# The Limitations of 5f Delocalization and Dispersion
## 5 X-ray Absorption Spectroscopy: Angular Momentum Coupling and Perturbations
### Crystal Field Modifications of Spherically Symmetric Systems  
At this point, it is a useful exercise to consider how a potential of non-spherical symmetry would affect angular momentum coupling. To begin, consider a hypothetical situation with a single 5f electron, having both orbital and spin angular momentum, to which is applied a crystal field potential. In this hypothetical case, the crystal field is defined such that \(\mathrm{E_{cf}}^{\mathrm{m}}=1\,\mathrm{m}\,\mathrm{E_{cf}}\), where m is the magnetic quantum number of the 5f spherical harmonic. This case has been solved before in Reference [42] and utilized in Reference [43]. Briefly, the non-spherically-symmetric crystal field potential and the spherically symmetric spin-orbit interaction are treated on an equal basis within a single particle model. The wave functions are orthogonalized, producing solutions that are dependent upon the relative size of the two potentials. In Figure 9 are a series of plots for several different values \(\Delta\mathrm{E_{so}}\) and \(\mathrm{E_{cf}}\), where \(\Delta\mathrm{E_{so}}=(7/2)\zeta\), and \(\zeta\) (see) is the 5f spin-orbit splitting parameter. Again, this is a hypothetical case with a simplistic perturbation. In the upper left-hand panel, the case for \(\Delta\mathrm{E_{so}}=1\) eV is shown. This roughly corresponds to U, as can be seen in Figure 2, where a spin orbit splitting of about 1 eV can be observed. It is of interest that in this case, as \(\mathrm{E_{cf}}\) increases to values on the scale of 1 eV, there is substantial mixing of the pure 5f\({}_{7/2}\) and 5f\({}_{5/2}\) states. This means that reasonable values of a crystal field splitting should ruin the selectivity of the electric dipole selection rules that underlie the XAS results of Figures 2 and 7. This result would seem to suggest that crystal field splitting must be much smaller than 1 eV. However, there is a complication. In the lower panels, the cases associated with a much smaller \(\zeta\) (\(\Delta\mathrm{E_{so}}=0.1\) eV) and larger \(\zeta\) (\(\Delta\mathrm{E_{so}}=10\) eV) are shown. For a smaller \(\zeta\), the convergence to the crystal field driven states is almost immediate. The small \(\zeta\) case would correspond to the conditions of Russell-Saunders (RS) Coupling, going towards the limit of a vanishingly small \(\zeta\). This brings home an important point: in general, RS coupling generates states that are consistent with crystal field splitting and require less mixing; but jj coupling is inconsistent with crystal field symmetries and substantial mixing may be required, with a concomitant change in spectral features. For the larger \(\zeta\), it is clear that even for the larger \(\mathrm{E_{cf}}\) values near 2 eV, the states are still in the first order perturbation limit, where the state energies vary linearly with \(\mathrm{E_{cf}}\) and there is not yet significant state mixing between the 5f\({}_{7/2}\) and 5f\({}_{5/2}\) states. The large \(\zeta\) case has the advantage that in the case of the conventional XAS shown in Figure 7, the electric dipole selection rules would continue to hold within the intermediate coupling model. That is because, with the large lifetime broadenings, the intensity measurements will average over the entire spectrum and the underlying fine  
Figure 8: Here is a comparison of the measured absorption coefficient of Ce, \(\mu\) (solid line) to the calculated relative positions and line strengths (histogram), in the region near the N4,5 edge. Taken from [39]. See Reference [39] for further details. This extended peak structure is the pre-peak region described in Figure 7 and elsewhere [6,38,40,41]. structure will be lost but the cross-sectional dependences would survive. Nonetheless, from the results of Figure 2, it is clear that \(\zeta\sim\)1 eV, not 10 eV. However, it is not the value of \(\Delta\)E\({}_{\rm so}\) that E\({}_{\rm cf}\) must overcome, but rather the coulombic repulsion energies, such as those that drive the energy separation in Figure 8 [43].