# The Limitations of 5f Delocalization and Dispersion
## 5 X-ray Absorption Spectroscopy: Angular Momentum Coupling and Perturbations
### Evidence of Crystal Field Splittings in HERFD/XAS  
Returning to the central argument, the hypothetical case in Figure 9 very roughly corresponds to the situation in UF\({}_{4}\), at least in a qualitative fashion. The monoclinic symmetry of UF\({}_{4}\) allows only doubly degenerate groupings of states and UF\({}_{4}\) is clearly localized and without the dispersion that one might expect from a metallic system. Thus, for illustrative purposes only, it is useful to compare the new high energy resolution fluorescence detection (HERFD) spectra for UF\({}_{4}\)[43] with the projection from Figure 9. (Note: HERFD is able to achieve higher resolution by monitoring other decay channels than those that dominate conventional XAS. The result is very high resolution XAS. See Reference [43] for more detail.) This comparison is presented in Figure 11.  
The selection rule for the M\({}_{4}\) (3d\({}_{3/2}\)) peak is very strong: 3d\({}_{3/2}\)\(\rightarrow\) 5f\({}_{5/2}\) only. There is a similar but weaker selection rule for the M\({}_{5}\) (3d\({}_{5/2}\)) peak: 3d\({}_{5/2}\)\(\rightarrow\) 5f\({}_{7/2}\) mainly [43]. Thus the M\({}_{5}\) peak is broader and more featureless than the M\({}_{4}\) peak. The M\({}_{4}\) peak clearly shows the underlying structure predicted by the model from Figure 9; not so for the M\({}_{5}\) peak. However, even the broader M\({}_{5}\) peak is consistent with the envelope suggested by the first order perturbation model.  
Figure 10: **(Left Panel)** The spin-orbit splittings in the RE’s and some other atoms are shown here. For the rare earths, the spin-orbit splitting used here is the energy separation of the 4f\({}_{7/2}\) and 4f\({}_{5/2}\) peaks. The table uses data from Reference [44]. The Fano spectral data in the inset are from References [45; 46]. They are consistent with the extrapolated value of \(\sim\)0.2 eV for the 4f spin orbit splitting in Ce. The panel was taken from Reference [37]. **(Middle Panel)** Fano effect results for Au 4f core states with circularly polarized photons and true electron spin detection. The experimental spin resolved results are shown in color. The experimental geometry diagram is in the inset in the upper left and the results of a simple theory are in the insets in the upper right. See References [45; 46] for details. (Right Panel) Pt 5d Fano from Reference [47], using unpolarized HeI and a chiral configuration of vectors, is shown here. Spin-resolved photoemission spectra obtained with unpolarized HeI (hv = 21.2 eV) light for normal emission from the valence-bands of Pt(001). The spin integrated total intensity I (black) and spin separated partial intensities I+ in red and I\(-\) in blue are shown for 5 \(\times\) 1 surface. The measured spin polarization P\({}_{Y}\) for a 5 \(\times\) 1 surface is also shown.  
If one considers the possible point group symmetries, in general the cases will correspond to situations similar to those above: a series of doubly degenerate states. The only exception appears to be octahedral/cubic symmetry as in UO\({}_{2}\). Thus, in general, the 5f electronic structure of uranium materials should have a series of doubly degenerate states or bands [42; 43]. This is also consistent with [PERSON]'s ODOS calculation in Figure 3 and the Opeil experiment in Figure 6, for U metal with \(n=3\): a sharp peak right at the Fermi energy (the half-filled second pair) and a broader peak nearby, \(\text{B}^{\text{F}}\sim\
icefrac{{1}}{{3}}\), with an occupancy of 2, for a total of 3 electrons of 5f character.