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Ising Model
Arrangement of Ising-spins in a 1d chain with pbc.
We will now examine collective magnetism. The simplest model for this purpose is the Ising model, which is described by the following Hamilton function
\[H = - J \sum_{\langle i,j\rangle} S_{i} S_{j} - B \hat M\qquad(1)\] \[\hat M = \mu \sum_{i} S_{i},\;
\bigg(\mu := - \frac{\mu_{B} g_{e}}{2}\bigg)\qquad(2)\] \[H= - J \sum_{\langle i,j\rangle} S_{i} S_{j} - \mu B \sum_{i} S_{i}\qquad(3)\]
where the spins \(S_{i}\) can only take the values \(S_{i}=\pm 1\) . \(\mu_{B}\) is the Bohr magneton and \(g_{e}\) the electronic Landé factor. \(J\) is the exchange coupling and the sum over sites is restricted to nearest neighbour sites, such that each neighbouring pairings are only counted once! \(B\) stands for the magnetic flux density. The Ising model is also used to describe binary alloys. However, the parameters then have a different meaning. The Ising model can be solved exactly in 1d and 2d and in \(2d\) it even has a phase transition. We consider periodic boundary conditions.