Application of FEFF analyses to actinide 5f systems Free

FEFF is a Green’s function based platform for spectral simulations, which has been shown to be an accessible, robust, and very effective tool for the analysis of x-ray measurements.^1–4 More specifically, FEFF is a data analysis program used in x-ray absorption spectroscopy and related techniques, including self-consistent real space multiple-scattering code for simultaneous calculations of x-ray absorption spectra and the electronic structure, particularly, for extended x-ray absorption fine structure (EXAFS). FEFF’s origins trace directly to the need to quantify the elastic electron scattering events in the EXAFS final state, which, in turn, gives rise to sensitivity of the technique to interatomic distances. In the case of actinides, the application of FEFF has been long standing and effective. For example, the EXAFS analysis of Pu utilized a FEFF simulation of EXAFS oscillations,⁵ and spectral simulations of the Pu L₃ near edge structure [x-ray absorption near edge structure (XANES)] were used to analyze Pu hydrates.⁶ Part of the allure of FEFF is its accessibility to the novice. Consider the results shown in Figs. 1 and 2. With only a fairly small amount of effort, it was possible to construct the models and generate the simulations, which mirror the earlier results in Refs. 5 and 6.

In Fig. 1, it is essentially a trivial exercise to generate the EXAFS oscillations shown. These results are in qualitative and even semiquantitative agreement with the earlier work by Cox et al.⁵ To get exact, quantitative agreement, further refinement would be required. Although a trivial exercise to generate these results with FEFF, it was not a trivial exercise to build the platform that allows the novice to succeed so easily. In any case, the essential physics is captured by FEFF: The backscattering of the electrons causes interference patterns in the electron wave-functions and thus cross sections, which then provide a direct measure of nearest neighbor distances.

Figure 2 shows a similar success story. Here, a particularly simple model is used: a PuO₂ molecule, which is an analog of CO₂, with Pu replacing C and the Pu-O distance set at 2.3 Å, which was taken from the bulk value. As can be seen, there is essentially quantitative agreement between the FEFF calculation and the total fluorescence yield (TFY) results. The higher resolution partial fluorescence yield (PFY) is in qualitative agreement. The broadening in TFY is due to the core-hole lifetime; the same broadening that is also there when XANES is measured in a transmission mode, where no fluorescence is involved. However, when detecting x-ray emission with high resolution in the PFY mode, this detection mode enables the suppression of core-hole lifetime broadening in the XANES spectrum. The experimental results were taken from Ref. 7. Further discussion of TFY, PFY, x-ray emission spectroscopy, and XANES is available in Ref. 7.

However, there is a situation in which FEFF does not fare so well. An example of this is shown in Fig. 3. Here, the experimental results for N_4,5 x-ray absorption spectroscopy (XAS) for Pu and U are compared to the FEFF predictions of XAS from the model shown in Fig. 1. Obviously, something is significantly wrong. Qualitatively, the Pu FEFF is similar to the U spectrum but does not match the Pu spectrum.⁸ Quantitatively, the branching ratios (BRs) are in disagreement [BR = I4d_5/2/(I4d_5/2+I4d_3/2)]. U BR = 0.68 and Pu BR = 0.82,^9–14 while Pu FEFF BR = 0.63.

Interestingly, it is useful to try to find a case that experimentally matches the theoretical situation of the Pu FEFF model. One might argue that Ce is such a case. The smaller spin-orbit splitting of the 4f states, relative to that of the 5f states, combined with 4f delocalization and mixing of the 4f_5/2 and 4f_7/2 components, generates a situation close to the operational configuration in the FEFF calculations. That comparison is shown in Fig. 4. While there is some qualitative agreement, the details of angular momentum coupling in the Ce¹⁵ produce a spectrum with far more structure and other complications.

In Fig. 5, the occupied and unoccupied partial densities of states (DOS) for the Pu crystal are presented. While the central atom (diamonds) is the source of spectroscopy, the surrounding atoms (solid lines) provide the environment. Nevertheless, it is very reassuring that the L-specific DOS for the central atom and surrounding atoms agree so well. Note that the Fermi edge region looks metallic, with Fermi cutoff going through 5f, 6d, 7p, and 7s non-zero density. The Pu 6p core level is also present near E = −28 eV. However, it appears that the 5f occupation in the FEFF calculation is substantially less than the expected n = 5. (The Fermi energy can be treated as an adjustable parameter in FEFF. That has not been done here.) While this has impact on simulation of the Pu N_4,5 spectra via the distribution between j = 5/2 and j = 7/2 manifolds, the effect of the magnitude by itself is lessened somewhat because both are Pu 5f. However, the reduction in the number of 5f electrons (n) will have an important impact when 5f peaks are compared to 6p peaks. Returning the issue of the branching ratio associated with Pu N_4,5 transitions, it is noteworthy that there is only one 6p peak in the Pu DOS in Fig. 5. One can argue that this is not unreasonable. For example, in a final state picture of spectroscopic transitions, the 6p level in the ground state configuration in the initial state is fully occupied, and therefore, all 6p electrons should be degenerated. That is, the 6p_3/2–6p_1/2 splitting happens once an electron is removed out of the previously fully occupied 6p state. Experimentally, of course, there are two peaks, 6p_3/2 and 6p_1/2, near −20 and −30 eV, respectively.^16–19 Other theoretical approaches provide DOS estimates and transition probabilities that include 6p_3/2–6p_1/2 splitting: For example, Ryzhkov’s models show two separate features of the actinide 6p’s in UO₂, UF₄, and Pu.^20–23 However, as noted in the FEFF manual, FEFF calculates only L-resolved DOS and no other quantum number can be taken into consideration at this time. Thus, although FEFF uses the correct relativistic electric dipole selection rules and cross sections, the j-specificity is lost. Without j-specificity, the cross sectional effects are averaged, consistent with a 5f BR value of near 0.6, the statistical value, which reflects the initial state occupation of the 4d states: BR_Stat = 6/(6 + 4) = 0.6.

The situation in Fig. 6 is slightly different. The O2s (heavy blue line, near E = −25eV) overlap with the Pu states, particularly, the Pu 6p (green diamonds with light line) and the O2p (heavy green line, near E = −10 eV) overlay the Pu 5f (red diamonds with thin line), 6d (brown diamonds with thin line), and 7s (blue diamonds with thin line). Note the Fermi energy is at −9.3 eV in this case. Here, the separation between the manifold of states near E = −10 eV and the manifold near E = −3 eV has the appearance of an oxide bandgap. It appears that the distribution of peaks in the manifold of states near E = −30 eV is not driven by spin-orbit splitting but rather the boundary conditions of the small PuO₂ molecule (i.e., one-dimensional particle in a box) and overlap of the Pu and Oxygen states. It is important to note that while the L-DOS will manifest such details, they will be far more difficult to see in the calculated spectra. The combination of lifetime and instrumental broadenings will often mask such effects. Finally, once again, the 5f population appears to be significantly lower than that expected in bulk PuO₂ (n = 4–5).²⁴ 

One aspect of the FEFF calculations which is readily apparent is the strong mixing of states. Compared to experimental XPS measurements, there is too much mixing of the core levels.^16–19 In those studies, there are distinct Pu6p_1/2, Pu6p_3/2, and O2s peaks, with a spread of over 10 eV. The overmixing is also historically a problem with the 5f states. The successful calculation of the correct volumes for the six solid phases of Pu was not achieved until 2004, when Söderlind and Sadigh²⁵ calculated the volumes using an artificial magnetic ordering. This magnetic ordering is a fair approximation for the strong spin-orbit splitting later utilized by Kutepov and co-workers.⁹ Without the application of either magnetic ordering or strong spin-orbit splitting, the 5f states have a propensity to overmix and overdisperse. This overmixing is also present in FEFF.

In many respects, the observations herein show that FEFF is consistent with its origins. Since FEFF is based on the elastic scattering amplitude f_eff, it is not surprising that FEFF works very well for EXAFS, quite well for the L₃ spectrum, and not as well for the N_4,5 spectra. The EXAFS processes are dominated by elastic scattering of the electrons off of nearby neighboring atoms. This matches up strongly with FEFF’s foundation. The L₃ spectrum includes both elastic and inelastic events. While the white line peak at threshold is an inelastic event with the absorption of a photon, the next main feature is the first EXAFS peak, as discussed in detail elsewhere.²⁶ Finally, the N_4,5 branching ratio focuses entirely upon the inelastic x-ray absorption white line peaks and is thus the most removed from FEFF’s origins. While the impact of this effect is to limit the utility of FEFF for the analysis of N_4,5 branching ratios, it is clear that FEFF can still make important contributions in x-ray spectroscopy of the actinides, such as described above and in 5f XES.^27,28

Regarding the N_4,5 spectra, the lack of angular momentum coupling is responsible for the failure of FEFF to obtain accurate branching ratios. Qualitatively, the spectra generated may work for the early actinides, U and before, but will fail in the later actinides, e.g., Pu and Am. While this angular momentum coupling can be treated in various ways, approximate calculations have been done in FEFF dynamic screening models of the transition matrix elements.¹ Perhaps it might be useful to consider such an approach for actinides in future work. However, care must be taken to ensure that the underlying physics is accurately represented.

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