Size dependent magnetic properties of magnetite (Fe$_{3}$O$_{4})$ nanoparticles.

Magnetism of magnetite (Fe$_{3}$O$_{4})$ nanoparticles was studied as a function of the particle size. Fe$_{3}$O$_{4}$ nanoparticles with different size from 3 nm to 10 nm were synthesized by high temperature organic solution phase method. Hysteresis loops of all the particles showed superparamagnetic behavior at room temperature. The blocking temperature (T$_{B})$ decreases with decreasing particle size. All hysteresis loops were fitted by the Langevin's function, where the saturation magnetization (M$_{s})$ was extracted. M$_{s}$ was further deduced by using the saturated moment and accurately measured mass of the particles. The two methods agree with each other excellently. M$_{s}$ decreases as the particle size is decreased, and is in general much smaller than that of bulk. M$_{s}$ shows a sharp drop with increasing temperature at low temperatures and deviates from the T$^{3/2}$-law. This behavior is attributed to competing ferromagnetic and antiferromagnetic exchange interactions which contribute differently at the surface and interior of the particles.