Nonadiabatic effects in phonon spectra of superconductors

Abstract

The nonadiabatic contribution of the electron-phonon interaction to the phonon self-energy Σ( q, ω) of superconductors is studied for different values of the phonon momentum q. It is demonstrated that the long range Coulomb field does not affect the nonadiabatic contribution of Σ( q, ω). It is shown in the framework of the BCS theory that the vertex corrections for q = 0 lead to the appearance of the pole in the phonon self-energy connected with the collective electron excitations inside the energy gap and can be an essential contribution to nonadiabatic effects. Σ( q, ω) has the logarithmic singularity near ω ≈ 2 Δ for large values of q but the vertex corrections are not important for this case. It is shown that for q = Q , where Q is the nesting vector, the zero order phonon self-energy Σ 0( Q , ω) has the square root singularity as well as for q = 0 . The vertex corrections lead again to the appearance of the pole in Σ( Q, ω) which has a strong influence on nonadiabatic effects. The results are compared with the recent neutron scattering measurements in borocarbides.