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14 "neel" - Ising model with dipolar interactions; ferromagnetic domains

"neel" generalizes the Ising model in two dimensions to include dipolar interactions among the Ising spins. This approximates the behavior of a magnetic system with very strong uniaxial anisotropy. In "neel" the easy axis may be oriented either in the plane of the array of spins or perpendicular to it. The interaction strengths, magnetic field, and temperature may be specified. The spin orientations of a 30x60 array of Ising spins are displayed as the system evolves in time after changes in the applied field and/or temperature. The display includes a graph of magnetization vs. time, and numerical output of average energies, magnetizations, and their mean square deviations.

"neel" was adapted by Joerg Draeger, with help from Dan Vernon, from the "ising" program by Russ Thompson.


Here are portions of three displays from "neel" showing three different equilibrium phases, two with antiphase grain boundaries and one with an orientational grain boundary. The system is a planar array of ``ising'' spins with the easy direction perpendicular to the plane of the array. In the left hand display there is only dipole-dipole interaction among the spins giving a simple antiferromagnetic phase (the checkerboard phase). In the middle, a magnetic field has been applied to give what we call the tablecloth phase. On the right, the magnetic field has been turned off, but a ferromagnetic nearest neighbor exchange has been added to give a lamellar phase.

. . . . . .


We have a 95K animation which shows a transition from the antiferromagnetic phase to the lamellar phase. The initial conditions in the simulation are set to establish a single domain antiferromagnet. The strength of the dipolar coupling and the ferromagnetic exchange are set to give a lamellar phase as the equilibrium phase. The temperature is well below the ordering temperature, but enough to provide some thermal activation for spin rearrangements. The first ten frames of the animation are taken at intervals of ten animation steps. In the later frames, the temperature is doubled and the sampling more sparse to illustrate the annealing and grain growth in the lamellar phase.

Table of contents for Chapter 14 of Simulations for Solid State Physics
  1. Introduction
  2. Antiferromagnetism
    1. Antiferromagnetic domains
    2. Magnetic susceptibility
    3. Mean field analysis*
    4. Other phases
  3. Ferromagnetic domains
    1. Domain size
    2. Hysteresis
  4. Summary
  5. Appendix: "neel" -- the program
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