General Superparamagnetic Behavior of an Aligned Assembly of Uniaxially Anisotropic Particles

The partition function which describes the thermal equilibrium behavior of an aligned assembly of uniaxially anisotropic particles in a field H applied along the direction of alignment is: Z = 2π ∫ 0π sinθ exp(−a sin2θ+b cosθ)dθ. In this expression a = KV/kT; and b = μH/kT, where K is the anisotropy energy density;V the particle volume; μ the particle magnetic moment; k the Boltzmann factor; and T the absolute temperature. It is shown that the tabulated function y = exp(−x2) ∫ 0x exp(l2)dtcan be used to obtain the fractional magnetization m∥ along the direction of alignment for general values of a and b. Numerical results for m∥ have been obtained for a = 0.01, 0.1, 1, 2, 5, 10, 100; and 0.01≤b≥100. Curves of m∥ vs b with a as a parameter are presented. For a<25 the curves are the magnetization curves for superparamagnetism. For a = 1 the maximum deviation from the Langevin function is about 30%. As a → ∝, m∥ approaches tanhb. For a = 10 the latter approximation is fairly good. The dependence on ashows that superposition of magnetization curves when H is normalized to b is allowable only for a≪1 and a≫1. Thus, when a is not known, the use of such superposition as a test for superparamagnetism will not be valid in samples in which there is significant particle alignment.