Simulated nuclear spin-lattice relaxation in Heisenberg ferrimagnets: Indirect observation of quadratic dispersion relations

In response to recent proton spin-relaxation time measurements on $\mathrm{NiCu}\left(\mathrm{pba}\right)({\mathrm{H}}_2\mathrm{O}{)}_3\cdot 2{\mathrm{H}}_2\mathrm{O}$ with $\mathrm{pba}=1,3\mathrm{}-\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{b}\mathrm{i}\mathrm{s}\left(\mathrm{oxamato}\right)\mathrm{},$ which is an excellent one-dimensional ferrimagnetic Heisenberg model system of spin $(1,\frac{1}{2}),$ we study the Raman relaxation process in spin-$\mathrm{}(S,s)\mathrm{}$ quantum ferrimagnets on the assumption of predominantly dipolar hyperfine interactions between protons and magnetic ions. The relaxation time $T_1$ is formulated within the spin-wave theory and is estimated as a function of temperature and an applied field H by a quantum Monte Carlo method. The low-temperature behavior of the relaxation rate $T_1^{-1}$ qualitatively varies with $\mathrm{}(S,s),$ while $T_1^{-1}$ is almost proportional to $H^{-1/2}$ due to the characteristic dispersion relations.