Spin-wave analysis of the sublattice magnetization of the quadratic double-layer antiferromagnet ${\mathrm{K}}_3$${\mathrm{Mn}}_2$${\mathrm{F}}_7$

The variation with temperature of the sublattice magnetization of the quadratic double-layer antiferromagnet ${\mathrm{K}}_3$ ${\mathrm{Mn}}_2$ ${\mathrm{F}}_7$ has been determined by measuring the NMR frequency of the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ nuclei adjacent to the Mn sites in the double layer. The data been analyzed in terms of a two-dimensional four sublattice spin-wave theory. Temperature-dependent and temperature-independent renormalization as formalized by Oguchi, as well as temperature variation of hte $k=0\mathrm{}$ energy gap have been included, while the integrations over the Brillouin zone have been carried out exactly. THe dispersion is 2 × 2 fold degenerate, where teh lower branches represent in-phase and the upper branches, with energy larger than $4\left|J\right|S$ out-of-phase precession of the spins in the paired layers, respectively. In a least-squares adjustment, spin-wave theory then appears to account for the sublattice magnetization up to 28 K, to be compared with $T_N=58\mathrm{}$ K. The least-squares adjustment yielded for the exchange constant $\frac{J}{k_B}=-7.59\mathrm{}\pm 0.03$ K and for the zero-temperature energy gap $T_G\mathrm{}\left(0\right)=5.99\mathrm{}\pm 0.06$ K (including effects of residual $c$-axis dispersion), which corresponds to an anisotropy field $H_A=1.4\mathrm{}$ kG. In contrast to the signle-layer structure ${\mathrm{K}}_2$Mn${\mathrm{F}}_4$, Oguchi renormalization resulted in a marked improvement, by 9 K, of the range of concurrence over the unrenormalized theory. The functional dependence of the sublattice magnetization of ${\mathrm{K}}_3$ ${\mathrm{Mn}}_2$ ${\mathrm{F}}_7$ on temperatuer is found to be close to that in the single-layer ${\mathrm{K}}_2$Mn${\mathrm{F}}_4$, at least at such low temperatures, that the upper magnon branches are not yet excited. Finally, evidence is found for zero-point spin reduction in good accord with the spin-wave value ${\Delta }_0=0.124\mathrm{}$.