  The fast luminescence component of barium fluoride (BaF$_2$) crystals with a
subnanosecond decay time can find wide application in particle physics and
nuclear physics. However, the slow luminescence component with the 630~ns decay
time could cause pile-up signals at a high rate environment. Doping of BaF$_2$
crystals with rare earth elements suppresses the slow emission component, but
at the same time the radiation hardness of the crystals deteriorates. This work
presents the results of studying crystal samples, both pure BaF$_2$ and those
doped with yttrium in a proportion of 1at.\%Y, 3at.\%Y and 5at.\%Y, irradiated
with a fast neutron fluence of about 2.3$\times$10$^{14}$~n/cm$^2$. Their light
output and decay kinetics were measured before and after irradiation. It is
found that the light output loss of a pure BaF$_2$ crystal after irradiation is
about 7\%, and the light output loss of yttrium doped samples after irradiation
is about two times higher. The measurement results demonstrate that after
irradiation the fast component of each sample has a relative light output loss
2-3\% larger than the slow one.
