DFT+U investigation of local configurations and oxidation states of Cr in Cr-doped UO₂

Abstract

Doping UO₂ with Cr modifies the material’s microstructure, enhancing its properties and making Cr-doped UO₂ a promising candidate as accident-tolerant nuclear fuel (ATF). Numerous studies have examined the oxidation state and localization of Cr in UO₂ but often yield inconsistent results, identifying either Cr²⁺ or Cr³⁺ as the most stable oxidation state. In the present study, DFT+U is employed to model the incorporation of Cr in the UO₂ matrix, providing insights into the oxidation state of Cr in UO₂, in relation to the local atomic configurations. In particular, we investigate the ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$${{{\rm{U}}}}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x local configuration recently proposed by EPR and XANES experiments, alongside other theoretical configurations. Cr³⁺ is found to be the most favorable oxidation state in this configuration, agreeing with the most recent experimental data. This work clarifies the controversy over Cr oxidation states and incorporation sites within UO₂, offering critical data for developing efficient and safer nuclear fuels.

Introduction

To continuously improve the performance and economic efficiency of nuclear fuel, research over the past several decades has focused on optimizing UO₂-based fuels for higher burnup, better manufacturability, and improved in-reactor behavior. One particularly effective strategy has been the addition of dopants during sintering to tailor the microstructure of UO₂ pellets and thus improve the physical properties of the fuel. Doped fuels were developed primarily driven by industrial needs, including more stable pellet fabrication, enhanced sintering kinetics, and improved mechanical performance at high burnup^1,2,3,4,5,6. These developments aimed to enable longer reactor residence times and more flexible reactor operation, while maintaining fuel integrity. More recently, after the Fukushima Daiichi nuclear accident, these doped fuels have attracted interest within the broader context of accident-tolerant fuels (ATFs). An ATF should be able to withstand severe accident conditions for longer durations than the UO₂ fuel currently used in most water reactors, while demonstrating similar performance under nominal operating conditions. In the short term, doping has the advantage of minimally affecting the industrial manufacturing process of the pellets⁷. Using modified UO₂ should also facilitate the qualification processes with nuclear safety authorities compared to other innovative fuel concepts⁷.

Among the various dopants investigated over the past decades (Cr, Mg, Nb, Ni, Ti, Al, Si, Mn, V, Fe, and their oxides^8,9,10), Cr and Cr₂O₃ have shown particular benefits. The addition of Cr results in a significant increase in the average grain size, reaching over 100 μm in UO₂-Cr compared to 10–20 μm in undoped UO₂^6,10,11. This grain size enhancement leads to improved plasticity of the fuel^12,13, which helps moderate the pellet-cladding interaction by allowing the pellet to deform more easily, thus reducing stress on the cladding. Additionally, for Cr concentrations near the solubility limit, improved fission gas retention has been observed under power ramping conditions¹². Consequently, UO₂-Cr fuels can achieve higher burnup rates⁵.

Over the past fifteen years, numerous studies have investigated the speciation of Cr in UO₂ to understand its role in grain growth. In 2014, ref. ¹⁴ proposed a thermochemistry solubility model for Cr in UO₂, highlighting that Cr solubility increases with its oxidation state in the solid phase, with Cr³⁺ being the most soluble. X-ray absorption near edge structure (XANES) by Riglet-Martial et al., and later, extended X-ray absorption fine structure (EXAFS) measurements by ref. ¹⁵ further confirmed the +3 oxidation state for Cr in UO₂ and showed that Cr substitutes uranium in an octahedral environment surrounded by six oxygen atoms.

These experiments support the simulation results obtained with empirical interatomic potentials by ref. ⁸ and ref. ¹⁶, both identifying uranium vacancies as the most favorable incorporation site for Cr in UO₂. Cardinaels et al. showed that in UO_2−x, charge neutrality is favorably maintained by pairs of Cr atoms bound to one oxygen vacancy, while in UO_2+x, configurations with U⁵⁺ ions become more stable⁸. This result was confirmed by Middleburgh et al. with Hubbard-corrected density functional theory (DFT+U) calculations, in which U⁵⁺ ions were shown to accommodate Cr³⁺ in UO_2+x¹⁶. In 2017, ref. ¹⁷ used a modified Morelon interatomic potential to study Cr doping effects on point defects, finding that the most stable configuration involved oxygen vacancies located in the first neighbor site of two Cr atoms substituting U, consistent with ref. ⁸. Cooper et al.⁹ combined empirical interatomic potentials and DFT+U calculations to study dopants in UO₂. Their study shows that Cr³⁺ substituting U, in an octahedral environment, is predominant at low temperature. However, at high temperature, Cr³⁺ transitions to Cr⁺ in an interstitial position.

In 2020, DFT+U calculations by ref. ¹⁸ challenged the consensus on Cr’s preference for the +3 oxidation state in UO₂, suggesting that Cr incorporates as Cr²⁺ in a U-O divacancy. Based on these results, Sun et al. also questioned the 2014 XANES spectra interpretation by ref. ¹⁴. They compared the pre-peak positions to a different Cr²⁺ reference and found that in Cr-doped UO₂, the second pre-peak at 5993 eV indicated a +2 oxidation state¹⁹. Note that in their DFT+U study, Sun et al. used different U parameters for Cr²⁺ and Cr³⁺. In general, the DFT+U studies by ref. ¹⁸, ref. ⁹, and ref. ¹⁶ each employed different U parameters. These differences in U parameters are one possible cause of the divergent predictions observed in these studies, with Cooper et al.⁹ and Middleburgh et al.¹⁶ predicting Cr³⁺, while Sun et al.¹⁸ predicted Cr²⁺.

In 2022, Smith et al. conducted XANES and EXAFS measurements at the Cr K-edge on commercial Cr₂O₃-doped UO₂ powders²⁰. They confirmed the presence of Cr²⁺ and Cr³⁺ in both calcined and sintered samples, with the average valence of Cr being 2.35+. The authors proposed a tentative explanation of the discrepancies observed in previous studies by suggesting Cr would be in a mixture of Cr²⁺ and Cr³⁺. In the sintering scenario proposed by the authors, Cr would be mainly in the form of Cr²⁺, dissolved in U vacancies, accompanied by O vacancies. For a material with a certain Cr concentration above the solubility limit, Cr would also be present in the form of Cr²⁺ or Cr³⁺ in interstitial positions, easily mobile during sintering to diffuse towards grain boundaries.

Finally, in 2023, ref. ²¹ conducted electron paramagnetic resonance (EPR) and XANES/EXAFS measurements at the Cr K-edge on single mechanically extracted grains from a Cr-doped UO₂ sample as well as a polycrystalline bulk sample. In the single grains, Cr was found only as Cr³⁺, substituting U and accompanied by O vacancies. EPR and XANES spectra at the U M₃ and L₃ edges ruled out the presence of U⁵⁺ ions. Furthermore, according to Murphy et al., EPR measurements could not be explained by the formation of Cr twin pairs (such as those suggested by the calculations by Guo et al.¹⁷ and Cardinaels et al.⁸). Thus, Murphy et al. proposed a ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$${{{\rm{U}}}}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration with one oxygen vacancy V_O for two Cr, maintaining crystal electroneutrality despite local charge imbalances. In the polycrystalline bulk sample, Cr⁰, Cr²⁺, and ${{{\rm{Cr}}}}_{2}^{3+}$O₃ were detected, present only at grain boundaries and in precipitates. Thus, previous studies identifying Cr⁰ and Cr²⁺ likely detected grain boundaries and precipitates rather than the true oxidation state of Cr dissolved in the matrix²¹.

Discrepancies in simulations of Cr-doped UO₂ are a well-documented challenge, reflecting the inherent complexities of accurately modeling actinide materials. In particular, for DFT simulations, challenges arise from the complex electronic structure of actinide materials, which involves strong electron correlations of localized orbitals. Accurate modeling of such systems demands careful parameterization to ensure reliable results. Over time, the evolution of best practices in DFT modeling of actinides has progressed from standard DFT to including a U parameter to address electron correlation effects^22,23. However, defining an appropriate U value remains a matter of significant debate. While there is more consensus on the U parameter for uranium, the optimal U value for chromium remains an open question. Moreover, it has been demonstrated that DFT+U simulations require occupation matrix (OM) control to prevent the convergence towards metastable states^24,25,26. Despite these advancements, up-to-date studies on Cr-doped UO₂ have employed varying U parameters. Therefore, a unifying study is essential to establish a consistent framework, addressing these discrepancies.

In this work, we employed state-of-the-art DFT+U calculations to model chromium-doped UO₂, focusing on the preferential incorporation site of chromium in the UO₂ lattice, charge equilibration, and the resulting oxidation states of U and Cr cations. We devoted special attention to address the discrepancies in previous studies due to the various U parameters. By comparing our results to prior studies under consistent conditions, we provided a unified framework for understanding the role of the U value in determining the oxidation state of Cr. Further, we simulated various configurations of the recently proposed configuration by Murphy et al. and compared them with those proposed by Guo et al. and Cardinaels et al. Our results resolve the long-standing debate on the oxidation state of Cr in UO₂ in favor of the thermochemistry model of ref. ¹⁴. Finally, we provide solid evidence regarding the local configuration of Cr in UO₂.

Results and discussion

Oxidation states and trapping sites of a diluted Cr atom in UO₂

Previous atomistic studies of Cr incorporation in UO₂ revealed three main possible incorporation sites for this species in the lattice: the U vacancy (V_U)^8,9,16,17, the U-O divacancy (V_UO)¹⁸ and the octahedral interstitial position⁹.

In particular, the DFT+U studies by ref. ¹⁶ and ref. ⁹ both identified the V_U to be the most favorable incorporation site for Cr, with a Cr³⁺ state. In contrast, using the same method, ref. ¹⁸ found the V_UO to be the most favorable trapping site for Cr, with a Cr²⁺ state. Interestingly, Sun et al. also investigated the case of a Cr atom substituting for U, and enforced the initial occupation of the d-orbitals of Cr to simulate this configuration with Cr²⁺, Cr³⁺, and Cr⁴⁺. In all cases, these calculations converged toward the Cr²⁺ state¹⁸.

We advocate here that such discrepancies in the predictions of Cr behavior, notably in the substitution of uranium, should be related to the differences in computational parameters used across the various studies^9,16,18. In particular, all authors employed markedly different Hubbard parameters in their respective DFT+U calculations. The Hubbard values applied to U and Cr atoms by ref. ¹⁸, ref. ⁹, ref. ¹⁶, and in this work are summarized in Table 1, along with the exchange-correlation functionals, cutoff energies, and the procedures adopted in each study to address metastable states in DFT+U. Notably, Sun et al. applied different Hubbard parameters for Cr²⁺ and Cr³⁺ oxidation states, which, as discussed in section “Solution and incorporation energies of Cr in UO₂”, complicates direct energy comparisons between the various Cr configurations investigated. In addition, these authors used a relatively low Hubbard value of 1.7 eV for uranium, in contrast to the 4.5 eV value predominantly found in the literature^23,27,28, and employed by Cooper et al., Middleburgh et al., and in the present study. Although the 1.7 eV value was derived from first-principles linear response calculations with the PBEsol functional²⁹, it yields a significantly underestimated band gap for UO₂ (0.4 eV compared to the experimental value of 2 eV²⁹), and was therefore not considered in this work.

Table 1 Calculation parameters used in previous DFT+U studies of Cr in UO₂^9,16,18 and in the present work

As a first step, we compute here the solution and incorporation energies of Cr⁺, Cr²⁺, and Cr³⁺ in the V_U, the V_UO, and the interstitial position, for all previously investigated values of Cr Hubbard parameter (i.e., 3.2, 4.0, and 6.5 eV), in a 2 × 2 × 2 UO₂ supercell (96 atoms). We use the occupation matrix control (OMC) scheme²⁴ to monitor the oxidation state of Cr in each incorporation site (see section “OMC scheme and oxidation states monitoring”).

Oxidation states of Cr in UO₂

The oxidation states that could be stabilized in each defect are summarized in Table 2. These results hold for all investigated values of the Hubbard parameter.

Table 2 Possible oxidation states of Cr in each neutral incorporation site

Cr⁺ is the only state that can be accommodated in the interstitial position, with the appearance of 1 U³⁺ cation in the supercell. This result is consistent with the findings of ref. ⁹, and rules out the hypothesis of Smith et al. that Cr could be found in the +2 or +3 state in interstitial position²⁰. All calculations involving the V_UO led to a Cr²⁺ state with no oxidized nor reduced U⁴⁺ cations, even when preconditioning the simulation with 1 Cr³⁺ and 1 U³⁺ or 1 Cr⁺ and 1 U⁵⁺. Finally, both Cr³⁺ and Cr²⁺ can be stabilized in the substitution of U. In each case, this configuration is accommodated by the oxidation of U⁴⁺ cations into U⁵⁺, in agreement with the findings of Middleburgh et al. for Cr³⁺¹⁶ and Sun et al. for Cr²⁺¹⁸ in the V_U. It is important to note that only neutral supercells were employed in this study; however, another mechanism of charge compensation involving unbound electrons and holes, or polarons, might be relevant. While we did not explore different charge states of the supercell due to the computational cost of considering multiple charge states across all configurations and Hubbard values, we note that ref. ⁹ investigated charged cells and also found Cr⁺ and Cr³⁺ as the most stable states in the interstitial and U vacancy sites, respectively, in agreement with our results.

Solution and incorporation energies of Cr in UO₂

The incorporation and solution energies of Cr in these various configurations are respectively given, for all investigated Hubbard parameters, in Fig. 1a, b. First, we can see that, for all Hubbard parameters investigated, the incorporation energies of Cr are negative in the V_UO and the V_U, the latter being the most favorable. As such, these two defects constitute favorable trapping sites for Cr, when directly available in the UO₂ crystal to incorporate the dopant. However, the quantity of potential trapping sites in the UO₂ crystal used during Cr-UO₂ pellets manufacturing is extremely low in comparison to the Cr quantity added to the fuel⁹. Thus, a much more relevant quantity to investigate in the context of UO₂ doping is the Cr solution energy, which considers the energy required to create the defects in which Cr is hosted. As illustrated in Fig. 1b, the solution energies of Cr are positive for all investigated sites, showing that Cr possesses an endothermic solubility in the pristine UO₂ matrix. This result is consistent with indications of a low solubility of Cr found by Cooper et al. for the dopant at low temperature⁹. We focus on comparison with low-temperature results here, as our DFT calculations correspond to 0 K temperature. Nevertheless, experimental studies at higher temperatures further support the limited solubility of Cr in UO₂, as evidenced by the evaporation of Cr during sintering and the formation of Cr₂O₃ precipitates even at doping levels below the nominal solubility limit⁶.

Fig. 1: Energetics of Cr in the various trapping sites of UO₂.

a Incorporation energies and b solution energies, for a 96 atoms supercell and for all values of Cr Hubbard parameters investigated.

Overall, regarding both incorporation and solution energies of Cr, the Hubbard parameter does not influence the stability order of the various types of defects. As such, the solution energy of the Cr atom is always the lowest in the V_UO, then goes to Cr in the V_U, and finally in the interstitial position. Having the most favorable solution energy for V_UO defect implies that only 2+ oxidation state can be stabilized (see Table 2), or, as we show later in this article (see section “Investigation of the Cr³⁺_x U⁴⁺_1−xO_2−0.5x configuration), the 3+ state can be stabilized if another Cr is present for charge compensation.

However, we stress that, in the V_U, the most favorable Cr oxidation state depends on the Hubbard parameter value that is used. This observation also holds for a 3 × 3 × 3 UO₂ supercell (324 atoms). The evolution of Cr³⁺ and Cr²⁺ solution energies in the V_U as a function of the Cr Hubbard parameter is given in Fig. 2a for a 2 × 2 × 2 supercell and in Fig. 2b for a 3 × 3 × 3 supercell.

Fig. 2: Evolution of Cr³⁺ and Cr²⁺ solution energies in the V_U as a function of the Cr Hubbard parameter.

a For a 96 atoms supercell, b for a 324 atoms supercell. Lines are included as guides to the eye.

For a Hubbard parameter of 3.2 eV, the Cr³⁺ state is slightly more favorable than Cr²⁺, which is consistent with the findings of Cooper et al., who used this value for all Cr oxidation states⁹. At 4 eV, the solution energies of Cr²⁺ and Cr³⁺ in the V_U are quasi-equal for 96 atoms, while the Cr³⁺ state is still slightly more favorable than Cr²⁺ for 324 atoms. Finally, for a Hubbard parameter of 6.5 eV, the Cr²⁺ appears much more favorable than Cr³⁺ in the V_U. These results demonstrate a strong influence of the Cr Hubbard parameter on the oxidation state that is perceived as most favorable in the uranium vacancy. In particular, using a value of 6.5 eV for both Cr²⁺ and Cr³⁺ introduces a strong bias in favor of Cr²⁺. Nevertheless, we cannot use this bias to explain the strong stability of Cr²⁺ reported by Sun et al. in the V_U, as the authors used a 4.0 eV Hubbard parameter for the Cr³⁺ state. In fact, our results show that, in the 96 atoms supercell (respectively 324 atoms), the solution energy of Cr²⁺ calculated at U = 6.5 eV is still 0.28 eV (respectively 0.23 eV) higher than the solution energy of Cr³⁺ calculated at 4.0 eV. Thus, the fact that Sun et al. could not stabilize Cr³⁺ in the V_U, even by enforcing the initial occupation of the d-orbitals of Cr to match this oxidation state, cannot be explained by the different Cr Hubbard values they used for Cr³⁺ and Cr²⁺. Another remaining avenue to explore in order to explain this discrepancy between the study by Sun et al. and previous DFT studies, as well as this work, is the influence of the Hubbard parameter associated with uranium 5f orbitals on the behavior of Cr in the V_U. While our results highlight the significant impact of U on oxidation state predictions, there is currently no consensus within the community regarding the optimal approach to determine U. We emphasize, however, that using different U values for distinct oxidation states is not desirable, as such an approach does not allow for direct energy comparisons between the different states and can introduce inconsistencies in the theoretical framework. Specifically, the oxidation state of U and Cr can evolve dynamically during a calculation, whereas the corresponding U values remain fixed. Consequently, it becomes impossible to determine the “correct" value of U to apply at the start of the calculation, further complicating the consistency and reliability of the theoretical predictions. Moreover, some discrepancies in the literature could also come, to a lesser extent, from the use of various exchange-correlation approximations across the different DFT studies, as Sun et al. used PBEsol³⁰, Cooper et al. used LDA, and Middleburgh et al. and we used PBE³¹ (see Table 1).

For the incorporation in a V_U, the transition between a 96 and a 324 atoms supercell preserves the order of stability of Cr³⁺ and Cr²⁺ for Cr Hubbard values of 3.2 and 6.5 eV, and allows to distinguish the energies of the two oxidation states at a Hubbard value of 4.0 eV. We note that the impact of the supercell size on the absolute values of the calculated energies is particularly strong, the solution energy of Cr in the V_U being overestimated in the 96 atoms supercell by 0.88 eV on average, compared to a 324 atoms supercell. We determined the formation energy of the V_U to be overestimated by 0.23 eV in the 96-atoms supercell, compared to 324 atoms. Thus, the average overestimation of the Cr incorporation energy in the V_U, in the 96 atoms supercell, is of 0.65 eV. In a previous study³², we found the solution energy (respectively incorporation energy) of the I atom substituting for the V_U to be underestimated by 0.01 eV (respectively 0.24 eV) in the 96-atom supercell, compared to 324 atoms. These results show that the impact of the supercell size on the solution and incorporation energies of a given species in a certain defect strongly depends on the nature of the considered species.

Atomic environments of Cr in UO₂

For each configuration of Cr that we could stabilize (see Table 2), we investigate the atomic environment of Cr after a full relaxation of the supercell. The resulting configurations of oxygen anions around the Cr atom are reported for each site in Fig. 3. We found these configurations for all the Hubbard parameters considered.

Fig. 3: Atomic environments formed by O atoms around Cr.

Oxygen atoms (in red) around a Cr⁺ in the interstitial position, b Cr²⁺ in the U-O divacancy, c Cr²⁺ in the U vacancy, d Cr³⁺ in the U vacancy, in a 96-atom cell. Similar environments are observed in a 324-atom cell. Cr-O atomic bonds are only shown for distances shorter than 2.5 Å, which is the minimum distance at which all Cr⁺-O bonds are visible in the interstitial position.

As illustrated in Fig. 3, each oxidation state of Cr corresponds to a specific environment. In the interstitial position, Cr⁺ is surrounded by eight O atoms, forming a cubic environment. In the V_UO and the V_U, Cr²⁺ is directly coordinated by four near O atoms, forming a (101) plane that includes the dopant. Finally, in the V_U, Cr³⁺ is coordinated by six O atoms. Thus, this latter configuration is the only one for which we get the octahedral environment that was reported by Riglet-Martial et al.¹⁴ and Mieszczynski et al.¹⁵ for Cr in UO₂, based on X-ray absorption spectroscopy (XAS) experiments. While Cooper et al. also reported a 6-oxygen environment of Cr³⁺ in substitution of U, the authors additionally observed the octahedral environment for Cr⁺ in an interstitial position⁹, which does not correspond to our findings. Note that ref. ¹⁸ did not report the environment of Cr²⁺ in the various configurations they investigated.

Table 3 summarizes the Cr-O distances obtained in each configuration.

Table 3 Cr-O distances found for each Cr stable configuration

For each distance indicated in Table 3, the corresponding uncertainty was calculated as the largest deviation from the mean of Cr-O distances that were found, considering Cr Hubbard values of 3.2, 4.0 and 6.5 eV. In the case of Cr³⁺ substituting for U, we found an average distance of 2.17 ± 0.09 Å between Cr and the 6 O atoms around it. This value is consistent with the 2.13 Å reported by Cooper et al. for the Cr³⁺-O distance in the octahedral environment⁹. Our result is also in reasonable agreement with the experimental value of 2.02 Å, determined through EXAFS measurements by ref. ¹⁵. Finally, in this configuration, we found an average U-O distance of 2.39 Å for all Hubbard parameters, in good agreement with the 2.38 Å calculated by Cooper et al. and the 2.36 Å measured by Mieszczynski et al. Our findings on the local configuration around each Cr oxidation state are consistent in both 96 and 324-atom supercells.

In the following section, we use a 324-atom supercell to investigate the configurations involving the incorporation of two Cr atoms in the UO₂ supercell, in particular those proposed by ref. ¹⁷ and ref. ²¹. We use a unique U value for both Cr²⁺ and Cr³⁺, as the oxidation state of Cr can change during the calculation, while the associated value of U cannot, as previously discussed. We chose the 3.2 eV Hubbard parameter for Cr, which is the value for which the energy difference between Cr²⁺ and Cr³⁺ is the smallest for this supercell size, thus favoring neither of the oxidation states (see Fig. 2).

Investigation of the ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$ ${{{\rm{U}}}}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration

In 2017, ref. ¹⁷ used empirical interatomic potentials to study the impact of Cr on the formation of defects in UO₂. The authors assumed that the Cr atoms existed in pairs within the lattice, that they maintained close proximity to each other, and that each pair was accommodated by one oxygen vacancy to ensure the electroneutrality of the crystal. By comparing different positions of the V_O around the two Cr atoms substituting for U, Guo et al. found that the most stable configuration was obtained when the V_O was located as a first nearest neighbor to both Cr atoms, acting as a bridge. Interestingly, this configuration was also found to be favorable in a previous empirical interatomic potentials study by ref. ⁸. Note that this simulation method does not allow for variations of oxidation states during the calculation. As such, the Cr³⁺ oxidation state was fixed during all computations in these two studies^8,17.

In 2023, ref. ²¹ directly probed the oxidation state of Cr in single Cr-UO₂ grains, using XAS and EPR characterization techniques. The authors unequivocally determined the Cr³⁺ oxidation state of Cr in the corresponding samples. Moreover, the interpretation of the EPR and XANES spectra ruled out the formation of U⁵⁺ ions in the crystal. Furthermore, EPR measurements dismissed the formation of Cr pairs such as those simulated by Guo et al. and Cardinaels et al. To explain these experimental findings, Murphy et al. proposed a ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$${{{\rm{U}}}}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration, with one V_O for two spatially separated Cr atoms in substitution of U, maintaining the crystal electroneutrality despite local charge imbalances.

In this section, we simulate various possible configurations of the ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$${{{\rm{U}}}}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration proposed by Murphy et al. using DFT+U. To that aim, we have explored several relative positions of the Cr atoms and the V_O in the lattice, using a 324-atom supercell. We compare the relative stabilities of Murphy’s configuration with various other configurations, including the aforementioned Cr twin pair bridged by one V_O. A summary of all investigated configurations, involving different Cr oxidation states, O vacancy positions, and Cr-Cr distances, is provided in Table 4.

Table 4 Summary of all investigated configurations involving two Cr atoms and one V_O in the 324-atom supercell

Note that, in a typical calculation workflow, the U and Cr occupation matrices are allowed to evolve during the relaxation of atomic positions. Thus, the oxidation states can change for configurations in which initial occupation matrices do not correspond to their ground state. This was the case for the configuration (iii)’, in which we found that oxidation states drifted away from the initial settings at the beginning of the calculation.

For each investigated configuration, the final oxidation states of Cr and U atoms are given in Table 5, along with a comparison of the energy of each configuration, relative to the most stable case, indicated in bold.

Table 5 Final oxidation states of Cr and U atoms, relative energies and final Cr-Cr distances for each investigated configuration involving two Cr atoms and one V_O in the 324-atom supercell

Four main conclusions can be drawn from Table 5:

1) The two Cr atoms cannot be considered as non-interacting within a supercell of 324 atoms:

Our calculations indicate that the positions (a), (b), (c), and (d) of the O vacancy relative to one of the Cr atoms are not equivalent, with the energies of configurations (i.1), (i.2), (i.3), and (i.4) spanning 0.15 eV. This position dependence is caused by the presence of the second Cr atom in the supercell, which introduces a specific axis of symmetry along the  <111> crystallographic direction that links the two Cr atoms in configurations (i). In fact, (b) and (c) V_O are symmetrically positioned on either side of the axis, resulting in a very similar distance of the V_O relative to the second Cr atom, which leads to a difference of energy of only 0.02 eV between (i.2) and (i.3). On the other hand, the distance between the second Cr atom and the V_O becomes maximum in (i.1) and minimum in (i.4), resulting in a difference of energy of 0.08 eV between the two configurations. Although very small, these energy differences reveal that a 9.47 Å Cr-Cr distance is insufficient to consider the Cr atoms as non-interacting. Indeed, we expect the four configurations (a), (b), (c), and (d) to become equivalent in a larger supercell, where both Cr atoms could be placed sufficiently far from each other so that the V_O would only be affected by the presence of the nearest Cr. Unfortunately, calculations in a larger supercell at this level of theory are hindered by computational cost.

2) It is energetically favorable for the two Cr atoms to come closer together:

Note that decreasing the Cr-Cr distance to 5.47 Å along the  <100> crystallographic direction leads to a more stable configuration (ii), relative to (i), suggesting that the stability of Cr pairs increases when both atoms become closer.

3) Cr²⁺ and U⁵⁺ ions are not stable in the ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$${{{\rm{U}}}}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration:

We found that all Murphy-like configurations, in which one or two Cr³⁺ were replaced with Cr²⁺ (and U⁴⁺ cations were oxidized into U⁵⁺ to accommodate the electroneutrality of the supercell), yield total energies significantly higher than the (i) configuration. This result further supports that neither Cr²⁺ nor U⁵⁺ should form in the ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$${\rm U}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration, in agreement with the interpretations of EPR and XAS spectra by Murphy et al.²¹.

4) Guo’s configuration is more stable than Murphy’s, and should also only contain Cr³⁺ ions, accompanied by U⁴⁺ ions:

Our results show that, in the 324 atoms supercell with two Cr atoms, Guo’s configuration¹⁷ yields the lowest energy, with one V_O bridging both Cr atoms. Even though the initial (Cr³⁺, Cr³⁺) configuration of local structure (iii)’ turned into (Cr²⁺, Cr³⁺, 1 U⁵⁺) during the relaxation of atomic positions, we were able to stabilize the (Cr³⁺, Cr³⁺) configuration in local structure (iii), using the preconditioning procedure described in section “OMC Scheme and Oxidation States Monitoring”. We found the latter configuration to be more stable than (iii)’ by 0.16 eV, and more stable than (iii)” by 0.65 eV. These differences in energy underscore that Cr³⁺ is also more favorable than Cr²⁺ in Guo’s configuration.

While EPR spectra interpretations by Murphy et al. clearly state that no twin pairs of Cr should exist in the UO₂ matrix, our results designate Guo’s configuration as the most favorable one. Our theoretical results seem to challenge the ${\rm Cr}_{{{{\rm{x}}}}}^{3+}$${\rm U}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration without Cr twin pairs, as proposed by ref. ²¹. Nevertheless, we demonstrated here that two Cr atoms could not be considered as non-interacting in a 324-atom supercell. Besides, Murphy et al. did not specify the minimal separation distance that should be expected between two Cr atoms in the lattice to consider them as a non-twin pair. Moreover, the authors noted that oxygen vacancies could be mobile in the crystal, and could potentially be located far away from the two Cr atoms (a case which is impossible to investigate in DFT for now, considering the prohibitive calculation cost in supercells larger than 324 atoms). Thus, our results call for further theoretical investigations of the ${\rm Cr}_{{{{\rm{x}}}}}^{3+}$${\rm U}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration with a larger-scale simulation method, such as interatomic potentials, which would allow placing Cr atoms and oxygen vacancies several hundreds of Å apart. If such a configuration is still found to be less favorable than Guo’s one, measurement interpretations may be revisited.

Relative stabilities of a diluted Cr atom in UO₂ and the ${\rm Cr}_{{{{\rm{x}}}}}^{3+}$ ${\rm U}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration

In order to compare the relative stabilities of configurations involving one Cr atom (investigated in section “Oxidation states and trapping sites of a diluted Cr atom in UO₂”) with structures involving two Cr atoms (investigated in section “Investigation of the Cr³⁺_x U⁴⁺_1−xO_2−0.5x configuration”), we calculate the solution energy of Cr per dopant atom in the simulation box. The same approach was previously employed by ref. ¹⁸. We perform calculations for all configurations that we explored using a 324-atom supercell and a 3.2 eV Cr Hubbard parameter. The corresponding results are given in Table 6.

Table 6 Solution energy per Cr atom in the various configurations investigated with a 3.2 eV Cr Hubbard parameter, in 324-atom supercells

Note that we did not calculate the solution energy of Cr⁺ in the interstitial position in the 3 × 3 × 3 supercell, as we showed in section “Oxidation states and trapping sites of a diluted Cr atom in UO₂” that it was the least favorable configuration when considering one Cr atom. For each type of configuration, we only included the most relevant cases in Table 6, thus excluding (i.1), (i.2), (i.4), (i.4)’, (iii)’, and (iii)”.

The single Cr atom substituting for the V_U is the least favorable configuration of Table 6, with solution energies 0.2 eV higher than the (2 Cr²⁺, 2 U⁵⁺) (i.4)” Murphy-like configuration. This latter configuration is the least stable among those involving two Cr atoms, with a 2.93 eV solution energy of Cr. Next, we find that the two most favorable Murphy’s configurations, (i.3) and (ii), yield very close Cr solution energies (i.e., 2.76 and 2.72 eV, respectively). Finally, the single Cr²⁺ atom incorporated in the U-O divacancy is the most stable configuration, 0.75 eV below Guo’s configuration, which is the most favorable when considering cases involving two Cr atoms per supercell. While our results align with those of ref. ¹⁸, it is important to note that simulations with a single Cr atom per supercell do not accurately represent experimental conditions. Simulations with one Cr atom per supercell simulate an isolated dopant in an infinite bulk material. Experimental conditions involve concentrations of Cr at the solubility limit, as well as temperature and diffusion effects that may favor the formation of Cr pairs. Although no migration energies have yet been reported in the literature for Cr in UO₂, we expect Cr diffusion to occur via uranium vacancy-assisted mechanisms. This expectation is supported by the findings of ref. ⁹, who predicted that the concentration of uranium vacancies increases significantly with Cr doping. Our results for configurations (i.3), (ii), and (iii) show that Cr pair stability increases as Cr atoms get closer. Therefore, we interpret our results to suggest that isolated non-interacting Cr atoms in the UO₂ matrix are more stable in the 2+ oxidation state. However, Cr atoms will likely interact under experimental conditions and form pairs of Cr³⁺, accompanied by oxygen vacancies. Finally, in order to better reflect on experimental conditions, a natural extension of this DFT study would be to investigate how the stability of Cr oxidation states evolves with temperature and oxygen partial pressure for the configurations considered. This would require the construction of a comprehensive point defect thermodynamic model for Cr in UO₂, similar to the one recently developed for UO_2±x³³. The calculations reported here provide a solid foundation for building such a model, which would also necessitate the inclusion of charged supercells and the evaluation of vibrational entropies for the relevant configurations.

Conclusions

We used state-of-the-art DFT+U calculations to simulate the incorporation of Cr in the UO₂ lattice, systematically investigating the various oxidation states and local configurations of Cr atoms previously proposed in the experimental and theoretical literature.

First, we used the OMC scheme to enforce the Cr⁺, Cr²⁺, and Cr³⁺ states in the three main incorporation sites reported for Cr in UO₂, namely the U vacancy (V_U), the U-O divacancy (V_UO), and the interstitial position. We showed that, for each oxidation state of Cr, only certain host sites are allowed in the UO₂ matrix. As such, Cr⁺ can only be stabilized in the interstitial position, Cr³⁺ only in the V_U, and Cr²⁺ in both the V_U and the V_UO. Moreover, we found that each Cr oxidation state is associated with a distinct characteristic atomic environment. Interestingly, Cr³⁺ is the only state for which we obtained the octahedral environment that has been experimentally reported for Cr in UO₂. Note that the corresponding Cr-O and U-O calculated distances agree with experimental values. Additionally, we calculated the solution and incorporation energies of Cr⁺, Cr²⁺, and Cr³⁺ in various configurations and systematically investigated the effect of the Hubbard parameter used in DFT+U for Cr across the literature. Although we found a significant impact of the Hubbard parameter on the preferred oxidation state of Cr in the V_U, our results show that the discrepancies observed between previous DFT investigations of Cr doped UO₂ cannot be fully explained by the different Cr Hubbard parameters that were used in the literature.

Secondly, we explored the stability of various configurations involving incorporating two Cr atoms in the UO₂ supercell. In particular, we simulated the recently proposed ${\rm Cr}_{{{{\rm{x}}}}}^{3+}$${\rm U}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration, for various Cr-Cr distances and oxygen vacancy positions in a 324-atom supercell. We find such a configuration to gain stability when both Cr atoms get closer. The most favorable configuration is indeed obtained when the two Cr atoms form a twin pair, bridged by one oxygen vacancy. These findings seem to question the interpretations of the latest EPR measurements of Cr-UO₂, which dismissed the formation of Cr twin pairs in the UO₂ matrix. Nevertheless, even though we used the largest tractable supercell size in our DFT+U calculations of UO₂, we showed that two Cr atoms cannot be considered as non-interacting in our simulations, even for a 9.47 Å Cr-Cr distance. Additionally, we only investigated configurations where the oxygen vacancy is positioned as the first nearest neighbor to a Cr atom. To clarify whether the formation of Cr twin pairs is more stable than spatially separated Cr atoms and oxygen vacancies, we call for further theoretical investigations of the ${{{\rm{Cr}}}}_{{{{\rm{x}}}}}^{3+}$${{{\rm{U}}}}_{1-{{{\rm{x}}}}}^{4+}$O_2−0.5x configuration, using a larger-scale simulation method such as interatomic potentials.

Finally, this DFT+U study clarifies the controversy over Cr oxidation states and incorporation sites within UO₂. For interacting Cr atoms in the lattice, Cr³⁺ substituting for U, with one V_O for two Cr atoms, was found to be the most stable configuration. Additionally, the most stable configuration for a limit of isolated Cr atoms is a single Cr²⁺ in the UO divacancy. By providing accurate trapping sites and oxidation states of Cr in UO₂, this study contributes to the understanding of Cr incorporation mechanisms in UO₂, enabling a better prediction of Cr-UO₂ redox properties as a promising accident-tolerant fuel.

Methods

DFT+U

DFT calculations were performed using the 6.2.1 version of the Vienna ab initio simulation package (VASP)^34,35,36, in the projector augmented wave (PAW) formalism^37,38. The generalized gradient approximation (GGA) was used to describe the exchange-correlation energy, with the Perdew, Burke, and Ernzerhof (PBE) functional³¹. The Hubbard correction was applied to account for the strong correlations of uranium 5f and chromium 3d electrons using the rotationally invariant DFT+U Liechtenstein scheme³⁹. The U and J parameters were respectively set to 4.50 and 0.54 eV for uranium 5f orbitals, as derived from the X-ray photoemission spectroscopy (XPS) measurements of ref. ⁴⁰ and in accordance with previous DFT+U studies of UO₂^{24,32,41,42,43,44,45,46,47}, as well as previous DFT studies of Cr-UO₂ by ref. ⁹ and ref. ¹⁶. For chromium 3d electrons, three different values of U were investigated, i.e., 3.2⁹, 4.0¹⁸, and 6.5 eV¹⁸. To avoid the metastable states arising from the DFT+U method and to ensure the convergence of the calculations toward the ground state, the OMC scheme^24,25,26 was used.

For calculations conducted in a 2 × 2 × 2 supercell (96 atoms), a 2 × 2 × 2 Monkhorst-Pack k-point mesh⁴⁸ was used, along with a plane-wave cutoff energy of 500 eV, and a Gaussian smearing of 0.1 eV. For computations conducted in a 3 × 3 × 3 supercell (324 atoms), we have only sampled the Γ-point of the Brillouin zone. The use of symmetries was switched off for all computations. The atomic positions were optimized until the Hellmann–Feynman forces acting on ions reached values below 10⁻³ eV/Å and the total energy converged to less than 10⁻⁵ eV/atom. Volumes were relaxed to achieve external pressures with absolute values below 0.05 kbar.

Treatment of magnetism

Following the approach we used in ref. ³², we modeled the magnetic order of UO₂ as a longitudinal 1k-AFM phase, which has been found to approximate well both the 3k-AFM and the paramagnetic orders of UO₂, regarding energetic properties^49,50,51. We did not include spin-orbit coupling (SOC) in VASP calculations, as several theoretical studies indicate a negligible influence of SOC on the properties of UO₂^52,53,54. Note that SOC was also neglected in the previous DFT studies of Cr-UO₂ by Middleburgh et al.¹⁶, Cooper et al.⁹, and Sun et al.¹⁸. All calculations are initialized in the fluorite structure observed above the Néel temperature of UO₂. We verified that none of the relaxed supercells converged toward a Jahn-Teller distortion.

OMC scheme and oxidation states monitoring

In the present work, we use the 5f electron occupation matrices reported in ref. ³² for U³⁺, U⁴⁺, and U⁵⁺ cations. The Cr⁺, Cr²⁺, and Cr³⁺ initial OMs are reported in the Supplementary Note 1 and were determined through a systematic investigation of the diagonal OMs of Cr in Cr₂O, CrO, and Cr₂O₃, respectively. The initial OMs were kept fixed during the first 30 electronic steps, then were allowed to relax individually for each uranium and chromium atom in the following iterations until convergence. The oxidation states of uranium cations were unequivocally determined based on their spin magnetic moments with typical values of  ±(2.8 to 3.0) μB,  ±(1.9 to 2.0) μB, and ±  (1.0 to 1.1) μB for U³⁺, U⁴⁺, and U⁵⁺, respectively. As to Cr, the Cr⁺, Cr²⁺, and Cr³⁺ oxidation states were associated with typical values of  ±(4.5 to 4.6) μB,  ±(3.9 to 4.0) μB, and  ±(3.0 to 3.1) μB, respectively. In each simulation, the number of U³⁺ and U⁵⁺ arising in the supercell was consistent with the determined oxidation state of Cr.

The literature has previously predicted a broad range of oxidation states of Cr. As such, we investigated the stability of various plausible oxidation states for each incorporation site for Cr. To impose a particular oxidation state, we use the two-step preconditioning procedure described in ref. ³², that we recall here:

• 1. The calculation is initiated with specific uranium and chromium OMs, which will favor the Cr to adopt the desired oxidation state. The initial OMs are imposed throughout the entire calculation while the atomic positions are optimized, until the forces and the energy are converged for this first step.

• 2. Then, a self-consistent calculation is restarted from the charge density and positions calculated in step 1, with the OMs allowed to be optimized.

This procedure allowed us to fully monitor the oxidation state of Cr in each investigated incorporation site.

Formation energies

The formation energy of a neutral defect X is given by Eq. (1)^55,56:

$${E}_{{{{\rm{form}}}}}(X)={E}_{{{{\rm{tot}}}}}(X)-({E}_{{{{\rm{tot}}}}}(\,{\mbox{bulk}})-\sum\limits_{i}{n}_{i}{\mu }_{i})$$

(1)

where E_tot(X) is the total energy of the supercell containing the defect X and E_tot(bulk) is the total energy of the pristine supercell. n_i is the number of atoms of species i added (n_i > 0) or removed (n_i < 0) from the pristine supercell to create the defect X, and μ_i is the chemical potential of species i in the bulk material.

The μ_i chemical potentials refer to those of oxygen and uranium atoms in UO₂, respectively denoted ${\mu }_{{{{\rm{U}}}}}^{{{{{\rm{UO}}}}}_{2}}$ and ${\mu }_{{{{\rm{O}}}}}^{{{{{\rm{UO}}}}}_{2}}$. These quantities were determined in ref. ³² with PBE, following the approaches of ref. ⁵⁷ and ref. ⁵⁸:

• ${\mu }_{{{{\rm{U}}}}}^{{{{{\rm{UO}}}}}_{2}}=-12.017$ eV

• ${\mu }_{{{{\rm{O}}}}}^{{{{{\rm{UO}}}}}_{2}}=-8.69$ eV

Note that these values ensure the validity of:

$${E}_{{{{\rm{tot}}}}}({{{\rm{bulk}}}})={N}_{{{{{\rm{UO}}}}}_{2}}\times \left({\mu }_{{{{\rm{U}}}}}^{{{{{\rm{UO}}}}}_{2}}+2{\mu }_{{{{\rm{O}}}}}^{{{{{\rm{UO}}}}}_{2}}\right)$$

(2)

where ${N}_{{{{{\rm{UO}}}}}_{2}}$ represents the number of UO₂ patterns in the pristine supercell.

Solution energies

The solution energy of a specific species in a defect refers to the energy needed to both create the defect and incorporate the impurity into it. The solution energy of a dopant (DP) in a defect X is expressed by ref. ⁵⁹:

$${E}_{{{{\rm{sol}}}}}({{{\rm{DP}}}},X)={E}_{{{{\rm{tot}}}}}({{{\rm{DP}}}},X)-\left({E}_{{{{\rm{tot}}}}}({{{\rm{bulk}}}})-\sum\limits_{i}{n}_{i}{\mu }_{i}\right)$$

(3)

where E_tot(DP, X) represents the total energy of the supercell containing the dopant in the defect X. The μ_i chemical potentials refer not only to ${\mu }_{{{{\rm{U}}}}}^{{{{{\rm{UO}}}}}_{2}}$ and ${\mu }_{{{{\rm{O}}}}}^{{{{{\rm{UO}}}}}_{2}}$ but also to the chemical potential μ_DP of the incorporated DP in its reference state ${\mu }_{{{{\rm{Cr}}}}}^{{{{{\rm{Cr}}}}}_{2}{{{{\rm{O}}}}}_{3}}$, that we chose based on the fact that Cr doping is mainly done in UO₂ by sinterring a powdered UO₂-Cr₂O₃ mixture.

This quantity was determined using the procedure detailed in the Supplementary Note 2, following the approaches by ref. ⁵⁷, which uses experimental formation enthalpies of Cr₂O₃ and other CrO_x compounds (see Supplementary Fig. 1), and ref. ⁵⁸, which avoid positive formation energies of charged vacancies in Cr₂O₃ (see Supplementary Fig. 2). We obtained the following average values of ${\mu }_{{{{\rm{Cr}}}}}^{{{{{\rm{Cr}}}}}_{2}{{{{\rm{O}}}}}_{3}}$, which depend on the Hubbard parameter used for Cr:

• With U = 3.2 eV: ${\mu }_{{{{\rm{Cr}}}}}^{{{{{\rm{Cr}}}}}_{2}{{{{\rm{O}}}}}_{3}}$ = −7.742 eV.

• With U = 4.0 eV: ${\mu }_{{{{\rm{Cr}}}}}^{{{{{\rm{Cr}}}}}_{2}{{{{\rm{O}}}}}_{3}}$ = −8.1542 eV.

• With U = 6.5 eV: ${\mu }_{{{{\rm{Cr}}}}}^{{{{{\rm{Cr}}}}}_{2}{{{{\rm{O}}}}}_{3}}$ = −7.9534 eV.

Incorporation energies

The incorporation energy of a dopant in a defect X is given by ref. ⁵⁹:

$${E}_{{{{\rm{inc}}}}}({{{\rm{DP}}}},X)={E}_{{{{\rm{sol}}}}}({{{\rm{DP}}}},X)-{E}_{{{{\rm{form}}}}}(X)$$

(4)

where E_sol(DP, X) and E_form(X) are respectively expressed by equations (3) and (1).

The explicit expression of E_inc(DP, X) is given by:

$${E}_{{{{\rm{inc}}}}}({{{\rm{DP}}}},X)={E}_{{{{\rm{tot}}}}}({{{\rm{DP}}}},X)-({E}_{{{{\rm{tot}}}}}(X)-{n}_{{{{\rm{DP}}}}}{\mu }_{{{{\rm{DP}}}}})$$

(5)

Here, n_DP is negative as the dopant atom is added to the simulation box.