We have investigated the volume collapse occurring in ${\text{LaMnO}}_3$ unit cell using the anisotropic Potts model modified by two types of anisotropic interactions which has been used to study the behavior of the orbital-ordering configurations with temperature. The bond lengths are related to the occupation of the orbits. We have shown that the collapse is due to the change of Mn-O bonds as the temperature is raised through the transition. These results are in good agreement with the recently published experimental results.

The Jahn-Teller distortive transition of ${\mathrm{LaMnO}}_3$ is described by a modified three-state Potts model. The interactions between the three possible orbits depends both on the orbits and their relative orientation on the lattice. Values of the two exchange parameters which are chosen to give the correct low temperature phase and the correct value for the transition temperature are shown to be consistent with microscopy theory. The model predicts a first order transitions and also a value for the entropy above the transition in good agreement with experiment. The theory with the same parameters also predicts the temperature dependence of the order parameter of orbital ordering agreeing well with published experimental results. Finally, the type of the transition is shown to be close to one of the most disordered phases of the generalized Potts model. The short range order found experimentally above the transition is investigated by this model.

Hexaferrites SrAl_xFe_12−xO₁₉ (x = 0, 1, 2, 3, and 4) doped by Al³⁺ in an account of Fe³⁺ were synthesized using microwave digestion. X-ray diffraction pattern analysis revealed that the compound is polycrystalline with a hexagonal M-type structure of SrAl_xFe_12−xO₁₉ and P63/mmc space group is predominant for all dopants of Al³⁺ ion. The crystallite size of the SrAl_xFe_12−xO₁₉ phase has a maximum value for the un-doped sample (94.5 nm) and then decreases for all values of Al-doping. The SEM results showed that the average grain size increased with Al-doping till x = 2 and was almost stable. The magnetic properties were investigated by vibrating sample magnetometer. The results showed a continuous decrease in both saturation (Ms), and remnant (Mr), magnetization with Al-doping. At the same time, the coercive magnetic field, H_c, increases continuously with Al-doping except for x = 4, H_c slightly decreased. These results make these compound candidates to be used in data storage and magnetic memories. Dielectric properties investigation showed low-frequency dispersion in the dielectric results due to a space-charge effect. It has been demonstrated that enhancements in conductivity, dielectric constant (ε′), and dielectric loss (ε′′) rise with an increase in the concentration of Al³⁺ ions, suggesting that SrAl_xFe_12−xO₁₉ can be used for microwave absorption and other high-frequency applications.