1.2 Spin systems

Basic model of ferromagnetic solids: atoms are fixed on the vertices of a lattice. Their dipole vectors (spins) may have varying directions: either up/down (Ising model) or any direction (Heisenberg model).

Microscopic configuration $\mbox{$\bf\Gamma$}_{c}$: given by the $N$ spins on the lattice.

Example: Two-dimensional square Ising lattice; only the four nearest spins contribute to the energy of spin $\sigma_{i}$ ($= \pm 1$). (Three dimensions: six nearest neighbors). The total energy is

$$E=-\frac{A}{2}\sum_{i=1}^{N} \sum_{j(i)=1}^{4\; or \;6} \sigma_{i} \sigma_{j(i)}$$

($A$ ... coupling constant).

$\Longrightarrow$Magnetic polarization

$$M \equiv \sum_{i=1}^{N} \sigma_{i}$$

may be determined as a function of temperature. If a magnetic field $H$ is applied, the additional potential energy is $E_{H}=-H \sum_{i} \sigma_{i}$.