Physics of Magnetism

56 CHAPTER 5. CRYSTAL FIELDS

5.6. A SIMPLIFIED VIEW OF 4f-ELECTRON ANISOTROPY

For the case of a simple uniaxial crystal field, we have derived in Section 5.2 that the

leading term of the crystal-field interaction is given by the expectation value of

In this section, we will show that the crystal–field interaction expressed in Eq. (5.6.1) can
be looked upon in a different way, at the same time providing a simple physical picture for
this type of crystal–field interaction. If the exchange interaction is much stronger than the
crystal–field interaction, we showed in the previous section that ground state at zero Kelvin
is One then has

is the second-order term of symmetry in the spherical harmonic expansion of the
electrostatic crystal-field potential. This quantity can be looked upon as the gradient of the
electric field.
Equation (5.6.1) then represents the interaction of the axial quadrupole moment associ­
ated with the 4f-charge cloud with the local electric-field gradient. It is good to bear in mind
that a nonzero interaction with an electric quadrupole moment requires an electric-field
gradient rather than an electric field.
The shape of the 4f-charge cloud resembles a discus if It resembles a rugby
ball when Examples of both types of charge clouds are shown in Fig. 5.6.1.
It has already been mentioned that the molecular field in a magnetically ordered com­
pound is isotropic and has the same strength in any direction if the exchange
coupling between the moments is the only interaction present. Alternatively, one may say
that the magnetically ordered moments are free to rotate coherently into any direction.
This directional freedom of the collinear system of moments is exploited by the interaction
between the 4f-quadrupole moment and the electric-field gradient to minimize the energy
expressed in Eq. (5.6.2). If the crystal field is comparatively weak, one may neglect any
deformation of the 4f-charge cloud and the aspherical 4f-electron charge clouds shown in