Resonant inelastic x-ray spectroscopy on UO2 as a test case for actinide materials

Resonant inelastic x-ray spectroscopy at the uranium N4 absorption edge at 778 eV has been used to reveal the excitations in UO2 up to 1 eV. The earlier (1989) studies by neutron inelastic scattering of the crystal-field states within the 3H4 multiplet are confirmed. In addition, the first excited state of the 3F2 multiplet at ∼520 meV has been established, and there is a weak signal corresponding to the next excited state at ∼920 meV. This represents a successful application of soft x-ray spectroscopy to an actinide sample, and resolves an open question in UO2 that has been discussed for 50 years. The technique is described and important caveats are drawn about possible future applications.


Introduction
The large 5f spin-orbit parameter in the actinides (ζ 5 f ≈ 0.23 eV for uranium) implies that the appropriate manyelectron interaction requires the intermediate coupling scheme, in which J is still a good quantum number, but L and S are not [1]. The Hamiltonian to describe the properties must contain, in addition to the spin-orbit coupling, the Slater integrals describing the Coulomb and exchange interactions. These interactions are normally scaled from their free-atom * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. values by a Slater reduction factor g f f due to the interactions in the solid. Finally, the crystal-field (CF) interaction must be included to account for the site symmetry of the actinide ion in the lattice and the electrostatic interactions with its neighbors. Depending on these parameters, the level structure of the actinide ion is stretched over several eV. These spectral features were first observed by optical absorption measurements and can be seen in the pioneering work of Carnall and Wybourne [2] on 5 f 3 , and work on 5 f 2 was reported later [3]. These optical techniques fail when the system is metallic, although they give a rough range for the excited states. In the U 4+ (5 f 2 ) ionic uranium compounds, the ground state is a 3 H 4 multiplet, and the excited states 3 F 2 and 3 H 5 are expected to lie at about 540-640 meV and 750-850 meV, respectively. In the U 3+ (5 f 3 ) case there is considerable mixing of L Comparison of calculated and experimental energies (in meV) of excited states up to ∼1 eV in UO 2 . The energy of the Γ 1 level given in parenthesis is a theoretical value, as the Γ 5 → Γ 1 transition has a vanishing intensity for inelastic neutron scattering. The values of the CF parameters V 4 and V 6 (in meV) are given for each model. For the reader's convenience the irreducible representations of the CF levels are given in Bethe's Γ notation as well as Mulliken notation (A 1 , E, T 1,2   and S values from intermediate coupling, so that a pure spectroscopic state becomes less meaningful. The J mixing is relatively small and the first excited states are J = 11/2 and J = 13/2 manifolds, respectively lying at about 500 and 940 meV above the 4 I 9/2 ground state. The range of values arises because of the CF splitting of the various ground states, and, of course, will vary for different compounds. In general, it is known that the CF interactions are larger for the ionic like compounds, since in metallic systems the screening by the conduction electrons reduces the CF potentials. The effective Coulomb and exchange interactions are also reduced in metallic systems. The CF states are a unique fingerprint of the ground J state, and for metallic rare-earth materials neutron spectroscopy has been the tool of choice for the last 40 years [4]. Unfortunately, with some rare exceptions, such as UPd 3 [5], studies of metallic actinide systems with neutron scattering have a poor record of establishing the CF arrangement [6]. This is thought to be related to the strong hybridization of the 5 f and conductionband states, which washes out [7] the CF states so that they can no longer be observed. A relatively new technique of nonresonant inelastic x-ray scattering (NIXS) needs the detailed electronic structure to fit the x-ray data, so is a useful tool to understand the symmetry of the ground-state wave-functions, for example in URu 2 Si 2 [8], and is also sensitive to the strength of the CF, as recently demonstrated in UO 2 [9]. However, none of these methods provide direct information of the excited-state levels.
At higher energies (up to ∼1 eV) the technique of neutron scattering can observe intermultiplet transitions, and studies of the rare-earth metals, e.g. have been very successful [10,11]. Efforts have been made on uranium systems UPd 3 [12], UPt 3 [12], and USb [13]. Later, problems of noise at high energies were found with one of the spectrometers at the ISIS neutron source [14]. However, the results for UPd 3 (but not UPt 3 ) were verified with a transition 3 H 4 → 3 F 2 at 395 meV. A similar transition was also found in URu 2 Si 2 [15]. Experiments on systems with light elements (such as present in UO 2 ) are handicapped by the strong multiphonon background present at these higher energies and the small non-dipolar cross section for the 3 H 4 → 3 F 2 transition. Further experiments have not been reported in the last 20 years or so.
Recently, the technique of resonant inelastic x-ray scattering (RIXS) has been advanced to achieve resolutions of <50 meV, so it can be considered an additional technique to examine the actinides. Examples on Ce compounds [16,17] have shown considerable promise up to energy transfers of ∼0.5 eV, by using the Ce M 5 absorption edge at 882 eV as incident energy, and with a resolution of 30 meV [17]. We present in this paper such experiments at the uranium N edge in uranium dioxide (UO 2 ). UO 2 is probably the most studied of any actinide compound with extensive work going back to the 1960s. The first attempt to calculate the spectroscopic levels was by Rahman and Runciman [18] in 1966. This was followed by a study with optical techniques in 1980 [19]. In 1989 a neutron inelastic scattering experiment was able to unambiguously establish [20,21] the CF transitions within the 3 H 4 ground state, but could not observe the first excited multiplet state at higher energies. The CF parameters were confirmed by Nakotte et al using inelastic neutron scattering [22]. A Raman scattering investigation was reported by Livneh in 2008 [23]. In 2016 a new theoretical study was reported [24]. Recently, we completed a NIXS study of a single crystal of UO 2 , and the values giving the best fit [9] to the observed spectra are listed, along with those from references [18][19][20]24], in table 1.
There are considerable differences in these values in table 1, which is perhaps surprising considering the extent of our knowledge of UO 2 [25,26], but illustrates the general uncertainty about the excited energy levels in all light actinide systems.

Experimental details and results
The UO 2 sample used was an atomically flat epitaxial film of thickness ∼100 nm deposited on a substrate of yttriumstabilised zirconia with the growth axis [001]. The sample mass was ∼100 μg, which is almost six orders of magnitude smaller than the sample of 80 g used for the 1989 neutron experiments [20]. After fabrication of the sample in the sputtering chamber at the University of Bristol [27] it was transferred in a 'vacuum suitcase' to the Diamond Light Source. By using an interlock system, the sample was transferred into the vacuum chamber of beamline I21 [28] without any contact with air. At an incident x-ray energy of 778 eV (λ ≈ 16 Å) the 1/e attenuation length into UO 2 at an angle of incidence of ∼ 20 • is given by Henke et al [29] as 40 nm. However, these calculations do not take account of the strong white-line resonance at the N 4 absorption edge, common for soft x-ray energies [30], so the effective penetration is certainly far less than 20 nm, probably just a few nm, making this a surface sensitive experiment. This demand of a high-quality surface is a major disadvantage in the technique, and the role this plays in assessing the overall power of the technique is, as yet, challenging.
The RIXS experiment was performed by tuning to the uranium N 4 absorption peak (4d 3/2 → 5 f 5/2 transition), which according to our calculation and experiment has a higher intensity than the N 5 peak. The resolution at this incident energy was 35 meV. The sample temperature was held at 15 K, i.e., below the transition temperature T N of UO 2 [25]. The possible variation of the signal with temperature and with variation of the scattering vector (i.e., any dispersion) were only briefly examined, but no significant changes were observed. The results of the 16 h runs on the instrument are shown in figure 1. The spectra very clearly show the CF excitations peaked around 180 meV, the first excited multiplet of 3 F 2 at 520-580 meV, and a weak signal at ∼920 meV corresponding to the excited state multiplet of 3 H 5 .

Calculation
The N 4 edge RIXS is a photon-in photon-out process, 5 f n + hν 1 → 4d 9 5 f n+1 → 5 f n * + hν 2 , in which an electron in the 4d 3/2 core shell is promoted to the unoccupied 5 f 5/2 subshell of uranium, after which a 5 f electron can decay back to the 4d core shell, so that it carries the spectral information of the 5 f excited states.
For better interpretation of the spectra and extraction of the CF parameters, we performed a full-multiplet calculation using the Kramers-Heisenberg formulation with Quanty [31], a simulation code that includes Coulomb and spin-orbit interactions. The calculation is based on a U 4+ ion with two 5 f electrons. In intermediate coupling the ground state is J = 4 in the cubic CF potential defined by the parameters V 4 and V 6 , as described in reference [9]. The atomic parameters are calculated using Cowan's atomic multiplet code [32] and the 5 f -5 f (4d-5 f) Coulomb interactions are reduced to 50% (80%) to account for intra-atomic relaxation effects. Resonant spectra are calculated from a third order Green's function [31]: with T and H operators given in second quantization and ψ i being a many particle wavefunction. At the N edge, T 1 excite a 4d core electron into a 5 f empty state and T 2 de-excite a 5 f electron into the 4d core hole. The Slater parameters for the initial and final states have been set to F 2 = 9.5 eV, F 4 = 6.2 eV, and F 6 = 4.6 eV. For the intermediate XAS state these values have been increased by about 5%. The atomic Coulomb interaction was reduced to 50% (80%) for 5 f -5 f (4d-5 f). As shown in table 1, the obtained set of parameters shows good agreement with the neutron and NIXS study. The calculated spectra, shown by red lines in figure 1, reproduce well both the energies and the polarization dependencies of the experimental results. Note, that the splitting of the CF states in the 3 H 4 and 3 F 2 multiplets are related by the same parameters V 4 and V 6 hence limiting the quality of the fit.

Discussion
RIXS is described as consisting of two radiative transitions, absorption and emission, each one of which is regulated by the dipole-selection rules. The total RIXS transition, however, does not follow these rules. f f excitations provide a typical example: the transition 5 f → 5 f would be forbidden by dipole, since Δ = 0, but the two steps 5d → 5 f (Δ = +1) and 5 f → 5d (Δ = −1) are allowed, making the whole transition possible. RIXS can therefore access transitions forbidden by dipole selection rules.
As each step follows the selection rules ΔJ = 0, ±1, the RIXS process allows also ΔJ = ±2. In fact, we observe a strong excitation for ΔJ = 2, for 3 H 4 → 3 F 2 at ∼520 meV, but a weak one for 3 H 4 → 3 H 5 at ∼920 meV (a slightly stronger signal, shown in figure 1 inset, was observed at this position when the incident energy was tuned at the N 5 edge at 736 eV). This is exactly the opposite of the situation with neutrons, where the first excited level is non-dipolar, so is predicted to be weak [13,14], and the second excitation should be stronger; however, energy transfers up to 1 eV face difficulties due to the restrictions of the form factor in neutron studies [11]. The resonant process has no form factor, and x-rays are not subject to the kinematical restrictions present in neutron scattering due to the finite mass of the neutron. On the other hand, only the 3 H 4 and 3 F 2 multiplets are expected to yield a strong RIXS signal at the N 4 edge (4d 3/2 ), because only transitions into 5 f 5/2 are allowed, but not into 5 f 7/2 . The other multiplets, however, do not necessarily give vanishing contributions to the spectra due to the Coulomb interaction. Experimentally, this is confirmed by the observation of the transition to the 3 H 5 being stronger at the N 5 energy (see inset of figure 1) than at the N 4 shown in figure 1.
RIXS has also been shown to be very powerful in mapping dispersive excitations, especially of high-T c materials [33] and systems in which the dispersion relationships exceed ∼100 meV [34]. Such studies are rare in the actinide systems, e.g. in UO 2 the magnetic excitations do not exceed 20 meV [35], but in some systems, such as UFe 2 they may indeed continue to high energy [36].
We emphasize that this is an experimental paper showing how the technique of RIXS may allow further understanding of the electronic parameters in actinide systems. A more realistic model for UO 2 including configuration interaction would capture the electron-electron interactions to some extent and need less reduction of the 5 f -5 f Coulomb interactions. Such calculation (not shown) appears to differ at most by a larger energy splitting in the 3 F 2 multiplet and explains the deviation we find with the ionic model. In table 1 we therefore also list the experimental peak positions, which have well separated 3 F 2 states in the data. The results from the optical work of Schoenes [19] are difficult to assess because of the multiphonon contributions. Although this reference claims the work supports the energies given by reference [18], it clearly does not. However, it does support (by the observation of a major line at 525 meV) later energy estimates of the 3 F 2 excitation. The neutron work of reference [20] has a limited range (<800 meV) of energy, but the values of the CF excitations are given in bold as they are reliable (±3 meV).
Two caveats should be stressed. (i) Success with UO 2 does not necessarily imply that the technique will work for metallic actinide compounds. The question of how the hybridization between the 5 f states and those of the conduction electrons affects the observation of ff excitations is not yet answered for the actinides. Indeed, in recent experiments (at the same beamline) on a localized system UNi 2 Si 2 [37] no excitations were observed. The RIXS technique does work for Ce metallic systems [16,17], but the hybridization of the 5 f states is more complicated than that of the 4 f states in cerium. There are also difficulties in observing CF transitions in actinide intermetallic systems [7] with neutron scattering. (ii) The RIXS technique is exceedingly surface sensitive, probably at the level of a few nm, so that excessive care has to be taken in sample preparation. The UO 2 sample used here was atomically flat and was never exposed to air. Whether such care has to be taken for the technique to be successful remains an open question.

Conclusions
Our experiments have elucidated the higher 5 f related excitations in UO 2 that have been a source of speculation for at least 50 years. The use of soft x-ray spectroscopy (RIXS) thus opens up the actinides to such studies, in the same way that it has allowed studies of cerium compounds [16,17], and the observation of dispersion at high-energy transfers in transitionmetal systems [33,34]. The technique has confirmed the neutron experiments on the CF levels on UO 2 performed 30 years ago [20], and extends that study to clearly show the ff level structure up to 1 eV, which is difficult with neutron scattering. The reported excitations in UPd 3 (at 395 meV) [14] and URu 2 Si 2 (at 363 meV) [15] still remain to be understood. The ∼25% reduction of these excitations from UO 2 seems a very large amount. Further experiments and theory will be needed to understand these differences, and their significance to fundamental parameters describing materials with 5 f electrons.