Theory of High-Temperature Susceptibility of Heisenberg Ferromagnets Having Nearest-Neighbor Bilinear and Biquadratic Exchange Interactions

The high-temperature susceptibility series is derived for the nearest-neighbor Heisenberg ferromagnet when nearest-neighbor biquadratic exchange is included. The general diagrammatic technique developed by Rushbrooke and Wood for the bilinear interaction is extended to include the biquadratic interaction. The complete coefficients of terms through $\frac{1}{T^2}$ are computed, and since it is expected that the value of the ratio of the biquadratic ($j$) to bilinear exchange ($J$) constants is quite small, only terms linear in the biquadratic exchange in the coefficients of the terms $\frac{1}{T^3}$ and $\frac{1}{T^4}$ are determined. The coefficients are computed for arbitrary spin and general lattice structure. The series expansion has been applied to the susceptibility of KMn${\mathrm{F}}_3$ to determine the quality of information which can be obtained from the experimental data. KMn${\mathrm{F}}_3$ was selected since the biquadratic exchange between ${\mathrm{Mn}}^{++}$ ions has been extensively studied and the second-neighbor bilinear interaction is expected to be negligible. The experimental data, corrected for the temperature-independent diamagnetism, was root-mean-square-analyzed to determine the values of $J$, $j$, and $C$, the Curie constant, which give the best fit to the data. A value of $\left|\frac{j}{J}\right|=0.015\mathrm{}$ was thus found, in good agreement with the results of previous studies.