Upper bound to a single-domain behavior of a ferromagnetic cylinder

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Abstract
For a right circular, finite ferromagnetic cylinder with a radius R, a parameter, Rc1, is calculated so that for R>Rc1 there is always a nonuniform magnetization state whose free energy is smaller than that of the uniform magnetization state. Numerical computations for the cases of permalloy and of magnetite give Rc1, which is quite close to the recently calculated Rc0 in a cylinder of these materials. The latter is defined, as in the fundamental theorem of Brown on spherical particles, so that the lowest free-energy state is that of a uniform magnetization, if R<Rc0. The closeness of Rc0 and Rc1 makes them a sufficiently good estimate for the critical size at which the particle changes from uniform to nonuniform magnetization state, without going through the complicated numerical computation of the exact value.