Magnons and fractons in the diluted antiferromagnet ${\mathrm{Mn}}_{\mathrm{x}}$${\mathrm{Zn}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{F}}_2$

We report high-resolution inelastic-neutron-scattering studies of the spin dynamics in the diluted near-Heisenberg antiferromagnet ${\mathrm{Mn}}_{\mathrm{x}}$ ${\mathrm{Zn}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{F}}_2$ with x=0.75 and 0.50. The x=0.75 experiments reproduce previous results by Coombs et al. [J. Phys. C 9, 2167 (1976)], albeit with much higher resolution. In that case the excitations may be described as spin waves which broaden progressively as the wave vector approaches the zone-boundary value $q_{\mathrm{ZB}}$. However, even at $q_{\mathrm{ZB}}$ the excitation is underdamped. More interesting behavior is observed in the x=0.50 sample. At long wavelengths, the response function S(q,ω) is dominated by a sharp spin-wave peak; however, there is a weak ${\omega }^{\mathrm{-}3}$ tail extending to high energies. With increasing wave vector the sharp peak diminishes in intensity while a broad overdamped component, which is well described by a damped-harmonic-oscillator (DHO) function, grows in intensity. The crossover from a dominant spin wave to a dominant DHO response occurs for q∼0.3$q_{\mathrm{ZB}}$. In energy space, this phenomenon manifests itself as a crossover from propagating low-energy spin waves to localized high-energy excitations. An independent measurement of elastic diffuse magnetic scattering from the x=0.50 sample yields the percolation correlation length ${\xi }_p$ associated with the dilution as ${\xi }_p^{\mathrm{-}1}$=0.3$q_{\mathrm{ZB}}$. This demonstrates that the crossover in the dynamics occurs at the length scale characteristic of the static geometrical disorder. The results are thence related to the magnon-fracton crossover predicted by recent theories for percolation networks.