On the failure of the Bloch-Kubo-Dyson spin wave theory

Further investigations on magnets with pure spin moments are presented show ing that the temperature dependence of the order parameter can accurately b e described over a large temperature range by a single T-epsilon power term . The exponent epsilon is found to be independent of the spin order type bu t it depends on whether the spin quantum number is integral or half-integra l and, of course, on the dimensionality of the magnetic interactions. The s ix empirical spin wave exponents epsilon defined in this way are 9/2, 2 and 3 for isotropic, anisotropic and axial interactions and integral spin quan tum number but 2, 3/2 and 5/2 for isotropic, anisotropic and axial interact ions and half-integral spin quantum number. Thermodynamic crossover between neighbouring exponents is frequently observed: antiferromagnetic NiO havin g S = 1 is cubic above TN but undergoes a progressive trigonal lattice dist ortion with decreasing temperature. For this material a crossover from isot ropic (epsilon = 9/2) to anisotropic (epsilon = 2) interactions is observed as a function of decreasing temperature (see Table I). The hexagonal ferro magnet gadolinium having S = 7/2 exhibits a crossover from epsilon = 3/2 to epsilon = 5/2 indicative for a gradual change from anisotropic to predomin antly axial interactions with decreasing temperature. Also the itinerant fe rromagnets Fe, Ni and Co show the same exponents e as insulators with S = 1 /2. While cubic Fe and Ni exhibit epsilon = 2 the same crossover from epsil on = 3/2 to epsilon = 5/2 as for hexagonal Gd is observed for hcp cobalt.