The Fundamental Theorem of Fine-Ferromagnetic-Particle Theory

Abstract

The theory of fine‐particle magnets is based on the theorem that the state of lowest free energy of a ferromagnetic particle is one of uniform magnetization for particles of less than a certain critical size and one of nonuniform magnetization for larger particles. The theorem is inferred from several approximate calculations and has not been proved rigorously. Rigorous statements can be made if one is content to replace equalities by inequalities and exact values of critical radii by upper and lower bounds. For a sphere of radius a with uniaxial anisotropy, it can be shown that the lowest‐free‐energy state is one of uniform magnetization if a<a_c0 and one of nonuniform magnetization if a>a_c1 (for low anisotropy) or a_c2 (for high anisotropy), where a_c0, a_c1 (>a_c0), and a_c2(>a_c0) are determined by the exchange and anisotropy constants and the spontaneous magnetization. These bounds locate the critical radius to within about 12% at low anisotropy but only to within an order of magnitude or more at high.