Direct Analytical Method to Calculate Photopeak Efficiency and Photopeak Attenuation Coefficient of NaI ( Tl ) Well-Type Detector

In this paper full-energy peak (photopeak) efficiency and photopeak attenuation coefficient of 3'' × 3'' NaI(Tl) well-type scintillation detector were calculated using gamma-rayisotropic radiating point sources (with photon energy: 0.245, 0.344, 0.662, 0.779, 0.964, 1.1732, 1.333 and 1.408 MeV) placed outside the detector well. These energies were obtained from 152Eu, 137Cs and 60Co. The relations between the full energy peak efficiency and photopeak attenuation coefficients, were plotted vs. photon energy at different sources to detector distance, and it found that the full energy peak efficiency decreased by increasing the distance between the source and the detector.


Introduction
In this work direct mathematical formulae for measuring the full-energy peak efficiency of HPGe well-type detector are found and the values of the measured efficiencies are compared with the published works of the experimental and theoretical old methods which have a good agreement.In this new approach the path length d(θ, ϕ) is derived as a function in the polar angle θ, and the azimuthal angle ϕ.This will reduce the mathematical formulae to an easiest and compact shape.In low level gamma-ray spectroscopy, NaI(Tl) well-type scintillation detectors are extremely useful since they offer almost 4π solid angle detection.The efficiency of such detectors can be obtained, using experimental [1]- [3], semi-empirical [4] [5] and Monte Carlo methods [6]- [9].The present work relies on a direct mathematical method reported by Selim and Abbas [10]- [22], which is simple and accurate.It was used successfully in the calibration of well-type and cylindrical detectors using point, plane and volumetric sources.The efficiency is calculated in terms of the geometrical parameters of the source-detector configuration and the photopeak attenuation coefficient of the incident photon.Here, we consider 8 gamma-ray energies based on three different point sources.

Mathematical Treatment
In our analysis, we consider only an axial point source P, as shown in Figure 1.The efficiency in this case is given by [15] ( ) where, d is the path length traveled by a photon through the detector active medium, μ is the attenuation coefficient of the detector material, and f att is the attenuation factor, which is given by [17] e .
Here, μ n is the attenuation coefficient of the n th absorber for the gamma-ray photon [16], and δ n is the path length of the gamma-ray photon through the n th absorber read [23].

Results and Discussions
Using the above formalism, the full-energy peak efficiency and the photopeak attenuation coefficient of 3'' × 3'' NaI(Tl) well-type scintillation detector are calculated for in the case of point sources placed outside the detector well.The 3'' × 3'' NaI(Tl) well-type model number is 802-Canberra.The dimensions of this detector are, outer radius, 3.81 cm, cavity radius, 0.858 cm, outer height, 7.62 cm, and cavity depth, 4.987 cm.The full-energy peak efficiencies are calculated using the present work and compared with those obtained by theoretical and ex-perimental data, for various values of the source to detector distance, h', above the detector starting approximately at 20 cm up to 50 cm in a step of 5 cm.The results of the calculations are given in Tables 1-7, along with the experimental data and the theoretical calculations based on the transfer method presented in Ref. [23].It is clear that our theoretical results are identical to the experimental data.The error in the theoretical results of reference [23] based on the transfer method is approximately 1%.So, our present formalism consistently produces more accurate results than the other methods.
Comparing the various data in Tables 1-7, we can see that the full-energy peak efficiency decreases as the distance h' increases for the same energy of the gamma-rays.Obviously, it can be seen that the efficiency depends on the position of the source as will as it depends on the activity of the radioactive source.In Figure 2, we summarize this point by plotting the full-energy peak efficiency versus the distance h' for the eight energies of the gamma-rays.We notice the tendency that the efficiency decreases as the energy increases for the same distance as seen in Figure 2. In addition, the efficiency curves are converging at high energies.To explain these observations, we calculate the energy dependence of the attenuation coefficient.
In our formalism, the energy of the photon only enters through the dependence of the attenuation coefficient.From Equation ( 14), we see that the function ( ) increases.So for the efficiency to decrease with increasing gamma energy, the attenuation coefficient should diminish.We confirm this analysis by plotting the photopeak attenuation coefficient (μ P ) against the energy of the gamma-rays in Figures 3-9            the different values of the distance considered so far.The figures also include polynomial fits up to 6 th order.A common trend in μ P is that it decreases rapidly at lower energies and approximately levels of at high energies about 0.05 cm −1 .The sharp decrease in μ P is reflected in the large change of the efficiency at low energies.However, the negligible variation of the attenuation coefficient with the gamma energy plays a smaller rule in determining the efficiency at high energies.Thus the efficiency curves converge as indicated in Figure 2.

Conclusion
The full-energy peak efficiency (ε p ) of 3'' × 3'' NaI(Tl) well-type scintillation detector, using axial point sources is calculated and compared with both the experimental and the theoretical data in Ref. [23].In addition, the photopeak attenuation coefficients (μ p ) have been calculated as a function of the gamma-ray photon energy.Our formalism gives more accurate data than that of the transfer method presented in Ref. [23].The dependence of the photopeak attenuation coefficient on the gamma energy is used to explain the variations in the efficiency as the distance between the point source and the detector changes.From the result it is clear that the full-energy peak efficiency ε p decreases by increasing the distance between the detector and the source.

Figure 1 .
Figure 1.The well type detector configuration.The point source P, is placed on the axis of the well detector.

Figure 2 .
Figure 2. The relation between the full-energy peak (photopeak) efficiency (ε p ) and the source to detector distance (h').

Figure 3 .
Figure 3.The relation between the photopeak attenuation coefficient (μ p ) and the photon energy for h' = 20.442cm with fitting polynomial curve.