Towards the Quantiﬁcation of 5f Delocalization

: By using M 4,5 X-ray Emission Spectroscopy (XES) in the tender X-ray regime, it is possible to quantify 5f delocalization in the actinides. Previous analyses, utilizing the Branching Ratio (BR) in the N 4,5 X-ray Absorption Spectroscopy (XAS), could not discriminate between the cases of localized n = 2 and delocalized n = 3, in uranium materials, where n is the number of 5f electrons on the U entity. Here, it is shown that, by employing the ubiquitous 6p → 3d XES as a point of normalization, the localized n = 2 and delocalized n = 3 cases can be easily distinguished and quantiﬁed.

A particularly effective approach has been to measure the Branching Ratios (BR) in the N 4,5 X-ray Absorption Spectroscopy, where BR = I 4d5/2 /(I 4d5/2 + I 4d3/2 ), I x being the intensity of the respective level x. (Figure 1) The combination of strong electric dipole selection rules in the d to f transitions and a firm theoretical foundation for localized systems provided by van der Laan and Thole [22] has permitted the determination of the quantity of 5f electrons (n) in a number of localized actinide systems. While this may seem trivial, it was in fact a topic of serious disagreement for a number of years. For example, early on there was a viewpoint that championed 6d filling instead of 5f filling [23,24] and the reconciliation between the accepted n = 6 in atomic Pu and n = 5 in solid Pu was only realized completely in the recent past [25,26]. Nevertheless, there remained a very troublesome "fly in the ointment": the indistinguishability of the BR for the localized n = 2 U systems and the delocalized n = 3 of uranium metal, as shown in Figure 2. This issue was first reported by Kalkowki, Kaindl, Brewer and Krone in 1987 [21] and addressed more recently elsewhere [27]. Figure 1. The N4,5 X-ray Absorption Spectroscopy (XAS) of uranium dioxide and δ-Pu(Ga) are shown here. Note the significant reduction in the intensity of the Pu 4d3/2 peak relative to that of the U in uranium dioxide. (The spectra are normalized with the 4d5/2 level.) This has been shown to be due to the population of the 5f levels: n = 2 for UO2 and n = 5 for Pu. [2,10,11] The UO2 spectrum is courtesy of DK Shuh, Lawrence Berkeley National Laboratory (LBNL).

Figure 2.
The N4,5 XAS spectra of UF4 and α-U are shown here, analogous to Figure 1. UF4 is a localized n = 2 system, very similar to UO2. [2] The data are originally from Kalkowski, Kaindl, Brewer and Krone. [21]. The figure is reproduced from [27]. Note that the two spectra cannot be distinguished; they have the same BR. Figure 1. The N4,5 X-ray Absorption Spectroscopy (XAS) of uranium dioxide and δ-Pu(Ga) are shown here. Note the significant reduction in the intensity of the Pu 4d 3/2 peak relative to that of the U in uranium dioxide. (The spectra are normalized with the 4d 5/2 level.) This has been shown to be due to the population of the 5f levels: n = 2 for UO 2 and n = 5 for Pu [2,10,11]. The UO 2 spectrum is courtesy of DK Shuh, Lawrence Berkeley National Laboratory (LBNL).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 15 Figure 1. The N4,5 X-ray Absorption Spectroscopy (XAS) of uranium dioxide and δ-Pu(Ga) are shown here. Note the significant reduction in the intensity of the Pu 4d3/2 peak relative to that of the U in uranium dioxide. (The spectra are normalized with the 4d5/2 level.) This has been shown to be due to the population of the 5f levels: n = 2 for UO2 and n = 5 for Pu. [2,10,11] The UO2 spectrum is courtesy of DK Shuh, Lawrence Berkeley National Laboratory (LBNL).

Figure 2.
The N4,5 XAS spectra of UF4 and α-U are shown here, analogous to Figure 1. UF4 is a localized n = 2 system, very similar to UO2. [2] The data are originally from Kalkowski, Kaindl, Brewer and Krone. [21]. The figure is reproduced from [27]. Note that the two spectra cannot be distinguished; they have the same BR.  Figure 1. UF 4 is a localized n = 2 system, very similar to UO 2 [2]. The data are originally from Kalkowski, Kaindl, Brewer and Krone [21]. The figure is reproduced from [27]. Note that the two spectra cannot be distinguished; they have the same BR.  [18] with the Unoccupied density of states (UDOS) derived from the theory by Kutepov [10]. In the lower panel, the unsmoothed 5f5/2 calculation is shown in red, the unsmoothed 5f7/2 calculation is shown in green, the partially smoothed total is the jagged black line and the UDOS is shown by the very smooth, heavier black line. The UDOS was derived by applying an inverse Fermi function and more extensive smoothing to the total.
The essence of the problem is that the BR approach works very well for localized systems but cannot rule out the possibility of delocalization with a slightly different number of 5f electrons. However, there now appears to be a solution to this "blind spot." Recently, it has been shown that the M4,5 X-ray Emission Spectroscopy of UF4 [30] exhibits results consistent with the Intermediate Coupling Model of van der Laan and Thole. [22] Normalizing to the 6p → 3d peaks, it can be seen that there is a tremendous differential in the intensities of the 5f peaks in the two spectra, a factor of 5. Figure 4 (The X-ray Emission Spectroscopy (XES) results for UO2 are very similar to those of UF4 [30] and are omitted here.) It has been shown that this is consistent with the almost pure n5/2 nature of the two 5f electrons in UF4 and strong, albeit not complete, adherence to electric dipole select rules. It will be demonstrated here that this type of data will provide (1) a direct pathway to the  [18] with the Unoccupied density of states (UDOS) derived from the theory by Kutepov [10]. In the lower panel, the unsmoothed 5f 5/2 calculation is shown in red, the unsmoothed 5f 7/2 calculation is shown in green, the partially smoothed total is the jagged black line and the UDOS is shown by the very smooth, heavier black line. The UDOS was derived by applying an inverse Fermi function and more extensive smoothing to the total.
The essence of the problem is that the BR approach works very well for localized systems but cannot rule out the possibility of delocalization with a slightly different number of 5f electrons. However, there now appears to be a solution to this "blind spot." Recently, it has been shown that the M 4,5 X-ray Emission Spectroscopy of UF 4 [30] exhibits results consistent with the Intermediate Coupling Model of van der Laan and Thole [22]. Normalizing to the 6p → 3d peaks, it can be seen that there is a tremendous differential in the intensities of the 5f peaks in the two spectra, a factor of 5. Figure 4 (The X-ray Emission Spectroscopy (XES) results for UO 2 are very similar to those of UF 4 [30] and are omitted here.) It has been shown that this is consistent with the almost pure n 5/2 nature of the two 5f electrons in UF 4 and strong, albeit not complete, adherence to electric dipole select rules. It will be demonstrated here that this type of data will provide (1) a direct pathway to the experimentally driven discernment of the localized n = 2 and delocalized n = 3 cases in particular and (2) the degree of delocalization in 5f systems in general.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 15 experimentally driven discernment of the localized n = 2 and delocalized n = 3 cases in particular and (2) the degree of delocalization in 5f systems in general. spectrum corresponds to the 3d3/2 (3d5/2) hole. The normalization of the spectra is via the 6p peaks: M4 6p1/2 (p1/2 → d3/2) and M5 6p3/2 (p3/2 → d5/2), both Δj = 1. Red = M4. Green = M5. The black horizontal line is at unity (1). The intensity ratio of the 6p1/2 :6p3/2 is 0.8, following the electric dipole cross sections. In order to align the peaks on the M5 energy scale, the M4 spectrum has been shifted to −181 eV. There is a very large enhancement of the 5f peak in the M4 spectrum, versus that in the M5 spectrum. This figure is similar to that of Figure 1 in [30]. The details of the normalization are available in [30].

Experimental
The X-ray Emission Spectroscopy experiments were done at the Stanford Synchrotron Radiation Lightsource, using Beamline 6-2a. These were performed utilizing both input photons from a Si(111) monochromator and a photon detector, a high-resolution Johansson-type spectrometer [31,32], operating in the tender X-ray regime (1.5-4.5 keV). For the UF4 M5 and UF4 M4 experiments, the excitation photon energies were, respectively, 3650 and 3820 eV. Each was chosen to be significantly above the threshold for the transition under consideration. Instrumentally, the total energy bandpass of this experiment is about 1 eV. However, the lifetime broadening of the 3d core holes (several eV) dominates the spectral widths. The sample used was the same as used in earlier studies [2,3]. Uranium samples can be affected by oxidation and sample corruption, but these were not a problem here, as described earlier [30].

Results and Discussion
Below, three problems will be addressed sequentially, using a single electron picture: (a) X-ray Absorption Spectroscopy (XAS) and the Branching Ratio (BR); (b) X-ray Emission Spectroscopy (XES) and the 5f:5f Peak Ratios; and (c) the 6p:5f Peak Ratios in XES. The prior success of the BR analysis in XAS, discussed briefly above, argues that the electric dipole selection rules work very well for the 4d → 5f transitions and that they should hold for 5f → 4d transitions as well. However, spectrum corresponds to the 3d 3/2 (3d 5/2 ) hole. The normalization of the spectra is via the 6p peaks: M 4 6p 1/2 (p 1/2 → d 3/2 ) and M 5 6p 3/2 (p 3/2 → d 5/2 ), both ∆j = 1. Red = M 4 . Green = M 5 . The black horizontal line is at unity (1). The intensity ratio of the 6p 1/2 :6p 3/2 is 0.8, following the electric dipole cross sections. In order to align the peaks on the M 5 energy scale, the M 4 spectrum has been shifted to −181 eV. There is a very large enhancement of the 5f peak in the M 4 spectrum, versus that in the M 5 spectrum. This figure is similar to that of Figure 1 in [30]. The details of the normalization are available in [30].

Experimental
The X-ray Emission Spectroscopy experiments were done at the Stanford Synchrotron Radiation Lightsource, using Beamline 6-2a. These were performed utilizing both input photons from a Si(111) monochromator and a photon detector, a high-resolution Johansson-type spectrometer [31,32], operating in the tender X-ray regime (1.5-4.5 keV). For the UF 4 M 5 and UF 4 M 4 experiments, the excitation photon energies were, respectively, 3650 and 3820 eV. Each was chosen to be significantly above the threshold for the transition under consideration. Instrumentally, the total energy bandpass of this experiment is about 1 eV. However, the lifetime broadening of the 3d core holes (several eV) dominates the spectral widths. The sample used was the same as used in earlier studies [2,3]. Uranium samples can be affected by oxidation and sample corruption, but these were not a problem here, as described earlier [30].

Results and Discussion
Below, three problems will be addressed sequentially, using a single electron picture: (a) X-ray Absorption Spectroscopy (XAS) and the Branching Ratio (BR); (b) X-ray Emission Spectroscopy (XES) and the 5f:5f Peak Ratios; and (c) the 6p:5f Peak Ratios in XES. The prior success of the BR analysis in XAS, discussed briefly above, argues that the electric dipole selection rules work very well for the 4d → 5f transitions and that they should hold for 5f → 4d transitions as well. However, it will be seen that a correction term is required for the 5f → 3d transitions, which is not unexpected. The discussion will begin with the 4d-5f case, because the BR literature deals primarily with 4d → 5f transitions. For the XES, first, the 5f → 4d transitions will be considered, then the 5f → 3d transitions. The selection rules and cross sections are for d-f and should apply to all of these cases.

A Revisitation of XAS and the Branching Ratio
Let us begin by first considering the processes of X-ray Absorption Spectroscopy. Of course, van der Laan and Thole dealt with this in their earlier, ground-breaking work, including not just electric dipole transitions but also the intricacies of angular momentum for the three cases of Russell-Saunders or LS coupling, jj coupling and the Intermediate coupling case [22]. The goals here are more modest: to (1) better understand the underlying cause of the n = 2 localized/n = 3 delocalized dilemma and (2) to prepare to address the XES question.
The foundational single electron equations are below. A photon is absorbed and an electron moves from the 4d state into an empty 5f state. In Equation (1), there is only the 5f 5/2 final state. Under electric dipole selection rules, the transition into the 5f 7/2 state is forbidden. This selection rule, combined with preferentially filling the 5f 5/2 states, is the driving force behind the reduction in the 4d 3/2 peak in Pu, as seen in Figure 1 [10,11]. 4d 3/2 + hv → 5f 5/2 (1) 4d 5/2 + hv → 5f 5/2 and 5f 7/2 (2) If one applies the electric dipole operator to transitions between the states in the 4d manifolds [j =3/2 or 5/2, m j > and the states in the 5f manifolds [j = 5/2 or 7/2, m j >, one can obtain the relative cross sections shown in Table 1 [30,32]. Note that these cross sections are between completely filled 4d states and completely empty 5f states. (They would also apply for the reverse transitions, photon emission, between completely full 5f states and completely empty 4d states.) Table 1. Shown here are the relative 5f electric dipole cross sections. See Ref. [30] for details. Note that the relative 5f cross section total = 28/3. However, the 5f states are NOT completely empty: the 5fs are partially occupied. The cross sections in Table 1 can be combined with the partial occupation to derive relative intensities, I 5/2 and I 3/2 .
where N is the total number of 5f holes, N 5/2 is the number of holes in the 5f 5/2 manifold and N 7/2 is the number of holes in the 5f 7/2 manifold. Obviously, N = N 5/2 + N 7/2 and n = 14 − N, n 5/2 = 6 − N 5/2 and n 7/2 = 8 − N 7/2 , where n x is the corresponding number of electrons in each 5f manifold. If the appropriate filling pattern is used for each case, Equation (5) can generate the correct branching ratio for every case and equation in Table 1 in Reference [10], including the statistical, jj and intermediate cases as well as the experimental values. This then brings the discussion back to the degeneracy of the n = 2 localized case and the n = 3 delocalized case. While the two cases have significantly different occupations, they both have the same percentage un-occupations in the 5/2 and 7/2 manifolds of the 5f states. ( Table 2) This suggests that a technique that was dependent upon occupation, not un-occupations, would be able to differentiate the two cases. Such techniques would include X-ray Emission Spectroscopy between the 5f and 4d or 3d manifolds. That will be considered next. Values extracted from [2,10] and the equations above.

XES and 5f:5f Peak Ratios
It is possible to apply a parallel analysis to X-ray Emission Spectroscopy. The cross sections are closely related, as seen in Table 3, but now there is a partial occupation of the 5f states as before but only one hole in the d level. = 21n 5/2 (n 5/2 ) + (15n 7/2 ) Now, consider some limiting case behavior. In the ultimate limit, the material would be pure n 5/2 : n = n 5/2 , n 7/2 = 0 and PR = 21. This would require a jj coupling and has not been observed experimentally. Experimentally, UO 2 is an example of localized case with a strong spin-orbit splitting. For UO 2 , it is n = 2, n 5/2 = 1.96 and n 7/2 = 0.04, from Table 2 above. From Equation (8), the result would be PR (UO 2 ) = 16. From the data in Figure 5 [33], the ratio of the N4: N5 intensity is 9, although the situation is complicated by the presence a very strong satellite. (It is not clear that the satellite and main peaks would have the same cross sectional dependences.) Nevertheless, there is a qualitative, perhaps even semi-quantitative confirmation of Equation (8).
Another limiting case would be a small spin-orbit splitting with a statistical distribution of electrons in the f states. Ce metal is a good approximation to this limiting case, with n = 1, a small spin-orbit splitting and significant delocalization, which tends to mix the 4f 5/2 and 4f 7/2 states even further. (Ce is a 4f material and the transitions would be 4f → 3d, but the dependences should be the same. The 4f splitting of the Rare Earths is significantly smaller than the 5f splitting of the actinides [27]). The M 4,5 spectrum of Ce metal is shown in the inset in Figure 5 [34,35]. The predicted PR for the statistical distribution (n 5/2 /n 7/2 = 3 4 ) would be 1. As can be seen in the inset in Figure 5, the M 4 and M 5 intensities are approximately equal based upon peak heights.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 15 satellite and main peaks would have the same cross sectional dependences.) Nevertheless, there is a qualitative, perhaps even semi-quantitative confirmation of Equation (8).
Another limiting case would be a small spin-orbit splitting with a statistical distribution of electrons in the f states. Ce metal is a good approximation to this limiting case, with n = 1, a small spin-orbit splitting and significant delocalization, which tends to mix the 4f5/2 and 4f7/2 states even further. (Ce is a 4f material and the transitions would be 4f → 3d, but the dependences should be the same. The 4f splitting of the Rare Earths is significantly smaller than the 5f splitting of the actinides. [27]) The M4,5 spectrum of Ce metal is shown in the inset in Figure 5. [34,35] The predicted PR for the statistical distribution (n5/2/n7/2 = ¾) would be 1. As can be seen in the inset in Figure 5, the M4 and M5 intensities are approximately equal based upon peak heights. Figure 5. Main: presented here are the XES peaks for the N4,5 transitions in UO2. A peak fitting of the features with asymmetric Gaussian functions produces an area ratio of 8.9 between the sum of the two N5 peaks and the N4 peak, similar to the peak height ratio of 9 for these same features. Inset: the M4,5 XES of Ce metal is shown here.
These two limiting cases provide a significant confirmation of Equations (6)- (8), with an order of magnitude variation in the PRs of these two limits. On the other hand, as reported earlier [30], the measured peak ratio for the M4,5 XES of UF4 is only ~5, down a factor of two from the N4,5 results for UO2. Both UO2 and UF4 are highly localized, n = 2 cases [2] with BR values of 0.68 each and with the M5 XES spectra of UO2 and UF4 being very similar [30]. Thus, it is expected that the UO2 and UF4 results should be similar if not identical. However, the discrepancy between U N4,5 and U M4,5 results is not unanticipated. It is reasonable to expect that the U N4,5 (and Ce M4,5) situation would fall inside the long wavelength approximation used in the derivation of the electric dipole approximation [36,37]. The stunning success of the body of work based upon the application of the Intermediate Coupling Model to the experimentally measured N4,5 BRs supports this contention strongly [3,[10][11][12]. Nevertheless, the higher energies of the U N4,5 transitions mean shorter photon wavelengths, which translates into a larger impact for the higher order terms, such as magnetic dipole and electric quadrupole [30]. These two limiting cases provide a significant confirmation of Equations (6)- (8), with an order of magnitude variation in the PRs of these two limits. On the other hand, as reported earlier [30], the measured peak ratio for the M 4,5 XES of UF 4 is only~5, down a factor of two from the N 4,5 results for UO 2 . Both UO 2 and UF 4 are highly localized, n = 2 cases [2] with BR values of 0.68 each and with the M 5 XES spectra of UO 2 and UF 4 being very similar [30]. Thus, it is expected that the UO 2 and UF 4 results should be similar if not identical. However, the discrepancy between U N 4,5 and U M 4,5 results is not unanticipated. It is reasonable to expect that the U N 4,5 (and Ce M 4,5 ) situation Appl. Sci. 2020, 10, 2918 8 of 14 would fall inside the long wavelength approximation used in the derivation of the electric dipole approximation [36,37]. The stunning success of the body of work based upon the application of the Intermediate Coupling Model to the experimentally measured N 4,5 BRs supports this contention strongly [3,[10][11][12]. Nevertheless, the higher energies of the U N 4,5 transitions mean shorter photon wavelengths, which translates into a larger impact for the higher order terms, such as magnetic dipole and electric quadrupole [30].
Empirically, it is possible to correct for these higher order terms by adding a term to the numerator and denominator in Equation (8). Because the PR is being reduced and the denominator value is much smaller than the numerator, it is reasonable to begin by adding a corrective term to the denominator only.
Plugging in the values for UF 4 (PR = 4.6), the following value is found: a ≈ 6.4. Assuming that "a" is a constant, the following equation is obtained. PR = 21n 5/2 (n 5/2 ) + (15n 7/2 ) + 6.4 (10) For the two cases in Table 2, the simple model would predict these PRs: PR (UF 4 ) ≈ 5 and PR (U metal) ≈ 2.3. Obviously, it will be possible to easily distinguish these two cases with M 4,5 XES. However, owing to the large separation between the M 4 and M 5 edges in U, the spectra are collected separately and then normalized through the accompanying 6p XES, as can be seen in Figure 4. Thus, the best approach is to build the model around the 6p normalization. This issue will be addressed next.

XES and 5f:6p Peak Ratios
It is possible to calculate the electric dipole relative cross sections for the 6p → 3d transitions, following the same procedures as discussed earlier for the 5f → 3d transitions. The results are shown in Table 4 [30]. In this case, the 6ps are completely full and, because the 6ps should sit inside the 5f's [30], the electric dipole selection rules and cross sections should hold. One piece of information supporting this assertion is the absence of the 6p 1/2 peak in the M 5 spectrum in Figure 4, consistent with the cross section of zero in Table 4. For the 6p → 3d transitions, the ∆j = +1 transitions are the strongest and will be utilized for the normalization of the 5f intensities to the 6p intensities. At this point, it is possible to write out the relative intensity equations, including some constants for effects, such as the radial matrix elements.
Finally, substituting, the equations become as follows: Setting n = 2 and applying Equations (16) and (17), the results in Figures 6 and 7 were generated. Here, the spectra are normalized to a value of one at the 6p maximum, as shown below. Note the strong agreement of the experiment and the n 5/2 = 2 curve in each, as expected.  Setting n = 2 and applying Equations (16) and (17), the results in Figures 6 and 7 were generated. Here, the spectra are normalized to a value of one at the 6p maximum, as shown below. Note the strong agreement of the experiment and the n5/2 = 2 curve in each, as expected. Figure 6. Shown here is the experimental M 4 XES spectrum of UF 4 and the simulated spectra for n = 2 and various values of n 5/2 . The simulation used normalized Lorentzian line shapes, with a half-width at half-max of Γ. For the 6p 1/2 , Γ = 3.5 eV. For the 5f peak, Γ = 4 eV. The 5f/6p 3/2 peak is neglected in the simulation. Figure 6. Shown here is the experimental M4 XES spectrum of UF4 and the simulated spectra for n = 2 and various values of n5/2. The simulation used normalized Lorentzian line shapes, with a half-width at half-max of Γ. For the 6p1/2, Γ = 3.5 eV. For the 5f peak, Γ = 4 eV. The 5f/6p3/2 peak is neglected in the simulation. The real point of this exercise is not to confirm the analysis for the localized n = 2 case but rather to project what the n = 3 delocalized case would be. To that end, setting n = 3 and applying the equations again, the plots in Figures 8 and 9 were obtained.
Interestingly, the M4 spectrum corresponding to the n = 3, delocalized case (n5/2 = 2.1, Green or possibly n5/2 = 2.4, yellow) is not substantially different to the experimental spectrum for the n = 2 localized case, the UF4 result. This is because the n5/2 value for the delocalized n =3 case would only be about n5/2 = 2.23, not very different from the n = 2 localized case. (Table 2) One the other hand, the M5 projection shows a very clear and substantial change. The n5/2 = 2.1 (green) spectrum, and even the n5/2 = 2.4 (yellow), are very different to the experimental M5 spectrum for UF4. This is because the major change in the delocalization is to significantly populate the n7/2 states, which in turn is manifested in the M5 spectrum, not the M4. The real point of this exercise is not to confirm the analysis for the localized n = 2 case but rather to project what the n = 3 delocalized case would be. To that end, setting n = 3 and applying the equations again, the plots in Figures 8 and 9 were obtained.  Interestingly, the M 4 spectrum corresponding to the n = 3, delocalized case (n 5/2 = 2.1, Green or possibly n 5/2 = 2.4, yellow) is not substantially different to the experimental spectrum for the n = 2 localized case, the UF 4 result. This is because the n 5/2 value for the delocalized n =3 case would only be about n 5/2 = 2.23, not very different from the n = 2 localized case. (Table 2) One the other hand, the M 5 projection shows a very clear and substantial change. The n 5/2 = 2.1 (green) spectrum, and even the n 5/2 = 2.4 (yellow), are very different to the experimental M 5 spectrum for UF 4 . This is because the major change in the delocalization is to significantly populate the n 7/2 states, which in turn is manifested in the M 5 spectrum, not the M 4 . Figure 8. Shown here is the experimental M4 XES spectrum of UF4 and the simulated spectra for n = 3 and various values of n5/2. The simulation used normalized Lorentzian line shapes, with a half-width at half-max of Γ. For the 6p1/2: Γ = 3.5 eV. For the 5f peak, Γ = 4 eV. The 5f/6p3/2 peak is neglected in the simulation. Figure 9. Shown here is the experimental M5 XES spectrum of UF4 and the simulated spectra for n = 3 and various values of n5/2. The simulation used normalized Lorentzian line shapes, with a half-width at half-max of Γ. For the 6p3/2, Γ = 3.5 eV. For the 5f peak, Γ = 4 eV.

Summary and Conclusions
By using M 4,5 X-ray Emission Spectroscopy (XES) in the tender X-ray regime, it has been shown that it is possible to quantify 5f delocalization in the actinides. Previous analyses, utilizing the Branching Ratio (BR) in the N 4,5 X-ray Absorption Spectroscopy (XAS), could not discriminate between the cases of localized n = 2 and delocalized n = 3, in uranium materials, where n is the number of 5f electrons on the U entity. Here it is shown that, by employing the ubiquitous 6p XES as a point of normalization, the localized n = 2 and delocalized n = 3 cases can be easily distinguished and quantified via the M 5 X-ray emission spectrum.
As can be seen from the Introduction, XAS, XES, BIS and XPS each provide different types of information. Of course, all spectroscopies provide a measure of the joint density of states of the initial and final manifolds, connected by the appropriate operator. BIS and XPS can give a picture of the unoccupied and occupied DOS of a material, respectively, generally without elemental specificity but possibly with linear (translational) momentum resolution [38][39][40]. On the other hand, XAS and XES provide information about the unoccupied and occupied DOS, respectively, but with elemental and generally angular momentum resolution. In the case of the 5f states, the XES and XAS of the d-f transitions can provide a measure of the 5f state occupations, including a breakdown into the 5f 5/2 and 5f 7/2 distributions. The discussions today have been made within the framework of the n = 3 delocalization problem for metallic U. However, there are also other perturbations that can mix the 5f 5/2 and 5f 7/2 states, e.g., magnetic effects [41] and crystal field effects [42]. XES and XAS together should produce a measure of the magnitude of the number and distribution of the 5f electrons, but to assign the distribution as being due to delocalization will require a demonstration that other mixing effects are not significantly in effect. Obvously, all of this is played out under the conditions of the competition between angular momentum and delocalization effects, i.e., 5f duality [10,43,44].