Physics of Magnetism

50 CHAPTER 5. CRYSTAL FIELDS

In the case of 3d electrons, we have to proceed differently. First, we have to deal with the

turbation. Before application of the

crystal–field interaction. Subsequently, we can introduce the spin–orbit interaction as a per-

spin–orbit interaction, and and the corresponding

z components and are constants of the motion and hence L, S, and are good

quantum numbers. Because the crystal–field interaction is of electrostatic origin, it affects

only the orbital motion. Therefore, the crystal–field calculations can be made by leaving

the electron spin out of consideration and using the wave functions as basis set.

When calculating the matrix elements of the Hamiltonian given in Eq. (5.2.7), one has

to bear in mind that only even values of n need to be retained. It can also be shown that

terms with n > 2l vanish (l = 2 for 3d electrons).

As an example, let us consider the crystal-field potential due to a sixfold cubic (or

octahedral) coordination. Owing to the presence of fourfold-symmetry axes, only terms

with n = 4 and m = 0, ± 4 are retained, which leads to

where the coefficients of the terms have been calculated with the help of Eq. (5.2.4),

keeping as a constant depending on the ligand charges and distances. The calculations

are summarized in Table 5.2.2 for a 3d ion with a D term as ground state.

If one calculates the expectation value of for the various crystal-field-split eigen-

states, one finds that for all of them. In other words, the crystal–field interaction

has led to a quenching of the orbital magnetic moment. This is also the reason why the

experimental effective moments in Table 2.2.2 are very close to the corresponding effective

moments calculated on the basis of the spin moments of the various 3d ions.

5.3. EXPERIMENTAL DETERMINATION OF

CRYSTAL-FIELD PARAMETERS

In order to assess the influence of crystal fields on the magnetic properties, let us
consider again the situation of a simple uniaxial crystal field corresponding to a level
splitting as in Fig. 5.2.2. If we wish to study the magnetization as a function of the field
strength, we cannot use Eq. (3.1.9) because this result has been reached by a statistical
average of based on an equidistant level scheme (see Fig. 3.1.1). Such a level scheme
is not obtained when we apply a magnetic field to the situation shown in Fig. 5.2.2. The
magnetic field will lift the degeneracy of each of the three doublet levels. Since a given
magnetic field lowers and raises the energy of each of the sets of doublet levels in a different
way, one may find a level scheme for as shown in Fig. 5.3.1c. In order to calculate
the magnetization, one then has to go back to Eq. (3.1.4).
Further increase of the applied field than in Fig. 5.3.1c would eventually bring the
level further down to become the ground state, so that close to zero Kelvin one would
obtain a moment of Again measuring at temperatures close to zero Kelvin,
we would have obtained for applied fields much smaller than corresponding to
Fig. 5.3.1c. This means that the field dependence of the magnetization at temperatures close
to zero Kelvin looks like the curve shown in Fig. 5.3.2. The field required to reach
and hence the shape of the curve, depends on the energy separation between the crystal-field