Critical Properties of the Heisenberg Ferromagnet with Higher Neighbor Interactions (S=12)

The method of exact power-series expansions has been extended to include both nearest-neighbor and next-nearest-neighbor interactions in the Heisenberg model. The series expansions for the susceptibility in zero magnetic field and the free energy in zero magnetic field have been derived to the fifth power in reciprocal temperature for the simple cubic, body-centered cubic, and face-centered cubic lattices. For the special case when all interactions are equal (equivalent-neighbor model), an additional term has been obtained in these expansions. For purposes of discussing the susceptibility and magnetic specific heat, the series expansions have been derived for lattices in which third-neighbor interactions are included, but only for the equivalent-neighbor model. Estimates of critical points are given, and the Padé-approximant method is used to study the dependence of the critical properties (temperature, energy, and entropy) on the relative strength of the first- and second-neighbor interactions. It is found that the variation in the critical point is well represented by $T_c\left(\alpha \right)=T_c\mathrm{}\left(0\right)[1+m_1\alpha ],$ where $\alpha =\frac{J_2}{J_1}$ and lies in the range $0<~\alpha <~1$, and $T_c\mathrm{}\left(0\right)\mathrm{}$ is the critical temperature of the nearest-neighbor model. The values of $m_1$ are 0.76, 0.99, and 2.74 for the fcc, bcc, and sc lattices respectively. Both the second-neighbor model and the equivalent-neighbor model are used to investigate the behavior of ${\chi }_0$ for values of $T$ near $T_c$. It is found that all the coefficients in the magnetic-specific-heat series expansion are positive for the equivalent-neighbor model, and that for lattices with large coordination numbers, reliable estimates of the critical point may be obtained using this function.