Physics of Magnetism

SECTION 5.2. QUANTUM-MECHANICAL TREATMENT 45

and the unpaired electron, respectively. The summation is carried out over all ligand

ions in the crystal, taking the center of the magnetic ion considered as origin.

In a more rigorous treatment, the electric charges associated with the on-site valence

electrons of the magnetic ion also have to be included in the crystal-field potential and

the charges associated with the ligand atoms have to be included in the form of charge

densities. The crystal-field potential then takes the form of an integration in space over all

on-site and off-site charge densities around We will return to this point later and use

as given above for introducing the operator equivalent method, without loss in generality.

More rigorous treatments of crystal-field theory have been presented by Hutchings (1964),

White (1970), and Barbara et al. (1988).

The crystal-field Hamiltonian of the magnetic ion is obtained from Eq. (5.2.1) by

summing over all unpaired electrons

The Hamiltonian may be expanded in spherical harmonics since the charges causing the

crystal field are outside the shell of the unpaired electrons (4f electrons in the case of

rare-earth atoms):

Here, are the coefficients of this expansion. Their values depend on the crystal structure
considered and determine the strength of the crystal–field interaction. For instance, if the
point-charge model would be applicable, in which the ions of the crystal are described by
point charges located at the various crystallographic positions, the coefficients can be
calculated by means of