The 4π NaI(Tl) γ-ray detectors are consisted of the well cavity with cylindrical cross section, and the enclosing geometry of measurements with large detection angle. This leads to exceptionally high efficiency level and a significant coincidence summing effect, much more than a single cylindrical or coaxial detector especially in very low activity measurements. In the present work, the detection effective solid angle in addition to both full-energy peak and total efficiencies of well-type detectors, were mainly calculated by the new numerical simulation method (NSM) and ANGLE4 software. To obtain the coincidence summing correction factors through the previously mentioned methods, the simulation of the coincident emission of photons was modeled mathematically, based on the analytical equations and complex integrations over the radioactive volumetric sources including the self-attenuation factor. The measured full-energy peak efficiencies and correction factors were done by using ^{152}Eu, where an exact adjustment is required for the detector efficiency curve, because neglecting the coincidence summing effect can make the results inconsistent with the whole. These phenomena, in general due to the efficiency calibration process and the coincidence summing corrections, appear jointly. The full-energy peak and the total efficiencies from the two methods typically agree with discrepancy 10%. The discrepancy between the simulation, ANGLE4 and measured full-energy peak after corrections for the coincidence summing effect was on the average, while not exceeding 14%. Therefore, this technique can be easily applied in establishing the efficiency calibration curves of well-type detectors.

## I. INTRODUCTION

Low activities of radionuclides in environmental and occupational samples demand to lower the detection limits of the measuring system, which can be achieved by minimizing the source-to-detector distance. The preparation of standard radioactive sources is costly and time consuming, especially if the laboratory is required to measure samples with different geometries.^{1} The experimental efficiency calibration is restricted to several measurement geometries and cannot be applied directly to all measurement configurations. An alternative possibility of being able to compute the efficiencies is thus highly desirable.^{2} The environmental studies often include the measurement of very little amounts of material. This is partially due to sampling limitations or because the sampled material is generally shared by multiple scientists that require different type of measurements. The well-type detectors, with their high efficiency, are the type of detectors most often used for the measurement of small activity samples.^{3–5}

When γ-ray samples spectra, that are to be interpreted quantitatively are counted in highly efficient geometries, such as a small sources placed in the cavity of the well-type detector. The calibration should either be performed with the radionuclide of interest in the same geometry, to be used and the coincidence summing corrections should be applied, where the problem becomes even more complex when voluminous samples are measured.^{4,5} In the last one, the knowledge of the full-energy peak efficiency as well as the total efficiency curves are required, and these curves are obtained by independent measurements.^{6} In order to perform accurate and reliable activity measurements, periodical metrological calibrations of the gamma-ray spectrometry installations of the laboratories are required, where uncorrected efficiency curves will lead to erroneous activity calculations.^{7–9}

The true coincidence-summing (TCS) is at the present one of the main sources of systematic errors in γ-ray spectrometry. The coincidence summing occurs with radionuclides emitting two or more cascading photons within the resolving time of the spectrometer.^{10,11} Therefore, if the first photon spends its total energy in the crystal and if the second photon is also detected, a sum pulse is recorded. The event is lost from the full energy peak of the first and second photon. The probability for such summing effects increases with decreasing source-to-detector distance. It is dependent of counting rate and particularly high in low-level measurements, where the samples are usually positioned close to the detector surface.^{12–14} Of course, no corrections have to be applied, if a sample is measured relative to a standard of the same radionuclide. However, if the efficiency is determined as a function of energy by means of a set of standard or multi-gamma sources, then the problem of coincidence summing corrections arises makes experimental efficiency calibration difficult by limiting radionuclides available to coincidence free sources.^{15–18}

In this study complex procedure for coincidence-summing correction in γ-ray spectrometry were investigated. These procedures are based on common theoretical expressions of the correction factors for determination of the coincidence-summing correction factors for nuclides, with the complex decay scheme of ^{152}Eu radionuclide according to the methodology proposed before in Ref. 19, and modified it to deal with well-type detectors. A computer program has been developed to analytically calculate of the integrated effects of all coincidence summing in a complex spectrum; it incorporates a new method for calculating the full-energy and the total detection efficiencies at any distance, which have been obtained by using the numerical simulation method (NSM) and the ANGLE4 software as well.^{19} The method validity has been checked by comparison with experimental efficiencies determined for well-type detectors, with standard radioactive sources measured inside the detector cavity after correction for the true coincidence-summing (TCS). These comparisons were done with the free full-energy peak efficiency values, which were produced by the numerical simulation method (NSM) and the ANGLE4 software including the source self-attenuation factor.

This article is arranged as follows: Section II is the mathematical treatment and presents in details the new analytical calculation of the effective solid angle, the full-energy peak and the total efficiency using radioactive volumetric sources measured inside the well-type detector cavity. This section included also the ANGLE4 description and the coincidence summing model for ^{152}Eu radionuclide as a complex decay scheme. Section III includes the details of the experiment’s setup, which were conducted in the Radiation Physics Laboratory (RPL), Faculty of Science, Alexandria University, Egypt. Section IV contains the values for the true coincidence-summing (TCS), a comparison between the calculated full-energy peak efficiency values of well-type NaI(Tl) detectors, that were obtained by the recent numerical simulation method (NSM) and ANGLE4 software with the experimental results after performing the coincidence summing corrections, showing the applicability of the current approach. Conclusions are presented in Section V.

## II. MATHEMATICAL TREATMENT

### A. Total and full-energy peak efficiencies

In this section, a new numerical simulation method (NSM) based on the direct method^{20–31} will be functional to obtain the total and the full-energy peak efficiency of the well-type detector. The well-type detector of outer radius R_{2}, cavity radius R_{1}, outer height L and cavity depth S will be considered as shown in Figure. 1. The detector effective solid angle $\Omega $_{Eff(Cylinder)} for using an axial cylindrical radioactive source of height h_{s} and radius R_{s}, which placed inside the well-type detector cavity at distance h_{o} as shown in Figure. 1 can be given by:

The cylindrical radioactive source was treated as a group of point sources, each of them has an effective solid angle Ω_{Eff (Point Inside)} inside the detector cavity at different lateral distances ρ started from 0 up to R_{s} and calculated at different heights started from h_{o} up to (h_{o}+h_{s}). Based on the geometry between the source and the detector presented in Figure. 2. Depend on the relation between the different extreme polar angles, there are two possible cases of non-axial radioactive point source inside the well-type detector to determine the effective solid angle Ω_{Eff} _{(Point Inside)}. The first case at θ_{2}>θ_{3}>θ_{4}>θ_{1} and the second case at θ_{2}>θ_{3}>θ_{1}>θ_{4}. Each case depends on the height h between the source and the cavity base surface and the detector dimensions, beside it’s depends on the lateral distance ρ. The effective solid angle Ω_{Eff (Point Inside)} in this case can be given by:

Where:

It is clear to observe that, as the azimuthal angle φ increase as the cases transferred from case 1 to case 2 as shown in Figure. 2, also from the equation (2 and 3) and Figure. 2 as well an important remark is observed, where the angle $\theta 4$ plays an important role in determining the path lengths inside the detector active material.

The polar angles take four different steps depend on the value of R_{1}, h, φ and ρ, which can be given by:

There are five probabilities to be considered for the emitted photon from the isotropic radiating point source placed at certain position inside the crystal cavity to enter the detector and covering the distances, d_{1}, d_{2}, d_{3}, d_{4} and d_{5} as shown in Figure. 2. These distances can be given as follows:

According to equations (2 and 3), the factor f_{i} will be introduced to represent the probability fraction of interactions of photons with the detector material as given in equation (6).

Where f_{i} will take expressions from f_{1} to f_{4} also it can be changed according to possible path lengths d_{1}, d_{2},…, d_{n} traveled by the photon depending on the position of the incidence photon within the detector active volume from f_{1}, f_{2},…, f_{n}, where μ is the total attenuation coefficient of the detector active medium for a γ-ray photon with energy $E\gamma $ (the coherent scattering part excluded) as reported in Ref. 28. The attenuation factor f_{att} for that absorbers layers with attenuation coefficients $\mu 1$_{,}$\mu 2$,…, $\mu n$ and with the thickness t_{1}, t_{2},…, t_{n} between the source and the detector system can be given by:

In the case of radioactive volumetric source, not all the emitted photons leave the source vial with the same energy, a part of them is absorbed in the source matrix itself. This self-absorption factor S_{f} is defined by:

where: μ_{s} is the source matrix attenuation coefficient and d_{s} is the distance traveled by the photon within the source material as a function of the polar and the azimuthal angles, (θ, φ), as shown in Figure. 1, where the azimuthal angle φ changed from 0 up to 2π.^{23} The radioactive volumetric source polar angles θ take the steps:

In this case the distance traveled inside the source d_{S}, will vary upon the variation of the polar and azimuthal angles, (θ, φ), inside the source itself and can be given by:

The total efficiency of well-type detector ε_{T(Cylinder)}^{20,21} for using an axial cylindrical radioactive source of height h_{s} and radius R_{s}, which placed inside the well-type detector cavity at distance h_{o} as shown in Figure. 1 can be given by:

The efficiency transfer method (ET) was applied for the computation of the full-energy peak efficiency ε_{p(Cylinder)} of NaI(Tl) Well-type detectors for volume sources measured inside its cavity on the basis of the reference efficiency measured for a point source located out the cavity at 50 cm distance from the detector end cap surface according to the methodology proposed before.^{29–31} The full-energy peak efficiency ε_{p(Cylinder)} of the well-type detectors can be given based on the following equation:

where, ε_{P(Cylinder)} and ε_{Ref} are the full-energy peak efficiency (FEPE) of the target (a volume source measured inside the detector well cavity) and the reference geometry (an isotropic radiating axial point source measured out the detector well cavity), respectively. While Ω_{EFF(Cylinder)} and Ω_{Eff(Ref)} are the effective solid angles subtended by the detector surface with the volumetric and the reference geometry sources, respectively. In order to use the efficiency transfer technique (ET), the fitting curve for the measured reference efficiency ε_{ref} must be as essentially done.^{29–31}

The double integrals in the both methods, the direct mathematical method and the numerical simulation method (NSM)^{20–31} for the total and the full-energy peak efficiency of the well-type detector can be evaluated numerically using the trapezoidal rule. A computer program has been written in BASIC programming language to evaluate the efficiency of a well-type detector with respect to the source-to-detector separation. The accuracy of the integrations improved with increased the interval number n and convergence very good at n=20.

### B. ANGLE4 software

Angle4 is the latest version (released in 2016) of the well known advanced quantitative gamma-spectrometry software. Angle has been in use for over 20 years in numerous gamma-spectrometry based analytical laboratories worldwide. Originally developed for semiconductor (Ge) detectors, Angle4 currently supports scintillation (NaI) devices as well. The vast majority of gamma-spectrometry systems in operation nowadays are built around these two detector types. Angle allows for the accurate determination of the activities of gamma-spectroscopic samples, and thus is used for the quantification (i.e. spectrometry) of measurements. This is achieved by the calculation of detection efficiencies (see further), using the so-called “efficiency transfer” (ET) method. ET is a semi-empirical approach, which means that it is a combination of both experimental evidence and mathematical elaboration: detection efficiency for any “unknown” sample and its activity can be determined by a calculation from the measurement of a standard source with a known activity (detector calibration). The two (standard and sample) do not need to match whatever – by shape, size, composition or positioning vs. the detector, etc. – offering practically unlimited flexibility in application.^{19}

Angle4 comes with many new options, bringing brand new functionalities which make working with Angle easier. The most important new features are briefly highlighted as follows. NaI detectors. One of the most important new functionalities in Angle4 is support for two new scintillation (NaI) detector types: cylindrical and well types. Together with the existing six semiconductor (Ge) detector types, Angle is able to cover the vast majority of gamma-spectrometry systems in operation nowadays, which are built around these detector types. Demo detectors. Angle4 comes with a set of eight predefined demo detectors, one for each supported type.^{19} Demo detectors can be used for calculations even in demo mode, that is, even if Angle is not registered. This enables the users to run calculations and evaluate the results, so as to become more familiar with the scope and capabilities of the software without/before being registered.

In Angle4 one can define a “discrete curve”, meaning that you actually do not need to interpolate the curve, but to use the efficiencies for the exact energies which exist in the experimental points. The user is not being able to calculate efficiencies for energies other than those defined by their experimental points, but their exact efficiency values will be used (instead of interpolated ones), leading to better accuracy for the given energies. Previews. Using Angle4, the user is able to see the scaled image of your detectors, containers, geometries and even complete counting arrangements. The user can see and export the graph of the calculated efficiency curve. It shows both the calculated efficiencies for selected energies and the fitted efficiency curve, together with the reference efficiency curve.^{19}

It is now also possible to import parameters for detectors, containers, geometries, reference efficiency curves and energies from calculation parameter files and calculation results files. Additionally, now it is possible to export the results to ORTEC GammaVision – both to efficiency calibration (EFT) and geometry correction (GEO) files. Calculation reports. Calculation results can be easily printed. It is possible to print the calculated results with the summary of all input parameters, counting arrangements schematics, as well as a calculated efficiency curve. A detailed report can be printed as well. The report contains both the results and all the input parameter data used in calculations.^{19} This makes the results traceable to the input values. Command line parameters. It is possible to automate your calculations in Angle from another application or batch files, using a set of command line parameters. Together with the new XML-based file format, this allows easy integration with third-party applications (e.g. in-house developed software). This enables, for instance, thousands of calculations to be done in a couple of hours’ batch, making it suitable for research and scientific purposes (e.g. error propagation studies, control and optimization of spectrometry parameters, tracking sources of systematic errors, etc.).

### C. Coincidence summing model

In general practice, scientists prefer to use the multi-lines γ-ray sources for many different reasons such as: The calibration process has become much easier for using only one, two or three sources as a function of energy as well as of efficiency, the periodic checks are simple as the spectrum of a multi-lines γ-ray is constant in time and the half life time of the most of these sources is long (i.e. ^{152}Eu). These multi-lines sources have a little bit drawbacks due to their complex decay scheme. The analysis of its spectrum is not easy as well and the measured calibration points show a noticeable deviation from the ideal curve, which can be obtained if the mono-energetic sources were used. These deviations are caused due to count gained or lost by coincidence summing effect.

The true coincidence-summing (TCS) takes place when two or more photons are emitted in cascades from the radioactive source and detected simultaneously within the resolving time of the detector. These effects of coincidence produce the so-called summing-in or summing-out effect, which caused the coincidence gained or lost respectively from the full energy peaks and enhancement of sum peaks occur, the true coincidence-summing (TCS) can given by:

Considering that: N(E) and N^{o}(E), are the measured count rates of gamma-rays in absence of coincidence and in presence of coincidence respectively. The code proposed before in Ref. ^{32} is valid only for cylinder shape detectors, due to this drawback the total and full-energy peak efficiencies for well detector has been calculated and inserted into a spreadsheet contains all rest of the parameters of this code, in order to estimate the corresponding correction factors for ^{152}Eu at different energies. The Angular correlations were neglected within the model because of the large solid angle of detection inside the well-type detectors that cancels the effect.

This code becomes very complex for radioactive volumetric source with several cascade transitions. The general expression of the correction (TCS)_{IJ} to γ_{IJ}-ray transition emitted energy E_{IJ} from energy level I to energy level J in ^{152}Eu decay scheme with (N+1) levels started from 0 up to N is given by:

Where it’s conditions given as:

The first parameter A represents the coincidences with γ-ray lines preceding the γ_{IJ} line and can be given by:

Where:

The symbols K, L, B, Q and U in equations (16 and 17) used to refer to the limitations of the series, While T_{L}, T_{I} and T_{B} represent the sum of the probabilities of γ-ray transitions incoming to level L, I and B respectively and the formula T_{U} can be used to calculate all of them. X_{KL}, X_{LB} and X_{QU} represent the probability of the transition happen from (K to L), (L to B) and (Q to U) respectively, while the factor $\alpha KLT$refers to the total internal conversion coefficient and $\epsilon T(Cylinder)KL$ is the detector total efficiency of energy E_{KL} using equation (11) based on cylindrical vial dimensions filled with liquid ^{152}Eu as a radioactive material and measured inside the cavity.

The second parameter B represents the coincidences with γ-ray lines following the γ_{IJ} line and can be given by:

Where:

The symbols P, M, Q, Y and Z in the equations (18 and 19) used to refer to the limitations of the series, While _{TP} and T_{J} represent the sum of the probabilities of γ-ray transitions leaving from level P and J respectively and the formula T_{Y} can be used to calculate all of them. X_{PM,} X_{JP,} X_{JQ and} X_{y(Z-1)} represent the probability of the transition happen from (P to M), (J to P) and (J to Q) and (Y-(Z-1)) respectively, while the factor $\alpha PMT$ refers to the total internal conversion coefficient and $\epsilon T(Cylinder)PM$ is the detector total efficiency of energy E_{PM} using equation (11) based on cylindrical vial dimensions filled with liquid ^{152}Eu as a radioactive material and measured inside the cavity.

The third parameter C includes the coincidences of γ-ray lines and the sum of its energies is equal to the transition energy EIJ from I to J level, this parameter can be given by:

Where:

And,

The symbols G, Z and Q in the equations (20, 21 and 22) used to refer to the limitations of the series, While T_{G} represents the sum of the probabilities of γ-ray transitions incoming (or leaving) level G and the formula T_{G} can be used to calculate it. X_{GJ}, X_{IG}, _{XIJ}, X_{G(Z-1)} and X_{QG} represent the probability of the transition happen from (G to J), (I to G) and (I to J), (G-(Z-1)) and (Q to G) respectively, while the factors $\alpha IJT$, $\alpha IGT$ and $\alpha GJT$ refer to the total internal conversion coefficient. The notations $\epsilon P(Cylinder)GJ$, $\epsilon P(Cylinder)IG$ and $\epsilon P(Cylinder)IJ$ are the full-energy peak efficiency of the well-type detectors using equation (12) based on cylindrical vial dimensions filled with liquid ^{152}Eu as a radioactive material and measured inside the cavity.

## III. EXPERIMENTAL SETUP

### A. Detectors

There are two different detectors 2″×2″ (D1) and 3″×3″ (D2) NaI(Tl) well-type scintillation detectors from Canberra were used in the present work for measuring the γ-ray spectra of the radioactive sources. The main technical features were provided by the company are: The detector 2″×2″ dimensions were given as: 50.8 mm crystal length and 25.4 mm crystal radius, 0.25 mm Al end cap thickness, 5.16 mm and 2.71 mm Al_{2}O_{3} thickness of face and side reflector layers respectively, 8.58 mm crystal cavity radius and 34.11 mm crystal cavity depth. The detector 3″×3″ dimensions were given as, 76.2 mm crystal length and 38.1 mm crystal radius, 0.25 mm Al end cap thickness, 2.5 mm and 2 mm Al_{2}O_{3} thickness of face and side reflector layers respectively, 8.58 mm crystal cavity radius and 49.87 mm crystal cavity depth. The energy resolution (FWHM) of the detector was 9% at the 661 keV γ-ray line of ^{137}Cs source for both detectors based on the manufactory certificate and the shaping mode was Gaussian. The detectors were coupled to a Canberra data acquisition system (Osprey Base) applying a Genie-2000 analysis software, with many functions including peak area determination, background subtraction together with both γ-ray energy and radionuclide identification.

### B. Sources

There are two sets of sources were used in that work, the first one is ^{152}Eu radioactive point source, which was purchased from the Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig and Berlin-National Institute for Science and Technology and the highest technical authority of the Federal Republic of Germany in the field of metrology and certain sectors of safety engineering. The energy of incident photons was varied by (121.78, 244.69, 344.28, 778.90, 964.13 and 1408.01 keV). The activity of the previous sources in kBq was 290.0±4.0 respectively on 1 June 2009. The values of the photon emission probabilities per decay for the ^{152}Eu which used in the calibration process are available from the National Nuclear Data Center Web Page or on the IAEA website.^{33} The radioactive material was a very small, surrounding deposit with about 5 mm in diameter, in the center of two polyethylene foils, each having a mass per unit area of (21.3 ± 1.8) mg.cm^{-2}. Under pressure and by heating, the both foils were welded as one over the whole area so that they were leaked-proofed. To make possible handling, the foil 26 mm-diameter is mounted in a circular aluminum ring (external diameter: 30 mm, height: 3 mm) from which it could simply be removed if and when required.

The second set of radioactive sources was a homemade volumetric sources prepared from standard solution. The details of the preparation source and dimensions are listed in Table I and Figure. 3, where three HDPE vials of geometry types K, D and R from Posthumus Plastics Company in Beverwijk, Netherlands were filled with small amount of ^{152}Eu aqueous solution of a well known activities, where the symbols D, d, L1 and L2 in Table I are the dimensions of that vials and its cap as well.

HDPE Volumetric Sources Description in (mm) . | ||||||||
---|---|---|---|---|---|---|---|---|

Vial Type . | Cap Type . | D . | d . | L1 . | L2 . | Filled with (cm^{3})
. | Activity (Bq) . | Height . |

K (0.4 cm^{3}) | L | 7.2 | 5.7 | 16.0 | 15.4 | 0.33 | 67.26 | 13.05 |

D (0.8 cm^{3}) | E | 9.3 | 8.0 | 18.3 | 17.5 | 0.70 | 142.23 | 14.01 |

R (1.5 cm^{3}) | S | 11.5 | 10.0 | 22.0 | 21.3 | 1.26 | 254.00 | 16.01 |

Cap Type Details | Type | D | d | L1 | L2 | Company | ||

L | 7.2 | 5.7 | 1.7 | 1.2 | ||||

E | 9.3 | 8.0 | 1.7 | 1.2 | Posthumus Plastics Company in Beverwijk, Netherlands | |||

S | 11.5 | 10.0 | 2.8 | 1.9 |

HDPE Volumetric Sources Description in (mm) . | ||||||||
---|---|---|---|---|---|---|---|---|

Vial Type . | Cap Type . | D . | d . | L1 . | L2 . | Filled with (cm^{3})
. | Activity (Bq) . | Height . |

K (0.4 cm^{3}) | L | 7.2 | 5.7 | 16.0 | 15.4 | 0.33 | 67.26 | 13.05 |

D (0.8 cm^{3}) | E | 9.3 | 8.0 | 18.3 | 17.5 | 0.70 | 142.23 | 14.01 |

R (1.5 cm^{3}) | S | 11.5 | 10.0 | 22.0 | 21.3 | 1.26 | 254.00 | 16.01 |

Cap Type Details | Type | D | d | L1 | L2 | Company | ||

L | 7.2 | 5.7 | 1.7 | 1.2 | ||||

E | 9.3 | 8.0 | 1.7 | 1.2 | Posthumus Plastics Company in Beverwijk, Netherlands | |||

S | 11.5 | 10.0 | 2.8 | 1.9 |

The homemade sources were prepared by using the original aqueous source activity from (PTB), where the needed activity can be calculated by knowin how much grams requsted to extract from the parent solution.^{19} The total activity of the parent solution was 202±4 kBq with total mass 1000.41 g at 1 Jan, 2010. The parent solution activity was divided by the total of its mass in order to get the activity of each gram using the specific activity law:

where, A is the activity (Bq), a is the specific activity (Bq.g^{-1}) and m is the mass of the solution in gram.

### C. Measurements, data recording and processing

The ^{152}Eu radioactive point source was measured and recorded by using a special Plexiglas holder offer a broad free solid angle between the source and the detectors. The holder has enough thick base placed straightforwardly on the detector entrance window to protect the detector heads. In order to prevent the dead time, the pile up effects and staying away from the correction for coincidence summing effects, the large source-to-detectors axial distance was considered from the detector end cap. The sources were kept fixed at 50 cm in the measurement geometry from the detector outer end cap. The radioactive volumetric sources were measured inside the well-type detectors cavity using an especial Plexiglas holder to avoid the cavity contamination as shown in Figure. 4.

To get good calibration and measurement process, the detector with the smallest amount of electronic noise effects and unwanted pulses in the absence of the radioactive sources must be considered. The experimental measurement of both processes was done by using a good stabilizer and a good system configuration was kept away from the walls and the floors of the room.

The measured time was as long as required to get high and sufficient counts under each peak of attention with a statistical uncertainty less than 1%. Spectrum analysis is done by selecting the Region of Interest (ROI). This is done by selecting the Region of Interest or ROI. ROI will be selected between the start and stop channels as desired. The Genie-2000 analysis software gives integral counts, background counts and background subtracted counts under ROI. In addition, the program has a large facility to do automatic peak search and peak area calculations, along with changes in the peak fitting, using an interactive peak fitting interface, when necessary, to decrease the residuals and error in the peak area values. The measured full-energy peak efficiency ε(E) of the 4π NaI(Tl) γ-ray Detectors at energy E, can be determined by using the following equation:

where P(E) is the absolute γ-ray emission probability, N^{o}(E) is the net peak area in counts in presense of coincidence, A_{S} is the activity of source at the time of standardization, t is the elapsed time since standardization, λ is the decay constant, TCS is the correction factor for the coincidence summing and T is the measuring time (in seconds). The uncertainty $\sigma \epsilon $ of the measured full energy peak efficiency is given by:^{34,35}

where $\sigma A$, $\sigma P$ and $\sigma N$ are the uncertainties associated with the quantities A_{S}, P(E) and N(E) respectively. All the previous parameters in equation (24) and (25) was inserted in microsoft office excel worksheet to calculate the measured efficiency with the associated uncertainty.^{19}

## IV. RESULTS AND DISCUSSION

The true coincidence effects, in case of extended volumetric sources are considered to be more complicated in comparing with the situation of the point sources. This phenomenon due to the interaction of the γ-ray within the source matrix itself must be considered. In this case, the total and the full-energy peak efficiencies of the well-type detectors must be known as a function of the photon energy and as a function of space vector inside the source volume as well. In order to calculate the both efficiencies for using volumetric sources measured inside the cavity of the well-type NaI(Tl) detectors, equations (11) and (12) are used based on the direct mathematical method^{20–22,24–26} and the efficiency transfer method (ET).^{29–31}

In order to use the efficiency transfer technique (ET), the fitting curve for the measured reference efficiency $\epsilon ref$ must be as essentially done,^{29–31} which represents ^{152}Eu radioactive point source and was kept fixed at 50 cm in the measurement geometry from the detector outer end cap as shown in Figure. 5 for both detectors. While Table II contains the measured values of the reference efficiency for a point source fixed at 50 cm as a function of photon energy, with its uncertainty for NaI(Tl) well-type scintillation detectors D1 and D2, as well as the effective solid angles for radioactive volumetric and point sources.

Detector D1 . | |||||||
---|---|---|---|---|---|---|---|

. | . | Point Source (50 cm) . | Vial Inside ΩEff(Cylinder) . | ||||

Radionuclide . | Energy (keV) . | εRef . | Unc . | ΩEff(Ref) . | K . | D . | R . |

^{152}Eu | 121.78 | 4.62E-04 | 3.75E-06 | 4.92E-03 | 9.96 | 10.31 | 10.68 |

244.69 | 3.13E-04 | 4.41E-06 | 4.96E-03 | 8.04 | 8.36 | 8.70 | |

344.28 | 2.35E-04 | 1.84E-06 | 4.63E-03 | 6.45 | 6.72 | 7.01 | |

778.9 | 8.34E-05 | 8.63E-07 | 3.80E-03 | 4.30 | 4.49 | 4.69 | |

964.13 | 5.90E-05 | 6.47E-07 | 3.61E-03 | 3.93 | 4.11 | 4.30 | |

1408.01 | 5.18E-05 | 4.36E-07 | 3.27E-03 | 3.38 | 3.53 | 3.70 |

Detector D1 . | |||||||
---|---|---|---|---|---|---|---|

. | . | Point Source (50 cm) . | Vial Inside ΩEff(Cylinder) . | ||||

Radionuclide . | Energy (keV) . | εRef . | Unc . | ΩEff(Ref) . | K . | D . | R . |

^{152}Eu | 121.78 | 4.62E-04 | 3.75E-06 | 4.92E-03 | 9.96 | 10.31 | 10.68 |

244.69 | 3.13E-04 | 4.41E-06 | 4.96E-03 | 8.04 | 8.36 | 8.70 | |

344.28 | 2.35E-04 | 1.84E-06 | 4.63E-03 | 6.45 | 6.72 | 7.01 | |

778.9 | 8.34E-05 | 8.63E-07 | 3.80E-03 | 4.30 | 4.49 | 4.69 | |

964.13 | 5.90E-05 | 6.47E-07 | 3.61E-03 | 3.93 | 4.11 | 4.30 | |

1408.01 | 5.18E-05 | 4.36E-07 | 3.27E-03 | 3.38 | 3.53 | 3.70 |

Detector D2 . | |||||||
---|---|---|---|---|---|---|---|

. | . | Point Source (50 cm) . | Vial Inside ΩEff(Cylinder) . | ||||

Radionuclide . | Energy (keV) . | εRef . | Unc . | ΩEff(Ref) . | K . | D . | R . |

^{152}Eu | 121.78 | 1.26E-03 | 1.08E-05 | 1.26E-02 | 10.14 | 10.51 | 10.90 |

244.69 | 8.39E-04 | 9.12E-06 | 1.30E-02 | 9.57 | 9.95 | 10.37 | |

344.28 | 6.87E-04 | 5.64E-06 | 1.27E-02 | 8.42 | 8.77 | 9.15 | |

778.9 | 3.90E-04 | 4.09E-06 | 1.12E-02 | 6.19 | 6.48 | 6.78 | |

964.13 | 3.15E-04 | 3.31E-06 | 1.08E-02 | 5.75 | 6.02 | 6.30 | |

1408.01 | 2.68E-04 | 2.29E-06 | 9.97E-03 | 5.04 | 5.28 | 5.54 |

Detector D2 . | |||||||
---|---|---|---|---|---|---|---|

. | . | Point Source (50 cm) . | Vial Inside ΩEff(Cylinder) . | ||||

Radionuclide . | Energy (keV) . | εRef . | Unc . | ΩEff(Ref) . | K . | D . | R . |

^{152}Eu | 121.78 | 1.26E-03 | 1.08E-05 | 1.26E-02 | 10.14 | 10.51 | 10.90 |

244.69 | 8.39E-04 | 9.12E-06 | 1.30E-02 | 9.57 | 9.95 | 10.37 | |

344.28 | 6.87E-04 | 5.64E-06 | 1.27E-02 | 8.42 | 8.77 | 9.15 | |

778.9 | 3.90E-04 | 4.09E-06 | 1.12E-02 | 6.19 | 6.48 | 6.78 | |

964.13 | 3.15E-04 | 3.31E-06 | 1.08E-02 | 5.75 | 6.02 | 6.30 | |

1408.01 | 2.68E-04 | 2.29E-06 | 9.97E-03 | 5.04 | 5.28 | 5.54 |

In addition, the ANGLE4 software was used for the same purpose, the energy range of the both methods were covered by ^{152}Eu nuclide included 105 value of energy, which started from 121.78 keV up to 1647.41 keV for that complex decay scheme. All this range was required to solve the complex system of equations from (14) up to (22) for the coincidence summing factors, in order to correct the measured full-energy peak efficiencies for the most probable energies which appeared in the measured NaI(Tl) detector spectra (121.78, 244.69, 344.28, 778.90, 964.13 and 1408.01 keV) and to obtain the true efficiency.

A special computer program has been written in Microsoft office excel worksheets to allow the calculation of the coincidence effects factors for this complexity degree of decay scheme and geometry. The relevant nuclear and atomic data in that case were taken into account (intensity of transitions, total and K-shell internal conversion coefficient, electronic capture probability in the K-shell, K-shell fluorescence yield, etc.$\u2026$).^{36,37}

Once the total and the full-energy peak efficiencies of the 2″×2″ (D1) and 3″×3″ (D2) NaI(Tl) well-type scintillation detectors have been calculated by the two methods, the overall efficiency curves are obtained by fitting a polynomial logarithmic function of fourth and third order for the total and the full-energy peak efficiencies points respectively, using a nonlinear least square fit based on the following equation:

where: a_{i} are the coefficients to be determined by the calculations, ε is the full-energy peak efficiency of the well-type NaI(Tl) detectors at energy E.

As shown in Figures 6 and 7, the variation of the calculated total and the full-energy peak efficiencies of the 2″×2″ (D1) and 3″×3″ (D2) NaI(Tl) well-type scintillation detectors as a function of photon energy can come into view.

The behavior of these curves was based on using three dimensions’ vials K, D and R from Posthumus Plastics Company, which were filled with small amount of ^{152}Eu aqueous solution of a well known activities [67.26 Bq, 142.23 Bq and 253.93 Bq respectively] and measured inside the well-type detectors cavity.

The discrepancies $\Delta 1$% between the calculated total efficiency values $\epsilon T(Cylinder)NSM$ and $\epsilon T(Cylinder)ANGLE4$ for using the numerical simulation method (NSM) and ANGLE4 software, the discrepancies $\Delta 2$% between the calculated full-energy peak efficiency values $\epsilon P(Cylinder)NSM$ and $\epsilon P(Cylinder)ANGLE4$ based on the same previous methods as well are given by the following equations:

By comparison, the percentage differences $\Delta 1$% and $\Delta 2$% are being less than 10% for both detectors, using the numerical simulation method (NSM) and ANGLE4 software as shown in Tables III and IV. The true coincidence summing corrections (TCS)_{IJ} to γ_{IJ}-ray transition emitted energy E_{IJ} from energy level I to energy level J in ^{152}Eu decay scheme were calculated as shown in Tables III and IV based on equation (14).

Vial . | Detector D1 . | ||||||||
---|---|---|---|---|---|---|---|---|---|

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.490 | 1.485 | −5.95 | −3.28 | 32.87 | 32.65 | 73.28 | 74.22 | |

244.69 | 2.201 | 2.265 | −5.71 | −2.15 | 54.57 | 55.85 | 75.37 | 75.19 | |

K | 344.28 | 1.333 | 1.324 | −2.31 | −1.51 | 24.97 | 24.48 | 5.72 | 7.73 |

778.90 | 1.889 | 1.929 | 3.00 | 0.85 | 47.05 | 48.16 | −4.89 | −8.07 | |

964.08 | 1.488 | 1.524 | 3.11 | 1.77 | 32.80 | 34.37 | 2.88 | −1.24 | |

1408.01 | 1.556 | 1.612 | 2.33 | 3.79 | 35.72 | 37.98 | −3.61 | −11.63 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.482 | 1.480 | −7.41 | −4.64 | 32.51 | 32.42 | 73.87 | 75.06 | |

244.69 | 2.153 | 2.234 | −4.49 | −1.26 | 53.55 | 55.23 | 76.06 | 75.47 | |

D | 344.28 | 1.329 | 1.321 | −1.60 | −0.43 | 24.74 | 24.29 | 4.47 | 5.45 |

778.90 | 1.879 | 1.908 | 2.82 | 0.77 | 46.78 | 47.58 | −6.74 | −9.21 | |

964.08 | 1.486 | 1.514 | 2.67 | 1.14 | 32.70 | 33.93 | 1.23 | −1.77 | |

1408.01 | 1.527 | 1.595 | 0.45 | 2.09 | 34.50 | 37.30 | −2.00 | −8.82 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.477 | 1.474 | −7.41 | −4.54 | 32.28 | 32.15 | 72.09 | 73.35 | |

244.69 | 2.125 | 2.200 | −5.46 | −2.95 | 52.95 | 54.54 | 74.06 | 73.92 | |

R | 344.28 | 1.324 | 1.317 | −3.72 | −1.78 | 24.47 | 24.07 | 1.10 | 3.33 |

778.90 | 1.824 | 1.884 | 2.97 | 1.01 | 45.16 | 46.93 | −5.45 | −10.06 | |

964.08 | 1.469 | 1.502 | 3.63 | 1.56 | 31.92 | 33.44 | 1.92 | −1.93 | |

1408.01 | 1.511 | 1.576 | 0.44 | 2.21 | 33.82 | 36.54 | −4.86 | −11.84 |

Vial . | Detector D1 . | ||||||||
---|---|---|---|---|---|---|---|---|---|

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.490 | 1.485 | −5.95 | −3.28 | 32.87 | 32.65 | 73.28 | 74.22 | |

244.69 | 2.201 | 2.265 | −5.71 | −2.15 | 54.57 | 55.85 | 75.37 | 75.19 | |

K | 344.28 | 1.333 | 1.324 | −2.31 | −1.51 | 24.97 | 24.48 | 5.72 | 7.73 |

778.90 | 1.889 | 1.929 | 3.00 | 0.85 | 47.05 | 48.16 | −4.89 | −8.07 | |

964.08 | 1.488 | 1.524 | 3.11 | 1.77 | 32.80 | 34.37 | 2.88 | −1.24 | |

1408.01 | 1.556 | 1.612 | 2.33 | 3.79 | 35.72 | 37.98 | −3.61 | −11.63 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.482 | 1.480 | −7.41 | −4.64 | 32.51 | 32.42 | 73.87 | 75.06 | |

244.69 | 2.153 | 2.234 | −4.49 | −1.26 | 53.55 | 55.23 | 76.06 | 75.47 | |

D | 344.28 | 1.329 | 1.321 | −1.60 | −0.43 | 24.74 | 24.29 | 4.47 | 5.45 |

778.90 | 1.879 | 1.908 | 2.82 | 0.77 | 46.78 | 47.58 | −6.74 | −9.21 | |

964.08 | 1.486 | 1.514 | 2.67 | 1.14 | 32.70 | 33.93 | 1.23 | −1.77 | |

1408.01 | 1.527 | 1.595 | 0.45 | 2.09 | 34.50 | 37.30 | −2.00 | −8.82 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.477 | 1.474 | −7.41 | −4.54 | 32.28 | 32.15 | 72.09 | 73.35 | |

244.69 | 2.125 | 2.200 | −5.46 | −2.95 | 52.95 | 54.54 | 74.06 | 73.92 | |

R | 344.28 | 1.324 | 1.317 | −3.72 | −1.78 | 24.47 | 24.07 | 1.10 | 3.33 |

778.90 | 1.824 | 1.884 | 2.97 | 1.01 | 45.16 | 46.93 | −5.45 | −10.06 | |

964.08 | 1.469 | 1.502 | 3.63 | 1.56 | 31.92 | 33.44 | 1.92 | −1.93 | |

1408.01 | 1.511 | 1.576 | 0.44 | 2.21 | 33.82 | 36.54 | −4.86 | −11.84 |

Vial . | Detector D2 . | ||||||||
---|---|---|---|---|---|---|---|---|---|

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.798 | 1.816 | −8.10 | −7.15 | 44.39 | 44.93 | 68.74 | 70.54 | |

244.69 | 2.588 | 2.744 | −4.97 | −9.15 | 61.35 | 63.55 | 60.54 | 61.67 | |

K | 344.28 | 1.536 | 1.552 | −4.96 | −8.84 | 34.88 | 35.56 | −3.39 | 4.00 |

778.90 | 2.794 | 3.073 | −1.86 | −5.84 | 64.20 | 67.46 | 1.60 | −2.27 | |

964.08 | 1.693 | 1.747 | −0.76 | −4.76 | 40.94 | 42.77 | −5.37 | −3.80 | |

1408.01 | 1.532 | 1.603 | −0.29 | −2.76 | 34.72 | 37.63 | 4.68 | 2.91 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.821 | 1.809 | −4.82 | 1.06 | 45.09 | 44.71 | 74.55 | 74.45 | |

244.69 | 2.646 | 2.706 | −3.75 | −1.86 | 62.21 | 63.05 | 62.71 | 62.56 | |

D | 344.28 | 1.556 | 1.547 | −1.86 | −1.26 | 35.72 | 35.36 | −1.95 | −0.12 |

778.90 | 2.898 | 3.006 | 1.56 | 2.60 | 65.49 | 66.74 | −6.60 | −13.52 | |

964.08 | 1.696 | 1.733 | 2.03 | 3.65 | 41.03 | 42.30 | 1.18 | −7.32 | |

1408.01 | 1.541 | 1.587 | 2.78 | 5.07 | 35.10 | 37.01 | −4.37 | −13.27 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.855 | 1.800 | 0.61 | 6.29 | 46.08 | 44.43 | 75.86 | 75.00 | |

244.69 | 2.755 | 2.666 | 2.24 | 3.50 | 63.71 | 62.49 | 63.90 | 63.80 | |

R | 344.28 | 1.580 | 1.541 | 2.72 | 3.92 | 36.69 | 35.10 | −2.03 | −3.59 |

778.90 | 3.113 | 2.934 | 5.36 | 6.48 | 67.88 | 65.92 | −7.48 | −8.32 | |

964.08 | 1.746 | 1.718 | 5.76 | 6.95 | 42.74 | 41.78 | −0.43 | −6.15 | |

1408.01 | 1.573 | 1.571 | 4.62 | 7.04 | 36.43 | 36.34 | −3.90 | −11.61 |

Vial . | Detector D2 . | ||||||||
---|---|---|---|---|---|---|---|---|---|

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.798 | 1.816 | −8.10 | −7.15 | 44.39 | 44.93 | 68.74 | 70.54 | |

244.69 | 2.588 | 2.744 | −4.97 | −9.15 | 61.35 | 63.55 | 60.54 | 61.67 | |

K | 344.28 | 1.536 | 1.552 | −4.96 | −8.84 | 34.88 | 35.56 | −3.39 | 4.00 |

778.90 | 2.794 | 3.073 | −1.86 | −5.84 | 64.20 | 67.46 | 1.60 | −2.27 | |

964.08 | 1.693 | 1.747 | −0.76 | −4.76 | 40.94 | 42.77 | −5.37 | −3.80 | |

1408.01 | 1.532 | 1.603 | −0.29 | −2.76 | 34.72 | 37.63 | 4.68 | 2.91 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.821 | 1.809 | −4.82 | 1.06 | 45.09 | 44.71 | 74.55 | 74.45 | |

244.69 | 2.646 | 2.706 | −3.75 | −1.86 | 62.21 | 63.05 | 62.71 | 62.56 | |

D | 344.28 | 1.556 | 1.547 | −1.86 | −1.26 | 35.72 | 35.36 | −1.95 | −0.12 |

778.90 | 2.898 | 3.006 | 1.56 | 2.60 | 65.49 | 66.74 | −6.60 | −13.52 | |

964.08 | 1.696 | 1.733 | 2.03 | 3.65 | 41.03 | 42.30 | 1.18 | −7.32 | |

1408.01 | 1.541 | 1.587 | 2.78 | 5.07 | 35.10 | 37.01 | −4.37 | −13.27 | |

Energy (keV) | (TCS)_{NSM} | (TCS)_{ANGLE4} | Δ_{1}% | Δ_{2}% | Δ_{3}% | Δ_{4}% | Δ_{5}% | Δ_{6}% | |

121.78 | 1.855 | 1.800 | 0.61 | 6.29 | 46.08 | 44.43 | 75.86 | 75.00 | |

244.69 | 2.755 | 2.666 | 2.24 | 3.50 | 63.71 | 62.49 | 63.90 | 63.80 | |

R | 344.28 | 1.580 | 1.541 | 2.72 | 3.92 | 36.69 | 35.10 | −2.03 | −3.59 |

778.90 | 3.113 | 2.934 | 5.36 | 6.48 | 67.88 | 65.92 | −7.48 | −8.32 | |

964.08 | 1.746 | 1.718 | 5.76 | 6.95 | 42.74 | 41.78 | −0.43 | −6.15 | |

1408.01 | 1.573 | 1.571 | 4.62 | 7.04 | 36.43 | 36.34 | −3.90 | −11.61 |

The true coincidence summing corrections (TCS)_{NSM} and (TCS)_{ANGLE4} was depended on efficiency values obtained using the numerical simulation method (NSM) and ANGLE4 software respectively.

The discrepancies $\Delta 3$% between the corrected measured full-energy peak efficiency values $\epsilon P(Cylinder)Meas(NSM-Corr)$ using the true coincidence summing corrections (TCS)_{NSM} and the uncorrected measured full-energy peak efficiency values $\epsilon P(Cylinder)Meas(Uncorr)$ are given by the following equations:

By comparison, the percentage differences are being around 68% for both detectors at some energy points as shown in Tables III and IV based on the values of the coincidence summing corrections (TCS)_{NSM}. While the discrepancies $\Delta 4$% between the corrected measured full-energy peak efficiency values $\epsilon P(Cylinder)Meas(ANGLE4\u2013Corr)$ using the true coincidence summing corrections (TCS)_{ANGLE4} and the uncorrected measured full-energy peak efficiency values $\epsilon P(Cylinder)Meas(Uncorr)$ are given by the following equations:

By comparison, the percentage differences are being around 68% for both detectors at some energy points as shown in Tables III and IV based on the values of the coincidence summing corrections (TCS)_{ANGLE4}. While the discrepancies $\Delta 5$% between calculated full-energy peak efficiency values $\epsilon P(Cylinder)NSM$ and the corrected measured full-energy peak efficiency values $\epsilon P(Cylinder)Meas(NSM-Corr)$ using the true coincidence summing corrections (TCS)_{NSM} are given by the following equations:

By comparison, the percentage differences are being around 77% for both detectors at some energy points as shown in Tables III and IV based on the values of the coincidence summing corrections (TCS)_{NSM}. While the discrepancies $\Delta 6$% between calculated full-energy peak efficiency values $\epsilon P(Cylinder)ANGLE4$ and the corrected measured full-energy peak efficiency values $\epsilon P(Cylinder)Meas(ANGLE4\u2013Corr)$ using the true coincidence summing corrections (TCS)_{ANGLE4} are given by the following equations:

By comparison, the percentage differences are being around 76% for both detectors at some energy points as shown in Tables III and IV based on the values of the coincidence summing corrections (TCS)_{ANGLE4}. One of the reasons for such high discrepancy is due to low energy lines were not well resolved by the well-type NaI(Tl) detector, beside neglecting the contributions coming from X-rays to coincidence summing effects and the efficiency transfer method was applied in such unfavorable conditions different from the reference is at the origin of some absurd results inside the cavity of the NaI(Tl) well-type detector.

Direct comparison between the different types of the full-energy peak efficiency based on using three vials K, D and R inside the detectors cavity ($\epsilon P(Cylinder)Meas(Uncorr)$, $\epsilon P(Cylinder)Meas(NSM\u2013Corr)$, $\epsilon P(Cylinder)Meas(ANGLE4\u2013Corr)$, $\epsilon P(Cylinder)NSM$ and $\epsilon P(Cylinder)ANGLE4$) can be presented in Figure. 8 for the most probable energies, which appeared in the measured NaI(Tl) detector spectra (121.78, 244.69, 344.28, 778.90, 964.13 and 1408.01 keV), due to the large number of overlapping energy lines in a ^{152}Eu, where from inspection in the results. The NaI(Tl) well-type detectors not good quality enough to identify particularly some of ^{152}Eu lines inside the detector cavity with standard peak-fitting routines, especially between 122 keV and 244 keV in the spectrum results,^{35} which leads to increase the notice drooping in the detector efficiency for the both energies, while the rest of the energies were provide evidence of good and accepted discrepancies $\Delta 5$% and $\Delta 6$% within 14 % as a maximum at some energy points in Tables III and IV based on corrected efficiencies by the values of the coincidence summing corrections (TCS)_{NSM} and (TCS)_{ANGLE4} respectively, which is considered as fine results for the measured efficiency curve inside the well-type detectors. Also, its easy to remark from the vaules of the coincidence summing corrections (TCS) of D1 is smaller than that for D2 as presented in Tables III and IV , this behavior due to that the detector D2 shows large efficiency than detector D1 due to large detector size as shown in Figure. 8.

## V. CONCLUSIONS

This work is mainly aimed to calibrate NaI(Tl) well-type detectors. This has been supported by the numerical simulation method (NSM) and the experimental measurements. The coincidence summing corrections (TCS) of ^{152}Eu was calculated using very small volumetric source measured at a very small source-to-detector distance. The obtained (TCS) correction factors have been used to correct the measured full-energy peak efficiency values for the NaI(Tl) well-type detectors. A computer program was developed for that model and validated through laboratory experiments. The necessary parameters for running the program such as the total and the full-energy peak efficiency were calculated by using the numerical simulation method (NSM) and ANGLE4 software; these represent two different methods and both take into account detailed descriptions of the detectors and the sources employed. Results based on ^{152}Eu sources indicate a good agreement between the corrected measured full-energy peak efficiency values and the simulated one by using (NSM) and ANGLE4; there is an exception for 121.78 keV to 244.69 keV region, as several lines could not be conclusively identified here, neglecting the contributions coming from X-rays to coincidence summing effects and the efficiency transfer method was applied in such unfavorable conditions different from the reference is at the origin of some absurd results. The above procedure allows reducing the efforts required to correct γ-ray spectrometer measurements for coincidence summing effect. However, continued study is required for more testing in order to validate the described procedure for commercially available software packages.

## REFERENCES

^{152}Eu sources