Estimating the efficiency and purity for detecting annihilation and prompt photons for positronium imaging with J-PET using toy Monte Carlo simulation

The positronium imaging technique represents a potential enhancement of the PET imaging method. Its core principle involves employing a β$^{+}$ radiation source that emits additional gamma (γ) quanta referred to as prompt gamma. Our aim is to evaluate the capability to differentiate between annihilation and prompt gamma emissions, a vital aspect of positronium imaging. For this purpose, the selected isotopes should enable high efficiency and purity in detecting both prompt gamma and annihilation gamma. The assessment of the efficiency in identifying prompt and annihilation photons for various isotopes, which are potentially superior candidates for β$^{+}$+γ emitters, is conducted through toy Monte-Carlo simulation utilizing the cross-section formula for photon-electron scattering. In this article, we have performed calculations for efficiency and purity values across different isotopes under ideal conditions and examined how these values evolve as we incorporate the fractional energy resolution into the analysis. Ultimately, the primary goal is to determine the energy threshold that optimizes both efficiency and purity, striking a balance between accurately identifying and recording events of interest while minimizing contamination from undesired events.