No abstract

In the cluster-variation method of cooperative phenomena and also in the quasichemical method, the bottleneck step has been to solve simultaneous equations. This paper proposes a new iterative procedure for solving the equations. This iteration does not use differentiation nor matrix inversion and may be called the natural iteration method. The free energy always decreases as the iteration proceeds, with a consequence that the iteration always converges to a stable solution (a local minimum of free energy) as long as the initial state is a physically acceptable one. The method derives in its introductory step a superposition approximation which writes the distribution variables of the basic cluster as a product of those of subclusters. The method is first explained with the pair approximation of the Ising ferromagnet, and then is applied to the fcc binary alloys to derive a phase diagram which is compared with the one reported recently by van Baal.

It is shown that efficient convergence of the cluster-variation method (of cooperative phenomena in statistical mechanics) for the two-dimensional problem can be achieved by increasing the size of the basic cluster one dimensionally, rather than two dimensionally, and by formulating the degeneracy factor anisotropically. The method is illustrated with the Ising model in a square lattice, and the following are presented: (a) an angle-shaped (or a V-shaped) basic cluster of three points can give the same result as a square basic cluster; (b) a general case is formulated in which a zigzag shape of n V's is used as the basic cluster; and (c) the case of the W cluster (two V's) is calculated using the general formulation mentioned above. It is shown that, when they are plotted against the reciprocal of the number of points in a cluster, the Curie points calculated by different methods lie very close to a straight line.

A modified Landau-Lifshitz equation is solved for a single-domain sphere and an infinitely-wide thin single-domain sheet of ferromagnetic material neglecting anisotropy. The external magnetic field is switched from one direction to its opposite instantaneously at the initial time and the behavior of the magnetization vector is investigated thereafter. It is shown that there is a critical value of the damping constant corresponding to the minimum value of the (repetitive) magnetization reversal time. If the damping constant is larger than the critical value, the magnetization vector moves slower; if it is smaller, the magnetization vector moves faster but oscillates so that it takes longer time until it comes to a rest at the final position. The critical values of the Landau-Lifshitz damping constant λ are γM for the sphere and 0.013γM for the thin sheet, where γ and M are the gyromagnetic ratio and the magnetization, respectively. The computed minimum switching time for the thin sheet of 4–79 molybdenum Permalloy is of the order of 10−9 sec.