Sublattice Magnetization of Yttrium Iron Garnet at Low Temperatures

The energies and eigenvectors for long-wavelength acoustic spin waves in yttrium iron garnet are calculated neglecting spin-wave interactions, but including the effects of anisotropy and dipolar interactions. The sublattice magnetization is found to be $S_{{α}}^{}{}_{}{}ᶻ(T)=S{-}{{δ}}_{{α}}{-}cT{-}{A}_{{α}}{T}3/2{-}{B}_{{α}}{T}5/2{-}{C}_{{α}}{T}7/2{⋯}$. Here ${α}$ labels the sublattice $a$ or $d$, ${{δ}}_{{α}}$ expresses the effect of zero-point motion, and $cT$ is the Holstein-Primakoff correction for dipolar interactions. Expressions in terms of the exchange integrals ${J}ₐₐ$, ${J}dd$, ${J}ad$, and $Jad^{}{}_{}{}^{{'}}$, where $Jad^{}{}_{}{}^{{'}}$ describes interactions between next-nearest-neighbor $a$ and $d$ sites, are given for ${A}_{{α}}$, ${B}_{{α}}$, and, when $Jad^{}{}_{}{}^{{'}}=0$, for ${C}_{{α}}$. The spin-wave spectrum of some substituted garnets and the effect of spin-wave interactions on the zero-point disordering are treated in appendices.