Determination of the detector efficiency using volumetric cylindrical sources is very important in various scientific and industrial fields, especially in the field of quantitative analysis. To calculate the absolute activity of any sample, the full-energy peak efficiency (FEPE) of the detector is needed. By applying the efficiency transfer method, the FEPE of the detector would be determined easily without using the standard sources. This approach depends on two main factors. The first one, is the reference efficiency of the reference source, which is determined experimentally, and the second one, is the calculation of the effective solid angle ratio between the sample and the reference source geometries. This work introduces an empirical formula for calculating the second factor for using two different sizes of NaI(Tl) detectors. The validity of this empirical formula was successfully demonstrated by comparing the calculating values with the experimental values.

The scintillation counters are used to measure the radiation in different applications such as, radiation survey meters, medical imaging, nuclear plant safety, measuring radon levels, oil well logging and monitoring for radioactive contamination. In the gamma-ray spectroscopy, one usually needs to know the full-energy peak efficiency for any specific source-to-detector configuration of concern. Traditionally, measurements are performed in gamma-ray spectrometry by the relative method, according to which the measured sample is first prepared, that should match the used standard source in all the important characteristics, such as its size, chemical composition and density [1]. This method is tedious and time consuming process. In order to overcome the problems of the experimental method, several non-experimental methods [2-6] have been proposed and applied, depending on the photon energy, source-to-detector geometry and volume. One of the most common approaches is called the efficiency transfer method. In this technique, the detector efficiency of using various source dimensions is derived from the known efficiency for the reference source-to-detector geometry. The efficiency transfer method is particularly useful due to, its insensitivity to the inaccuracy of the input data, such as the uncertainty of the detector characterization [7,8].

Badawi, et al. [9-11] were introduced an approach to calculate the full-energy peak efficiency for NaI(Tl) and HPGe detectors, with respect to different volumetric sources. This approach stated that, the detector efficiency using a certain cylindrical radioactive source, $\epsilon \left(\text{E,Cyl}\right)$ , equal the reference efficiency of using reference radioactive point source, $\epsilon \left(\text{E},{\text{P}}_{\text{ο}}\right)$ , with the same detector multiplied by the effective solid angle ratio, R, between the two geometries and expressed by the following equation

$\epsilon \left(\text{E,Cyl}\right)=\epsilon \left(\text{E},{\text{P}}_{\text{ο}}\right).\text{ R (1)}$

Calculations of the effective solid angle are based on the direct mathematical method which reported by Selim and Abbas [12-16] and used successfully before to calibrate different detectors with different sources. The present work will introduce empirical equations to calculate the effective solid angle ratios of two NaI(Tl) detectors with different geometries. The effective solid angle ratio can be used as a conversion factor from using the radioactive point source case to the case in which the cylindrical radioactive sources were used. Consequently, the corresponding full-energy peak efficiency can be calculated simply.

The full-energy peak efficiency (FEPE) values were determined for two NaI (Tl) detectors with resolutions 8.5% and 7.5% at the 662 keV peaks of ¹³⁷Cs labeled as D1 and D2 respectively. The manufacturer parameters and the setup values are shown in table 1. The experimental measurements were carried out by using point and cylindrical radioactive sources.

The radioactive standard point sources (²⁴¹Am, ¹³³Ba, ¹⁵²Eu, ¹³⁷Cs, and ⁶⁰Co) are used for the calibration of gamma spectrometers. The radioactive substance is a very thin, compact grained layer applied to a circular area about 5 mm in diameter, in the middle of the source between two polyethylene foils and each having a mass per unit area of (21.3±1.8) mg.cm^-2. By heating under pressure, the two foils are welded together over the whole area so that they are leak-proofed. To facilitate handling, the foil 26 mm in diameter is mounted in a circular aluminum ring (outer diameter: 30 mm, height: 3 mm) from which it can easily be removed if and when required. These point sources were purchased from the Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig and Berlin, which is the 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 sources activities and their uncertainties, half-lives, photon energies, and photon emission probabilities per decay for all of PTB sources are listed in table 2.

The homemade Plexiglas holder was used to measure these standard point sources, each at seven different axial distances starting from 20 cm up to 50 cm from the surface of the detector (with a 5 cm as a step). The measurements started from a source-to-detector distance equals 20 cm to minimize the effect of the coincidence summing effect. Spectra were recorded as, P4D1, where P refers to the source type (point) measured at the detector (D1) at position number (4), which equal 20 cm.

The cylindrical radioactive sources were in polypropylene plastic vials form with radius greater than the radius of the detectors, and volumes of 200 ml, 300 ml and 400 ml filled with an aqueous solution containing ¹⁵²Eu radionuclide, which used for the calibration process. The ¹⁵²Eu source emits γ-ray in the energy range from 121.78 keV up to 1408.01 keV. Table 3 shows the source dimensions. In order to minimize the dead time, the activity of the sources is prepared to be a few kilo Becquerel (5048±49.98 Bq).

The radioactive volumetric cylindrical sources were measured on a 0.36 cm thickness Plexiglas cover and placed directly on the detector end-cap. These measurements were done using two cylindrical detectors with numbers (D1 & D2). Figure 1 shows a diagram of a cylindrical detector with cylindrical source. Spectra were recorded as V1D2, where V1 is the volume (V1) measured at the detector (D2). The angular correlation effects can be neglected for the low source-to-detector distance [17,18].

All the measurements are carried out to obtain statistically significant main peaks in the spectra that are recorded and processed by winTMCA32 software made by ICx Technologies. Measured spectrum, which saved as spectrum ORTEC files can be opened by the Genie 2000 data acquisition and analysis software made by Canberra. The acquisition time is high enough to get at least the number of counts 20,000, which make the statistical uncertainties less than 0.1%. The spectra are analyzed with the program using its automatic peak search and peak area calculations, along with changes in the peak fit using the interactive peak fit interface when necessary to reduce the residuals and error in the peak area values. The peak areas, the live time, the run time and the start time for each spectrum were entered in the spreadsheets that are used to perform the calculations necessary to generate the efficiency curves.

The efficiency transfer theoretical method (ETTM) has been used to convert the (FEPE) curve for using radioactive point source at positions start from P4 up to P10 to the (FEPE) for using radioactive cylindrical sources, which represented in V1, V2, and V3. These calculations extended for two cylindrical NaI(Tl) detectors (D1 & D2). By using equation (1) and the experimental efficiency values for using point and cylindrical radioactive sources, that published before in 2012 [19], the one can calculate the effective solid angle ratio, R, values for both detectors experimentally as tabulated in table 4.

The analytical expressions presented in [19] were used to calculate the effective solid angle ratio as presented in table 5, these values were tested before to obtain the detector FEPE and it was accepted by comparison with the experimental values. The percentage deviations between the effective solid angle ratio values obtained by the two methods are shown in figure 2. A remarkable agreement between them was achieved with discrepancies less than 10%.

By plotting a three dimensional relation between the Log values of the point source position, P (cm), the effective solid angle ratio, R, and the photon energy, E (keV) for the two detectors (D1 & D2) was done as shown in figure 3. The plotted data for each source volume (ml) with the two detectors have shown semi plane shape and the empirical formulae that represent these shapes are described below to calculate the effective solid angle ratios, R, for both detectors.

The empirical formula for the detector (D1) is given by:

$\text{Log}\left(\text{E}\right)-\text{26}.\text{77 Log}\left(\text{R}\right)+\text{ 49}.\text{18 Log}\left(\text{P}\right)-0.0\text{176 V}-\text{3}0.\text{62 = }0\text{ (2)}$

while, the empirical formula for the detector (D2) is given by:

$\text{Log}\left(\text{E}\right)-\text{26}.\text{77 Log}\left(\text{R}\right)+\text{ 49}.\text{18 Log}\left(\text{P}\right)-0.0\text{166 V}-\text{33}.\text{63 }=0\text{ (3)}$

By knowing the photon energy and the reference position, the effective solid angle ratio, R, for both detectors was calculated using equations (2) and (3). The obtained values were tabulated in table 6. Therefore, these equations provide a simple method to calculate the full-energy peak efficiency (FEPE) of two different cylindrical NaI(Tl) scintillation detectors. These two formulae are valid through a wide energy range and different radioactive volumetric source geometries. The percentage deviations between the calculated effective solid angle ratio, that obtained experimentally and that obtained from equations (2) and (3) were shown in figure 4. A remarkable agreement between them was achieved with discrepancies less than 7%.

The main advantage of this process is the simplicity of obtaining the effective solid angle ratios, R, especially in between any two measured positions, without using analytical or experimental calculations. These ratios are considered to be the efficiency conversion factor between any two different geometrical conditions, and used to save the time in absent the standard calibration sources.

The present work leads to a simplified method to calculate the effective solid angle ratio empirical, which can be used to calculate the conversion factors of the detector efficiency, in the case of using point and cylindrical radioactive sources. The efficiencies can be determined at any calibration position or any energy situated in the domain of the study based on these conversion factors. These formulas are valid through a wide energy range and different source-to-detector geometries. Therefore the corresponding full-energy peak efficiency can be calculated simply, and the activity of unknown samples measured in the same conditions can be determined easily.

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