Approximate explicit formulas for Stoner–Wohlfarth hysteresis loops

Approximate explicit formulas for the hysteresis loops in the Stoner–Wohlfarth model are derived. We consider the hysteresis loops both for a single particle with a fixed easy-axis direction and for an ensemble of particles with randomly oriented anisotropy axes. The physical assumption used to derive the formulas is that the particle’s magnetic moment lies in the vicinity of the easy axis at low fields, and in the vicinity of the external field direction at high fields. Surprisingly, the low-field formula is approximately valid even near the Stoner–Wohlfarth astroid, where the magnetic field is not very small. The general piecewise formula is obtained by an appropriate matching of the functions defined on different intervals of the magnetic field, which are chosen to maximize the accuracy of the formula. For the averaged hysteresis loop, the maximal, but reasonably small, deviation of our formula from the numerically calculated magnetization occurs near the coercivity field, where the magnetization slope changes sharply.