Linear excitations and the stability of the hard sphere glass

We extend Fixman’s self-consistent phonon theory of the thermodynamic properties and fluctuations of the hard sphere crystal to an amorphous lattice. In doing so we develop a theory of lattice vibrations in amorphous materials via a perturbation theory around the Einstein oscillator approximation. This development is analogous to the mean spherical model description of electronic excitations in liquids. The theory exhibits a mechanical instability rather close to computer simulated glass transitions, suggesting the possibility that the simulations probe only linear stability because of the short simulation times.