Neutron scattering and theB1gphonon in the cuprates

The momentum-dependent line shape of the out-of-phase oxygen vibration as measured in recent neutron-scattering measurements is investigated. Starting from a microscopic coupling of the phonon vibration to a local crystal field, the phonon line shift and broadening is calculated as a function of transferred momentum in the superconducting state of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7.$ The absolute magnitude of the shift is obtained as a function of momentum and is found to be in excellent agreement with experiment. It is shown that the anisotropy of the density of states, superconducting energy gap, and the electron-phonon coupling are all crucial in order to explain these experiments.