  In a nuclear medium, spin-dependent forces cause the nucleon spins to
fluctuate with a rate $\Gamma_\sigma$. We have previously shown that as a
consequence the effective axial-current neutrino-nucleon scattering cross
section is reduced. Here, we calculate this reduction explicitly in the
perturbative limit $\Gamma_\sigma\ll T$. By virtue of an exact sum rule of the
spin-density structure function, we express the modified cross section in terms
of the matrix element for neutrino-nucleon scattering in the presence of a
spin-dependent nuclear potential. This representation allows for a direct
comparison with and confirmation of Sawyer's related perturbative result. In a
supernova core with a typical temperature $T=10\,\rm MeV$, the perturbative
limit is relevant for densities $\rho\alt10^{13}\,\rm g\,cm^{-3}$ and thus
applies around the neutrino sphere. There, the cross-section reduction is of
order a few percent and thus not large; however, a new mode of energy transfer
between neutrinos and nucleons is enabled which may be important for neutrino
spectra formation. We derive an analytic perturbative expression for the rate
of energy transfer.
