The relation between elasticity and the superprotonic phase transition temperature for M3H(XO4)2

We have investigated the rule concerning the superprotonic phase transition temperature T c for M3H(XO4)2-type superprotonic conductors. It was found that the component of spontaneous strain a 0 ( = ( b m − a m / 3 ) / 2 b m ) , where a m and b m are lattice constants far below T c , is closely related to the phase transition temperature and that the superprotonic phase transition temperature obeys the equation a 0 2 = α − β T c ( α = 9.02 × 1 0 − 5 , β = 1.74 × 1 0 − 7 K − 1 ) . This result indicates that the spontaneous strain in the a m – b m plane is strongly correlated with the transition temperature T c . Moreover, it is deduced from analysis of the free energy that this scaling rule is describing the competition between the spontaneous strain energy and the kinetic energy of the protons, that is, the energy accompanying the increase in entropy of protons in the superprotonic phase.