Ferromagnetism
Iron, Nickel, Cobalt, etc. are ferromagnets
Unpaired electrons
Qualitative description:
–Permanent magnetization at low T
–Above Curie Temperature, no longer
magnetized, but are paramagnetic
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Ferromagnetism: Ising Model Solutions and Simulations, Slides of Computational Physics
An overview of ferromagnetism, focusing on the ising model and its solutions in one and two dimensions. It also discusses the lack of an exact solution in three dimensions and the use of simulations to study the system. Information on the heisenberg model, the curie temperature, and applications of the ising model to various physical systems.
Typology: Slides
2011/2012
1 / 9
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Ferromagnetism
Iron, Nickel, Cobalt, etc. are ferromagnets
Unpaired electrons
Qualitative description:
- Permanent magnetization at low T
- Above Curie Temperature, no longer magnetized, but are paramagnetic
The Model (1925)
Lattice of particles of spin 1/
Interact by a spin-spin coupling and with
external B-field:
- First sum over all lattice sites
- Second sum over nearest neighbors
H
J
si s j
j i
^ B si
i
Solution (2D) - Onsager 1944
Also solved exactly in 2D for zero B-field
Expression for energy is very complicated
Where kBTC ≈ 2.269185*(J/4)
Now there is a (2nd^ order) phase transition
M ( T ) 1 ^ sinh(
J )
^ ^4
1/ 8 T TC
0 T TC
What about 3D?
There is no exact solution
(Proved by Istrail 2000)
So does the 3D model exhibit properties
similar to real ferromagnets?
Software
MCStep Performs the Monte Carlo step. Returns a new lattice and the change in energy. (Recalculating is computationally expensive) ThemoNewState Used by MCStep. Picks a new state based on the criteria just mentioned. (Though it works for a general thermodynamic problem, not just the Ising Model) SiteEnergy Returns the energy of a single site of the lattice with a certain spin.
In Action
Demo_IsingPlots1D Demo_IsingPlots2D Show the results of the simulation and compare with the exact results Demo_IsingPlots3D Show the results of the simulation Demo_IsingAnimation Actually shows the system being simulated as it is cooled Demo_IsingHysteresis Shows response to varying B field at constant T