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

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. *Corresponding author: Chaohui He, Department of Nuclear Science and Technology, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China


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][3][4][5][6][7][8][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][13][14][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  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 ( Or, as a summation, Where i ρ are the radial positions of the point sources from the beaker axis. Eq 14 and Eq 15 can be written as For h 1, in the active volume of the detector, were consid-ered for the evaluation of full energy peak efficien-cy. The simulated full energy peak efficiencies are obtained from where Ɛ 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 → with their respective probabilities as P 1 , P 2 , and P 3 .
In absence of coincidence summing, the count rate is given by; (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 1 N * 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 Where ε T2 is the total detection efficiency for γ 2 . The CSF Simu for γ 1 is given by Similarly for γ 2,  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 (<J>) values obtained with our simulation approach is simple and precise to be used to calculate the CSF. The <J> values of the nuclides 60 Co and 88 Y for the various source volumes are listed in Table 2. The <J> values for each source volumes are smaller at low energies and significantly increase at high energy range as shown in Table 2. The computed <J> value depends on the source volumes. In volumes (50-300 ml) and (450-1000 ml), the <J> values decrease with the increase of source volumes for each photon energy. For Marinelli beaker source the <J> value is greater because of the close contact and the small distance of the source inside in the Marinelli beaker to the detector is shown in Table 2.
The CSF values were simulated for cylindrical and Marinelli beaker sources filled with aqueous solution of density 1 g/cm 3 . 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 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 i

Results and Discussion
In order to simulate the CSF, the total efficiency     nuclide, the CSF simu value is somewhat greater at low photon energy because of the greater <J> value at high photon energy, which means that there is an compared with the experimental and calculated results and obtained good agreement with the relative deviation equal to 2%. For each multi γ-ray  tively. 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 inverse relationship between <J>and CSF simu values. To observe the sample density effect on the CSFsimu 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 ) respec-  ulated 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 scattered in the samples itself at greater density. This analysis shows that the CSF value increased with the self-absorption of the source matrix.