Abstract
Cooperative phenomena are introduced via the simple Ising model in which spins having two states occupy a lattice and interact with nearest neighbors and an applied magnetic field. We study this model in the mean field approximation. Correlations among spin states are neglected, so each spin interacts with a self-consistent mean field. With no applied magnetic field, the model predicts ordering of spins below some critical temperature for lattices of all dimensionalities, 1, 2, 3, …, and enables properties such as heat capacity and magnetic susceptibility to be calculated. Exact solutions for a one-dimensional lattice show no ordering transition; the mean field model fails badly in that case but otherwise shows reasonable trends. Exact solutions exist in two dimensions and show ordering. Better approximate solutions (Boethe cluster model) or numerical solutions can be obtained for lattices of all dimensionalities. We introduce Monte Carlo simulation for numerical solution of the Ising model as well as for models involving interacting classical particles.
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