  In view of the application to supernova simulations, we calculate neutrino
reaction rates with nucleons via the neutral and charged currents in the
supernova core in the relativistic random phase approximation (RPA) and study
their effects on the opacity of the supernova core. The formulation is based on
the Lagrangian employed in the calculation of nuclear equation of state (EOS)
in the relativistic mean field theory (RMF). The nonlinear meson terms are
treated appropriately so that the consistency of the density correlation
derived in RPA with the thermodynamic derivative obtained from EOS by RMF is
satisfied in the static and long wave length limit. We employ pion and rho
meson exchange interactions together with the phenomenological Landau-Migdal
parameters for the isospin-dependent nuclear interactions. We find that both
the charged and neutral current reaction rates are suppressed from the standard
Bruenn's approximate formula considerably in the high density regime. In the
low density regime, on the other hand, the vector current contribution to the
neutrino-nucleon scattering rate is enhanced in the vicinity of the boundary of
the liquid-gas phase transition, while the other contributions are moderately
suppressed there also. In the high temperature regime or in the regime where
electrons have a large chemical potential, the latter of which is important
only for the electron capture process and its inverse process, the recoil of
nucleons cannot be neglected and further reduces the reaction rates with
respect to the standard approximate formula which discards any energy transfer
in the processes. These issues could have a great impact on the neutrino
heating mechanism of collapse-driven supernovae.
