Characterization of SABRE crystal NaI-33 with direct underground counting

Ultra-pure NaI(Tl) crystals are the key element for a model-independent verification of the long standing DAMA result and a powerful means to search for the annual modulation signature of dark matter interactions. The SABRE collaboration has been developing cutting-edge techniques for the reduction of intrinsic backgrounds over several years. In this paper, we report the first characterization of a 3.4 kg crystal, named NaI-33, performed in an underground passive shielding setup at Gran Sasso National Laboratory. NaI-33 has a record low $^{39}$K contamination of 4.3$\pm$0.2 ppb as previously determined by Inductively Coupled Plasma Mass Spectrometry. We measured a light yield of 11.1$\pm$0.2 photoelectrons/keV and an energy resolution of 13.2% (FWHM/E) at 59.5 keV. From the measurement of the $\alpha$ rate due to $^{210}$Po decays, we evaluated a $^{210}$Pb activity at equilibrium of 0.53$\pm$0.04 mBq/kg and an $\alpha$ quenching factor of 0.63$\pm$0.01 at 5304 keV. This paper also illustrates the analyses techniques developed to reject electronic noise in the lower part of the energy spectrum: a traditional cut-based strategy and an innovative Boosted Decision Tree approach. Both of these indicated a signal attributed to the intrinsic radioactivity of the crystal with a rate of $\sim$1 count/day/kg/keV in the [5-20] keV region.