^{1}

^{2}

^{*}

^{3}

^{3}

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
^{152}Eu,
^{137}Cs and
^{60}Co. 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.

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 [

In our analysis, we consider only an axial point source P, as shown in

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 [

Here, μ_{n} is the attenuation coefficient of the n^{th} absorber for the gamma-ray photon [_{n} is the path length of the gamma-ray photon through the n^{th} absorber read [

The polar angles, θ, are given by (see

The distance

But, if they emerge from the side of the outer well,

Next, when the gamma-ray photons enter from the inner side of the well and emerge from the bottom of the outer well, we have

If they emerge from the side of the outer well, the photon path length is given by

Finally, when the gamma-ray photon enters from the upper surface of the well and emerge from the bottom of the outer well, then

If emerging from the side of the outer well, we have

The resulting efficiency depends on the magnitudes of the polar angles. This results in the following three cases. When θ_{3} > θ_{2} > θ_{4} > θ_{1}, using Equation (1) the efficiency is given by

with

Similarly, if θ_{3} > θ_{2} > θ_{1} > θ_{4}

with _{3} > θ_{4} > θ_{2} > θ_{1}, we have

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 experimental 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. [

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

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 _{P}) against the energy of the gamma-rays in Figures 3-9 for

Photon energy (MeV) | ε_{p} (present work) | ε_{p} (experimental [ | ε_{p} (theoretical [ |
---|---|---|---|

2.45E−01 | 4.790E−03 | 4.794E−03 | 4.765E−03 |

3.44E−01 | 3.818E−03 | 3.815E−03 | 3.799E−03 |

6.62E−01 | 2.355E−03 | 2.358E−03 | 2.353E−03 |

7.79E−01 | 2.107E−03 | 2.105E−03 | 2.086E−03 |

9.64E−01 | 1.675E−03 | 1.677E−03 | 1.676E−03 |

1.17E+00 | 1.521E−03 | 1.522E−03 | 1.523E−03 |

1.33E+00 | 1.441E−03 | 1.444E−03 | 1.438E−03 |

1.41E+00 | 1.427E−03 | 1.429E−03 | 1.417E−03 |

Photon energy (MeV) | ε_{p} (present work) | ε_{p} (experimental [ | ε_{p} (theoretical [ |
---|---|---|---|

2.45E−01 | 3.151E−03 | 3.153E−03 | 3.149E−03 |

3.44E−01 | 2.548E−03 | 2.546E−03 | 2.531E−03 |

6.62E−01 | 1.575E−03 | 1.577E−03 | 1.581E−03 |

7.79E−01 | 1.417E−03 | 1.416E−03 | 1.404E−03 |

9.64E−01 | 1.136E−03 | 1.138E−03 | 1.130E−03 |

1.17E+00 | 1.027E−03 | 1.026E−03 | 1.028E−03 |

1.33E+00 | 9.713E−04 | 9.715E−04 | 9.713E−04 |

1.41E+00 | 9.607E−04 | 9.605E−04 | 9.572E−04 |

Photon energy (MeV) | ε_{p} (present work) | ε_{p} (experimental [ | ε_{p} (theoretical [ |
---|---|---|---|

2.45E−01 | 3.151E−03 | 3.153E−03 | 3.149E−03 |

3.44E−01 | 2.548E−03 | 2.546E−03 | 2.531E−03 |

6.62E−01 | 1.575E−03 | 1.577E−03 | 1.581E−03 |

7.79E−01 | 1.417E−03 | 1.416E−03 | 1.404E−03 |

9.64E−01 | 1.136E−03 | 1.138E−03 | 1.130E−03 |

1.17E+00 | 1.027E−03 | 1.026E−03 | 1.028E−03 |

1.33E+00 | 9.713E−04 | 9.715E−04 | 9.713E−04 |

1.41E+00 | 9.607E−04 | 9.605E−04 | 9.572E−04 |

Photon energy (MeV) | ε_{p} (present work) | ε_{p} (experimental [ | ε_{p} (theoretical [ |
---|---|---|---|

2.45E−01 | 1.675E−03 | 1.677E−03 | 1.672E−03 |

3.44E−01 | 1.348E−03 | 1.345E−03 | 1.356E−03 |

6.62E−01 | 8.490E−04 | 8.493E−04 | 8.546E−04 |

7.79E−01 | 7.595E−04 | 7.598E−04 | 7.599E−04 |

9.64E−01 | 6.177E−04 | 6.179E−04 | 6.124E−04 |

1.17E+00 | 5.579E−04 | 5.577E−04 | 5.578E−04 |

1.33E+00 | 5.303E−04 | 5.301E−04 | 5.276E−04 |

1.41E+00 | 5.183E−04 | 5.185E−04 | 5.200E−04 |

Photon energy (MeV) | ε_{p} (present work) | ε_{p} (experimental [ | ε_{p} (theoretical [ |
---|---|---|---|

2.45E−01 | 1.291E−03 | 1.294E−03 | 1.296E−03 |

3.44E−01 | 1.055E−03 | 1.058E−03 | 1.055E−03 |

6.62E−01 | 6.649E−04 | 6.647E−04 | 6.678E−04 |

7.79E−01 | 5.997E−04 | 5.995E−04 | 5.943E−04 |

9.64E−01 | 4.797E−04 | 4.796E−04 | 4.796E−04 |

1.17E+00 | 4.365E−04 | 4.367E−04 | 4.370E−04 |

1.33E+00 | 4.138E−04 | 4.137E−04 | 4.135E−04 |

1.41E+00 | 4.058E−04 | 4.056E−04 | 4.075E−04 |

Photon energy (MeV) | ε_{p} (present work) | ε_{p} (experimental [ | ε_{p} (theoretical [ |
---|---|---|---|

2.45E−01 | 1.020E−03 | 1.022E−03 | 1.031E−03 |

3.44E−01 | 8.366E−04 | 8.369E−04 | 8.417E−04 |

6.62E−01 | 5.380E−04 | 5.382E−04 | 5.347E−04 |

7.79E−01 | 4.773E−04 | 4.771E−04 | 4.758E−04 |

9.64E−01 | 3.801E−04 | 3.804E−04 | 3.842E−04 |

1.17E+00 | 3.513E−04 | 3.511E−04 | 3.505E−04 |

1.33E+00 | 3.324E−04 | 3.322E−04 | 3.315E−04 |

1.41E+00 | 3.245E−04 | 3.247E−04 | 3.268E−04 |

Photon energy (MeV) | ε_{p} (present work) | ε_{p} (experimental [ |
---|---|---|

2.45E−01 | 8.390E−04 | 8.392E−04 |

3.44E−01 | 6.873E−04 | 6.871E−04 |

6.62E−01 | 4.378E−04 | 4.376E−04 |

7.79E−01 | 3.895E−04 | 3.897E−04 |

9.64E−01 | 3.149E−04 | 3.147E−04 |

1.17E+00 | 2.874E−04 | 2.871E−04 |

1.33E+00 | 2.715E−04 | 2.717E−04 |

1.41E+00 | 2.676E−04 | 2.679E−04 |

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

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. [_{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. [_{p} decreases by increasing the distance between the detector and the source.

K. S. Al-Mugren,Mahmoud I. Abbas,Eman M. El-Bayoumi,N. S. Aly, (2016) Direct Analytical Method to Calculate Photopeak Efficiency and Photopeak Attenuation Coefficient of NaI(Tl) Well-Type Detector. World Journal of Nuclear Science and Technology,06,115-124. doi: 10.4236/wjnst.2016.62012