Efficiency calibration is an important prerequisite for γ-spectroscopy measurement, and its accuracy directly affects the reliability of measurement results. The commonly used efficiency calibration methods are active efficiency calibration based on experimental measurement and passive efficiency calibration based on Monte Carlo (MC) simulations.

This study aims to propose a sourceless efficiency calibration method for γ spectrum based on numerical analysis.

Based on theoretical analysis and numerical calculation, a passive efficiency calibration method based on numerical analysis was established, and applied to a constructed geometric model for typical cylindrical LaBr₃(Ce) and NaI(Tl) detector with a diameter and length of 3.8 cm. The γ-ray path trajectory inside the detector, relative position of the source and detector, and peak-to-total ratio were comprehensively analyzed. Efficiency formulas for radioactive point, surface, and volume radioactive sources were formulated. Then, a numerical integration program based on Simpson's algorithm was implemented in MATLAB to solve the numerical analytical formula, and applied to the calculation of detection efficiencies of the 3.8 cm cylindrical LaBr₃(Ce) and NaI(Tl) detectors for isotropic ¹³⁷Cs point, surface, and volume sources. The calculation was performed again by changing the position of the point source and the size of the surface and volume sources. Simultaneously, the Monte Carlo (MC) simulation software MCNP5 was employed to establish the physical model of the detector, and the F8 pulse amplitude card was used to calculate the detection efficiency of the detector for specific energy γ-rays. Finally, detection efficiency obtained by numerical calculation was compared with that of MC simulation to verify the efficiency of proposed sourceless efficiency calibration method for γ spectrum.

Compared with the MC simulation results, the maximum error of numerical calculation does not exceed 3.5%, and the maximum relative errors of the point source efficiency, surface source efficiency and volume source efficiency are 3.26%, 3.33% and 3.36%, repectively, which proves the reliability and accuracy of the numerical analysis method in calculating the detection efficiency.

The passive efficiency calibration method proposed in this study is accurate and fast, avoids complicated MC simulation software, which tends to be difficult to understand and operate. Additionally, this work can be extended to the efficiency calibration of nuclear radiation detectors with different shapes, providing a new pathway for the efficiency calibration of detectors.