  We study in detail the barrier distributions extracted from large-angle
quasi-elastic scattering of heavy ions at energies near the Coulomb barrier.
Using a closed-form expression for scattering from a single barrier, we compare
the quasi-elastic barrier distribution with the corresponding test function for
fusion. We examine the isocentrifugal approximation in coupled-channels
calculations of quasi-elastic scattering and find that for backward angles, it
works well, justifying the concept of a barrier distribution for scattering
processes. This method offers an interesting tool for investigating unstable
nuclei. We illustrate this for the $^{32}$Mg + $^{208}$Pb reaction, where the
quadrupole collectivity of the neutron-rich $^{32}$Mg remains to be clarified
experimentally.
