Low field hysteresis in disordered ferromagnets

We analyze low field hysteresis close to the demagnetized state in disordered ferromagnets using the zero temperature random-field Ising model. We solve the demagnetization process exactly in one dimension and derive the Rayleigh law of hysteresis. The initial susceptibility a and the hysteretic coefficient b display a peak as a function of the disorder width. This behavior is confirmed by numerical simulations d=2,3 showing that in limit of weak disorder demagnetization is not possible and the Rayleigh law is not defined. These results are in agreement with experimental observations on nanocrystalline magnetic materials.