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Numéro
Radioprotection
Volume 54, Numéro 2, April–June 2019
Page(s) 133 - 140
DOI https://doi.org/10.1051/radiopro/2019006
Publié en ligne 21 mars 2019

© EDP Sciences 2019

1 Introduction

γ-ray spectrometry with high purity germanium detectors (HPGe) is widely used for determining the concentration and identification of unknown radionuclides in environmental samples. To determine the activity of each radionuclide, it is necessary to calculate the full energy peak efficiency at the energy of γ-ray emissions for a given measuring geometry. Most of the authors used different approaches and methods to calibrate HPGe detector efficiency (Helmer et al., 2004; Hurtado et al., 2004; Budjáš et al., 2009; Conti et al., 2013). But the major problem for HPGe detector efficiency calibration with environmental samples is the extended source dimension and self-absorption of the source matrix. However, the extended source dimension is not a significant problem because the average path length traveled by a photon inside the source matrix but photon absorption within the sample itself is difficult to achieve. For this reason, several procedures were developed to determine self-absorption correction factors. The most accurate method to determine the correction factor is the experimental method (Aguiar et al., 2006; Pilleyre et al., 2006; El-Khatib et al., 2014), where there is no need to make approximations. However, the experimental method is time-consuming and it is difficult to measure the full energy peak efficiency curve with sample preparation of different densities. So, the determination of the full energy peak efficiency is difficult by experimentally for the extended sources. To overcome these difficulties, Monte Carlo (MC) methods were used. The importance of such MC methods is that they enable one to quickly calculate a new efficiency value for changes in the measuring conditions. Different theoretical and MC approaches were used to calculate the full energy peak efficiency value including the effect of source self-absorption (Hardy et al., 2002; Vargas et al., 2002; Mostajaboddavati et al., 2006; Abbas, 2007; Khater and Ebaid, 2008; Badawi et al., 2012; Ababneh and Eyadeh, 2015). However, these methods required approximations and simplifications in the internal structure of the Ge crystal and source-detector geometries calculations (Shizuma et al., 2016).

In addition, when the source is positioned close to the detector, a coincidence summing effect arise in those nuclides which emit cascade γ-rays. Many authors observed some strong deviation from experimental results without including the effect of coincidence summing in the simulation (Rodenas et al., 2003; Conti et al., 2013). For the correction of such effect, the total efficiency is also required with the full energy peak efficiency. Debertin and Schötzig (1979) used the total efficiency (the ratio of the total number of counts observed to the number of photons emitted by the source) and calculated coincidence summing correction factor in measurements. Abbas et al. (2001) used the analytical approach to calculate the correction factor with total efficiency. Most of the authors used total efficiencies in the simulation for the calculation of coincidence summing correction factor (Wang et al., 2002; Vidmar et al., 2007; Ababneh and Eyadeh, 2015), and in recent years some authors used GEANT4 code and obtained good agreement with experimental results (Quintana and Montes, 2014; Giubrone et al., 2016). But these approaches required elaborate work in its implementation, especially for the close geometry measurements and large volume samples (Lee et al., 2008).

In this work, a simple and accurate method was developed in GEANT4 code to simulate the full energy peak efficiency (εsimu) of a co-axial HPGe detector, including the self-absorption (SAFcal) and coincidence summing (CSFcal) correction factor of the extended environmental source. Simulated results were compared to experimental measurement for a typical cylindrical measuring geometry with different volumes in the energy range 60–1836 keV.

2 Materials and methods

The MC code GEANT4 (Khan et al., 2018) can directly determine the energy deposited in the simulated volume of a detector, enabling the determination of εsimu. The code provides a realistic and fast procedure for the accurate assessment of self-absorption correction in samples without any optimizations and approximations. It can handle complex source geometries with any sample density and composition. The code follows the history of each individual primary photon until its energy dissipated in the detector and produces secondary particles as a result of the photoelectric effect, Compton effect, and pair production interaction. Only the γ-rays, which deposit their full energy in the active volume of the detector, were considered for the evaluation of the full energy peak efficiency. The secondary electrons formed by photon interaction processes were also taken into consideration in the simulation. G4EMStandardPhysics class was used for the low-level γ-rays spectrometry in the simulation since the energy limit for the electromagnetic process is 10 to 100 TeV. Therefore, Ge X-rays of energy below 10 keV cannot be processed. The number of total histories (106 primary photons) was considered for the simulation to obtain a statistical uncertainty of no more than 0.1%. The εsimu values were obtained by the VMware workstation 15.0.1 using i5-3570K 3.40 GHz Intel core processor. The GEANT4 calculation CPU times, for 12 εsimu values, i.e. εsimu values for 12 γ-ray energies, were ∼0.2 s for point sources and ∼1 min for cylindrical sources.

The detector considered for MC simulation was a p-type coaxial HPGe detector (Canberra) with an active volume of approximately 18960.18 mm3 (detector crystal with a length (l) of 89.7 mm, radius (r) of 34.95 mm, a core cavity with a height (h) of 80 mm and a radius (r) of 10 mm). The detector geometry was a closed-end coaxial and its relative efficiency at 1332 keV (60Co) is 44.3%. The detector has an aluminum end-cap window of thickness (w) of 1 mm, placed at a distance (d) of 4 mm from the crystal and a nominal dead-layer thickness (t) of 0.7 mm. A scheme of the detector is shown in Figure 1. No information was available by the manufacturer about whether the Ge crystals had rounded edges. Sharp edges of the crystals were assumed in the simulation. A cylindrical beaker source of radius (S = 34.8 mm) filled with γ radionuclides aqueous solution of volumes V1 (100 mL) and V2 (500 mL) were used to obtain the full energy peak efficiency values. The radionuclides contained in the source solution are (241Am, 85Sr, 137Cs, 109Cd, 114Sn, 88Y, 57Co, 139Ce, 60Co, 203Hg), covering the energy range from 60 to 1836 keV. The cylindrical beaker source was placed in contact with the detector end-cap window. Besides the source self-absorption, it is also necessary to consider the photon attenuation in the germanium dead-layer and entrance aluminum window. The thickness of the dead layer has a large influence on the detector efficiency, especially for low energy range, where the low energy photons are increasingly absorbed. Regarding the effect of aluminum material surrounding the detector, there is the probability of low energy photons attenuation in this region. In our model, the radiation enters only through the upper face of the crystal and therefore, the sidewall thickness has no influence on full energy peak efficiency value.

The simulated full energy peak efficiency () is obtained from: (1) where εsimu is the full energy peak efficiency, Q is the number of events that deposit their full energy in the active detector volume, and M is the number of total simulated events for a given energy, E.

The calculated self-absorption correction factor (SAFcal) is obtained from: (2) where and εsimu are the simulated efficiencies in the presence (water) and absence (vacuum) of the source attenuation, respectively.

In the absence of coincidence summing, the count rate (N1) is given by: (3) where A is the source activity, γ1 is the emission probability with energy E1 and ε1 is the peak efficiency for γ1 with E1.

If the energy of γ1 is totally absorbed, the sum count rate is observed at an energy between E1 and El + E2 and the event is lost from the full energy peak of γ1. The observed full energy peak would become: (4) where is the count rate in the presence of coincidence summing, εT2 is the total detection efficiency for γ2. For a point source, the calculated coincidence summing correction factor (CSFcal) for γ1 is given by: (5)or (6)

Similarly, for (7) (8) (9) or (10)where and are the calculated coincidence summing correction factors, and are the simulated total efficiencies of 1173 keV (a) and 1332 keV (b) respectively, similarly for 88Y. is the ratio between the emission probabilities for each multi γ-ray nuclide. Equations (6) and (10) show that the correction factors depend only on the total efficiencies and γ emission probabilities. The γ-ray emission probability (γ) values are listed in Table 1.

The coincidence summing effects become more complicated for the extended volume source. In this case, the correction factor not only depends on the peak and total efficiencies but also on the source volume and the differential efficiency distributions within the source. For a volume source, equations (3) and (4) have to be rewritten in a differential forms: (11) (12)

ρ dρ is considered as a volume element, r denotes the position of ρ dρ, ε1(r) and εT2(r) are the peak and total efficiencies at r, n1(r) ρ dρ and represent N1 and respectively, but the γ-rays emitted from volume element ρ dρ at r in this case. a(r) is the activity concentration at r. Integration on both sides of equations (11) and (12) over the source volume, we obtain: (13) (14) (15)

or (16) or (17)

Similarly, for (18) (19) (20) (21) (22) or (23)or (24)or, as a summation, equations (17) and (24) can be written as: (25) (26)where ρi is the point source positions from the detector axis in volume source. Equations (25) and (26) can be written as: (27)

For H1, (28)

For the whole volume source height, (29) where Hi are the different distances from the beaker bottom. Similarly, (30) (31) (32)where ⟨J1⟩ and ⟨J2⟩are the 10-point integration of efficiencies for each nuclide. To calculate the coincidence summing correction for both source volumes, first, the beaker volume is divided into two volumes (H1 and H2) and then further subdivided into 5 volume elements (ρi) for each (H1 and H2). Four single nuclide point sources with photon energies (60Co (1173 keV, 1332 keV) and 88Y (898 keV, 1898 keV)) were placed at 10 positions within the source volume with two different distances (H1 and H2) from the beaker bottom. To get for 898 or 1173 keV, first computed the and (1836 or 1332 keV) values at 5 different positions in the source volume and then computed the 5-point integration (i.e., multiplied each value by ρi, summed them, and divided by the sum of the ρi). Similarly, calculated (5-point integration of efficiencies) at height H2 and averaged them to get ⟨J1⟩ at 10 volume elements. The and values does not change with the further subdivision of the beaker volume. Similarly, computed the and (898 or 1173 keV) values at 10 different positions to obtain ⟨J2⟩ for 1836 or 1332 keV.

The true simulated full energy peak efficiency () is obtained from: (33)

thumbnail Fig. 1

Schematic of the source-detector.

Table 1

Multi γ-ray nuclides with emission probability.

3 Results and discussion

3.1 Analysis of self-absorption correction factor

The full energy peak efficiency was simulated for a cylindrical water composition source of density (ρ = 1 g/cm3) with two different volumes. First, the simulated result was obtained without source self-absorption of the source matrix (water) and there was no attenuation of a γ-ray photon from the source matrix but attenuated from the source container material. As given in Tables 2 and 3, the εsimu results without source self-absorption are compared with the experimental full energy peak efficiency (εExp) values. Obviously, the non-inclusion of the source self-absorption caused an increase in εsimu values. For various source volumes, large deviations in values without source self-absorption were observed with the experimental results. So to obtain the correct simulated results, the source self-absorption must be taken into consideration. Tables 2 and 3 show the SAFcal values for different source volumes. These tables clearly show the effect of the source self-absorption on the εsimu value, especially for the low photon energy. The SAFcal value is somewhat great and it’s more effective for the low photon energy. The relative deviations (RD) are somewhat greater at high energies for both source volumes. Tables 2 and 3 show good agreement between simulated () and experimental (εExp) results with RD less than 2% at low energies due to the inclusion of SAFcal.

For the extended sources, different samples with different chemical composition and density caused significant variations in the full energy peak efficiency value. But for most environmental samples, the full energy peak efficiency with source self-absorption greatly depends on the density of the samples. To observe the sample density effect on the full energy peak efficiency value, we simulated the full energy peak efficiency value for four samples with different density (0.8, 1, 1.5 and 1.9 g/cm3). As shown in Tables 4 and 5, the comparison of the simulated results for various samples volumes, show the dependence of value on different sample density. The tables show that when the density of the sample increases the value decreases because the minimum number of γ-rays scattered in the samples itself at greater density.

Table 2

Relative deviation between experimental and simulated () full energy peak efficiency values with self-absorption correction factor for V1.

Table 3

Relative deviation between experimental and simulated () full energy peak efficiency values with self-absorption correction factor for V2.

Table 4

Variation of the full energy peak efficiency value with density for V1.

Table 5

Variation of the full energy peak efficiency value with density for V2.

3.2 Analysis of coincidence summing correction factor

For the energy range (60 to 662 keV), a good agreement was achieved with SAFcal. However, for the high energy range (898 to 1836 keV), maximum discrepancies were obtained due to the non-inclusion of coincidence summing effect in values. In the case of extended sources, the εTsimu value needs to be taken into account to find CSFcal for 60Co and 88Y. Table 6 shows the 10-point integration of efficiency values obtained with GEANT4 for the extended volumetric sources.

The 10-point integration of efficiency values obtained with our simulation approach is simple and precise to be used to calculate the coincidence summing correction factor. The values of the correction factor for 60Co and 88Y are shown in Table 7. The CSFcal is independent of the detector count rate but it is strongly dependent on the full energy peak and total efficiency. By comparison, there is an inverse relationship between ⟨ J ⟩ and CSFcal values for nuclides 60Co and 88Y. Results indicated that the CSFcal also depends on the different source volume. Table 7 shows that the CSFcal decreases with the source volume, which means that the probability of such summing effects decreases with increasing of the source to detector distance.

The true values were obtained by applying the CSFcal in the simulation for nuclide 60Co and 88Y. Table 8 shows a good agreement with the experimental results, with discrepancies less than 2% for both extended volumetric sources.

Table 6

Calculated 10-point integration of efficiency values for different source volumes.

Table 7

Calculated coincidence summing correction from 10-point integration efficiency values.

Table 8

Comparison of experimental and simulated full energy peak efficiency values with CSFcal.

4 Conclusions

GEANT4 simulation toolkit was used to simulate the full energy peak efficiency of a coaxial HPGe detector for the extended volumetric sources. The self-absorption correction factors were calculated and obtained accurate full energy peak efficiency values for the low energy range. The simulation was performed and observed the dependence of the full energy peak efficiency value on different sample densities. A new method was used in GEANT4 to calculate the coincidence summing correction factors and obtained accurate simulated results; the discrepancies between the experimental and simulated efficiencies were found less than 2%. The proposed simulated method avoids the preparation of the great variety of radioactive samples with several isotopes and has added the advantages to improve the detection efficiencies for the measurement of the activity of various samples.

Acknowledgments

This work at Xian Jiaotong University was fully supported by the Chinese government. The authors would like to thank the entire staff of the Nuclear Science and Technology department for the very valuable information in the completion of this work.

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Cite this article as: Khan W, He C, Cao Y. 2019. Calculation of self-absorption and coincidence summing correction factors for the extended sources using GEANT4. Radioprotection 54(2): 133–140

All Tables

Table 1

Multi γ-ray nuclides with emission probability.

Table 2

Relative deviation between experimental and simulated () full energy peak efficiency values with self-absorption correction factor for V1.

Table 3

Relative deviation between experimental and simulated () full energy peak efficiency values with self-absorption correction factor for V2.

Table 4

Variation of the full energy peak efficiency value with density for V1.

Table 5

Variation of the full energy peak efficiency value with density for V2.

Table 6

Calculated 10-point integration of efficiency values for different source volumes.

Table 7

Calculated coincidence summing correction from 10-point integration efficiency values.

Table 8

Comparison of experimental and simulated full energy peak efficiency values with CSFcal.

All Figures

thumbnail Fig. 1

Schematic of the source-detector.

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