  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.

Disordered Systems and Neural Networks;Materials Science