Introduction

Nuclear reactors use mainly UO2-based fuel elements. Additions of other trace elements, such as chromium, are considered to improve the microstructural and mechanical properties of fuel matrices1,2. Among other desired effects, chromium is known to significantly increase the grain sizes, which due to the elongation of bulk diffusion path limits the release of fission gas products such as xenon or krypton2, and increases the fission products retention capability by the fuel elements3. However, how Cr incorporates into the UO2 matrix on the atomic level is not conclusively understood. The impact of Cr doping on the performance of UO2 matrix depends also on the oxidation state of Cr in the nuclear fuel, which is claimed to be Cr3+ 4,5. Other oxidation states, however, are not excluded6,7. In particular, by analyzing X-ray absorption spectrum of Cr-doped UO2 Mieszczynski et al.7 suggest existence of Cr2+, but express a need for the electronic structure calculations to confirm such a possibility.

The most common oxidation states of chromium are Cr3+ and Cr6+. However, the Cr0, Cr1+, Cr2+, Cr4+, or Cr5+ could be also realized8. The previous experimental studies of Cr-doped UO2 show evidence of bulk Cr incorporation as Cr3+ 4. At the same time, only small amounts of Cr could be incorporated into UO2, with maximum solubility at high temperature (T ~ 2300 K) of up to 0.1 wt%∕UO23,4,5,9. The formed diluted solid solution shows a linear decrease in the matrix lattice parameter with content of Cr1,9, which however, is significantly smaller than the predictions by most theoretical models1.

Besides the experimental effort, the atomistic modeling studies have been used to understand the incorporation of Cr into the UO2 matrix at atomic level. Most of these studies used force-fields approach, in which interatomic interactions are represented by analytical functions and charge/oxidation states of the involved species are fixed10, for chromium atom to Cr3+ 1,11,12. Middleburgh et al.12 and more recently Guo et al.11 found that Cr incorporates as a pair of Cr atoms with the formation of an oxygen vacancy (on the oxygen site that bridges the two Cr atoms) as a charge balance mechanism. The density functional theory (DFT)-based first principles studies of Middleburgh et al.12 consider formation of Cr3+ as dominant Cr species in the same structural arrangement as resulted from the force-field simulations, and discuss a less energetically favorable scenario of Cr4+. The more recent ab initio studies of Cooper et al.6 show the possibility of Cr in “+1” oxidation state. Interestingly, the “+2” state has never been considered as dominant at ambient or high-temperature synthesis conditions (T> 1500 K). The oxidation state chemistry of uranium oxides is very interesting with different oxides realizing uranium in different oxidation states (U4+, U5+, or U6+), for instance U3O8 and U4O7 as mixtures of (U6+,U5+) and (U5+, U4+) pairs, respectively13,14,15. Potential change in the cations redox states would have a significant impact on the evolution of nuclear fuel, including, for instance, its dissolution rates16,17. The real oxidation states are not always well constrained, but could be conclusively deducted by a combination of X-ray absorption near edge structure (XANES) measurements and first-principle atomistic modeling15. It is also not conclusively clear, how different impurities or defects affect oxidation states of the UO2 matrix atoms.

Here, we present the atomistic modeling studies combined with an extensive analysis of the relevant experimental data (XANES, the maximum solubility and the lattice parameter), aimed at understanding the atomic structure and redox chemistry of Cr-doped UO2 material. We have been especially interested in the resulting oxidation state of Cr and U cations, derivation of the conclusive structural model of Cr incorporation and testing the ability of such a model structure to match the measured solubility limit of Cr in UO2 and the lattice parameter of Cr-doped UO2 matrix.

Results

The computed Cr-doped UO2

In order to perform the DFT+U calculations, we first derived the Hubbard U parameter for the d and f elements in the considered compounds (Cr and U oxides, see Fig. 1 for the computed structure of UO2). The computed value of Hubbard U parameter for Cr in Cr2O3 (i.e., Cr3+) is 4.0 eV, which is consistent with the one used in previous studies of Cr-bearing systems (~3.5 eV, e.g., refs. 6,18,19,20,21). For Cr2+ we used the value of 6.5 eV, which we derived for CrO. The same value we were getting for Cr2+ in UO2 and we observed that the derived Hubbard U parameter depends strongly on the Cr oxidation state, but is weakly dependent on the local environment of Cr. Therefore, we decided to use the oxidation state-dependent values derived for Cr-oxides in all the calculations of Cr-doped UO2. For UO2 we used 1.7 eV derived for UO2 by Beridze and Kowalski22, which is also representative for U4+ species15,22,23. The detailed information and discussion on the lattice parameter, the band gap and the X-ray related spectral signatures obtained for UO2, as well as other uranium oxides with the applied computational methodology is provided in the previous studies15,22.

Fig. 1: The structure of computed UO2 supercell.
figure 1

Arrows indicate the computed transverse 1k anti-ferromagnetic spin arrangement, which was found by Pegg et al.52 as the preferred magnetic structure in UO2 and adopted in our studies.

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Initially, we computed the two most plausible cases of Cr-doped UO2, namely: (1) the cation exchange with the formation of U5+ species and (2) the two neighboring site cation exchange with the formation of O-vacancy on the bridging oxygen site attached to the Cr atoms as proposed by Guo et al.11. Both configurations are visualized in Fig. 2 and are marked as 1Cr and 2Cr + 1Ov, respectively. Following the interpretation of XANES measurements by Riglet-Martial et al.4, in both cases we initially assumed formation of Cr3+ species. In the case (1) we assumed the charge balance through formation of one U5+ per one Cr atom. In addition, for the case (1) we tried to compute systems with Cr2+ and Cr4+ species, by enforcing the initial occupation of d-orbitals of Cr (4, 3, and 2 d electrons for Cr2+, Cr3+, and Cr4+, respectively). Surprisingly, in all the calculations, the resulting electronic configuration was always that of Cr2+ (i.e., four electrons), with charge balanced by the formation of U5+ species. For instance, in case (2) with the formation of two Cr2+ species we got a pair of U5+.

Fig. 2: The computed supercells representing different considered Cr incorporation configurations.
figure 2

In cases of 1Cr+1Ov (a) and 2Cr+2Ov (a) Cr is shifted towards empty cation interstitial site and the two structures resulted from classical molecular simulation runs.

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Such an unexpected theoretical result requires sounding justification and support from the experimental data. The simplest thing to do, when considering oxidation state of a species substituting another host matrix element, i.e., chromium substituting uranium atom in our case, is to check the sizes of involved cations. This is because it is easier to exchange cations of similar sizes due to minimization of the associated strain resulting from sizes mismatch. In Table 1, we list the ionic radii and volumes of the considered species24. Cr3+ is much smaller than U4+ (by 66% in volume) and even U5+ (by 51% in volume). The size of Cr2+ is much similar to that of U4+ (smaller by only 32 % in volume) and nearly the same as that of U5+. As excess strain effects arise as quadratic function of volume mismatch (e.g., Kowalski and Li25), considering the sizes of cations, the scenario with Cr2+ is more probable. However, even with appealing theoretical arguments in hand, one can not easily neglect the results of XANES studies of Riglet-Martial et al.4 that seem rather conclusively indicate formation of Cr3+ in Cr-doped UO2. Therefore, we decided to perform re-analysis of these data using the reference XANES for Cr2+, the case that is missing in the analysis of Riglet-Martial et al.4.

Table 1 The ionic radius (r), and volume (V) of Cr and U cations in different oxidation state.
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The re-evaluation of the experimental XANES

The oxidation state of an atom in a solid matrix is usually measured with the help of XANES technique26. For that purpose, the measured data are compared to a set of reference data of a species in different oxidation state. In the previous studies of Riglet-Martial et al.4, the references for Cr0 (Cr metal), Cr3+ (Cr2O3, FeCrO4), and Cr6+ (CaCrO4) were used for analysis of XANES of Cr-doped UO2. The authors concluded that their XANES of Cr-doped UO2 resembles best the reference for Cr3+, which they use as a definitive evidence for the existence of Cr3+ in UO2. The conclusion is mainly supported by the best match of the positions of two pre-edges, which follows from the in-depth characterization of XANES spectra of various Cr-bearing compounds by Farges27. However, the match to the reference XANES is not ideal, which suggest possible other interpretation. In similar studies, Mieszczynski et al.7, although assumed Cr3+, also discuss possibility of existence of Cr2+ state, which could be indicated by a shoulder feature at 5996 Å, marked by the arrow in Fig. 3. In order to confirm or reject the possible Cr2+ state scenario they expressed a need for ab initio calculations. As such simulations performed here show definitely the Cr2+ state, we made an extensive literature search for a relevant reference for Cr2+ state. We found such data on Cr2+ in synthetic enstatite (Mg0.9Cr0.1SO3)28. With these, in Fig. 3 we plot the XANES of Cr-doped UO2 of Riglet-Martial et al.4, which are consistent with the data of Mieszczynski et al.7, together with the reference data for Cr3+ (Cr2O3) and for the aforementioned Cr2+. The comparison of the entire spectra shows that the XANES measured by Riglet-Martial et al.4 resembles as closely the Cr3+ as Cr2+ references. This refers to the spectral shape and the distribution of peaks and dips, that are, anyway, not perfectly reproduced by the references. This “imperfectness” is rather expected and, as explained by Farges27, comes from the different local environments of Cr in all the considered phases. Following analysis of Farges27, applied also by Riglet-Martial et al.4, it is the positions of the two pre-edges, or weighted centroids (of the peaks located at  ~5991 eV and  ~5994 eV in the case of Cr2O3, Fig. 3) that discriminate between the different oxidation states of Cr. There is a significant and progressive shift in the second pre-edge peak (the higher energy peak) and centroid positions as a function of Cr oxidation state (~1 eV per increase in the oxidation state). For different Cr species, the centroid should be located as follows:  ~5993 eV (Cr2+),  ~5994 eV (Cr3+),  ~5995 eV (Cr4+),  ~5995.7 eV (Cr5+) and  ~5996.7 eV (Cr6+) (see Fig. 4 of Farges27). We thus have checked the position of that centroid in XANES of Cr-doped UO2 and compared it to the references for Cr2+ and Cr3+. The positions of the first, lower energy pre-edge peak are very similar for both oxidation states of Cr7,27 (See Fig. 3 and Supplementary Fig. 1). We note that the comparison to the Cr2+ reference was not performed by Riglet-Martial et al.4.

Fig. 3: The XANES spectra of the Cr-doped UO2.
figure 3

a The spectra of the Cr-doped UO2 (thick solid blue line)4 and (thin solid blue line)7, and reference phases for Cr2+ 28 and Cr3+ 4; b the pre-edge region of the spectra. The black solid and dotted lines in b represent the linear baselines used for extraction of the second pre-edge feature peak and positions of the two pre-edges peaks as indicated by Riglet-Martial et al.4 (their Fig. 2b). We note very good agreement between the two independent measurements of the XANES spectra of the Cr-doped UO24,7. The arrow indicates the pre-edge 5996 Å feature speculated by Mieszczynski et al.7 to be a signature of the Cr2+.

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As shown in Fig. 3, the second pre-edge peak that discriminates the different Cr species is located at  ~5993 eV for Cr2+ and  ~5994 eV for Cr3+ species, and the offset of  ~0.7 eV is clearly visible. On the other hand, that peak is barely visible, but clearly detectable (Fig. 3) in the spectra of Cr-doped UO2 of Riglet-Martial et al.4. In order to check the right position of that pre-edge peak, we isolated the relevant peak from the spectra by subtracting out a linear baseline that spans the base of the feature. The obtained profiles of the peaks for the Cr2+ and Cr3+ references and the Cr-doped UO2 XANES spectra are compared in Fig. 4. The three peaks extracted from XANES are symmetric, which is demonstrated with good fits by the Gaussian (see Supplementary Table 1 for the fitted parameters). This also validates the applied data processing procedure. It is clearly visible that the position and height of the peaks of Cr2+ and the Cr-doped UO2 are similar and the position of the peak in the case of Cr3+ is significantly shifted rightwards by  ~0.7 eV. The so derived second pre-edge positions obtained by fitting with Gaussian function are given in Table 2. Here, we also report the positions of centroid (i.e., a weighted average of the all pre-edge peaks) obtained using the multi-peaks fitting procedure proposed by Farges27 and Wilke et al.29 (see Supplementary Note 1 and Supplementary Tables 2 and 3 for details).

Fig. 4: The extracted XANES pre-edges peaks for Cr-doped UO2, and reference Cr2+  and Cr3+  structures.
figure 4

The energy resolution of the measured spectra of Cr-doped UO2 is estimated at 0.3 eV4. This permits accurate fits of the pre-edge peak positions and conclusive identification of the clear offsets from the reference spectra.

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Table 2 The position of the second pre-edge peak feature obtained by fits by Gaussian function (\(A\exp (B{(x-C)}^{2})\)) as illustrated in Fig. 4 and the position of the centroid.
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The obtained positions of the pre-edge centroid were then compared with the results obtained by Farges27 for the set of Cr-materials containing Cr in different oxidation states ranging from “+2” to “+6”. The results are given in Fig. 5, which resembles Fig. 4 of Farges27. The positions of centroids we got for the Cr2+ and Cr3+ references (Table 2) fell within the given ranges of Farges27, indicating agreement with that data. The centroid position in the case of Cr-doped UO2 falls in the range corresponding to Cr2+. The careful analysis of the pre-edge XANES spectrum of Cr-doped UO2 shows thus rather clearly the “2+” oxidation state of chromium in Cr-doped UO2.

Fig. 5: The pre-edge centroid positions for the Cr in different oxidation states varying from “+2” to “+6”.
figure 5

The data represented by black triangles come from ref. 27. The blue diamond is the result for Cr-doped UO2 and the green circle and red square are the result for Cr3+ and Cr2+ references.

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Both, the above-discussed atomistic simulations and the experimental XANES data thus clearly indicate incorporation of chromium into UO2 matrix as Cr2+ species. This must be associated with the formation of U5+ species or O vacancies. This may affect the overall performance of such material, including its dissolution rate, which strongly depends on the U oxidation states30,31. In order to predict the most probable structural arrangement of Cr in UO2 matrix we computed a set of simple, different Cr incorporation arrangements involving single, as well as pair of dopants with charge balancing through formation of oxygen vacancies (Ov) or U5+ species. The eight computed configurations are visualized in Fig. 2 and some of them (e.g., 1Cr+1Ov (a) and 2Cr+2Ov (a)) were obtained with the aid of classical molecular dynamics simulations. All the constructed supercells were then carefully analyzed regarding the enthalpy and the free energy of relevant cation exchange reaction and the change in the matrix lattice parameter. The obtained results were analyzed against the available experimental data on the maximum solubility (requiring the reaction enthalpy and the free energy as parameters)1,4,9,32,33 and change in the lattice parameter of Cr-doped UO21,9.

The Cr solubility limit in UO2

Riglet-Martial et al.4 derived a simple thermodynamic model of maximum Cr solubility in UO2, which when fitted to the measured temperature dependent maximum Cr solubility in UO24,9,32,33 resulted in the estimation of Cr solution enthalpy and entropy (Table 5 of Riglet-Martial et al.4). The result depends on the Cr-oxide reference phase and for Cr2O3 they obtained the reaction enthalpy of 92.4 kJ mol−1, assuming Cr3+ incorporation into UO2. In Table 3 we provide the enthalpy of single Cr atom solution computed ab initio, assuming the following cation exchange reactions (having the size of computed supercell as 32 UO2 units), realizing eight considered structural arrangements of Cr in UO2 (Fig. 2):

$$\frac{{\rm{1}}}{{\rm{2}}}{\rm{C}}{{\rm{r}}}_{{\rm{2}}}{{\rm{O}}}_{{\rm{3}}}+{({\rm{U}}{{\rm{O}}}_{{\rm{2}}})}_{{\rm{32}}}\to {(({\rm{Cr}}:{\rm{U}}){{\rm{O}}}_{{\rm{2}}})}_{{\rm{32}}}+{\rm{U}}{{\rm{O}}}_{{\rm{2}}}+q{{\rm{O}}}_{{\rm{2}}}$$
(1)
Table 3 The computed solution enthalpy, ΔH, and free energy, ΔG, of Cr in UO2.
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In addition, in Table 3 we report an estimate for the Cr solution free energy (ΔG), which we derived by adding/subtracting the high-temperature entropy of molecular oxygen gas (S(O2) = 270 J mol−1 K−1, Chase34) and the oxygen partial pressure contribution to the free energy. Depending on the involvement of molecular oxygen as product or reactant, the free energy is estimated from:

$${\rm{\Delta }}G={\rm{\Delta }}H-T{\rm{\Delta }}S+q(RT{\mathrm{ln}}\,10){\mathrm{log}\,}_{10}{P}_{{{\rm{O}}}_{{\rm{2}}}},$$
(2)

where ΔS ≃ qS (O2) and ΔH is the enthalpy of single cation exchange reaction (Eq. (1)). The temperature dependent oxygen partial pressure was estimated using the model of Toker et al.35 and the values used in the calculations are reported in caption of Table 3.

The derived reaction enthalpies suggest that the 2Cr+1Ov incorporation mechanism is the most preferable, which is in line with the prediction of Guo et al.11. However, the reaction free energy—a more relevant parameter—points towards 1Cr+1Ov and 2Cr+2Ov scenarios, with the 1Cr+1Ov being the most preferable. We thus assume that Cr incorporates as an associated pair of Cr and Ov, and focus on checking the performance of the two structures (1Cr+1Ov (a) and 1Cr+1Ov (b), Fig. 2) in matching the measured solubility data. For that case we consider the following cation exchange reaction:

$$\frac{1}{2}{\rm{C}}{{\rm{r}}}_{{\rm{2}}}{{\rm{O}}}_{{\rm{3}}}+{({\rm{U}}{{\rm{O}}}_{{\rm{2}}})}_{{\rm{32}}^-}> {(({\rm{Cr}}:{\rm{U}}){{\rm{O}}}_{{\rm{2}}})}_{{\rm{32}}}+{\rm{U}}{{\rm{O}}}_{{\rm{2}}}+\frac{1}{4}{{\rm{O}}}_{{\rm{2}}}.$$
(3)

The activity, a, of Cr in UO2 is related to the oxygen gas partial pressure and the cation exchange reaction enthalpy and entropy as (Riglet-Martial et al.4, Eq. (6)):

$${\mathrm{log}\,}_{10}(a({\rm{Cr}}:{\rm{U}}{{\rm{O}}}_{{\rm{2}}}))=q \, {\mathrm{log}\,}_{10}{P}_{{{\rm{O}}}_{{\rm{2}}}}+\frac{1}{Rln10}({\rm{\Delta }}S-{\rm{\Delta }}H/T).$$
(4)

The 1Cr+1Ov structure represents a pair of Cr atom and associated Ov. In that case, for cubic fluorite structure, the activity of Cr in UO2 is a(Cr:UO2) = (y∕8), where y is the concentration of Cr in UO2. This is because an associated Ov has eight possibilities to arrange around the Cr atom (the case of a cube with Cr atom in the center and eight oxygen atoms in the corners), This gives the following equation for the maximum solubility of Cr in the case of 1Cr+1Ov structural arrangements:

$${\mathrm{log}\,}_{10}(y)=-\frac{1}{4}{\mathrm{log}\,}_{10}{P}_{{{\rm{O}}}_{{\rm{2}}}}+\frac{1}{R \, ln10}({\rm{\Delta }}S-{\rm{\Delta }}H/T)+{\mathrm{log}\,}_{10}8.$$
(5)

The 2Cr+2Ov structure represents a pair of associated Cr atoms with removed two oxygen atoms that bridge the Cr atoms. In that case, for cubic fluorite structure, the activity of Cr in UO2 is a(Cr:UO2) = (y∕12)1∕2. This is because for a pair of associated Cr atoms, second Cr atom has 12 possibilities to be arranged around the first Cr atom (the two Cr atoms are connected by a cube edge and there are 12 such arrangements (edges)), and the activity of a single atom is just the square root of the activity of a pair. This gives the following equation for the maximum solubility of Cr in the case of 2Cr+2Ov structural arrangement:

$${\mathrm{log}\,}_{10}(y)=-\frac{1}{2}{\mathrm{log}\,}_{10}{P}_{{{\rm{O}}}_{{\rm{2}}}}+\frac{2}{Rln10}({\rm{\Delta }}S-{\rm{\Delta }}H/T)+{\mathrm{log}\,}_{10}12.$$
(6)

The resulted Cr solubility, y, estimated for the conditions of various measurements reported in the literature, by applying the two discussed structural arrangement models and the computed cation exchange reaction enthalpies (Table 3) along with the experimental oxygen partial pressures, are given in Table 4. We note that such an estimate in the case of the preferred 1Cr+1Ov model, with the computed here ΔH, only slightly overestimates the measured solubilities and that the perfect match could be obtained with the reaction enthalpy increased by only ~21 kJ  mol−1. This is an excellent agreement, keeping in mind computational uncertainty of  ~50 kJ mol−1 22. On the other hand, in the case of 2Cr+2Ov model, the computed ΔH is by  ~90 kJ mol−1 larger than the value that matches the maximum solubility data. The good description of the Cr solubility data with the 1Cr+1Ov model is yet another argument supporting the Cr2+ scenario found in our atomistic modeling studies and by the re-evaluation of the XANES data, pointing towards 1Cr+1Ov structural arrangements as a realistic case. In the next step, we will check how that structure matches the measured lattice parameters of Cr-doped UO2.

Table 4 The computed and measured maximum solubility (y) of Cr in UO2 reported for the experimental conditions.
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Lattice parameter of Cr-doped UO2

It is established experimentally that the UO2 matrix doped with Cr undergoes contraction. The two available experimental studies of Leenaers et al.9 and Cardinaels et al.1 show linear decrease of the lattice parameter with content of Cr, with the amount of incorporated chromium of up to 1000 μg g−1. The measured relative decrease of lattice parameter, reported in Table 5, varies by a factor of 2 between the two experimental studies, but in both cases is smaller than the one expected from consideration of sizes of the involved species9. In Table 5 we also report the simulated change in the lattice parameter obtained for all the considered structures (Fig. 2). Interestingly, the 1Cr+1Ov (a) and 2Cr+2Ov (a) structures result in very small shift and the result obtained for 1Cr+1Ov (a) structure matches the measured data. The other structures, including the 2Cr+1Ov configuration, result in significantly, at least order of magnitude larger change. We note however, that most of the configurations, except 2Cr+1Uv and 2Cr+2Ov (a) cases, predict decrease of the lattice parameter. The small, and well consistent with the measurements, lattice contraction with 1Cr+1Ov (a) structure is thus yet another argument that this is the correct Cr incorporation model.

Table 5 The computed and measured relative change of lattice parameter a (Δa∕a) in Cr-doped UO2 with Cr content of 1000 μg g−1.
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Discussion

Using the accurate first principles method we performed intensive ab initio calculations of chromium-doped UO2 matrix that represents an enhanced nuclear fuel case. In spite of trying to obtain the Cr3+ oxidation state, either by co-formation of U5+ or creation of oxygen vacancies, we always ended up with the Cr2+ state. The consideration of U4+, Cr3+, and Cr2+ cations sizes shows that Cr2+ has much similar size to U4+ and Cr3+ is much smaller, indicating plausibility of Cr2+ scenario. The goodness of such, nevertheless, rather-unexpected prediction has been further demonstrated by the re-evaluation of the measured XANES with the previously omitted reference for Cr2+ species. The calculations show that the most thermodynamically favorable structure is with an associated pair of Cr atom and oxygen vacancy. In such a configuration, Cr atom is shifted towards cations sublattice vacant site, which results in reduced, pseudo sixfold coordination, as opposite to the eightfold coordinated uranium site. The thermodynamic consideration of such a structure results in a good match to the measured maximum solubility of Cr in UO2 and the small lattice contraction of Cr-doped UO2. This strongly indicates that we not only obtained a correct description of Cr oxidation state as “+2” valence state, but also good understanding of the structural incorporation of Cr into the UO2 matrix. In these aspects the obtained results should be very helpful for correct interpretation of the measurements of Cr-doped UO2 nuclear fuel. We hope that our studies will trigger supplementary investigations of not always intuitive redox solid state chemistry and structural/thermodynamic parameters of doped nuclear fuel matrices.

The reported results come from the synergy of atomistic modeling and data collected by different experimental techniques. A careful choice of the accurate computational method allowed for making an unexpected prediction regarding the oxidation state of Cr, which initiated the re-evaluation of the experimental data. This shows the need of atomistic modeling support for correct interpretation of the measurements and thus in-depth, correct understanding of challenging electronic systems such as, for instance, the doped UO2 matrices.

Methods

Density functional theory (DFT) simulations

The DFT-based ab initio calculations were performed with the DFT-based plane-wave Quantum-ESPRESSO package36. In DFT calculations, we specifically applied the gradient-corrected PBEsol exchange correlation functional37 because it results in a good description of structures of f-element-bearing materials22,38,39 and a good prediction of their vibrational40,41 and thermodynamic parameters25,42. We applied the scalar relativistic ultrasoft pseudopotentials to mimic the presence of core electrons of the considered atoms43. We note that in the case of uranium, the spin-orbit effects are of secondary importance and can be neglected22,44. As a result of our previous extensive tests studies, we applied the plane-wave cut-off energy of 50 Ryd22. The computed 2 x 2 x 2 supercell of UO2 contained 96 atoms and the Brillouin zone was sampled with the 2 x 2 x 2 Methfessel-Paxton k-points grid45. The size of the computed system and the k-points sampling are similar to the simulation setup used in the previous studies6,46,47,48. The lattice parameters and the atomic positions in all the computed structures were relaxed to the equilibrium positions, assuming constant pressure of P = 0 GPa with a tolerance of 0.1 GPa. The computed atomic structure is visualized in Fig. 1. It is of cubic fluorite-type with eightfold coordinated U atoms. As U and Cr atoms contain strongly correlated electrons, 5f and 3d, respectively, we applied the DFT+U method and derived the Hubbard U parameter from first principles using the linear response method49. This method is known to give accurate values for 3d elements50,51 and has been extensively tested by us for f elements and subsequently applied to describe the electronic structure of uranium oxides, including UO2 (e.g., refs. 15,22,38). Following the recent studies of Pegg et al.52, all the calculations were performed assuming anti-ferromagnetic transverse 1k spins arrangement in UO2. This resulted in a small distortion towards tetrahedral symmetry (~0.3% of the lattice parameter value). Computation of the correct ground state of UO2 is a challenge by itself and to assure the convergence to the correct ground state we applied the occupation matrix control technique53. After the tests performed by Beridze and Kowalski22 for uranium-based compounds, we estimate the uncertainty of the so computed solid state reaction enthalpies at the level of  ~50 kJ mol−1.

Molecular dynamic simulations

In order to probe the possible structural arrangements of Cr atoms within the UO2 matrix, focusing on the potential distortions of the computed atomic arrangements, we performed a set of test, qualitative level classical molecular dynamics simulations runs using GULP code54. For such purpose we applied the simple Buckingham-type pair potentials of Lewis and Catlow55 for the Cr2+–O, U4+ –O and O– O interactions. We also performed simulations with the more accurate Embedded Atom Model (EAM) force fields developed specifically for UO2 system by Cooper and Grimes56. In the second case we adjusted the parameters of Cr2+–O Buckingham-type pair potential to match the lattice parameters and the atomic arrangements of CrO solid, for the charge of Cr2+ of  +1.1104 imposed by the charges of U (+2.2208) and O (−1.1104) in the EAM potential of Cooper and Grimes56. We note that simple Buckingham-type potentials can not be expected to provide accurate (e.g., at the ab initio calculations level) structural parameters or energies and that mixing of the interaction potentials of different types is not always well grounded. However, we used the molecular dynamics simulations only to explore potential structural distortions of the computed structural arrangements of Cr in UO2 and these simulations were not used to derive any numbers or structural/thermodynamic parameters discussed in the paper. The molecular dynamics NPT (constant pressure-temperature) runs were performed using standard approach (e.g., ref. 12). These were 20 ps long, with time step of 0.001 ps, and were performed at T = 1000 K and P = 0 GPa, both controlled by the thermostat.