  Aspects of nuclear structure in alpha-conjugate nuclei have long been
associated with alpha clustering, including the existence of superdeformed
bands. In this paper, an alpha-particle inelastic scattering experiment
investigating the location of $0^+$ states in $^{28}$Si is reported in an
attempt to locate possible cluster states. The results are compared to a
semi-microscopic model which shows good agreement with the data, and supports
the assignment of a newly discovered $0^+$ state as the band-head of a
previously observed superdeformed band in $^{28}$Si
