The Ising model as a pedagogical tool

Though originally developed to analyze ferromagnetic systems, the Ising model also provides an excellent framework for modeling alloys. The original Ising model represented magnetic moments (up or down) by a $+1$ or $-1$ at each point on a lattice and allowed only nearest neighbors interactions to be non-zero. In alloy modeling, the values $\pm1$ represent A and B atoms. The Ising Hamiltonian can be used in a Monte Carlo approach to simulate the thermodynamics of the system (e.g., an order-disorder transition occuring as the temperature is lowered). The simplicity of the model makes it an ideal starting point for a qualitative understanding of magnetism or configuration ordering in a metal. I will demonstrate the application of the Ising model in simple, two-dimensional ferromagnetic systems and alloys.