Temperature dependence of the nuclear spin-magnon relaxation time in the two-dimensional antiferromagnetsK2MnF4andK2NiF4

The relaxation time $T_1$ of the out-of-layer ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ nuclei against decay to the magnons by mediation of hyperfine coupling has been studied as a function of temperature in the nearly ideal two-dimensional (2-D) quadratic-layer (QL) Heisenberg antiferromagnets ${\mathrm{K}}_2$Mn${\mathrm{F}}_4$ ($S=\frac{5}{2}$) and ${\mathrm{K}}_2$Ni${\mathrm{F}}_4$ ($S=1\mathrm{}$). By virtue of the tetragonal symmetry, the lowest-order relaxation process involves three magnons; energy is conserved, but $\overrightarrow{\mathrm{k}}$ is not. Theoretical expressions applicable to the QL structures are derived for $T_1$ due to the three-magnon process, including enhancement by four-magnon exchange scattering. In the numerical evaluation, the multiple summations over the Brillouin zone are carried out rigorously with the exact density of states, dispersion relation, and Bogoliubov coefficients, rather than the commonly used small-$\overrightarrow{\mathrm{k}}$ approximation. Exchange scattering turns out to play a significant role. The enhancement of the direct three-magnon relaxation rate is dependent on temperature, and amounts to up to a factor of 15, about four times larger than estimated previously. Then, generally quantitative agreement, ranging over three decades, is obtained with experiment, with the spin-wave and hyperfine parameters taken from other sources. At very low temperatures, the relaxation is impurity dominated. Residual effects due to multiple exchange scattering are estimated to add by some tens of percent. Another possible process, relaxation induced by electronic dipolar interaction, is worked out in detail for the 2-D case, only to be found negligible. Finally, breaking the symmetry by a magnetic field transverse to the tetragonal axis invokes a two-magnon process, starting off with $T_1^{-1\mathrm{}}$ quadratically dependent on field, and dominating the three-magnon processes by two orders of magnitude at ∼35 kG. Again, the rigorous spin-wave result is found to be in excellent agreement with experiment.