Institute of Metals Division - The Measurement of Superparamagnetic Particle Shapes and Size Distribution

The magnetization curves for specimezs containing superparmagnetic particles are considered. It is shown that the curves may differ from a Langezlin function because of particle anisotropy, particle interactions , and the presence of a size distribution of particles. The various causes of particle anisotropy are considered, and magnetization curves for specimens with cubic and uniaxial anisotropy are presented. The magnetization curves depend on the In the particle-size below 100A it is quite difficult to obtain quantitative size and shape information on precipitate particles. All of the available techniques have their limitations; however, quantitative applications of superparmagnetism have not been developed as fully as possible due to a lack of analysis of the complete magnetization curve. With such an analysis it appears possible to obtain from a single magnetization curve of a specimen containing randomly oriented SPM (superparamagnetic) particles a quantitative volume distribution and a description of the particle shape.* sign of the anisotropy energy and in the case of a negative uniaxial anisotropy (prolate spheroids) the resulting curves may differ very considerbly from a Langevin function. A graphical technique is developed whereby it is possible to separate out the effects of the size distribution, the particle interaction, and the particle anisotropy. Thus it is possible to obtain quantitative values for both the particle shape and the size distribution. Langevin function2 I/Is = ctnh(HVIs/k T)-k T/HVIS [ 1 ] where I, is the saturation magnetization, V the particle volume, H the applied field, T the particle temperature, k the Boltzmann constant, and I the magnetization in the direction of the applied field. Such behavior is rarely if ever observed in actual systems. One explanation for this is that the SPM particles do possess a significant anisotropy and thus the magnetic energy associated with the particle depends on the direction of magnetization in a more complicated way than for the isotropic particle of Eq. [I]. In general the magnetic energy E of an isolated SPM particle embedded in a nonmagnetic matrix can be expressed as a sum of several contributions: E = Eh + Ed+Es+Ec [2] where Eh is the external field energy, Ed the energy resulting from the particle demagnetizing factor, E, the magnetostriction energy, and E, the crystalline anisotropy energy. In each term we consider only the part dependent on the orientation of the magnetization vector and the particle axes as the shape of the magnetization curve will be determined by these contributions. Following the same procedure used in deriving the Langevin function,