54 CHAPTER 5. CRYSTAL FIELDS
5.5. CRYSTAL-FIELD-INDUCED ANISOTROPY
As will be discussed in more detail in Chapter 11, in most of the magnetically ordered
materials, the magnetization is not completely free to rotate but is linked to distinct crys
tallography directions. These directions are called the easy magnetization directions or,
equivalently, the preferred magnetization directions. Different compounds may have a dif
ferent easy magnetization direction. In most cases, but not always, the easy magnetization
direction coincides with one of the main crystallographic directions.
In this section, it will be shown that the presence of a crystal field can be one of the
possible origins of the anisotropy of the energy as a function of the magnetization directions.
In order to see this, we will consider again a uniaxial crystal structure and assume that the
crystal–field interaction is sufficiently described by the term. Since we are discussing
the situation in a magnetically ordered material, we also have to take into account a strong
molecular field as introduced in Section 4.1.
The energy of the system is then described by a Hamiltonian containing the interaction
of a given magnetic atom with the crystal field and with the molecular field
The exchange interaction between the spin moments, as introduced in Eq. (4.1.2), is
isotropic. This means that it leads to the same energy for all directions, provided that
the participating moments are collinear (parallel in a ferromagnet and antiparallel in an
antiferromagnet). So the exchange interaction itself does not impose any restriction on the
direction of The two magnetic structures shown in Fig. 5.5.1 have the same energy
when only the exchange term in the Hamiltonian is considered.
The examples shown in Fig. 5.5.1 are ferromagnetic structures
and the same
reasoning can be held for antiferromagnetic structures in which the moments are
either parallel and antiparallel to or parallel and antiparallel to a direction perpendicular
to c. Also in these cases, the two antiferromagnetic structures have the same energy.
After inclusion of the term in the Hamiltonian, the energy becomes anisotropic
with respect to the moment directions. This will be illustrated by means of the two fer
romagnetic structures shown in Fig. 5.5.1. We assume that is sufficiently large and