# Towards the Quantification of 5f Delocalization
## 3 Results and Discussion
### 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. From these, it is possible to calculate relative intensities and a Peak Ratio (PR). \[I_{5f}^{N5}=\frac{n_{5/2}}{6}\bigg{(}\frac{4}{90}\bigg{)}+\frac{n_{7/2}}{8} \bigg{(}\frac{16}{18}\bigg{)} \tag{6}\]  
\[I_{5f}^{N4}=\frac{n_{5/2}}{6}\bigg{(}\frac{56}{60}\bigg{)} \tag{7}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **n** & **BR** & **n\({}_{5/2}\)** & **n\({}_{7/2}\)** & **N** & **N\({}_{5/2}\)** & **N\({}_{7/2}\)** & **N\({}_{5/2}\)/N** & **N\({}_{7/2}\)/N** \\ \hline Int. Coupling, & \multirow{2}{*}{2} & \multirow{2}{*}{0.68} & \multirow{2}{*}{1.96} & \multirow{2}{*}{0.04} & \multirow{2}{*}{12} & \multirow{2}{*}{4.04} & \multirow{2}{*}{7.96} & \multirow{2}{*}{0.337\(\sim\)0.34} & \multirow{2}{*}{0.663\(\sim\)0.66} \\ UO\({}_{2}\) and UF\({}_{4}\) & & & & & & & & \\ \hline U metal & 3 & 0.68 & 2.23 & 0.77 & 11 & 3.77 & 7.23 & 0.343\(\sim\)0.34 & 0.657\(\sim\)0.66 \\ \hline \end{tabular}
\end{table} 
Table 2: XAS branching ratios and 5f populations. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & & \multicolumn{2}{c}{5f\({}_{5/2}\)} & \multicolumn{2}{c}{5f\({}_{7/2}\)} & \multirow{2}{*}{**5f\({}_{5/2}\)**} & \multirow{2}{*}{**5f\({}_{5/2}\)**} \\  & & **Full** & **Full** & & & & **Full** \\  & & **(Empty)** & **(Empty)** & & & & & \\ \hline N\({}_{5}\) (M\({}_{5}\)) & d\({}_{5/2}\) & \multirow{2}{*}{\(\frac{4}{15}\)} & \multirow{2}{*}{\(\frac{16}{3}\)} & d\({}_{5/2}\) & \multirow{2}{*}{\(\frac{4}{90}\)} & \multirow{2}{*}{\(\frac{16}{18}\)} \\ (Fully) & & & & & & & & \\ \hline N\({}_{4}\) (M\({}_{4}\)) & d\({}_{3/2}\) & \multirow{2}{*}{\(\frac{56}{15}\)} & \multirow{2}{*}{0} & d\({}_{3/2}\) & \multirow{2}{*}{\(\frac{56}{60}\)} & \multirow{2}{*}{0} \\ (M\({}_{4}\)) & (Full) & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 3: Shown here are the 5f Electric dipole cross sections. Note that (1) the total 5f cross section is 28/3; and (2) the 5f cross section per 3d hole is 14/15. Leftmost are the results corresponding to completely empty 3d states. Rightmost are the results per 3d hole. See text for details. \[PR=\frac{I_{5f}^{\text{N4}}}{I_{5f}^{\text{N5}}}=\frac{\left(\frac{n_{5/2}}{6} \right)\left(\frac{56}{60}\right)}{\left(\frac{n_{5/2}}{6}\right)\left(\frac{4} {50}\right)+\left(\frac{n_{7/2}}{8}\right)\left(\frac{16}{18}\right)}=\frac{21n _{5/2}}{\left(n_{5/2}\right)+\left(15n_{7/2}\right)} \tag{8}\]  
Now, consider some limiting case behavior. In the ultimate limit, the material would be pure n5/2: n = n5/2, n7/2 = 0 and \(PR=21\). This would require a j coupling and has not been observed experimentally. Experimentally, UO2 is an example of localized case with a strong spin-orbit splitting. For UO2, it is n = 2, n5/2 = 1.96 and n7/2 = 0.04, from Table 2 above. From Equation (8), the result would be \(PR\) (UO2) = 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 4f5/2 and 4f7/2 states even further. (Ce is a 4f material and the transitions would be 4f \(\rightarrow\) 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].