Magnetically Ordered Crystals

There are some crystals that exhibit a non-zero magnetic moment below a certain temperature and in the absence of an external magnetic field. This property is called spontaneous magnetization and it may suggest the existence of magnetic ordering in crystals possessing it. Such crystals are called magnetically ordered.
Using the nomenclature of solid state physics, magnetically ordered crystals are such that in the absence of an external magnetic field, the mean magnetic moment of at least one of the atoms in each unit cell of the crystal is non-zero. The atomic magnetic moments may or may not add up to a net magnetization density for the crystal as a whole.
At T=0K the atomic moments of magnetically ordered crystals are essentially completely ordered. But as the temperature increases the magnetization (only for magnetically ordered crystals that exhibit spontaneous magnetization) decreases towards zero at a critical temperature, as the atomic moment gain enough thermal energy to create a random ordering and sum up to zero magnetization.
The simplest type of magnetically ordered crystals is ferromagnets, in which the magnetic moments of all atoms have the same orientation and hence they sum up to produce a non-zero macroscopic magnetic moment even in the absence of an external magnetic field. This definition is valid provided that the temperature of the ferromagnet is below a critical temperature. In the case of ferromagnets it's called the Curie temperature. Here are some examples of ferromagnetic chains (1D):

More common magnetically ordered crystals are antiferromagnets in which the magnetic moments of all atoms sum up to zero macroscopic moment. The simplest example of an antiferromagnet is of anti-parallel alignment of the atomic magnetic moments with each other like in the next example:

One can treat antiferromagnets as consisting of a set of sublattices, each of which has a non-zero mean magnetic moment but of opposite orientations and such magnitude that they cancel out each other. Check this example of antiferromagnetic ordering on a simple cubic lattice:

click to view animation

In this example spins of the same kind form two interpenetrating face-centered cubic lattices
(the red cubic contour doesn't represent a unit cell).

Antiferromagnets, of course, don't possess the property of spontaneous magnetization. Their magnetic order is probed using more subtle means like neutron scattering and nuclear resonance. The above definition is valid provided that the temperature of the antiferromagnet is below a critical temperature called the Néel temperature.
Ferrimagnets produce macroscopic magnetic moment although in a sense they resemble the antiferromagnets. Ferrimagnets are also treated as consisting of a set of sublattices, each of which has a non-zero mean magnetic moment of opposite orientation but different magnitude so they don't cancel out each other. Hence Ferrimagnets like the ferromagnets will exhibit spontaneous magnetization below a critical temperature. Here are some examples of ferrimagnetic chains (1D):

Theortical model is needed to explain the magnetic order of such crystals. There must be some interaction between the atomic moments that depends on their relative orientation to produce such ordered structures.
One may at first consider the dipolar interaction between two magnetic moments to be the main factor underlying these ordered structures:

Where m1 and m1 are two magnetic dipoles seperated by r.
Ignoring the angular dependence, then substituting the approximate magnitude of atomic magnetic dipole moments m1 ≈ m2 ≈ gμB ≈ eh/2πmc and r ≈ 2Å we get U ≈ 10-4eV.
10-4eV corresponds to 1 Kelvin degree, which is too low, since there are ferromagnets that exhibit spontaneous magnetization at much higher temperatures and have critical temperature of hundreds of degrees Kelvin degrees. If the magnetic moments (spins) were aligned by magnetic dipolar interaction, one would expect ferromagnetic alignment to be obliterated above few Kelvin degrees.
The next step would be considering the spin orbit coupling as the main source for this magnetic interaction, but still spin orbit coupling isn't sufficient since it usually superseded by purely electrostatic effects. What we are looking for is an interaction of the order of 1eV or a fraction of an electron-Volt, which is satisfied by electrostatic interactions.In what follows we find out that the main source for the magnetic interaction is electrostatic and is a consequence of electron exchange.