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The radioactive contamination of a BaF$_2$ scintillation crystal with mass of1.714 kg was measured over 101 hours in the low-background DAMA/R&D set-up deepunderground (3600 m w.e.) at the Gran Sasso National Laboratories of INFN(LNGS, Italy). The half-life of $^{212}$Po (present in the crystal scintillatordue to contamination by radium) was measured as $T_{1/2}(^{212}$Po) =298.8$\pm$0.8(stat.)$\pm$1.4(syst.) ns by analysis of the events' pulseprofiles. The $^{222}$Rn nuclide is known as 100% decaying via emission of$\alpha$ particle with $T_{1/2}$ = 3.82 d; however, its $\beta$ decay is alsoenergetically allowed with $Q_\beta = 24\pm21$ keV. Search for decay chains ofevents with specific pulse shapes characteristic for $\alpha$ or for$\beta/\gamma$ signals and with known energies and time differences allowed usto set, for the first time, the limit on the branching ratio of $^{222}$Rnrelatively to $\beta$ decay as $B_\beta < 0.13$% at 90% C.L. (equivalent tolimit on partial half-life $T_{1/2}^\beta > 8.0$ y). Half-life limits of$^{212}$Pb, $^{222}$Rn and $^{226}$Ra relatively to $2\beta$ decays are alsoimproved in comparison with the earlier results.

Investigation of rare nuclear decays with BaF$_2$ crystal scintillator contaminated by radium