Random Anisotropy Model for Nanocrystalline Soft Magnetic Alloys with Grain-Size Distribution

Abstract

A simple model considering grain-size distribution is proposed based on the random anisotropy model. When the maximum grain size (D_m) is less than the exchange correlation length and induced anisotropies are sufficiently small, the effective magnetic anisotropy constant (⟨K₁⟩) is given by using a distribution function (f(D)) for the grain size (D) as ⟨K₁⟩≈K₁⁴{∫₀^D_mD³f(D)dD}²⁄(\\varphi⁶A_c³), where K₁ is the magnetocrystalline anisotropy constant, \\varphi is a parameter which reflects both the symmetry of ⟨K₁⟩ and the total spin rotation angle over the exchange-correlated coupling chain and A_c is the exchange stiffness constant. The log-normal distribution function reproduces well the observed grain-size distribution and yields ⟨K₁⟩≈K₁⁴⟨D⟩⁶exp(6σ_D²)⁄(\\varphi⁶A_c³), where ⟨D⟩ is the mean grain size and σ_D is the geometric standard deviation for the distribution. This result satisfies the well-known ⟨D⟩⁶ law. However, ⟨K₁⟩ increases with increasing σ_D even if ⟨D⟩ is constant. Our model has been extended by taking into account the effect of the coherent induced anisotropies on the exchange correlation length. The coercivity (H_c∝⟨K₁⟩⁄J_s, where J_s is the saturation magnetization) of the nanocrystalline Fe-Nb-B(-P-Cu) alloys with different grain-size distribution have been calculated. Our model explains well the dependence of H_c on the grain-size distribution. These results suggest that one should pay attention on not only the mean grain size but also on the grain-size distribution since the inhomogeneity of the grain size increases H_c.