Surface anisotropy broadening of the energy barrier distribution in magnetic nanoparticles

The effect of surface anisotropy on the distribution of energy barriers in magnetic fine particles of nanometer size is discussed within the framework of the $Tln(t/tau_0)$ scaling approach. The comparison between the distributions of the anisotropy energy of the particle cores, calculated by multiplying the volume distribution by the core anisotropy, and of the total anisotropy energy, deduced by deriving the master curve of the magnetic relaxation with respect to the scaling variable $Tln(t/tau_0)$, enables the determination of the surface anisotropy as a function of the particle size. We show that the contribution of the particle surface to the total anisotropy energy can be well described by a size--independent value of the surface energy per unit area which permits the superimposition of the distributions corresponding to the particle core and effective anisotropy energies. The method is applied to a ferrofluid composed of non-interacting Fe$_{3-x}$O$_{4}$ particles of 4.9 nm in average size and $x$ about 0.07. Even though the size distribution is quite narrow in this system, a relatively small value of the effective surface anisotropy constant $K_{s}=2.9times 10^{-2}$ erg cm$^{-2}$ gives rise to a dramatic broadening of the total energy distribution. The reliability of the average value of the effective anisotropy constant, deduced from magnetic relaxation data, is verified by comparing it to that obtained from the analysis of the shift of the ac susceptibility peaks as a function of the frequency.