Parapositronium – a singlet positronium ground state, has the total angular momentum of electron and positron forming the atom J_s = 0, magnetic quantum moment m = 0, and its annihilation spectrum is dominantly created by the 511 keV discrete photons. A process of cooperative emission of annihilation photons (cooperative annihilation) by a system of positronium atoms, i.e., annihilation superradiance, had been considered by other researchers, and the theory of annihilation superradiance in a system of parapositronium atoms for two-photon annihilation was constructed (two interacting parapositronium atoms; emission of the 1022 keV photons flying apart at an angle of 180°). The 32-crystal spectrometer ARGUS, with 16 detector pairs at an angle of 180° capable of registering double gamma coincidences, with lead collimators (80 mm in diameter) mounted on each detector – was used to test the phenomenon. Parapositronium annihilation spectra were acquired using ²²Na (A = 4×10⁵ Bq) as a positron source, and SiO₂ (as „positronium forming“ medium; probability: 32 %), as well as Al (as „positronium not forming“ target) used as a blank – for estimation of the background events. In the case when after emission of two starting positrons from ²²Na (i.e., the 1275 keV nuclear photons) coincident registration (respecting the spectrometer time resolution) of four annihilation photons should be considered as a six-fold coincidence event, experimentally obtained counting rates were – 0.25 s^-1 (SiO₂) and 0.23 s^-1 (Al), while theoretically predicted – 0.34 s^-1. The main background process competitive to a registration of the parapositronium cooperative annihilation is four-fold coincidence event (the two 180° detectors register 1275 keV photons, and two – summing of the 511 keV annihilation photons), with experimental counting rates – 0.026 s^-1 (SiO₂) and 0.023 s^-1 (Al), as theoretically predicted – 0.023 s^-1. On the other hand, cooperative annihilation should be a four-fold coincidence event (the two 180° detectors register 1275 keV photons, and two – 1022 keV photons), which has not been registered by the ARGUS spectrometer (theoretically predicted counting rate – 1.7×10^-5 s^-1). The analyses showed that probability of detecting the parapositronium cooperative annihilation will increase significantly with increasing positron source activity, but also with decreasing diameter of lead collimators.

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