Journal of Magnetism and Magnetic Materials 118 (1993) 359-364 North-Holland Magnetic anisotropy in fine magnetic particles Sami H. Mahmood Physics Department, Yarmouk University, Irbid, Jordan Received 8 April 1992; in revised form 22 June 1992 In this work, the magnetization curves and the anisotropy energy constants K are calculated for three systems of Fe,O, fine particles using a simple model for the particle-size distribution function. The model gives a value for the mean particle diameter which is in good agreement with the value obtained from transmission electron micrographs. Also, the results for the anisotropy constant are in good agreement with those obtained from the temperature dependence of the remanent magnetization. Furthermore, Mijssbauer spectra are used, together with the calculated anisotropy constants, to determine the particle-size distribution. 1. Introduction Fine magnetic particles (FMP) have received considerable attention due to their technological and scientific importance [ll. FMPs are single +omain particles with diameters typically = 100 A. The magnetic easy axes of the particles are randomly oriented, and the magnetic anisotropy is usually assumed uniaxial with an effective anisotropy constant K [2]. For such particles, the thermal energy is comparable to the magnetic anisotropy energy, and the magnetic moment of the particle could relax during the measuring time giving a zero net magnetization in the ab- sence of an applied magnetic field. Thus, an assembly of such particles exhibits superparamag- netic behavior, and hence, no remanence or coer- civity would be observed. The classical theory of paramagnetism is employed in the interpretations of the magnetic properties of fine particle sys- tems. This theory proved successful for such sys- tems. For details of the theory of superparamag- netism the reader is referred to ref. [3]. In an applied magnetic field H, the classical theory predicts that the magnetization A4 of a Correspondence to: Dr. S.H. Mahmood, Physics Department, The anisotropy constant K for magnetic sys- Yarmouk University, Irbid, Jordan. tems was calculated by workers using different sample of fine particles has the form of a Langevin function, i.e., =&[coth(PpW - I/PcLH], (1) where p = M,V’ is the magnetic moment of the particle, I/ is the volume of the particle, MS is the saturation magnetization of the bulk material, p = l/k,T and ‘M, is the saturation magnetiza- tion of the sample. However, in real systems, the size of the particles is not constant. To account for the effect of the variability of the particle size, one assumes a convenient distribution function for the particle size. Chantrell et al. [4] employed a log-normal volume distribution, with the me- dian particle diameter 0, and its standard devia- tion a, as fitting parameters. In this method, the values of 0, obtained are usually smaller than those obtained by electron micrographs. It is usu- ally suggested that the difference is due to the non-magnetic coating layer. Further, the log-nor- mal distribution does not give a good fit to the experimental data over the entire magnetization curve unless corrected for inter-particle interac- tions [5]. 0304-8853/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved