  In this paper, we consider the localization of a five-dimensional gravitino
field on $f(R)$-thick branes. We obtain the coupled chiral equations of the
Kaluza-Klein (KK) modes of gravitinos with the gauge condition $\Psi_z=0$. The
chiral equations of a gravitino's KK modes are found to be almost identical to
those of the Dirac fermion. However, their chiralities are exactly opposite.
The chiral KK modes of gravitinos could be localized in some types of
$f(R)$-thick branes on introducing a coupling term. We investigate the
localization of a gravitino on three types of $f(R)$-thick branes through a
Yukawa-like coupling term with background scalar fields. It has been shown that
all the KK modes of gravitinos cannot be localized in the pure geometric
$f(R)$-thick branes by adding a five-dimensional gravitino mass term. However,
for the $f(R)$-thick branes generated by one or two background scalar fields,
only the left- or right-handed zero mode could be localized in the branes, and
the massive KK resonant modes are the same for both left- and right-handed
gravitinos despite their opposing chiralities. All these results are consistent
with those of the five-dimensional Dirac fermion except their chiralities,
which may be an important sign to distinguish the gravitino field and the Dirac
fermion field.
