  We use the cell model to justify the use of a lattice model to study the
ideal glass transition. Based on empirical evidence and several previous exact
calculations, we hypothesize that there exists an energy gap between the lowest
possible energy of a glass (the ideal glass IG) and the crystal (CR). The gap
is due to the presence of strongly correlated excitations with respect to the
ideal CR; thus, one can treat IG as a highly defective crystal. We argue that
an excitation in IG requires energy that increases logarithmically with the
size of the system; as a consequence, we prove that IG must emerge at a
positive temperature T_{K}. We propose an antiferromagnetic Ising model on a
lattice to model liquid-crystal transition in a simple fluid or a binary
mixture, which is then solved exactly on a recursive (Husimi) lattice to
investigate the ideal glass transition, the nature of defects in the
supercooled liquid and CR analytically, and the effects of competing
interactions on the glass transition. The calculation establishes the gap. The
lattice entropy of the supercooled liquid vanishes at a positive temperature
T_{K}>0, where IG emerges but where CR has a positive entropy. The macrostate
IG is in a particular and unique disordered microstate at T_{K}, just as the
ideal CR is in a perfectly ordered microstate at absolute zero. This explains
why it is possible for CR to have a higher entropy at T_{K} than IG. The
demonstration here of an entropy crisis in monatomic systems along with
previously known results strongly suggests that the entropy crisis first noted
by Kauzmann and demonstrated by Gibbs and DiMarzio in long polymers appears to
be ubiquitous in all supercooled liquids.
