Magnetoelastic model for the relaxation of lanthanide ions inYBa2Cu3O7−δobserved by neutron scattering

Abstract

Established physical properties of ceramic superconductors and lanthanide ions suggest lattice vibrations are significant in the paths for relaxation of the ions as substitutional impurities in such high-${T}_{c}$ materials. Here, the $4f$ electrons of a lanthanide ion substituting for Y in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ are described with a crystal-field potential (symmetry ${D}_{2h})$ and a magnetoelastic interaction, which is linear in the normal modes of vibration of the paramagnetic ion and anions. The most significant matrix elements of the interaction are determined using selection rules and wave functions for the crystal-field states. Applied to ${\mathrm{Tb}}^{3+},$ ${\mathrm{Ho}}^{3+},$ and ${\mathrm{Tm}}^{3+},$ calculations for the ground state and first excited state indicate that the dynamic properties of the lanthanide ions are adequately described by a simple three-state model not unlike the one introduced by Orbach for the interpretation of electron paramagnetic resonance signals from a lanthanide ion in dilute concentration in a salt. The cross section for inelastic scattering of neutrons by the lanthanide ion is derived by constructing a pseudospin model $(\mathrm{spin}=1)$ and treating the magnetoelastic interaction as a perturbation on the three crystal-field states. In the case of ${\mathrm{Tb}}^{3+},$ the energy of the first excited state relative to the ground state is found to be very much smaller than the energy of any other state, the scattering of neutrons is thus a quasielastic process and the width in energy or, alternatively, the relaxation rate is proportional to ${\mathrm{exp}(\ensuremath{\Delta}{/k}_{B}T)\ensuremath{-}1{}}^{\ensuremath{-}1},$ where T is the temperature and \ensuremath{\Delta} is the energy of the third, intermediate crystal-field state at which the density of phonon states probed. The value of \ensuremath{\Delta} suggested by the calculation and the law predicted for the temperature dependence of the relaxation rate are in accord with measurements, on metallic and nonmetallic samples. Equally impressive accounts are given of data published on the relaxation rates of ${\mathrm{Ho}}^{3+}$ and ${\mathrm{Tm}}^{3+}$ in metallic ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$