The neutrino process that occurs in the outer stellar shells during a supernova explosion and involves neutrino-nucleus reactions produces a range of rare, stable and radioactive isotopes. We improve previous $ u$-process studies by using, for the first time, the time-dependent neutrino emission spectra, as predicted from supernova simulations, rather than a simplified parametric description modeled after the neutron-star cooling phase. In particular, our calculations use time-dependent neutrino spectra for all neutrino species, consider their deviation from a Fermi-Dirac distribution and account for the neutrino emission from the neutrino burst and accretion phases. We find that the time-dependent treatment of the neutrino emission spectra results in higher yields for the selected nuclei produced by the $ u$~process as compared to previous studies and also compared to the approximation of assuming constant neutrino energies corresponding to the time-averaged mean energy radiated in each species. The effect is largest for nuclides produced by charged-current reactions. Our results reflect the dynamical competition between neutrino-induced reactions and the effect of the shock passage through the star. By varying the neutrino burst luminosity and the duration of the accretion phase, we study the impact of these early emission phases and their uncertainties on the $ u$-process nucleosynthesis. We find that the deviation of the neutrino spectra from a Fermi-Dirac distribution calculated in supernova simulations has a negligible effect on the $ u$-process yields.
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