5

Crystal Fields

5.1. INTRODUCTION

Almost all magnetic phenomena described in the preceding two chapters depend on the
lifting of the degeneracy of the (2J + 1)-degenerate ground-state manifold by magnetic
fields (internal and external) and on the occupation of the levels of this manifold as a
function of magnetic-field strength and temperature.
Apart from magnetic fields, electrostatic fields are also able to lift the (2J + 1)-fold
degeneracy. In order to see this, we will consider first the comparatively simple case of an
atom with orbital angular momentum L =
of two positive ions located along the z- axis. In the free atom, the states

1 situated in a uniaxial crystalline electric field

0 have

identical energies and are degenerate. However, in the crystal lattice, the atom has a lower

energy when the electronic charge cloud is close to the positive ions as in Fig. 5.1.1a than

when it is oriented midway between the positive charges, as in Fig. 5.1.1b and c. The wave

functions which give rise to these electronic charge densities have the form

and yf(r) and are called the and orbitals, respectively. In the axially symmetric

electric field considered in Fig. 5.1.1, the and orbitals are still degenerate. The three

degenerate energy levels referred to the free atom are shown as a broken line in the right part

of Fig. 5.1.1. Had the symmetry of the electric field been lower than axial, the degeneracy

of the and orbitals would also have been lifted.

The crystalline electric field is able to orient the electronic charge cloud into an energet­

ically favorable direction (situation a in Fig. 5.1.1). This means that the associated orbital

moment also may have a preferred direction in the crystal. We have seen in Chapter 2 that

the spin moment is tied to the orbital moment by means of the spin–orbit interaction. This

implies that there also exists some directional preference for the spin moment.

In the next section, it will be shown how one can describe the effect of electrostatic

fields by means of a quantum-mechanical treatment.

The reader who is more materials oriented will be mainly interested in the magnetic

anisotropy resulting from the crystal–field interaction. This holds in particular for readers

interested in rare-earth-based permanent-magnet materials. For these readers it is not strictly

necessary to work through Sections 5.2–5.5. Instead, we offer in Section 5.6 a simple

physical picture by means of which the magnetic anisotropy induced by the crystal field in

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