  The inverse tritium beta decay (ITBD) reaction, $\nu_e + ^3$H $\to e^- +
^3$He, is a promising experimental tool for observing relic neutrinos created
in the early Universe. This reaction has been selected by the PTOLEMY
experiment for the search of relic neutrinos. Despite its potential, the ITBD
reaction induced by any sources of neutrinos has yet to be observed. We show
that an intense $^{51}$Cr radioactive neutrino source is suitable for observing
the ITBD reaction for the first time. As the Sun is another source of intense
electron neutrinos, we also examine the ITBD reaction rate from solar
neutrinos. Based on our recent studies on the evolution of the helicity of
relic neutrinos, we further present the ITBD rate for capturing relic neutrinos
as a function of neutrino mass hierarchy, the Dirac versus Majorana nature of
neutrino, and the mass of the lightest neutrino.
