Numerical toy-model calculation of the nucleon spin autocorrelation function in a supernova core - art. no. 023001

We develop a simple model for the evolution of a nucleon spin in a hot and dense nuclear medium. A given nucleon is limited to one-dimensional motion in a distribution of external, spin-dependent scattering potentials. We cal culate the nucleon spin autocorrelation function numerically for a variety of potential densities and distributions which are meant to bracket realist ic conditions in a supernova core. For all plausible configurations the wid th Gamma of the spin-density structure function is found to be less than th e temperature T. This is in contrast with a naive perturbative calculation based on the one-pion exchange potential which overestimates Gamma and thus suggests a large suppression of the neutrino opacities by nucleon spin flu ctuations. Our results suggest that it map be justified to neglect the coll isional broadening of the spin-density structure function for the purpose o f estimating the neutrino opacities in the deep inner core of a supernova. On the other hand, we find no indication that processes such as axion or ne utrino pair emission, which depend on nucleon spin fluctuations, are substa ntially suppressed beyond the multiple-scattering effect already discussed in the literature. Aside from these practical conclusions, our model reveal s a number of interesting and unexpected insights. For example, the spin-re laxation rate saturates with increasing potential strength only if bound st ates are not allowed to form by including a repulsive core. There is no sat uration with increasing density of scattering potentials until localized ei genstates of energy begin to form. [S0556-2821(99)00212-X].