Para- and ferromagnetism

Magnetism.exe

First select the model to be used for the calculation (left column) and the temperature. Enter the value J in eV for the coupling strength of neighboring spins. And specify whether the spin-spin coupling should be isotropic(alpha=1.0 and beta=1.0) or anisotropic. The values for alpha and beta specify a weighting for the coupling strength in the xy plane(alpha) and the z direction(beta). For the special case alpha=0, beta=1 the Ising model is obtained again and for alpha=1, beta=0 the xy model. (See below for details of the models and units used).

If an external magnetic field is to be applied, select in the right-hand column whether it is to be applied in the plane (screen plane = xy plane) or in a plane perpendicular to the crystal lattice(xz plane). The direction of the field can be set using the arrow on the colored disc (hold down the left mouse button and turn the disc). Then enter the magnitude of the magnetic field (magnetic field strength) as a numerical value. The unit used is the energy that a magnitic moment (spin) has when aligned parallel in the magnetic field E = g_µBB.

Now press the "Restart" button. The program displays the current direction of the magnetic moments for each grid position. The color scale is selected as follows: If the direction of the magnetic moment is in the plane of the grid, then the color changes starting with the y-direction from red to green and blue back to red, just as it is shown on the colored disk. If the magnetization points out of the plane, the z-component is shown as increasing brightness of the color (positive z-direction) or decreasing brightness (negative z-direction). Magnetizations exactly perpendicular to the plane are black (-z) or white (+z).

All parameters can be changed during the calculation so that it is possible to directly observe how the system reacts (e.g. to changes in temperature, rotation of the magnetic field, etc.). The program can also be paused and continued later with "Continue". If the model is changed during the calculation, a restart is automatically triggered. Changes to the size of the grid are not taken into account until the next restart.

The direction of the magnetic moments is described by a unit vector S. Due to the exchange interaction, the energy of the overall system depends on the angle between neighboring spins. The total energy is expressed by the Hamiltonian operator H. The coupling energy J of neighboring spins determines the essential properties of the system. This consideration within the framework of the Heisenberg model is subdivided into three cases, which allow different freedoms for the spin directions. The fourth model used is the molecular field approximation, in which each spin couples to the mean field of all other spins rather than to neighboring spins. (For a detailed description of the models see e.g. Czycholl: Theoretische Festkörperphysik)

The magnetic moments (spins) can only point in the z direction and therefore S can only assume the values +1 and -1. The Hamilton operator is

Only neighboring lattice sites are considered in the sum. g is the gyromagnetic ratio, _µB the Bohr magneton and B the external magnetic field. The Ising model in a 2-dimensional lattice shows a phase transition at the critical temperature _TC.

The magnetic moments (spins) lie in the xy plane. The Hamilton operator is

S are 2-dimensional vectors in the xy-plane. This model, which is 2-dimensional with respect to both the lattice and the spins, is also a widely used model in statistical physics. However, this is particularly due to the fact that it shows a special form of phase transition (Kosterlitz-Thouless transition).

In the three-dimensional classical Heisenberg model, the magnetic moments (spins) can point in any direction in three-dimensional space. Due to the classical treatment, there is no quantization with respect to the spin direction. The Hamilton operator is

where S are now 3-dimensional vectors. In the classical Heisenberg model, an anisotropy is often introduced in which the spins couple with a different strength in the z-direction than in the xy-plane. This anisotropy leads, for example, to the formation of distinct domains, as can also be found in the Ising model. The Hamilton operator is

In the isotropic case, alpha = beta = 1. For the limiting case alpha = 0, beta = 1, the Ising model results and for alpha = 1, beta = 0 the xy model.

The molecular field approximation shows a spontaneous magnetization below the Curie temperature. However, no spontaneous formation of domains can be observed.

For the statistical redistribution of the spins, a random lattice site is continuously selected and a random rotation of the spin is examined. If the energy of the overall system (cf. Hamilton operator) is smaller after the observed rotation(_En) than before(_Ev), the rotation is carried out in any case. If the energy _En is greater than before, the Boltzmann factor exp((_En-Ev)_/kBT) is calculated and only redistributed if a random number generated for this purpose is smaller than the Boltzmann factor.