Deviations from Bloch law in the case of surfacted nanoparticles

Bloch's law which describes the variation with temperature of the spontaneous magnetization at low temperatures no longer applies in the case of surfacted nanoparticles. The deviation is a result of the modification of the superexchange interaction in the surface layer of the nanoparticles, which leads to an increase of the magnetic diameter (D-m) ( attached to the core where the spins are ferrimagnetically aligned), and of their magnetic moment, when the temperature decreases below 300 K. This anomaly was studied for a ferrofluid containing particles of Mn0.6Fe2.4O4 with a mean physical diameter of [D] = 12.2 nm that were surfacted with oleic acid. Taking into account the contribution of the layer at the nanoparticles' surface, based on experimental results I have established the law for the variation of the saturation magnetization (M-sat( T)) with temperature ( T) in this case as an addendum to the Bloch law. The temperature dependence of chi(-1)(i) (chi(i) - initial susceptibility) of the nanoparticle system in the range T > T-b (T-b - blocking temperature), confirms on the one hand, the increase of the nanoparticles' magnetic diameter when they cool down, and on the other hand, confirms the law that I have found for the dependence M-sat( T).