# The Limitations of 5f Delocalization and Dispersion
## 6 X-ray Emission Spectroscopy: A New Approach for the Quantification of Mixing  
Recently, it has been shown that M\({}_{4,5}\) XES of UF\({}_{4}\) exhibits manifestations of the same powerful electric dipole selection rules as the N\({}_{4,5}\) XAS of the actinides [26]. Here, because of the wide separation of the M\({}_{4}\) and M\({}_{5}\) edges, the scaling must be done through a well-behaved spectator line, in this case the 6p XES. An example of the result is shown in Figure 12. The large difference between the 5f peaks is driven by the almost pure 5f\({}_{5/2}\) character of the two 5f electrons in UF\({}_{4}\). Figure 11: Comparison of the first order perturbation model with \(\Delta\)E\({}_{\text{so}}=1\) eV vs. the M\({}_{4,5}\) HERFD XAS spectra. The spin orbit splitting in the HERFD is not an adjustable parameter. The lowest pair of states (green) is filled and does not contribute to HERFD/XAS. It has also recently been shown that while the XAS branching ratio depends upon the 5f\({}_{5/2}\)/5f\({}_{7/2}\) mixing of the unoccupied 5f states, the XES cross sectional variation is driven by the 5f\({}_{5/2}\)/5f\({}_{7/2}\) mixing of the occupied 5f states [17]. It can be shown that for UF\({}_{4}\) and UO\({}_{2}\), where \(n=2\), and U metal, where \(n=3\), the 5f\({}_{5/2}\)/5f\({}_{7/2}\) mixing of the unoccupied 5f states is the same but the 5f\({}_{5/2}\)/5f\({}_{7/2}\) mixing of the occupied 5f states is distinctly different (see Reference [17] for the full derivation and other details). Using the model developed in Reference [17], it can be shown that the M\({}_{4}\) XES of UF\({}_{4}\) and U metal should be significantly different, as illustrated below in Figure 13.  
The important point here is that the effect of 5f delocalization is to further mix the pure 5f\({}_{5/2}\) and pure 5f\({}_{7/2}\) states beyond what is predicted with intermediate coupling model. The M\({}_{5}\) XES is sensitive to this mixing, as shown in Figure 13. Thus, the M\({}_{5}\) XES provides a direct measure of the mixing and delocalization. Of course, other types of 5f\({}_{5/2}\)/5f\({}_{7/2}\) mixing can also occur, e.g., magnetically driven or from electron correlation. These other possibilities will need to be eliminated before the assignment of delocalization-driven can be affixed to the measured mixing. Finally, a significant advantage of the XAS/XES approach is that the probe is intrinsically consistent with the dominant spherical symmetry of both the core holes and the 5f occupied states. Again, it is worth noting the different information that is extracted from XAS and XES. The XAS BR measurement allows a determination of the percentage un-occupation of the 5f states. The XES can be used to determine the percentage occupation of the 5f states. If only one of these is available, as was the case with the XAS BR measurements, then an additional outside piece of information is required to characterize the material. For example, an independent determination of the 5f occupation number, n. Having both  
Figure 12: The M\({}_{4}\) (3d\({}_{3/2}\) hole) and M\({}_{5}\) (3d\({}_{5/2}\) hole) XES spectra of UF\({}_{4}\) are shown here. The spectra have been normalized to the M\({}_{4}\) 6p\({}_{1/2}\) (p\({}_{1/2}\)\(\rightarrow\) d\({}_{3/2}\), \(\Delta\)j = 1) and M\({}_{5}\) 6p\({}_{3/2}\) (p\({}_{3/2}\)\(\rightarrow\) d\(5/2\), \(\Delta\)j = 1) peaks. Orange X = M\({}_{4}\). Purple line = M\({}_{5}\). Black line = unity (1) for 6p normalization. The M\({}_{4}\) spectrum has been shifted -181 eV to align the peaks on the M\({}_{5}\) energy scale. The 6p\({}_{1/2}\): 6p\({}_{3/2}\) intensity ratio = 0.8, following the electric dipole cross sections. Note the very large enhancement of the 5f peak in the M\({}_{4}\) spectrum relative to that of the M\({}_{5}\) spectrum. See text and Reference [26] for details. Similar to Figure 1 in Reference [26]. the XES and XAS together reduces, if not eliminates, the need for the outside information.