Green's-Function Theory of the Spin Wave in Ferromagnetic Rare-Earth Compounds with Cubic Structures

A Green's-function (GF) theory which is different from the usual two-time GF theory (canonically averaged products of the time-dependent spin operators) is given for the system composed of cubic crystalline fields, Zeeman energy for an applied field, and the exchange energy. When interactions between excited spin waves are neglected, the GF's can be solved. The $2J$ poles or energies of the GF correspond to $2J$ branches of excited spin-wave modes where $J$ is the total angular momentum of the rare-earth ion. The spontaneous magnetization $M$, the specific heat $C_v$, the parallel susceptibility ${\chi }_{\parallel }$, the second-order anisotropy constant $K_u$, the fourth-order cubic anisotropy constant $K_1$ and the sixth-order cubic anisotropy constant $K_2$ are evaluated by operating the corresponding spin operators to the GF. At very low temperatures and in the case of very small crystalline anistropy fields, $\frac{K_u}{K_u}\mathrm{}(T=0\mathrm{})\mathrm{}$, $\frac{K_1}{K_1}\mathrm{}(T=0\mathrm{})\mathrm{}$, and $\frac{K_2}{K_2}\mathrm{}(T=0\mathrm{})\mathrm{}$ are approximately proportional to ${\left\{\frac{M}{M}\mathrm{}\right(T=0\mathrm{}\left)\right\}}^{n_2}$, ${\left\{\frac{M}{M}\mathrm{}\right(T=0\mathrm{}\left)\right\}}^{n_4}$, and ${\left\{\frac{M}{M}\mathrm{}\right(T=0\mathrm{}\left)\right\}}^{n_6}$, respectively, in which $n_2=\frac{3(J_0^{}{}_{}{}^2-J_e^{}{}_{}{}^2)}{(3\mathrm{}J_0^{}{}_{}{}^2-J^2)}\{1-(\frac{J_e}{J_0}\left)\right\}$, $n_4=\frac{5\left\{7\right(J_0^{}{}_{}{}^4-J_e^{}{}_{}{}^4)-6\mathrm{}{J}^2(J_0^{}{}_{}{}^2-J_e^{}{}_{}{}^2\left)\right\}}{(35\mathrm{}J_0^{}{}_{}{}^4-30\mathrm{}{J}^{2}J_0^{}{}_{}{}^2+3\mathrm{}{J}^4)}\times \{1-(\frac{J_e}{J_0}\left)\right\}$, and $n_6=\frac{21\left\{11\right(J_0^{}{}_{}{}^6-J_e^{}{}_{}{}^6)-15\mathrm{}{J}^2(J_0^{}{}_{}{}^4-J_e^{}{}_{}{}^4)+5\mathrm{}{J}^4(J_0^{}{}_{}{}^2-J_e^{}{}_{}{}^2\left)\right\}}{(231\mathrm{}J_0^{}{}_{}{}^6-315\mathrm{}{J}^{2}J_0^{}{}_{}{}^4+105\mathrm{}{J}^{4}J_0^{}{}_{}{}^2-5\mathrm{}{J}^6)}\{1-(\frac{J_e}{J_0}\left)\right\}$, where $J_e^{}{}_{}{}^n$ and $J_0^{}{}_{}{}^n$ mean the magnetic quantum numbers ${\left(J^z\right)}^n$ averaged over the first excited state and the ground state, respectively. They reduce to the well-known third, tenth, and twenty-first laws when cubic crystalline fields disappear. All the thermodynamical properties mentioned above change rapidly at temperatures corresponding to excitation spectra of the spin-wave modes. In the case that the rare-earth ion has two levels, the formulas for $M$, ${\chi }_{\parallel }$, and $C_v$ become identical with the expressions previously reported by the author and others.