Ising Model of a Ferromagnet

1. Ising Model of a Ferromagnet Presented by Kathleen McNamara

2. A few definitions • Ferromagnet-a material in which neighboring dipoles align parallel to each other [1] • Antiferromagnet- a material in which neighboring dipoles align antiparallel to each other [1]

3. A few more definitions • Partition Function- the normalization constant in the Boltzmann distribution (this constant changes value when temperature increases. • Boltzmann Distribution- a measure of the probability of a given state • Monte Carlo- a way to randomly sample a large system to obtain a representative group

4. Ising Model • The Ising model assumes that dipoles only have interactions with their “nearest neighbor” dipoles, and long range interactions are ignored. Also, it is assumed that there is a preferred axis of magnetization so that the dipoles can only point up or down. [1]

5. … One-Dimensional Ising Model i= 1 2 3 4 5 6 7 8 9 … N si= 1 -1 -1 1 -1 1 1 -1 1 … -1 This is a one-dimensional Ising model with N elementary dipoles. If the dipole points up, and if the dipolepoints down,

6. … One-Dimensional Ising Model • The energy of the one-dimensional Ising Model is: • The partition function for this model is:

7. Two-Dimensional Ising Model • The two-dimensional Ising Model can be solved exactly for infinite lattices and for finite periodic lattices (imagine the top wrapping around to attach to the bottom and the left wrapping around to attach to the right.)

8. Two-Dimensional Ising Model • Since donut magnets and infinite magnets don’t exist, we need an approximate solution. • We can write a computer program that will use Monte Carlo methods to generate a representative group of lattice arrangements.

9. Two-Dimensional Ising Model • The program will start by randomly generating an lattice. • Then it will randomly select one dipole and either flip its spin, if that results in a lower energy, or not. • This continues through many iterations to produce many lattice configurations.

10. Two-Dimensional Ising Model These four lattices are examples of what some of the output might be from such a program. Obviously, for the data to be statistically significant, you would do many more iterations and average over all of these. 2-D Ising

11. Three-Dimensional Ising Model • Has never been solved exactly • Monte Carlo method requires much more computer time • Results are difficult to display • More than one lattice structure • Simple cubic • Body-centered cubic • Face-centered cubic

12. Class Problem • What is the energy for the particular state of the one-dimensional lattice shown to below? • Hint: recall that the formula for the energy is:

13. Solution to in-class problem S= +1 -1 -1 +1 -1 +1 +1 -1 +1 +1 +1 -1

14. References and Acknowledgements • Schroeder, Daniel V. An Introduction to Thermal Physics. Addison Wesley Longman: 2000 • Christopher Scilla, co-author of the two-dimensional Ising model program • The Physics Department for use of their Bravais Lattice Models and magnets