Spin-lattice relaxation in the mixed state of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ and Doppler-shifted $d-\mathrm{wave}$ quasiparticles

We present calculations of the rate of planar Cu spin lattice relaxation in the mixed state of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ due to (i) vortex vibrations and (ii) electron spin-flip scattering. We emphasize that both mechanisms give position-dependent rates due to the presence of vortices, and hence the magnetization recovery is characterized by a distribution of rates. We conclude that relaxation by vortex vibrations is too slow to be a significant factor in this material. Using a semiclassical model of Doppler shifted $d-\mathrm{wave}$ quasiparticles with a linear dispersion around the nodes, our calculation of the relaxation rate from electron spin-flip scattering shows partial agreement with experiment.