The statistical-mechanical properties of a spin-one Heisenberg ferromagnet with isotropic nearest-neighbor biquadratic interactions are discussed. First, conditions for having an exact ferromagnetic ground state are found and the dispersion relation for magnons is derived. Then the relevant thermodynamical properties are obtained in a generalized molecular-field approximation (MFA). It is shown that if three stable phases, paramagnetic, "quadrupolar, " and ferromagnetic, come into play in the discussion, ferromagnetic ordering is always preferred to "quadrupolar" ordering, preventing the latter from appearing as a separate phase transition. The critical ratio of bilinear to biquadratic exchange for the onset of a first-order ferromagnetic transition is obtained.

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The development of new synthetic methodologies of perovskite oxynitrides is challenging but necessary for the search of new compounds and the investigation of new properties. Here, we report a new method of preparation of the perovskite LaTaON 2 that has been investigated as a pigment and photocatalyst for water splitting. The synthesis proceeds through the solid-state reactions under N 2 at 1500 °C between La 2 O 3 , LaN, and Ta 3 N 5 or between LaN and TaON, which are completed after 3 h and lead to sintered, highly crystalline samples with particle sizes up to 1 μm. Nitrogen-deficient samples LaTaO 1+x N 2−x with x ≤ 0.35 are prepared by changing the N/O ratio in the mixture of reactants. Electron diffraction, synchrotron diffraction, and neutron diffraction studies on stoichiometric and nitrogen-deficient compounds indicate that they crystallize in the monoclinic space group I2/m with lattice parameters for LaTaON 2 of a = 5.71458( 7), b = 8.05987(10), c = 5.74772(6) Å, and β = 89.982(3)°. The three anion sites of the I2/m structure are partially occupied by oxygen and nitrogen, with a preference of nitride for two positions with occupancies of 77 and 88%. This anion distribution is different from that reported in previous studies of samples prepared by ammonolysis at lower temperature, suggesting that the synthesis conditions affect the anion order of this perovskite. Optical measurements indicate a band gap of about 1.9 eV, which is close to that observed in samples prepared by other methods. The determined dielectric permittivity for LaTaON 2 ε r ≈ 200, reported for the first time for a highly nitrided pseudocubic perovskite, is similar to that observed in perovskites with one nitrogen per formula.