We have investigated the spin-fluctuations at energy transfers up to $\sim 110\mathrm{meV}$, well above the resonance energy $\left(33\mathrm{meV}\right)$ in the $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.5}$ ortho-II superconductor using neutron time-of-flight and triple-axis techniques. The spectrum at high energies differs from the low-energy incommensurate modulations previously reported where the incommensurate wave vector is largely independent of energy. Well above the resonance the peak of the spin response lies at wave vectors that increase with energy. Within error the excitations at all energies above the resonance are best described by a ring around the $(\pi ,\pi )$ position. The isotropic wave-vector pattern differs from a recently reported square pattern in different but related systems. The spin spectral weight at high energies is similar to that in the insulator but the characteristic velocity is $\sim 40\%$ lower. We introduce a method of extracting the acoustic and optic weights at all energies from time-of-flight data. We find that the optic spectral weight extends to surprisingly low energies of $\sim 25\mathrm{meV}$, and infer that the bilayer spin correlations weaken with increase in hole doping. When the low-energy optic excitations are taken into account we measure the total integrated weight around $(\pi ,\pi )$, for energies below $120\mathrm{meV}$, to agree with that expected from the insulator. As a qualitative guide, we compare spin-wave calculations based on models with ordered and disordered stripes and describe the inadequacy of these and other stripe models for the high-energy fluctuations.

• Received 19 February 2004

DOI:https://doi.org/10.1103/PhysRevB.71.024522