Development of a semi-empirical method to determine the efficiency of a gamma radiation detector for point sources

The total intrinsic efficiency of a detector is defined as

where: E is the energy of the detected photons, $\overrightarrow{r}$ is the source position, $\lambda$ is the total linear attenuation coefficient for the energy E and d the path traveled by photons in the detector.

If we know the total intrinsic efficiency, ${\in }_{\text{int}}\left(E_{\text{1}},\overrightarrow{r}\right)$ and the total linear attenuation coefficient for an energy $E_{\text{1}}$, $\lambda \left(E_{\text{1}}\right)$ , the average path length can be determined by

Now if we know the linear attenuation coefficient for another energy, $\lambda \left(E_{\text{2}}\right)$ , we can determine the total intrinsic efficiency of the energy $E_{\text{2}}$ by the following expression

where the average traveled is known, because it was already determined experimentally in Equation (2).

Finally, we can determine the peak intrinsic efficiency of the detector using the relation between total efficiency and peak efficiency given by,

where $\left(P/T\right)$ is the peak to total ratio.

The peak to total ratio must be determined by simulation as Moens indicates in his work [4]. As our work is developed using simulations, this last part of the method is not performed yet.

Using the FLUKA code we simulated a bare cylindrical germanium detector and the isotropic monoenergetic radioactive sources. The dimensions of the simulated detector are 3,84 cm height and 2,375 cm radii.

First, we determined the total absolute efficiencies by statistical count for three reference energies well distributed all along the interest gamma energy range by simulation using FLUKA code [1]. These energies are 59,54 keV, 661,65 keV and 1 274 keV corresponding to the emissions of ²⁴¹Am, ¹³⁷Cs and ²²Na respectively. Then, considering the solid angle subtended for the source over the detector the total intrinsic efficiencies were determined for such energies. Finally, using total attenuation coefficient obtained from XCOM database [2], we determined the reference average traveled path of photons into the detector.

The average traveled path must be the same for all energies, but this can suffer variations as we can observe in table 1. For this reason, the results obtained in the first work were worse for farther energies from the reference energy, because we only used one energy reference (661,65 keV), and as we can see the average traveled path changes in table 1. In the table, we show the reference energies used for the extrapolation of the efficiency, together with the efficiency obtained in the simulation and the total attenuation coefficient. In the last column, we show the average traveled path of the photon into the detector.

For example, for the high energies range the multiple interactions into the detector increases the average traveled path. On the opposite case, for low energies, the average traveled path decreases due to the fact that photons rarely cross the detector, and cross a small path when these impact in the edge or the detector.

This can clearly be seen in the results presented in the next section in table 2. We have maintained the sign of relative bias to show such effect. The reason for the positive sign for all efficiencies obtained by the method for energies lesser than the reference energy is because these are greater than the expected value of the efficiency obtained by statistical count. Therefore the average traveled path of such photons must be less than the obtained from the reference efficiency. The opposite happens for energies greater than the reference energy. This is deduced from the change of sign when passing the reference energy.

In this table, we can see that the best result for e 53,16 keV energy is achieved when using the 59,54 keV reference energy. The best result for energies within the 121,8 - 867,4 keV range is achieved when using 661,65 keV reference energy, and for energies greater than 867.4 the optimal extrapolation is when using 1 274 keV as reference energy.