  We use an analytic criterion for vanishing of exponential damping of
correlations developed previously (Piasecki et al, J. Chem. Phys., 133, 164507,
2010) to determine the threshold volume fractions for structural transitions in
hard sphere systems in dimensions D=3,4,5 and 6, proceeding from the YBG
hierarchy and using the Kirkwood superposition approximation. We conclude that
the theory does predict phase transitions in qualitative agreement with
numerical studies. We also derive, within the superposition approximation, the
asymptotic form of the analytic condition for occurence of a structural
transition in the D->Infinity limit .
