4
The Magnetically Ordered State
4.1. THE HEISENBERG EXCHANGE INTERACTION AND
THE WEISS FIELD
It follows from the results described in the previous sections, that all N atomic moments
of a system will become aligned parallel if the conditions of temperature and applied field
are such that for all of the participating magnetic atoms only the lowest level (m = –J in
Fig. 3.1.1) is occupied. The magnetization of the system is then said to be saturated, no
higher value being possible than
This value corresponds to the horizontal part of the three magnetization curves shown
in Fig. 3.1.2. It may furthermore be seen from Fig. 3.1.2 that the parallel alignment of
the moments is reached only in very high applied fields and at fairly low temperatures.
This behavior of the three types of salts represented in Fig. 3.1.2 strongly contrasts the
behavior observed in several normal magnetic metals such as Fe, Co, Ni, and Gd, in which
a high magnetization is already observed even without the application of a magnetic field.
These materials are called ferromagnetic materials and are characterized by a spontaneous
magnetization. This spontaneous magnetization vanishes at temperatures higher than the
so-called Curie temperature Below the material is said to be ferromagnetically
ordered.
On the basis of our understanding of the magnetization in terms of the level splitting
and level population discussed in the previous section (Eq. 3.1.4; Fig. 3.1.1), the occurrence
of spontaneous magnetization would be compatible with the presence of a huge internal
magnetic field, This internal field should then be able to produce a level splitting of suf
ficient magnitude so that practically only the lowest level m = –J is populated. Heisenberg
has shown in 1928 that such an internal field may arise as the result of a quantum-mechanical
exchange interaction between the atomic spins. The Heisenberg exchange Hamiltonian is
usually written in the form
where the summation extends over all spin pairs in the crystal lattice. The exchange constant
depends, amongst other things, on the distance between the two atoms i and j considered.
19