coefficient with gamma energy for a typical soil or sediment (TSS) matrix is also shown in Fig. 8. Thus, considering the average composition of a soil or sediment it is possible to consider a general potential function (E ) for any sample, which can produce adequate results. The mass attenuation coefficients for different elements have been taken from Berger et al. (2005). The curve obtained by the fitting of the mass attenuation coefficients of the TSS matrix with gamma energies appropriately describes the behavior of the other elements. Table 1 shows the radionuclide activities determined by the SEM proposed in this work, using the relative efficiency curve of each sample, and the results obtained from the CM using as reference materials Soil-6 for 137 Cs, IAEA-RGK-1 for 40 K, IAEA-RGU-1 for 214 Bi (226 Ra) and IAEA-RGTh-1 for 208 Tl. With regard to the problem of coincidence summing, no corrections have to be applied when activities are determined by the CM since the sample is measured relative to a standard of the same radionuclide, in the same geometry. Deviations (D) between the specific activities for single energy line radionuclides (137 Cs and 40 K) were calculated as

D.F. Palacios et al. / Radiation Measurements 43 (2008) 77 – 84

83

Table 1 Comparison between results obtained by the comparative method (CM) and the semi-empirical method (SEM) presented in this work Sample

C1-LC O1-LC P7-LC P9-LC Y10-LC

137 Cs

(Bq kg−1 )

40 K

(Bq kg−1 )

226 Ra

(Bq kg−1 )

232 Th

(Bq kg−1 )

CM

SEM

D (%)

CM

SEM

D (%)

CM

SEM

CM

SEM

0 0 2.9 0 0

0 0 3.1 0 0

0 0 6.4 0 0

476.2 329.9 811.4 795.0 422.1

478.5 334.7 971.9 747.7 401.7

0.5 1.4 19.7 6.3 5.1

17.8 43.8 46.8 43.3 33.8

19.3 ± 3.3 47.5 ± 6.9 42.7 ± 4.1 41.8 ± 6.9 31.6 ± 3.7

45.3 43.9 53.8 60.2 62.8

46.7 ± 4.1 48.0 ± 3.5 54.7 ± 4.5 57.4 ± 4.6 58.7 ± 4.3

(|CMi −SEMi |/CMi )100(%). When the SEM was applied, the 232 Th and 226 Ra concentrations were calculated from the average values of the activities obtained from the most intense gamma ray of each daughter in each series. For these radionuclides the average values and uncertainties at one sigma level (1 ) are presented. As can be seen, the results were in good agreement with the activity measurement carried out by the CM. In general, it can be concluded that the deviations between calculated activities by the SEM and those determined by the CM are on average less than 10%. The 232 Th and 226 Ra concentrations obtained by the CM are almost all included in the range of variation of the activities calculated by the developed SEM. Practically the same results were obtained when in the SEM the average relative efficiency curve was employed. In fact, if one is only interested in accuracy between 10% and 15% the commercial Monte-Carlo programs with a simplified composition of the sample and the real density can be employed. The experimental method can then also be used, but not all laboratories have the required Monte-Carlo programs or the appropriate standards or reference materials. Some advantages of the proposed method are the simplicity in the implementation, the relative short time to obtain reliable and repeatable results determined by the calculation of geometric efficiency that is between 1 and 5 min with a standard PC. Furthermore, the geometric efficiencies for different sample dimensions would be available at any time if previously they were calculated and stored in a database. To obtain the FEPE curve, the proposed method does not require the use of reference materials or point sources. The measured data of the gamma radiation emitted from the sample contains the original information of the sample self-absorption and the gamma attenuation in the sample container without any other assumed physical approximation. In the case of soil or sediment samples, the present methodology does not require the determination of absorption coefficients for each measured material. Uncertainties related to the simulation of the performance of an HPGe detector and the effects of the incomplete collection of charges in Ge crystals are avoided. 4. Conclusion A general semi-empirical fit to the FEPE data for cylindrical samples measured on closed-end HPGe detector was developed, yielding an acceptable accuracy for routine

monitoring of environmental samples. The developed model employs a single trial function with a small number of physically well-defined parameters covering a sufficiently wide gamma-ray energy range (130–3000 keV) to suit most environmental applications. The method was tested with data sets from different sample thicknesses and different types of samples to verify its performance. Acknowledgment This work was supported by the National Foundation of Science, Technology and Innovation (FONACIT), Venezuela, under Contract number S1-2001000954. References Alfassi, Z.B., 2000. In-house absolute calibration of gamma-emitting radioactive voluminous samples: detector setups using natural radionuclides. J. Radioanal. Nucl. Chem. 245 (3), 561–565. Berger, M.J., Hubbell, J.H., Seltzer, S.M., Chang, J., Coursey, J.S., Sukumar, R., Zucker, D.S., 2005. http://physics.nist.gov/PhysRefData/Xcom/Text/ XCOM.html. Briesmeister, J.F., 2000. MCNP—a general purpose Monte Carlo n-particle transport code, version 4C. Los Alamos National Laboratory Report, LA-13709-M. Brun, R., Bruyant, F., Maire, M., McPherson, A.C., Zanarini, P., 1987. GEANT3, DDD/EE/84-1. CERN Data Handling Division, Geneva. Clouvas, A., Xanthos, S., Antonopoulos-Domis, M., Silva, J., 1998. Monte Carlo based method for conversion of in-situ gamma-ray spectra obtained with a portable Ge detector to an incident photon flux energy distribution. Health Phys. 74 (2), 216–230. Daza, M.J., Quintana, B., García-Talavera, M., Fernández, F., 2001. Efficiency calibration of a HPGe detector in the [46.54–2000] keV energy range for the measurement of environmental samples. Nucl. Instrum. Methods A 470, 520–532. De Corte, F., de Wispelaere, A., Vancraeynest, L., de Neve, P., Van den Haute, P., 1994. True-coincidence correction for field gamma-ray spectrometry in Auger hole counting geometry. Nucl. Instrum. Methods A 353 (1–3), 539–541. Eisenbud, M., 1987. Environmental Radioactivity: From Natural, Industrial and Military Sources, third ed. Academic Press, New York. El-Daoushy, F., García-Tenorio, R., 1995. Well Ge and semiplanar Ge(Hp) detectors for low-level gamma-spectrometry. Nucl. Instrum. Methods A 356, 376–384. GEANT, 1994. Detector Description and Simulation Tool. CERN Program Library Long Writeup W5013. Hossain, S.M., De Corte, F., Vandenberghe, D., Van den Haute, P., 2002. A comparison of methods for the annual radiation dose determination in the luminescence dating of loess sediment. Nucl. Instrum. Methods A 490 (3), 598–613.

84

D.F. Palacios et al. / Radiation Measurements 43 (2008) 77 – 84

Hurtado, S., García-León, M., García-Tenorio, R., 2004. Monte Carlo simulation of the response of a germanium detector for low-level spectrometry measurements using GEANT4. Appl. Radiat. Isot. 61, 139–143. Laborie, J.M., Le Petit, G., Abat, D., Girard, M., 2002. Monte Carlo calculation of the efficiency response of a low-background well-type HPGe detector. Nucl. Instrum. Methods A 479, 618–630. Oishi, T., Tsutsumi, M., Sugita, T., Yoshida, M., 2003. An EGS4 user code developed for design and optimization of gamma-ray detection systems. J. Nucl. Sci. Technol. 40 (6), 441–445. Pérez-Moreno, J.P., San Miguel, E.G., Bolívar, J.P., Aguado, J.L., 2002. A comprehensive calibration method of Ge detectors for low-level spectrometry measurements. Nucl. Instrum. Methods A 491, 152–162. Ranger, N.T., 1999. Radiation detectors in nuclear medicine. Radiographics 19, 481–502. Ridikas, D., Feray, S., 2003. Non-destructive method of characterization of radioactive waste containers using gamma spectroscopy. CEA Saclay, DSM/DAPNIA.

Schoenfeld, E., Janssen, H., Klein, R., Hardy, J.C., Iacob, V., Sanchez-Vega, M., Griffin, H.C., Ludington, M.A., 2002. Precise efficiency calibration of an HPGe detector: source measurements and Monte Carlo calculations with sub-percent precision. Appl. Radiat. Isot. 56, 215–222. Sima, O., Dovlete, C., 1997. Matrix effects in the activity measurement of environmental samples. Implementation of specific corrections in a gammaray spectrometry analysis program. Appl. Radiat. Isot. 48 (1), 59–69. Sima, O., Arnold, D., Dovlete, C., 2001. Gespecor: a versatile tool in gammaray spectrometry. J. Radioanal. Nucl. Chem. 248 (2), 359–364. Sima, O., Cazan, I.L., Dinescu, L., Arnold, D., 2004. Efficiency calibration of high volume samples using the GESPECOR software. Appl. Radiat. Isot. 61, 123–127. Vidmar, T., Korun, M., Likar, A., Lipoglavšek, M., 2001. A physically founded model of the efficiency curve in gamma-ray spectrometry. J. Phys. D: Appl. Phys. 34, 2555–2560.