Crossover from Spin Waves to Quasilocalized Excitations in the Diluted Antiferromagnet (${\mathrm{Mn}}_{0.5}$${\mathrm{Zn}}_{0.5}$)${\mathrm{F}}_2$

High-resolution neutron scattering from a single crystal of the randomly diluted antiferromagnet (${\mathrm{Mn}}_{0.5}$ ${\mathrm{Zn}}_{0.5}$)${\mathrm{F}}_2$ has been observed at $T=5\mathrm{}$ K. We find sharp spin waves near the zone center which broaden rapidly with increasing wave vector; at about $0.3q$ (zone boundary) there is a crossover to a very broad and continuous energy spectrum. Equivalently, the low-energy spin waves cross over to quasilocalized excitations at higher energies $\hslash \omega \gtrsim 1.5$ meV. These results are related to the recent conjecture of "fractons" in disordered media.