A mathematical approach to calculate the absolute total detection efficiency of a 4π NaI(Tl) sum spin spectrometer

The absolute total detection efficiency of a detector for gamma rays intensely depends on the solid angle subtended by the detector’s front surface at the gamma source. In the present work, we have developed, for the first time, a new mathematical formalism to calculate the solid angle subtended by an n-sided polygonal detector at the point source when the source is placed along the axis of the detector. The method is based on the subtraction of the solid angle in which gamma rays do not enter the detector from that of the circular cross-section. The solid angle formula, so derived, has been applied to calculate the absolute total detection efficiency of NaI(Tl) scintillation detectors of pentagonal and hexagonal cross-sections for monoenergetic gamma rays. These efficiencies are then used to calculate the absolute total detection efficiency of 4π sum-spin spectrometer for gamma rays up to energy 20 MeV. The spectrometer is an array of 32 conical NaI(Tl) detectors of pentagonal and hexagonal cross-sections. To validate the formalism developed, the theoretically calculated efficiencies are compared with those obtained from realistic GEANT4 simulations and experimental measurements considering 137Cs and 60Co gamma sources. The reasonably good agreement clearly validates the developed formalism.