Russell-Saunders terms crystal field

Example 7.3-1 (a) Into which states does the Russell-Saunders term d2 3F split in an intermediate field of Oh symmetry (b) Small departures from cubic symmetry often occur as a result of crystal defects, substituent ligands, and various other static and dynamic perturbations. If some of the IRs of O do not occur in the group of lower symmetry, then additional splittings of degenerate levels belonging to such IRs must occur. Consider the effect of a trigonal distortion of D3 symmetry on the states derived in (a) above. 

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. 

For the weak field case, we have the situation where the crystal field interaction is much weaker than the electronic repulsion. In this approximation, the Russell-Saunders terms 3F, 3P, 1G, lD, and 5 for the d2 configuration are good basis functions. When the crystal field is turned on, these terms split according to the results given in Table 8.4.2 ... 

The value of performing intermultiplet spectroscopy has been demonstrated by optical results on ionic systems. Well defined atomic spectra from intra-4f transitions have been measured up to 6 eV in all the trivalent lanthanides (except, of course, promethium) [Dieke (1968), Morrison and Leavitt (1982) see fig. 1 based on Carnall et al. (1989)]. Each level is characterised by the quantum numbers L, S, J, F), where L and 5 are the combined orbital and spin angular momenta of the 4f electrons participating in the many-electron wavefunctions, and J is the vector sum of L and 5. The quantum number F represents the other labels needed to specify the level fully. It is usually the label of an irreducible representation of the crystal field and we shall omit it. The Coulomb potential is responsible for separating the 4f states into Russell-Saunders terms of specific L and S, while the spin-orbit interaction is diagonal in J and so splits these terms into either 25-1-1 or 2L -I-1 levels with 7 = L - 5 to L -f 5. Provided the spin-orbit interaction is weaker than the Coulomb interaction, as is the case in the lanthanides, the resulting levels consist of relatively pure L, 5, J), or in spectroscopic notation states. These 27-1-1 manifolds are then weakly... 

The crystal-field splitting of the and Russell-Saunders terms can be unravelled from high-resolution emission spectra of both srid Di trarrsitions and excitation spectra of the (/ = 0-2) F (/ = 0 and 1) transi-... 

Table 7.6 Crystal field splitting of Russell-Saunders terms arising from d" configurations in an octahedral crystal field. The spin multiplicity, not included in this tabie, is the same for the crystal field terms as for the parent Russell-Saunders term...

So far it has been shown that sets of d and f orbitals (and therefore D and F terms) split into subsets in an octahedral crystal field but nothing has been said about the relative energies of these subsets. For the moment, the discussion will be restricted to orbitals because it is easy to give pictures of them. In subsequent sections the discussion will be extended to the corresponding terms. In preparation for this extension, it would be helpful if the reader has some idea of their derivation so that he or she is fully aware, for example, that an F state means seven spatial (as opposed to spin) functions, just like a set of f orbitals. One of the simplest ways of appreciating this is through the Russell-Saunders coupling scheme, that which is adopted to obtain the explicit functions themselves. This scheme is outlined in... 

In Russell-Saunders coupling S and I remain good quantum numbers. In intermediate coupling only 7 is a good quantum number. By 7-mixing, several 7 terms with different 7 values can mix under the influence of the crystal-field Hamiltonian. 

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