We consider the probability of a magnetic nanoparticle to flip its magnetisation near the blocking temperature, and use this to develop quasi-analytic expressions for the zero-field-cooled and field-cooled magnetisation, which go beyond the usual critical energy barrier approach to the superparamagnetic transition. The particles in the assembly are assumed to have random alignment of easy axes, and to not interact. We consider all particles to be of the same size and then extend the theory to treat polydisperse systems of particles. In particular, we find that the mode blocking temperature is at a lower temperature than the peak in the zero-field-cooled magnetisation versus temperature curve, in agreement with experiment and previous rate-equation simulations, but in contrast to the assumption many researchers use to analyse experimental data. We show that the quasi-analytic expressions agree with Monte Carlo simulation results but have the advantage of being very quick to use to fit data. We also give an example of fitting experimental data and extracting the anisotropy energy density K.
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