  Conventional coupled-channels analyses, that take account of only the
collective excitations of the colliding nuclei, have failed to reproduce the
different behavior of the experimental quasi-elastic barrier distributions for
the $^{20}$Ne + $^{90,92}$Zr systems. To clarify the origins of this
difference, we investigate the effect of non-collective excitations of the Zr
isotopes. Describing these excitations in a random-matrix model, we explicitly
take them into account in our coupled-channels calculations. The non-collective
excitations are capable of reproducing the observed smearing of the peak
structure in the barrier distribution for $^{20}$Ne + $^{92}$Zr, while not
significantly altering the structure observed in the $^{20}$Ne + $^{90}$Zr
system. The difference is essentially related to the closed neutron shell in
$^{90}$Zr.
