Rates of neutrino absorption on nucleons and the reverse processes in strong magnetic fields

The rates of ${\nu }_e+n\rightleftharpoons e^{-}+p$ and ${\overline{\nu }}_e+p\rightleftharpoons e^{+}+n$ are important for understanding the dynamics of supernova explosion and the production of heavy elements in the supernova environment above the protoneutron star. Observations and theoretical considerations suggest that some protoneutron stars may be born with strong magnetic fields. In a previous paper we calculated the above rates in supernova environments with magnetic fields up to $\sim {10}^{16}\mathrm{G}$ assuming that the nucleon mass $m_N$ is infinite. We also applied these rates to discuss the implications of such strong fields for supernova dynamics. In the present paper we take into account the effects of a finite $m_N$ and develop a numerical method to recalculate the above rates in similar environments. This method is accurate to $\mathcal{O}(1/m_N)$ and such an accuracy is required for application to supernova nucleosynthesis. We show that our results have the correct behavior in the limit of high neutrino energy or small magnetic field. Based on comparison of our results with various approximations, we recommend efficient estimates of the above rates for use in models of supernova nucleosynthesis in the presence of strong magnetic fields.