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A simple model considering grain-size distribution is proposed based on the random anisotropy model. When the maximum grain size (Dm) is less than the exchange correlation length and induced anisotropies are sufficiently small, the effective magnetic anisotropy constant (hK1i) is given by using a distribution function (f ðDÞ) for the grain size (D )a shK1 iK 4 f R D m 0 D 3 f ðDÞdDg 2 =ð' 6 A 3 cÞ, where K1 is the magnetocrystalline anisotropy constant, ' is a parameter which reflects both the symmetry of hK1i and the total spin rotation angle over the exchange-correlated coupling chain and Ac is the exchange stiffness constant. The log-normal distribution function reproduces well the observed grain-size distribution and yields hK1 iK 4 hDi 6 expð6 2Þ=ð' 6 A 3Þ, where hDi is the mean grain size andD is the geometric standard deviation for the distribution. This result satisfies the well-known hDi 6 law. However, hK1i increases with increasingD even if hDi 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 (Hc /h K1i=Js, where Js 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 Hc 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 Hc.

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