44 CHAPTER 5. CRYSTAL FIELDS
uniaxial rare-earth-based materials can be understood and by means of which the formulae
used in Section 12.4 become sufficiently transparent.5.2. QUANTUM-MECHANICAL TREATMENTIn most compounds, the magnetic atoms or ions form part of a crystalline lattice in which
they are surrounded by other ions, the symmetry of the nearest-neighbor coordination being
determined by the crystal structure. In ionic crystals, the metal ions are usually surrounded
by negatively charged diamagnetic ions. Also in metallic systems, the constituting atoms
carry an effective electric charge. This is due to the fact that they have donated all or at
least a substantial part of their valence electrons to the conduction band. The resultant
positive ions are screened to some extent by the conduction electrons, making the effective
charge smaller than the corresponding ionic charges. The electrostatic field experienced by
the unpaired electrons of a given magnetic ion is called crystal field or ligand field. The
neighboring ions, surrounding the atom with the unpaired electrons, are called the ligands.
A typical situation, where the atom carrying the unpaired electrons is situated in a uniaxial
crystal field, is shown in Fig. 5.2.1.
If J is the total angular-momentum quantum number of the magnetic atom, the (2J + 1)-
fold degeneracy of its ground state will be lifted in the presence of a magnetic as well in
the presence of a crystal field. This will result in changes in the magnetic properties of the
corresponding compound if a crystal field is present.
In order to derive the magnetic properties, it is necessary to solve the Hamiltonian of
the crystal–field interaction explicitly. The crystal-field potential due to the surrounding
ions at the location of the kth unpaired electron of the magnetic ion, iswhere jth ligand ion is
can be either positive or negative).is the absolute value of the electron charge. The charge of the
and are the positions of the jth ligand ion