Magnon gas and deviation from the Bloch law in a nanoscale Heisenberg ferromagnet

It is known that saturation magnetization of ferromagnetic nanoparticles and nanoclusters deviates strongly from the Bloch T 3 / 2 law. To describe the deviation one usually uses a modified power law T α with a size dependent exponent α , which is motivated simply by its flexibility in fitting the observed behavior. By considering the Heisenberg spin model, we derive an explicit expression in terms of a magnon gas which generalizes the Bloch formula to a finite size system. Comparison to the experimental data shows a good agreement with the observed behavior and gives a better understanding of its physics.