Crystal-field multiplets

Fig. 3 Scheme of the calculation of energy levels in atoms, ions and unimetal complexes. The weak-field path (a) + (b) + (c). The strong-field path (b) + (a) + (c). The multiplet path (a) + (c) + (b). The crystal-field multiplets represent an appropriate basis for involvement of the Zeeman perturbation in the magnetic field... 

The lowest crystal-field multiplets are separated by ( 66 = (T>) s( 6), which is large... 

The labeling of the crystal-field multiplets according to the irreducible representations of a double group can be performed as follows. 

The complex matrices are denoted by a wave-symbol. The diagonalization yields the crystal-field multiplets... 

Third, the transformation into the crystal-field multiplets is provided by a matrix product... 

The lowest crystal-field multiplets IR-Bethe(Mulliken) x degeneracy, energy (Zeeman coefficient gzMj) ... 

The crystal-field multiplets are classified according to the irreducible representations of the respective double-group where both, the Bethe and the Mulliken (in parentheses) notations are written. DsH means the spin-Hamiltonian D-value accounting for all excitations AmIi - the lowest energy levels difference using the model-Hamiltonian in the first iteration Affl - the... 

Due to the so-called /-mixing within the crystal field, multiplets with different / values are coupled. However, similar to the free-ion case, the levels are still designated by the principal 25+1L j component of the crystal-field wavefunction. For the further labeling of levels split by the crystal field, either the irreducible representation /j (Bethe, 1929) to which the particular wavefunction belongs or the crystal quantum number /i defined by Hellwege (1949) are most commonly used. 

Inspection of Equation 1.23 and consideration of the properties of 3-y and 6-j symbols confirm that only even A--values contribute to crystal field splitting. Further, it indicates that mixing between levels belonging to different / multiplets can only occur if terms with k site symmetry of the lanthanide, in much the same way as discussed above for the Stevens formalism. 

The inclusion of the crystal field destroys the rotational symmetry of the ion and lifts the degeneracy of J levels (except of course Kramer s degeneracy) the only good quantum numbers will be T s, the irreducible representations of the point-group symmetry operation. If the crystal field interaction is comparable to J-J splitting (and we see from Table 2 that this is the case of actinides) it will also cause an admixture of different J multiplets. 

When the ion is embedded in a matrix, the crystal field splits the ground multiplet and the crystal field levels give rise to the same problem as the free ion multiplets, to be solved in the same way. If several crystal field levels are involved, a complete treatment using Eq. (13) is needed. 

Localized d or/electrons retain their one-atom manifolds, except that states arising from different d" or f are split by crystal field and spin-orbit coupling. Multiplet splittings due to spin-orbit coupling are larger than crystal-field splittings of Af levels the converse is the case for 3d" levels. The difference in energy between d"(f) and d" (/ 1) manifolds corresponds to the amount of free atom U that is decreased due to interatomic interaction in the solid. 

Example 8.2-1 Examine the effect of spin-orbit coupling on the states that result from an intermediate field of O symmetry on the Russell-Saunders term 4F. Correlate these states with those produced by the effect of a weak crystal field of the same symmetry on the components produced by spin-orbit coupling on the 4F multiplet. 

Describe the splitting of the multiplet 4D under the conditions specified in (i)-(iv) of Problem 8.3, except that the crystal field is of 0 l symmetry. [Hint Since a crystal field does not affect the parity of a state, it is sufficient to work with the double group O.]... 

Classification of Crystal-field Terms and Multiplets Double Groups... 

---