Influence of anharmonicity on superconductivity

The method of functional differentiation is used to obtain equations for the Green function of electrons in a superconductor making full allowance for anharmonic interactions. It is shown that these equations are formally identical with the well-known Eliashberg equations. The main difference is that the equations allowing for the lattice anharmonicty have to be modified by replacing the spectral density of the one-phonon Green function occurring in the Eliashberg equation with the dynamic part of a function representing the correlation between this density and the ion concentration. The results obtained demonstrate that allowance for the anharmonicity alters the superconducting properties of a metal for the following reasons: 1) the existence of the Debye--Waller factors in the ionic potential; 2) a change in the spectral density of the one-phonon Green function because of the occurrence of phonon--phonon interactions; 3) many-phonon process; 4) interference effects between one- and many-phonon processes. An analysis is made of the superconductivity of PdH and of the influence of the anharmonicity on the critical temperature T/sub c/ of this compound. It is shown that in this case the greatest influence of the anharmonicity on T/sub c/ is due to the Debye--Waller factors and due to a changemore » in the spectral density of the one-phonon Green function. These two effects reduce the electron--phonon coupling constant of palladium hydride. The effect is greater in PdH than in PdD because of the greater amplitude of the zero-point vibrations of the H atoms. This gives rise to an anomalous isotopic effect in PdH(D).« less