48 CHAPTER 5. CRYSTAL FIELDS
Until now, we have used the 4f wave functions corresponding to the represen
tation to calculate the perturbing influence of the crystal field by means of the Hamiltonian
given in Eq. (5.2.7). This means that we have tacitly assumed that the crystal–field interaction
is small compared to the spin–orbit interaction introduced via the Russell–Saunders coupling
and Hund’s rules, and that J and m are good quantum numbers. Before applying this crystal-
field Hamiltonian to 3d wave functions, we will first briefly review the relative magnitude
of the energies involved in the formation of the electronic states. In the survey given below,
we have listed the order of magnitude of the crystal-field splitting relative to the energies
involved with the Coulomb interaction between electrons (as measured by the energy dif
ference between terms), and the LS coupling in various groups of materials, comprising
materials based on rare earths (R) and actinides (A ). The numbers listed are given per
centimeter.
These energy values may be compared with the magnetic energy of a magnetic moment
in a magnetic field B:
Using typical values for and B (1T), one finds with
a magnetic energy equal in absolute value to or
This then leads to the following sequences in energies:
For Fe-group materials: crystal field > LS coupling > applied magnetic field,
For rare-earth-based materials: LS coupling > crystal field > applied magnetic field.The physical reason for this difference in behavior is the following: The 3d-electron-charge
clouds reside more at the outside of the ions than the 4f-electron-charge clouds. Therefore,
the former electrons experience a much stronger influence of the crystal field than the latter.
The opposite is true for the spin-orbit interaction. This interaction is generally stronger,