# International Journal of Atomic and Nuclear Physics

# (ISSN: 2631-5017)

### Volume 4, Issue 1

### Research Article DOI: 10.35840/2631-5017/2511

# Coincidence Summing Correction for Cylinder and Marinelli Beaker Sources by Monte Carlo Simulation

Waseem khan, Chaohui He and Yu Cao

### Table of Content

### Figures

**Figure 1:** A typical decay scheme to show...

A typical decay scheme to show the coincidence summing effect.

**Figure 3:** Simulated peak and total efficiencies...

Simulated peak and total efficiencies for cylindrical beaker source.

**Figure 4:** Simulated peak and total efficiencies...

Simulated peak and total efficiencies for Marinelli beaker source.

### Tables

** Table 1:** Single line and multi gamma ray nuclides with emission probability.

** Table 2:** Computed 15-point integration of efficiency values for cylindrical and Marinelli beaker sources.

** Table 3:** Comparison between experimental and simulated coincidence summing correction factors for the cylindrical source.

** Table 4:** Comparison between experimental and simulated coincidence summing correction factors for Marinelli beaker source.

** Table 5:** Comparison of the simulated coincidence summing correction factors for different densities.

** Table 6:** Comparison of experimental and simulated coincidence summing correction factors of ^{133}Ba for cylindrical source.

** Table 7:** Comparison of experimental and simulated coincidence summing correction factors of ^{133}Ba for Marinelli beaker source.

** Table 8:** Comparison of experimental and simulated coincidence summing correction factors of ^{152}Eu for cylindrical source.

** Table 9:** Comparison of experimental and simulated coincidence summing correction factors of ^{152}Eu for Marinelli beaker source.

### References

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**Author Details**

Waseem khan, Chaohui He and Yu Cao

Department of Nuclear Science and Technology, Xi'an Jiaotong University, China

**Corresponding author**

Chaohui He, Department of Nuclear Science and Technology, School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China.

Accepted: May 18, 2019 | Published Online: May 20, 2019

Citation: Khan W, He C, Cao Y (2019) Coincidence Summing Correction for Cylinder and Marinelli Beaker Sources by Monte Carlo Simulation. Int J At Nucl Phys 4:011.

Copyright: © 2019 Khan W, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

## Abstract

Coincidence summing effects arises when two or more γ-rays are emitted in a cascade from an excited nucleus and are detected within the resolving time of the detector. Without correction of such effects, the activity of radionuclides cannot be accurately determined. For the correction of summing effects, a new simulation method in GEANT4 was established to simulate the coincidence summing correction factors (CSF_{simu}) for an HPGe detector. In the simulation, a cylindrical and Marinelli beaker source containing several radionuclides were used with different volumes, covering the energy range from 59.50 keV to 1836.01 keV. In the case of volumetric sources, the coincidence summing correction factors for two nuclides (^{60}Co and ^{88}Y) were calculated from the efficiencies at different points throughout the source volume. The dependence of the coincidence correction factor on the sample density was also carried out for some particular nuclide and photon energy. The same methodology of coincidence summing correction factor was applied for the complex decay scheme of ^{133}Ba and ^{152}Eu obtained a good agreement with the experimental results.

## Keywords

GEANT4, HPGe detector, Coincidence summing, Marinelli beaker sources

## Introduction

γ-ray spectrometry with HPGe detector is widely used to determine the activity of radionuclides in environmental samples. The accurate assessment of the activity of radionuclides would require a minimum source-detector distance to reduce the detection limit of the measuring system. The coincidence summing effect is more significant at a small source-detector distance because the probability of two γ-rays reaching the detector at the same time cannot be negligible at such distance. The coincidence summing effect changes in the count from the peaks corresponding to the two γ-rays and nuclides activity become inaccurate if no correction is performed. For the correction of such effects, the contribution of total efficiency is also required with the full energy peak efficiency. Various groups used different calibration techniques and obtained the coincidence summing correction factors (CSFs) from the total efficiency. Debertin & Schötzig [1] used the experimental technique and calculated the CSF from the total efficiency (the ratio of the total number of pulses recorded to the number of photons emitted by the source). Practically, the total efficiency curve is difficult to achieve due to the single γ-ray emitting nuclides and preparation of standard sources. Several authors used the analytical approaches for the calculation of the CSF from the total efficiency [2-9]. These approaches required information about the nuclear decay parameters such as the mode of parent nuclide decay, conversion factors, and the probability for the γ-ray transition from one energy level to another etc. Z Wang, et al. [10] used the Monte Carlo code MCNP and simulated the total efficiencies for the correction of coincidence summing effect. They used point source to test the coincidence summing correction method and observed a coincidence peak efficiency of at small source-detector distances. However such analysis is difficult to achieve for the close geometry measurements and large volume samples because in volumetric sources the contribution of the scattered γ-rays to the total efficiency cannot be neglected [11]. Many authors proposed an approach of point sources positioned in the matrix of the extended source for the calculation of peak, total efficiencies and CSF [4,12-15]. Tk Wang, et al. [4] include the effects of volume factor in the CSF values and observed a good agreement between calculated and experimental results. Recent techniques [16,17] in GEANT4 were good for the calculation of CSF, but such computational techniques required elaborate work in its implementation.

The aim of this paper is to develop a simplest and modest method in Geant4 for the coincidence summing correction factors (CSF_{simu}) of the extended sources. The CSF_{simu} values were compared with the calculated and experimental results reported by Wang, et al. [4] and obtained good agreements.

## Materials and Methods

GEANT4 [18] toolkit includes simulation of the electromagnetic interaction of charged particle, gamma, and optical photons. 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 photoelectric effect, Compton effect, pair production interaction, multiple scattering, bremsstrahlung, and ionization. The secondary electrons formed by photon interaction processes were also taken into consideration in the simulation. GEANT4 electromagnetic physics class was used in the simulation since the energy limit for the electromagnetic process is 10 keV to 100 TeV. Therefore, Ge X-rays of energy below 10 keV cannot be processed. GEANT4 also includes low-level electromagnetic processes that can simulate a particle down to 250 eV. The number of total histories (10^{7} primary photons) was considered for the simulation to obtain a statistical uncertainty of no more than 0.1%. All the photon energies emitted by the source were individually simulated for the source-detector geometries.

Only the γ-rays, which deposit their full energy in the active volume of the detector, were considered for the evaluation of full energy peak efficiency. The simulated full energy peak efficiencies are obtained from

$$\text{\epsilon =}\frac{\text{Q}}{\text{M}}\text{(1)}$$

where ε is the full energy peak efficiency, Q is the number of counts that deposit their full energy in the active detector volume, and M is the number of total simulated γ-rays counts for a given energy, E.

In order to simulate the total efficiencies and CSFs, a detailed decay scheme is considered as shown in Figure 1. The nuclide A decays to the two excited states of B. The two excited states deexcite by the emission of three γ-rays ${\text{\gamma}}_{\text{1}}\text{(3}\to {\text{2),\gamma}}_{\text{2}}\text{(2}\to {\text{1),\gamma}}_{\text{3}}\text{(3}\to \text{1)}$ with their respective probabilities as P_{}1, P_{2}, and P_{3}.

In absence of coincidence summing, the count rate is given by;

$${\text{N}}_{\text{1}}{\text{=Ap}}_{\text{1}}{\text{\epsilon}}_{\text{1}}\text{(2)}$$

Where A is the source activity, p_{1} is the emission probability with energy E_{1} and ε_{1} is peak efficiency for γ_{1} with E_{1}.

The count rate ${\text{N}}_{1}^{\ast}$ in the recorded full energy peak will be smaller than N_{1}. So the in presence of coincidence summing the count rate is given by

$${\text{N}}_{\text{1}}^{\text{*}}{\text{=Ap}}_{\text{1}}{\text{\epsilon}}_{\text{1}}{\text{-Ap}}_{1}{\text{\epsilon}}_{1}{\text{\epsilon}}_{\text{T2}}\text{(3)}$$

Where ε_{T2} is the total detection efficiency for γ_{2}. The CSF_{Simu} for γ_{1} is given by

$$\frac{{N}_{1}}{{N}_{1}^{\ast}}\text{=}\frac{1}{1-{\epsilon}_{T2}}\text{(4)}$$

$$\frac{{N}_{1}^{\ast}}{{N}_{1}}\text{=}1-{\epsilon}_{T2}\text{(5)}$$

or

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{a}}{\text{=1-\epsilon}}_{\text{Tsimu}}^{\text{b}}\text{(6)}$$

Similarly for γ_{2},

$${\text{N}}_{2}{\text{=Ap}}_{2}{\text{\epsilon}}_{2}\text{(7)}$$

$${\text{N}}_{\text{2}}^{\text{*}}{\text{=Ap}}_{2}{\text{\epsilon}}_{\text{2}}{\text{-Ap}}_{1}{\text{\epsilon}}_{2}{\text{\epsilon}}_{\text{T1}}\text{(8)}$$

$$\frac{{N}_{2}}{{N}_{2}^{\ast}}\text{=}\frac{1}{1-\frac{{p}_{1}}{{p}_{2}}{\epsilon}_{T1}}\text{(9)}$$

$$\frac{{N}_{2}^{\ast}}{{N}_{2}}\text{=}1-\frac{{p}_{1}}{{p}_{2}}{\epsilon}_{T1}\text{(10)}$$

Or

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{b}}\text{=1-}\frac{{\text{p}}_{\text{1}}}{{\text{p}}_{\text{2}}}{\text{\epsilon}}_{\text{Tsimu}}^{\text{a}}\text{(11)}$$

Where ${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{a}}$ and ${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{b}}$ are the simulated coincidence summing correction factors, ${\text{\epsilon}}_{\text{Tsimu}}^{\text{a}}$ and ${\text{\epsilon}}_{\text{Tsimu}}^{\text{b}}$ are the simulated total efficiencies of 1173.24 keV (a) and 1332.50 keV (b) respectively, similarly for ^{88}Y, ^{133}BA and ^{152}Eu.

The coincidence summing effects become more complicated for the extended volume sources. 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 volume sources, the CSF_{Simu} is given by

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{a}}\text{=}\raisebox{1ex}{$\int {\text{\rho \epsilon}}_{\text{1}}}\left({\text{1-\epsilon}}_{\text{Tsimu}}^{\text{b}}\right)\text{d\rho$}\!\left/ \!\raisebox{-1ex}{$\int {\text{\rho \epsilon}}_{\text{1}}\text{d\rho}$}\right.\text{(12)}$$

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{b}}\text{=}\raisebox{1ex}{$\int {\text{\rho \epsilon}}_{\text{2}}\left(\text{1-}\frac{{\text{p}}_{\text{1}}}{{\text{p}}_{\text{2}}}{\text{\epsilon}}_{\text{Tsimu}}^{\text{a}}\right)}\text{d\rho$}\!\left/ \!\raisebox{-1ex}{$\int {\text{\rho \epsilon}}_{\text{2}}\text{d\rho}$}\right.\text{(13)}$$

Or, as a summation,

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{a}}\text{=1-}\left[\raisebox{1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{\text{1}}}{\text{\epsilon}}_{\text{Tsimu}}^{\text{b}}\text{d\rho$}\!\left/ \!\raisebox{-1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{\text{1}}\text{d\rho}$}\right.\right]\text{(14)}$$

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{b}}\text{=1-}\left[\raisebox{1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{\text{2}}}\frac{{\text{p}}_{\text{1}}}{{\text{p}}_{\text{2}}}{\text{\epsilon}}_{\text{Tsimu}}^{\text{a}}\text{d\rho$}\!\left/ \!\raisebox{-1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{\text{2}}\text{d\rho}$}\right.\right]\text{(15)}$$

Where ${\text{\rho}}_{\text{i}}$ are the radial positions of the point sources from the beaker axis. Eq 14 and Eq 15 can be written as

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{a}}\text{=1-}\langle {\text{J}}_{\text{1}}\rangle \text{(16)}$$

For h_{1},

$${\text{J}}_{{\text{1h}}_{1}}\text{=}\raisebox{1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{1}}{\text{\epsilon}}_{\text{Tsimu}}^{\text{b}}\text{d\rho$}\!\left/ \!\raisebox{-1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{1}\text{d\rho}$}\right.\text{(17)}$$

For the whole volume source height,

$$\langle {\text{J}}_{\text{1}}\rangle \text{=}\frac{{\displaystyle \sum _{\text{i=1}}^{\text{3}}{\text{J}}_{{\text{1h}}_{\text{i}}}}}{\text{3}}\text{(18)}$$

Where h_{i} are the different distances from the beaker bottom. Similarly,

$${{\displaystyle \text{CSF}}}_{\text{Simu}}^{\text{b}}\text{=1-}\langle {\text{J}}_{2}\rangle \text{(19)}$$

$${\text{J}}_{{\text{2h}}_{\text{1}}}\text{=}\raisebox{1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{\text{2}}}{\text{\epsilon}}_{\text{Tsimu}}^{\text{a}}\text{d\rho$}\!\left/ \!\raisebox{-1ex}{$\sum {\text{\rho}}_{\text{i}}{\text{\epsilon}}_{\text{2}}\text{d\rho}$}\right.\text{(20)}$$

$$\langle {\text{J}}_{2}\rangle \text{=}\frac{{\displaystyle \sum _{\text{i=1}}^{\text{3}}{\text{J}}_{{\text{2h}}_{\text{i}}}}}{\text{3}}\text{(21)}$$

Where $\langle {\text{J}}_{\text{1}}\rangle $ and $\langle {\text{J}}_{2}\rangle $ are the average of 15-point integration of efficiencies.

To calculate the coincidence summing correction for all volumes, first, the cylinder and Marinelli beaker volumes are divided into three volumes (h_{1}, h_{2}, and h_{3}) and then further subdivided into 5 volume elements $\left({\text{\rho}}_{\text{i}}\right)$ for each (h_{1}, h_{2}, and h_{3}). Every single nuclide in ^{60}Co, ^{88}Y, and ^{152}Eu considered as a point source with their respective photon energies and placed at 15 positions within the source volume with three different distances (h_{1}, h_{2}, and h_{3}) from the beaker bottom. To get ${\text{J}}_{{\text{1h}}_{1}}$ for 898.02 keV or 1173.24 keV at volume source height h_{1}, first computed the ε and ${\text{\epsilon}}_{\text{Tsimu}}^{\text{b}}$ (at 1836.01 keV or 1332.50 keV) values at 5 different positions in the source volume and then computed the 5-point integration (i.e., multiplied each value by ${\text{\rho}}_{\text{i}}$, summed them, and divided by the sum of the ${\text{\rho}}_{\text{i}}$ ε). Similarly, calculated ${\text{J}}_{{\text{1h}}_{2}}$ (5-point integration of efficiencies) and ${\text{J}}_{{\text{1h}}_{3}}$ (5-point integration of efficiencies) at height h_{2} and h_{3} respectively and averaged them to get $\langle {\text{J}}_{\text{1}}\rangle $ at 15 volume elements except for the axial position of the beaker. The ε and ${\text{\epsilon}}_{\text{Tsimu}}$ value does not change with the further subdivision of the beaker volume. The same method was applied for 1836.01 keV and 1332.50 keV to obtain $\langle {\text{J}}_{2}\rangle $ but used ${\text{\epsilon}}_{\text{Tsimu}}^{\text{a}}$ (898.02 keV and 1173.24 keV) respectively in this case. The CSF values were also obtained for ^{133}Ba (276.39 keV, 302.85 keV) and ^{152}Eu (778.9 KeV, 964.0 keV and 444.0 KeV) nuclides using the same procedures.

The detector considered for MC simulation was a p-type coaxial HPGe detector (Canberra). The main parameters of the detector provided by the manufacturer are shown in Figure 2. 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. First, a cylindrical beaker source of diameter (D = 43.4 mm) filled with gamma radionuclides aqueous solution of volumes V1 (50 mL), V2 (100 mL), V3 (200 mL), and V4 (300 mL) was used to obtain the values. A Marinelli beaker source whose dimension is shown in Figure 2 with volumes V5 (450 mL), V6 (600 mL), V7 (800 mL) and V8 (1000 mL) was also used in the simulation. The cylinder beaker source was placed at a distance of 6.5 mm while the Marinelli beaker was placed in contact with the detector end-cap window. The radionuclides contained in the source solution with the γ-ray emission probability (P) are listed in Table 1. These nuclides were placed within the cylinder volumes at positions (h_{1} = 1.6 mm, h_{2} = 2.6 mm and h_{3} = 3.6 mm) and (5.32 mm, 16.5 mm, 25.5 mm, 34.5 mm and 42.5 mm) and Marinelli beaker volumes at (h_{1}= 30 mm, h_{2} = 60 mm and h_{3} = 90 mm) and (22 mm, 44 mm, 66 mm, 88 mm and 110 mm).

## Results and Discussion

In order to simulate the CSF, the total efficiency is always required with the full energy peak efficiency. The simulated full energy peak and total efficiency curves for cylindrical and Marinelli beaker sources with different volumes are shown in Figure 3 and Figure 4. The figures show that the full energy peak and total efficiency increases for the various volumes with the photon energy around 122.06 keV where the maximum values for the full energy peak and total efficiency were obtained. The full energy peak and total efficiency are close to each other at the low energy range because the absorption of the γ-rays in a single photoelectric interaction is predominated only for energies below about 145.44 keV as shown in figures. At high photon energy, the full energy peak efficiency drops off faster than the total efficiency because of the probability of Compton scattering followed by photoelectric absorption of the scattered photon is dominant than the absorption of the full photon energy in a single photoelectric event. As shown in figures the multiple scattering is the dominant contributor to the total efficiency over all but the lowest range of γ-ray energies. The total efficiency drops off slowly with the increased photon energy due to the less probability of scattered photon in the crystal active volume.

The 15-point integration of efficiency (

The CSF values were simulated for cylindrical and Marinelli beaker sources filled with aqueous solution of density 1 g/cm3. The values of the simulated coincidence summing correction factor (CSF_{simu}) obtained from Eq. 16 and Eq. 19 for (^{60}Co and ^{88}Y) are shown in Table 3 and Table 4. The CSF_{simu} is independent of the detector count rate but it is strongly dependent on the full energy peak and total efficiency. The CSF_{simu} values were compared with the experimental and calculated results and obtained good agreement with the relative deviation equal to 2%. For each multi γ-ray nuclide, the CSF_{simu} value is somewhat greater at low photon energy because of the greater _{simu} values.

To observe the sample density effect on the CSF_{simu} value, the simulation was performed for ethanol, water and sea sand sample (major component SiO_{2}) with densities (0.7, 1 and 2.5 g/cm^{3}) respectively. The comparison of the CSF_{simu} values for cylindrical and Marinelli beaker sources with different sample density are shown in Table 5. When the density of sample increases the CSF_{simu} value increases because the minimum number of γ-rays scattered in the samples itself at greater density. This analysis shows that the CSF value increased with the self-absorption of the source matrix.

The proposed simulated method was also applied to obtain the CSF values of ^{133}B and ^{152}Eu. The CSF_{simu} value for ^{133}B (276.39 keV) was calculated using Eq.16 with total efficiency of 160.61 keV. Similarly, CSF_{simu} value was calculated for 302.85 keV using Eq. 19 with emission probability ratio $\left(\frac{{\text{p}}_{\text{1}}}{{\text{p}}_{\text{2}}}\right)$ of (80.99 keV and 302.85 keV) and total efficiency of 80.99 keV. The simulated values were compared with the experimental results for cylindrical and Marinelli beaker sources as shown in Table 6 and Table 7. The simulated results agreed with the experimental values within 2% for all source volumes, except for the 50 ml and 300 ml where they are up to 3%. In the case of ^{152}Eu, Eq.16 was used to calculate the CSF_{simu} value for (778.9 KeV, 964.0 KeV and 444.0 KeV) with respect to the total efficiency of 344.3 keV, 1085.9 keV and 121.8 keV. The simulated results were compared with the experimental and calculated CSF values and obtained good agreements with experimental as shown in Table 8 and Table 9.

## Conclusions

A new method was used in GEANT4 to calculate the coincidence summing correction factors from the peak and total efficiencies and obtained accurate results for ^{60}Co, ^{88}Y, ^{133}Ba and ^{152}Eu, the average discrepancies between the experimental and simulated results were less than 1%. The simulation was performed and obtained the coincidence summing correction factors for various source volumes and observed the dependence of correction factors value on different samples densities. An easy technique developed in this study for the calculation of coincidence summing correction factor of complex nuclides. The suggested simulation method avoids the preparation of a great variety of gaseous samples with several isotopes and has added the advantages to improve the detection efficiencies for the measurement of the activity of environmental samples.

## Acknowledgments

This work at Xian Jiaotong University was fully supported by the Chinese government.