Spin-Wave Spectra of Yttrium and Gadolinium Iron Garnet

Abstract

The problem of deducing the values of the exchange integrals in yttrium and gadolinium iron garnet from measurements of the magnetization and the magnetic contribution to the specific heat at low temperatures is considered. For these garnets the spin-wave normal modes can be found by solving the semiclas sical equations of motion which give rise to a set of $n$ simultaneous linear equations, where $n$ is the number of magnetically inequivalent ions in the unit cell. Expressions for the thermodynamic functions at low temperatures in terms of the frequencies of the normal modes are given assuming the validity of the spin-wave approximation. It is argued that the temperature variation of the frequency of these normal modes on the macroscopic properties can be completely accounted for without considering the zero point energy explicitly. Due to the size of the unit cell, the equations for the frequencies of the normal modes can only be solved numerically for general values of k. Such solutions are obtained for k lying along a [111] direction for various values of the exchange integrals, and the thermodynamic functions corresponding to these choices of parameters are calculated. In the case of yttrium iron garnet, the value of $D$, the coefficient of ${a}^{2}{k}^{2}$ in the acoustic dispersion law, is reliably known and fixes one linear combination of ${J}_{\mathrm{aa}}$, ${J}_{\mathrm{ad}}$, and ${J}_{\mathrm{dd}}$. By comparing our calculations with the magnetization data of Solt, it was established that $\frac{{J}_{\mathrm{aa}}}{{J}_{\mathrm{ad}}}=0.2$, but since the magnetization was not very sensitive to variations of the ratio $\frac{{J}_{\mathrm{dd}}}{{J}_{\mathrm{ad}}}$ its value could not be estimated precisely. Taking $\frac{{J}_{\mathrm{dd}}}{{J}_{\mathrm{ad}}}=0.2$ gives ${J}_{\mathrm{aa}}={J}_{\mathrm{dd}}=6.35$ ${\mathrm{cm}}^{\ensuremath{-}1}$ and ${J}_{\mathrm{ad}}=31.8$ ${\mathrm{cm}}^{\ensuremath{-}1}$. For GdIG the specific heat data below 20\ifmmode^\circ\else\textdegree\fi{}K is not very much influenced by the exact values of the iron-iron exchange integrals which were taken to be those quoted above for yttrium iron garnet. Again one combination of ${J}_{\mathrm{ac}}$ and ${J}_{\mathrm{dc}}$ is known from the calorimetric determination of the single ion splitting. By comparing the specific heat data below 5\ifmmode^\circ\else\textdegree\fi{}K with calculations for various values of $\frac{{J}_{\mathrm{ac}}}{{J}_{\mathrm{dc}}}$ it was possible to determine ${J}_{\mathrm{ac}}$ and ${J}_{\mathrm{dc}}$ separately: ${J}_{\mathrm{dc}}=7.00$ ${\mathrm{cm}}^{\ensuremath{-}1}$ and ${J}_{\mathrm{ac}}=1.75$ ${\mathrm{cm}}^{\ensuremath{-}1}$. These values are about 25% larger than what one would expect using the Weiss molecular field approximation.