Effect of lattice instability on superconductivity in sodium tungsten bronze

The effect of lattice instability on the electronic properties of the tungsten bronzes, M/subx/WO$sub 3$(0 < x < 1), is considered. A model of the free energy which describes the various phases as local minima in configuration space is shown to provide a basis for understanding observed and transformations them where M is an alkali metal. For the case of Na/subx/WO$sub 3$, the effect on the superconducting transition temperature of a phonon which is assumed to soften as a function of x is explicitly calculated. Tunneling between the local free-energy minima is assisted by this soft phonon. Good agreement is obtained with recent experimental observations of a dramatic increase in T/subc/ as x decreases and approaches the critical value for transition between the superconducting and semiconducting tetragonal phases. Noteworthy features of this work are that the structural transformation does not correspond to a simple condensation of the soft phonon, and also that the phonon softening and configurational tunneling are considered simultaneously. (AIP)