Physics of Magnetism

46 CHAPTER 5. CRYSTAL FIELDS

where the summation again extends over all ligand charges and the corresponding

ligand positions in the crystal. Without going into details about numerical

computations of in terms of point charges, we will keep the treatment general and

consider them as numerical constants and focus our attention again on the Hamiltonian.

A relatively elegant form for this Hamiltonian can be obtained by using Stevens’ Opera­

tor Equivalents method. First, the spherical harmonics are expressed in Cartesian

coordinates, f (x, y, z), after which x, y, and z are replaced by and respectively.

In this way, an operator is formed with the same transformation properties under rotation

as the corresponding spherical harmonics. For instance

where is the expectation value of the 4f radius, is a constant (and where

may be replaced by Note that the introduction of the Operator Equivalents has

the obvious advantage that the summation over is no longer necessary. Equation (5.2.3)

may now be rewritten as

For a magnetic ion with a given J value, the operator equivalents are known.
A complete list of them and their relation to the spherical harmonics can be found in
the paper by Hutchings (1964). The quantities are so-called reduced matrix elements
that do not depend on the azimuthal quantum number m (but depend on J). Values of these
quantities are also listed in Hutchings’ paper. The latter constants are frequently indicated
by and for and 6, respectively.
Finally, it can be shown that for f electrons (l = 3), n cannot exceed 6
Furthermore, n must be even owing to inversion symmetry of the crystal-field potential.
This means that the above summation (for f electrons) is effectively only over n = 2, 4, 6,
since n = 0 gives an additive constant to the potential, which has no physical significance.
For crystal structures with uniaxial symmetry (tetragonal or hexagonal symmetry), it
is sometimes sufficient to consider only the n = 2 terms and neglect the higher order terms.
In this case, the crystal-field Hamiltonian takes the relatively simple form

In Table 5.2.1, an example of how the perturbation matrix may be obtained for the case
J = 5/2 is given. In uniaxial systems, it is obvious to choose the c- axis as quantization
axis or z- axis. The result is a lifting of the (2J + 1) six fold degeneracy of the ground
state. The perturbation leads to three doublet states that are linear combinations of the states
and