# Electronic structure investigation of the bulk properties of uranium–plutonium mixed oxides (U, Pu)O 2

Ibrahim Cheik Njifon, M. Bertolus, Roland Hayn, Michel Freyss

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Ibrahim Cheik Njifon, M. Bertolus, Roland Hayn, Michel Freyss. Electronic structure investigation of the bulk properties of uranium-plutonium mixed oxides (U, Pu)O 2. Inorganic Chemistry, 2018, 57 (17), pp.10974-10983. 10.1021/acs.inorgchem.8b01561. cea-02063607

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# Electronic Structure Investigation of the Bulk Properties of Uranium–Plutonium Mixed Oxides (U, Pu)O₂

Ibrahim Cheik Njifon,*,† Marjorie Bertolus,† Roland Hayn,‡ and Michel Freyss*,†

†CEA, DEN, DEC, Centre de Cadarache, 13108 Saint-Paul-Lez-Durance, France

‡IM2NP, Aix-Marseille Université, Campus Scientifique Saint-Jerôme, case 142, 13397 Marseille Cedex 20, France

**ABSTRACT:** We present electronic structure calculations of bulk properties of (U, Pu)O₂, in the whole Pu content range for which only very few experimental data are available. We use DFT+U and the vdW-DF functional in order to take into account the strong 5f electron correlations and nonlocal correlations. We investigate structural, elastic, electronic, and energetic properties of (U, Pu)O₂ in the approximation of the ideal solid solution as described by the special quasirandom structures (SQS) method. The results on electronic properties highlight the narrowing of the band gap due to the mixing of UO₂ and PuO₂. Results on the mixing enthalpy are used to describe the phase stability of (U, Pu)O₂ solid solutions, using both SQS configurations and a parametric method. In particular, the issue of an ideal solid solution on a limited supercell size is discussed.

## I. INTRODUCTION

Actinide oxide compounds are of wide interest in the nuclear industry. UO₂ and (U, Pu)O₂ with a Pu content of around 10% constitute the essential part of the currently used nuclear fuels. During the lifetime of UO₂ in the reactor, plutonium and minor actinides are produced. The presence of these radioactive elements results in very challenging safety issues for the long-term storage of nuclear waste, in particular for plutonium. Plutonium, however, can be reprocessed from the spent UO₂ fuel and used to fabricate mixed oxide (U, Pu)O₂ fuel. In this context, many studies are conducted on this compound either for current pressurized water reactors fuels (~10% Pu) or for generation IV sodium fast reactors fuels (~25% Pu) in order to describe well its basic properties. The study of structural, mechanical, and energetic properties is essential to improve the understanding and prediction of the behavior of the fuel in the reactor and its chemical or mechanical interaction with the cladding.¹

In the present work, we systematically investigate the structural, elastic, electronic, and energetic properties of (U, Pu)O₂ for various plutonium contents using electronic structure calculations in the framework of the density functional theory (DFT). The experimental studies of structural properties on a limited Pu content²,³ using X-ray diffraction suggest a linear evolution of the lattice constant, known as Vegard's law.⁴ As to the electronic structure, an experimental band gap of 2.1 eV was found for UO₂.⁵ This value has been reproduced with an acceptable accuracy by many electronic structure calculations.⁶,⁷ For PuO₂, the value of 1.8 eV was found through the determination of the activation energy for electronic conduction.⁸ Although this

value has been recently challenged,⁹ it remains the reference experimental value for many electronic structure calculations. On the contrary, no experimental value of the band gap of the uranium–plutonium mixed oxides has been published, even though the knowledge of electronic properties is important for heat transport, especially at high temperature. Moreover, there is no theoretical reference value to our knowledge, since the published values vary strongly from one author to another.¹⁰,¹¹ As to thermodynamic properties, the CALPHAD assessment by Guéneau et al.¹²⁻¹⁴ allows the determination of the phase diagram of the U–O, Pu–O, and U–Pu–O systems and some of its thermodynamic quantities such as formation enthalpies and entropies. There are no data concerning the enthalpy of mixing of the (U, Pu)O₂ system.

The article is organized as follows: In section II, we give the details of the methods used. In section III, we present the results on the bulk properties of (U, Pu)O₂, especially the structural, elastic, electronic, and energetic properties.

## II. METHODS

II.1. **DFT Functionals.** DFT calculations were performed using the projector augmented wave (PAW) method¹⁵,¹⁶ as implemented in the *Vienna Ab Initio Simulation Package* (VASP, version 5.3.5).¹⁷⁻¹⁹ The generalized gradient approximation (GGA) as parametrized by Perdew, Berke, and Ernzerhof (PBE)²⁰ was used for the description of the exchange-correlation interactions. A Hubbard-like term was added in order to take into account the strong 5f electron correlations in actinide cations,²¹ according to the DFT+U Liechtenstein scheme.²² The *U* and *J* parameters of DFT+U were set for uranium cations to
<table><caption>Table 1. Magnetic Stability of (U, Pu)O₂ for Various Pu Contentsᵃ</caption><thead><tr><th rowspan="2">Pu content</th><th colspan="3">AFM</th><th colspan="3">FM</th><th rowspan="2">E<sub>FM</sub> - E<sub>AFM</sub>/atom (meV)</th></tr><tr><th>a<sub>0</sub> (Å)</th><th>c/a</th><th>band gap (eV)</th><th>a<sub>0</sub> (Å)</th><th>c/a</th><th>band gap (eV)</th></tr></thead><tbody><tr><td>UO<sub>2</sub></td><td>5.543</td><td>0.987</td><td>2.5</td><td>5.547</td><td>0.986</td><td>2.2</td><td>+10.00</td></tr><tr><td>12.5%</td><td>5.532</td><td>0.990</td><td>1.2</td><td>5.532</td><td>0.990</td><td>0.9</td><td>+0.01</td></tr><tr><td>18.75%</td><td>5.526</td><td>0.992</td><td>1.2</td><td>5.526</td><td>0.992</td><td>0.9</td><td>-0.11</td></tr><tr><td>25%</td><td>5.520</td><td>0.994</td><td>1.2</td><td>5.520</td><td>0.994</td><td>0.9</td><td>-0.13</td></tr><tr><td>31.25%</td><td>5.514</td><td>0.996</td><td>1.2</td><td>5.514</td><td>0.995</td><td>0.9</td><td>-0.14</td></tr><tr><td>37.5%</td><td>5.509</td><td>0.997</td><td>1.2</td><td>5.509</td><td>0.997</td><td>0.9</td><td>-0.03</td></tr><tr><td>50%</td><td>5.497</td><td>1.001</td><td>1.2</td><td>5.495</td><td>1.001</td><td>0.9</td><td>+0.22</td></tr><tr><td>PuO<sub>2</sub></td><td>5.453</td><td>1.02</td><td>2.0</td><td>5.452</td><td>1.02</td><td>1.4</td><td>+3.31</td></tr></tbody></table>

4.50 and 0.51 eV, respectively. These values are in accordance with those used for UO₂ in earlier theoretical studies⁶,²³ and the values derived from experiments.²⁴ For plutonium cations, these values were set to 4.00 and 0.70 eV, in accordance with earlier DFT+U calculations by Jomard et al.²⁵ on PuO₂ and Dorado et al.¹⁰ on (U, Pu)O₂. The mixing of UO₂ and PuO₂ for the formation of (U, Pu)O₂ is expected to have a negligible impact on these parameters. We discuss this point in details in section II.5. In order to avoid the convergence to one of numerous metastable states induced by the DFT+U method and ensure as much as possible that the ground state is reached, we apply the occupation matrix control scheme.⁷,²⁵,²⁶ The strong 5f electron correlations can also be described using hybrid functionals,²⁷ self-interaction correction,²⁸ and dynamical mean field theory (DMFT),²⁹ but these approaches are more time-consuming compared to the DFT+U method which is found to be the best compromise between the computational effort and accuracy.

Furthermore, in order to improve the description of nonlocal interactions, we use the nonlocal van der Waals density functional (vdW-DF) in the vdW-optPBE parametrization.³⁰,³¹ In this approximation, the nonlocal correlation is calculated so that the exchange-correlation energy takes the form

$$E_{xc} = E_{x}^{GGA} + E_{c}^{LDA} + E_{c}^{nl} \qquad (1)$$

where EₓGGA is the GGA exchange energy, EₓLDA accounts for the local correlation energy, and Eₓnl is the nonlocal correlation energy. In the vdW-optPBE, the exchange energy is obtained by improving the enhancement factor in the PBE and the revised PBE (revPBE)³² exchange energies. The local correlation energy is obtained within the local density approximation (LDA). As to the nonlocal correlation energy, its expression is based on the electron densities interacting via a model response function. The accuracy of this method in the description of binding energies and solid structures has been demonstrated (see refs 30 and 31), especially for ionic systems. In the present study, vdW-DF is used in combination with DFT+U to account for the strong correlation of 5f electrons. This defines the vdW-optPBE+U approach. The vdW-optPBE+U results obtained for (U, Pu)O₂ are compared to those of GGA+U to assess the influence of the nonlocal interactions on the computed properties.

## II.2. Spin-Orbit Coupling and Disorder Description in (U, Pu)O₂.
We neglect the spin-orbit coupling (SOC) since taking it into account is computationally demanding and prevents the use of supercells sufficiently large for a more accurate description of random distribution of uranium and plutonium atoms. Moreover, SOC has been shown in many studies to have negligible impact on the ground-state properties of bulk UO₂ and PuO₂. Among them, we can cite LDA+U studies,³³ GGA+U investigations,⁶,³⁴⁻³⁷ as well as studies using hybrid functionals.³⁸ Furthermore, we tested the influence of SOC on point defect formation energies in UO₂³⁹ and found no significant effect. Thus, DFT+U calculations without taking into account the spin-orbit coupling are expected to yield good results on the properties of the bulk and defects of uranium and plutonium mixed oxides.

In order to study (U, Pu)O₂, we have also to take into account the spatial distribution of U and Pu in the cation sublattice. Despite the

miscibility gap observed for certain compositions in the phase diagram of the U-Pu-O system,³,¹⁴,⁴⁰,⁴¹ numerous experimental studies suggest that (U, Pu)O₂ is an ideal solid solution under stoichiometric conditions.²,⁴² This means that uranium and plutonium cations are randomly distributed in the cation sublattice. However, the modeling of an ideal solid solution requires a very large supercell in order to represent this random distribution in the mixture. Electronic structure calculations are currently limited to a few hundred atom supercells, especially for actinide oxides. We thus use the so-called “special quasirandom structures” (SQS) developed by Zunger et al.⁴³ to model the (U, Pu)O₂ solid solution. The principle of this method is to select in a set of binary structures generated the one for which the computed pair correlation function is the closest to the experimental value. In the 96-atom (U, Pu)O₂ supercells used in this study, one has to deal with 32 cations in the face-centered cubic (fcc) structure. The SQS configurations for 32-atom binary A-B alloy supercells in the fcc structure have been determined by Von Pezold et al.⁴⁴ for various A and B contents. We use the cation coordinates provided by these configurations for uranium and plutonium for various A and B contents. Even if these configurations cannot be the exact representation of an ideal solid solution, the relevance of this method to study bulk properties has been demonstrated. Other approaches aiming at modeling ideal solid solutions are the methods based on the coherent potential approximation (CPA)⁴⁵,⁴⁶ or the virtual-crystal approximation.⁴⁷ However, these models consider only the average occupations by 〈A〉 or 〈B〉 of sites in the case of A-B alloys, removing the information associated with the geometrical arrangements of atoms around a site.⁴³,⁴⁸ Furthermore, CPA cannot take into account the local lattice distortions induced by different alloying elements, which are observed in the oxygen sublattice in (U, Pu)O₂. These methods may therefore yield a less representative description of disordered structures in (U, Pu)O₂ than the SQS method.

## II.3. Magnetic Configuration in (U, Pu)O₂.
Another challenge in the modeling of the uranium-plutonium mixed oxide is the description of the magnetic order. Experimental studies⁴⁹⁻⁵¹ and first-principles calculations⁵² on UO₂ evidenced a 3k-antiferromagnetic order below the Neel temperature (30.8 K). Without taking into account the spin-orbit coupling effect, the properties are well described using a 1k-antiferromagnetic order.¹⁰ However, the first-principles calculations describe plutonium dioxide as antiferromagnetic at 0 K, while results of crystal-field calculations prove that PuO₂ is nonmagnetic Γ₁.⁴⁹,⁵³,⁵⁴ The anomaly in the magnetic response of PuO₂, however, suggests the existence of an antiferromagnetic exchange interaction between plutonium cations.⁴⁹ Owing to these contradictions on PuO₂, we have studied the magnetic stability in the mixed oxide (U, Pu)O₂ for various Pu contents taking into account two different magnetic configurations: the 1k-antiferromagnetic (AFM) and ferromagnetic (FM) orders. As shown in Table 1, the energy differences obtained between these magnetic orders are less than 1 meV. They can be positive or negative, except for the pure UO₂ and PuO₂ for which positive values are obtained (AFM configuration). There is also no difference on the lattice parameters, but almost 25% difference in the band gap. As observed on the uranium-neptunium mixed oxide⁵⁵⁻⁵⁷ where the magnetic field from

<footnote>ᵃa₀ is the lattice constant, and E<sub>FM</sub> - E<sub>AFM</sub>/atom is the difference in total energy between FM and AFM orders.</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Variation of lattice constant and band gap of UO₂, (U₀.₅Pu₀.₅)O₂, and PuO₂ as a function of the onsite Coulomb interaction parameter *U*. For (U₀.₅Pu₀.₅)O₂ only the *U* parameter of Pu is allowed to vary, and the values of the *J* parameter are kept constant.</figcaption></figure>

magnetic uranium ions induces a magnetic moment on neptunium ions, we adopt the AFM order for the uranium–plutonium mixed oxides for all compositions in order to have a consistent comparison of the ground-state properties with respect to the plutonium content.

**II.4. Computational Details.** For energy optimization, static calculations were performed in 96-atom (2 × 2 × 2) supercells, with a 2 × 2 × 2 k-point mesh, in accordance with the results of convergence tests. For the determination of elastic constants, we use the finite displacement method implemented in VASP, which makes finite distortions of the lattice to calculate the stress matrix response. This method is computationally very demanding. We therefore use a 12-atom supercell with a 6 × 6 × 6 k-point mesh. The use of 12-atom supercells only allows the calculation of elastic constants for a few Pu contents, namely, 25%, 50%, and 75%. In all calculations, a 0.1 eV Gaussian smearing was applied, and the energy cutoff of 500 eV was used. The full relaxation of the supercell was allowed until the

Hellmann–Feynman forces acting on ions become less than 0.01 eV Å⁻¹ and the convergence on energy less than 10⁻⁵ eV/atom.

**II.5. Effect of the Hubbard Parameter U of Pu⁴⁺ Cations on Bulk Properties of (U, Pu)O₂.** As mentioned in the previous section the use of PBE+U formalism requires the knowledge of two parameters. These are the strong onsite Coulomb interaction *U* and the exchange *J* parameters which are applied to the 5f electrons of the actinide cations. These parameters have been extracted from experiment for UO₂²⁴ and validated by many electronic structure calculations.⁶,¹¹,²³,⁵⁸ For PuO₂, these parameters have been adjusted from electronic structure calculations.²⁵

There is no indication in the literature, however, whether these values remain identical for both cations in the mixed oxides. Dorado et al.¹⁰ and Yang et al.,¹¹ both using LDA+U to evaluate structural and electronic properties of (U, Pu)O₂, applied different values of *U* for Pu⁴⁺ cations (*U* = 4.0 eV and *J* = 0.70 eV for Dorado et al., and *U* = 4.7 eV and *J* = 0.70 eV for Yang et al.). It is therefore important to
evaluate the effect of the variation of the *U* parameter for a Pu<sup>4+</sup> cation on the bulk properties of (U, Pu)O<sub>2</sub>. In this section, we present the effect of small variations of the onsite Coulomb interaction parameter *U* on the lattice parameter and the band gap of UO<sub>2</sub>, (U<sub>0.5</sub>Pu<sub>0.5</sub>)O<sub>2</sub>, and PuO<sub>2</sub>. The values of the *J* parameters are kept constant. The results are shown in Figure 1 and compared with experimental data.

The structural and electronic properties of actinide oxides presented in Figure 1 show a very small variation as a function of the *U* parameter. For *U* = 3 eV, an overestimation of 1% is observed in the lattice parameter of UO<sub>2</sub> compared to the experimental value. Further, when *U* increases from 3 to 5 eV, the lattice constant increases by 0.5%. A smaller variation is observed for (U<sub>0.5</sub>Pu<sub>0.5</sub>)O<sub>2</sub> and for PuO<sub>2</sub> when we allow the *U* parameter of Pu (U<sub>Pu</sub> in Figure 1) to vary from 3 to 5 eV. The overestimations compared to the experimental values are 1.0% and 1.2%, respectively. These are the order of magnitude of the usual PBE overestimation in interatomic distances. We can therefore conclude that a small variation of *U* parameter of Pu in (U, Pu)O<sub>2</sub> has a negligible impact on its structural properties.

As for the electronic properties, when *U* is increased from 3 to 5 eV, the band gap increases from 1.6 to 2.8 eV for UO<sub>2</sub>, from 1.6 to 2.3 eV for PuO<sub>2</sub>, but only from 0.9 to 1.3 eV for (U<sub>0.5</sub>Pu<sub>0.5</sub>)O<sub>2</sub>. The values calculated for UO<sub>2</sub> compare well to the available experimental value by Schoenes,<sup>59</sup> and the calculated values for PuO<sub>2</sub> compare well to the experimental value by McNeilly<sup>8</sup> for the optimal value of *U*. We note here that the variation is larger for both pure oxides than that of (U, Pu)O<sub>2</sub>, suggesting that, for a small variation of the *U* parameter on (U, Pu)O<sub>2</sub>, the electronic properties do not vary strongly.

These results on the impact of the *U* parameter for the Pulation show that the choice of the *U* values from UO<sub>2</sub> and PuO<sub>2</sub> is accurate enough for the electronic structure calculations of (U, Pu)O<sub>2</sub>.

### III. STRUCTURAL AND ELASTIC PROPERTIES OF URANIUM-PLUTONIUM MIXED OXIDES

#### III.1. Structural Properties.
The structural properties in terms of lattice parameters and interatomic distances are investigated as a function of the plutonium content using the PBE+U and vdW-optPBE+U functionals (cf., section II.1).

It is well-known that, for ideal solid solutions, the lattice parameter, and thus the interatomic distances, follows a linear evolution with respect to the composition, known as Vegard's law.<sup>4</sup> This behavior has been proven experimentally for the uranium-plutonium mixed oxide<sup>2,41,60-63</sup> and is confirmed by our DFT+U study, both for PBE+U and vdW-optBPE+U functionals, as shown in Figure 2, which shows the lattice constant for various Pu contents. The DFT+U results show a slight overestimation of around 1.1% (0.06 Å) for PBE+U and 1.4% (0.08 Å) for the vdW-optPBE+U calculations compared to the XRD results, as shown in Table 2 (values in parentheses).

This slight overestimation of lattice constant is within the error expected in GGA and should not affect the quality of the properties calculated. The values from vdW-optBPE+U are slightly larger than that from PBE+U, with a systematic difference of 0.3%. This could be due to the more repulsive character of the optPBE functional than the original PBE at short separation distances.<sup>31,64</sup>

Interatomic distances are also calculated to analyze the accuracy of the two functionals used in the modeling of the structural properties of the (U, Pu)O<sub>2</sub> compounds. To this aim we compare the DFT+U results to the X-ray absorption spectroscopy (XAS) experimental analysis of actinide-oxygen interatomic distances in the first coordination shell (d<sub>M-O</sub>) of (U, Pu)O<sub>2</sub>, UO<sub>2</sub>, and PuO<sub>2</sub> (see Table 3). The calculated U-O and Pu-O distances in (U, Pu)O<sub>2</sub> are different from their

<figure><img src="image_3.png" /><figcaption>Figure 2. Lattice parameter of (U, Pu)O<sub>2</sub> as a function of Pu content: Vegard's law. Comparison with XRD experiments.<sup>2</sup></figcaption></figure>

<table><caption>Table 2. Lattice Constant of UO<sub>2</sub>, (U<sub>0.5</sub>Pu<sub>0.5</sub>)O<sub>2</sub>, and PuO<sub>2</sub> Obtained from PBE+U and vdW-optPBE+U and Comparison with XRD Experiments<sup>2</sup></caption><thead><tr><th rowspan="2">oxide</th><th colspan="3">lattice constant (Å)</th></tr><tr><th>XRD</th><th>PBE+U</th><th>vdW-optPBE+U</th></tr></thead><tbody><tr><td>UO<sub>2</sub></td><td>5.47</td><td>5.543 (+1.3%)</td><td>5.559 (+1.6%)</td></tr><tr><td>(U<sub>0.5</sub>Pu<sub>0.5</sub>)O<sub>2</sub></td><td>5.434</td><td>5.497 (+1.1%)</td><td>5.513 (+1.4%)</td></tr><tr><td>PuO<sub>2</sub></td><td>5.398</td><td>5.453 (+1.0%)</td><td>5.469 (+1.3%)</td></tr></tbody></table>

<table><caption>Table 3. Interatomic Distances M-O (with M = U, Pu) in the First Coordination Shell and Comparison of Calculated Values with Experimental Data<sup>2,42</sup></caption><thead><tr><th>(U, Pu)O<sub>2</sub></th><th>compound</th><th>method</th><th>d<sub>U-O</sub> (Å)</th><th>d<sub>Pu-O</sub> (Å)</th></tr></thead><tbody><tr><td rowspan="2">DFT+U</td><td rowspan="2">(U<sub>0.75</sub>Pu<sub>0.25</sub>)O<sub>2</sub></td><td>PBE+U</td><td>2.38</td><td>2.37</td></tr><tr><td>vdW-optPBE+U</td><td>2.39</td><td>2.38</td></tr><tr><td rowspan="2">UO<sub>2</sub></td><td rowspan="2"></td><td>PBE+U</td><td>2.40</td><td></td></tr><tr><td>vdW-optPBE+U</td><td>2.41</td><td></td></tr><tr><td rowspan="2">PuO<sub>2</sub></td><td rowspan="2"></td><td>PBE+U</td><td></td><td>2.36</td></tr><tr><td>vdW-optPBE+U</td><td></td><td>2.37</td></tr><tr><td rowspan="3">expt<sup>2,42</sup></td><td rowspan="3">(U<sub>0.7</sub>Pu<sub>0.3</sub>)O<sub>2</sub></td><td>XAS</td><td>2.35</td><td>2.35</td></tr><tr><td>XAS</td><td>2.37</td><td></td></tr><tr><td>XAS</td><td></td><td>2.33</td></tr></tbody></table>

values in the pure oxides. The addition of Pu leads to a decrease in the U-O distance in comparison to its value in UO<sub>2</sub>, and to an increase in the Pu-O distance in comparison to its value in PuO<sub>2</sub>. The calculated U-O and Pu-O distances are then almost equal in the uranium-plutonium mixed oxides with 2.38 Å for a Pu content of 25%. These results are in good agreement with the experimental results. These comparisons show that the DFT+U method, using both PBE+U and vdW-optPBE+U functionals, yields a satisfactory description of the structural properties of (U, Pu)O<sub>2</sub> compounds.

#### III.2. Elastic Properties.
This section presents the results on elastic properties, especially the elastic constants C<sub>11</sub>, C<sub>12</sub>, C<sub>44</sub>, and bulk modulus B<sub>0</sub> of (U, Pu)O<sub>2</sub>. These properties are calculated using the density functional perturbation theory implemented in the VASP code, using a 12-atom supercell as mentioned in section II.4. A previous study in UO<sub>2</sub><sup>65</sup> has
<table><caption>Table 4. Elastic Constants C₁₁, C₁₂, and C₄₄ and Bulk Modulus B₀ of (U, Pu)O₂ᵃ</caption><thead><tr><th></th><th>functional</th><th>UO<sub>2</sub></th><th>(U<sub>0.75</sub>Pu<sub>0.25</sub>)O<sub>2</sub></th><th>(U<sub>0.5</sub>Pu<sub>0.5</sub>)O<sub>2</sub></th><th>(U<sub>0.25</sub>Pu<sub>0.75</sub>)O<sub>2</sub></th><th>PuO<sub>2</sub></th></tr></thead><tbody><tr><td rowspan="3">C<sub>11</sub> (GPa)</td><td>PBE+U<sup>b</sup></td><td>364</td><td>375</td><td rowspan="3">365</td><td rowspan="3">368</td><td rowspan="3">375</td></tr><tr><td>LDA+U</td><td>401<sup>d</sup></td><td>402<sup>d</sup></td></tr><tr><td>Exp<sup>c</sup></td><td>389<sup>e</sup></td><td></td></tr><tr><td rowspan="3">C<sub>12</sub> (GPa)</td><td>PBE+U<sup>b</sup></td><td>112</td><td>108</td><td rowspan="3">116</td><td rowspan="3">115</td><td rowspan="3">111</td></tr><tr><td>LDA+U</td><td>132<sup>d</sup></td><td>151<sup>d</sup></td></tr><tr><td>Exp</td><td>119<sup>e</sup></td><td></td></tr><tr><td rowspan="3">C<sub>44</sub> (GPa)</td><td>PBE+U<sup>b</sup></td><td>58</td><td>62</td><td rowspan="3">66</td><td rowspan="3">68</td><td rowspan="3">70</td></tr><tr><td>LDA+U</td><td>94<sup>d</sup></td><td>78<sup>d</sup></td></tr><tr><td>Exp</td><td>60<sup>e</sup></td><td></td></tr><tr><td rowspan="3">B<sub>0</sub> (GPa)</td><td>PBE+U<sup>b</sup></td><td>196</td><td>197</td><td rowspan="3">199</td><td rowspan="3">199</td><td>199</td></tr><tr><td>LDA+U</td><td>222<sup>d</sup></td><td>235<sup>d</sup></td><td>208<sup>f</sup></td></tr><tr><td>Exp</td><td>207<sup>g</sup></td><td>156.9–221<sup>h</sup></td><td>178<sup>g</sup></td></tr></tbody></table>

ᵃComparison with available experimental data. ᵇThis study. ᶜExp = experiments. ᵈRef 10. ᵉRef 66. ᶠRef 68. ᵃRef 67. ᵇSee ref 69.

established that vdW-optPBE+U yields elastic constants similar to those from the PBE+U functional. Therefore, in the present study, only PBE+U is considered. Table 4 shows the values obtained for the elastic constants C₁₁, C₁₂, C₄₄, and the bulk modulus B₀. We compare our results with available experimental data⁶⁶,⁶⁷ on UO₂ and PuO₂, as well as with LDA+U results available¹⁰,⁶⁸ on UO₂, (U, Pu)O₂, and PuO₂. Our C₁₁ value of 364 GPa from PBE+U on UO₂ is lower than the 389 GPa experimental one, with a difference less than 10%. This PBE+U value is also lower than the value of 401 GPa obtained by Dorado et al. (using LDA+U). The same tendency compared to Dorado's LDA+U values is observed for all Pu contents. For C₁₂ and C₄₄, our PBE+U results compare very satisfactorily with the experimental values of 119 and 60 GPa, respectively, for UO₂. The slight underestimation of elastic constants as compared to experimental data might be due to the overestimation of lattice constants by PBE+U, since the increase in lattice constant with temperature or pressure effects leads to the decrease in elastic constants.

On all compositions, we observe an increase of C₄₄ with respect to the Pu content, from 58 GPa for UO₂ to 70 GPa for PuO₂. Even though there are to our knowledge no experimental data on these quantities for the uranium-plutonium mixed oxides (U, Pu)O₂ and for PuO₂, the agreement between our PBE+U and the experimental results on UO₂ consolidates the fact that these results on C₁₁, C₁₂, and C₄₄ are good predictions for (U, Pu)O₂ and PuO₂.

Another quantity investigated in this study is the bulk modulus of uranium-plutonium mixed oxides for various plutonium contents. The PBE+U value for UO₂ of 196 GPa shows a good agreement with the experimental one of 207 GPa. Our calculated value for PuO₂ of 199 GPa is slightly overestimated as compared to the experimental value of 178 GPa. Thus, the calculated values for UO₂ and PuO₂ are in relatively good agreement with the experimental data.

For (U, Pu)O₂, only one composition on polycrystalline (U, Pu)O₂ has been studied experimentally (20% Pu), but the measured bulk modulus exhibits a large uncertainty since the two techniques used yield significantly different values (156.9 and 221 GPa) (see ref 69). Our calculated value of 197 GPa is between these results. Therefore, to conclude unequivocally on the quality of the DFT+U results concerning the elastic constants and bulk modulus, more experimental data are needed.

## IV. ELECTRONIC PROPERTIES

In this section, the electronic density of states and the band gap of the uranium-plutonium mixed oxides are described in order to provide a qualitative description of their electronic structure.

Figure 3 shows the projected density of states of antiferromagnetic UO₂, PuO₂, and (U₀.₇₅Pu₀.₂₅)O₂. We can first notice that all these oxides display an insulating character. UO₂ has been shown experimentally to be a Mott insulator⁷⁰ (since the gap is between f bands) with a band gap of 2.14 eV.⁵ This Mott-Hubbard character is reproduced using PBE+U calculations (Figure 3), with a band gap of 2.5 eV. This value is

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Projected density of states of antiferromagnetic UO₂, (U₀.₇₅Pu₀.₂₅)O₂, and PuO₂, respectively, for 5f orbitals and oxygen 2p orbitals.</figcaption></figure>
<table><caption>Table 5. Band Gap in (U, Pu)O₂ and Its Evolution as a Function of the Pu Content.</caption><thead><tr><th rowspan="2">functional</th><th colspan="5">band gap</th></tr><tr><th>UO<sub>2</sub></th><th>U<sub>0.94</sub>Pu<sub>0.06</sub>O<sub>2</sub>–U<sub>0.25</sub>Pu<sub>0.75</sub>O<sub>2</sub></th><th>U<sub>0.18</sub>Pu<sub>0.82</sub>O<sub>2</sub></th><th>U<sub>0.06</sub>Pu<sub>0.94</sub>O<sub>2</sub></th><th>PuO<sub>2</sub></th></tr></thead><tbody><tr><td>PBE+U*</td><td>2.5</td><td>1.2</td><td>1.4</td><td>1.7</td><td>2.0</td></tr><tr><td>vdW-optPBE+U *</td><td>2.6</td><td>1.4</td><td>1.5</td><td>1.8</td><td>2.0</td></tr><tr><td>LDA+U</td><td>2.3<sup>a</sup></td><td>1.5<sup>a</sup></td><td></td><td></td><td>1.8<sup>c</sup></td></tr><tr><td>Exp</td><td>2.1<sup>b</sup></td><td></td><td></td><td></td><td>1.8<sup>d</sup></td></tr></tbody><tfoot><tr><td colspan="6"><sup>a</sup>Ref 10. <sup>b</sup>Ref 59. <sup>c</sup>Ref 68. <sup>d</sup>Ref 8.</td></tr></tfoot></table>

<table><caption>Table 6. Enthalpy of Formation of UO₂, PuO₂, and Pu₂O₃ Using Both PBE+U and vdW-opt-PBE+U Functionals<sup>a</sup></caption><thead><tr><th rowspan="2">oxide</th><th rowspan="2">Exp</th><th colspan="3">ΔH<sub>f</sub> (eV)</th></tr><tr><th>CALPHAD</th><th>PBE+U</th><th>vdW-optPBE+U</th></tr></thead><tbody><tr><td>UO<sub>2</sub></td><td>-11.24<sup>75</sup></td><td>-11.23<sup>12</sup></td><td>-10.86</td><td>-11.27</td></tr><tr><td>PuO<sub>2</sub></td><td>-10.94<sup>76</sup></td><td>-10.99<sup>13</sup></td><td>-10.10</td><td>-10.90</td></tr><tr><td>Pu₂O₃</td><td>-17.14 ± 0.21<sup>77</sup> (T = 1373 K)</td><td>-17.17<sup>13</sup></td><td>-15.71</td><td>-17.11</td></tr></tbody><tfoot><tr><td colspan="5"><sup>a</sup>Comparison is made with data from Guéneau et al.<sup>12,13</sup> and experimental data.</td></tr></tfoot></table>

consistent with the experimental value presented above and agrees well with other electronic structure calculations.<sup>7,65</sup> For PuO₂, a band gap of 1.8 eV has been found experimentally through the determination of the activation energy for electronic conduction.<sup>8</sup> Although this value is widely used as reference for many theoretical studies, it has been recently challenged by McCleskey et al.<sup>9</sup> and Scott et al.<sup>71</sup> who argue that the optical band gap of 2.8 eV obtained by McCleskey et al. is more consistent, since the activation energy for electronic conduction is obtained for a temperature higher than 200 °C and is sensitive to defects and vacancies. Our DFT+U result shows a value of 2.0 eV, which is closer to the experimental value of 1.8 eV derived from the activation of electronic conduction. The charge-transfer insulating character of PuO₂ is found in our PBE+U calculations (Figure 3).

For the uranium–plutonium mixed oxides, the density of states is a superposition of those of UO₂ and PuO₂ 5f orbitals. In particular, the f orbitals of the uranium and plutonium in (U, Pu)O₂ keep the same sharp features and the hybridization character relative to the oxygen p orbitals as in the pure actinide dioxides UO₂ and PuO₂. Therefore, the uranium 5f states remain at a higher energy level compared to the oxygen 2p states as in UO₂. The plutonium 5f density of states overlaps with the oxygen 2p states at the top of the valence band as in PuO₂. As a consequence, there is a shift of the plutonium 5f valence band maximum within the valence band of uranium 5f states and a shift of plutonium 5f state conduction band minimum into the band gap of uranium 5f states.

The uranium 5f orbitals then form the top of the valence band while the bottom of the conduction band is made of the plutonium 5f orbitals. This leads to a narrowing of the band gap as shown qualitatively in Figure 3 and by the calculated band gap in Table 5. The band gap values obtained are 1.2 eV using PBE+U and 1.4 eV using vdW-optPBE+U for a Pu content ranging from 3% to 75%. These values increase from 1.2 or 1.4 eV to 2.0 eV for a Pu content from 75% to 100%. The values obtained using vdW-optPBE+U are slightly larger than that obtained using PBE+U.

Even if there is no experimental confirmation yet of a band gap narrowing in (U, Pu)O₂, this characteristic has already been observed in other semiconductor based alloys, such as InₓAl₁⁻ₓN, using first-principles calculations by Dridi et al.<sup>72</sup>

and confirmed experimentally by Iliopoulos et al.<sup>73</sup> The band gap of alloyed semiconductor AₓB₁⁻ₓC can be expressed as follows:

$$ \epsilon_g^{\text{ABC}} = x\epsilon_g^{\text{AC}} + (1-x)\epsilon_g^{\text{BC}} - C_g x(1-x) \quad (2) $$

where C_g is a bowing parameter. For binary semiconductors, the gap is a linear function of the lattice parameter. Since the lattice parameter varies linearly with the composition of the alloy (Vegard's law) as is the case for (U, Pu)O₂, the band gap is expected to evolve linearly with respect to the composition of alloys, which means that the bowing parameter C_g must be equal to zero. Iliopoulos et al. have however found an increase in C_g with the increase in Al content for InₓAl₁⁻ₓN. Moreover, Dridi et al. have obtained for InₓGa₁⁻ₓN a bowing factor of 0.71 eV, thus a narrowing of the band gap while the lattice parameter follows the Vegard's law.

In summary, the band gap in the uranium–plutonium mixed oxides is lower than those of the pure oxides UO₂ and PuO₂. Its density of states consists of uranium 5f states at the top of valence band and of plutonium 5f states at the bottom of conduction band, suggesting a Mott insulating character. These computed characteristics tend to be confirmed by DFT+U and XANES<sup>42,74</sup> results which show that the excess electrons are systematically localized on the plutonium cations to form Pu³⁺ for electron donor defects and the holes localized on the uranium cations to form U⁵⁺ for electron acceptor defects.

## V. ENERGETIC PROPERTIES

To predict the behavior of (U, Pu) mixed oxides at finite temperature, the energetic properties such as the enthalpy of formation and the mixing enthalpy must be accurately described. We present below the electronic structure calculation of the enthalpy of formation and the mixing enthalpy of the uranium–plutonium mixed oxides.

### V.1. Enthalpy of Formation of UO₂, (U₁⁻_yPu_y)O₂, and PuO₂
The enthalpy of formation is calculated from the results of the electronic structure calculations as follows:

$$ \Delta H_f^{\text{MOX}} = E_{\text{Tot}}^{\text{U}_{1-}^{\text{P}}\text{Pu}_{y}\text{O}_2} - (1-y)E_{\text{U}}^{\text{met}} - yE_{\text{Pu}}^{\text{met}} - 2E_{\text{O}}^{\text{O}_2} \quad (3) $$

where E<sub>Tot</sub><sup>U₁⁻_yPu_yO₂</sup> is the DFT+U total energy of (U, Pu)O₂, E<sub>U</sub><sup>met</sup> is the calculated chemical potential of metallic uranium in the
α-phase reference state, $E_{\text{Pu}}^{\text{met}}$ is the calculated chemical potential of metallic plutonium in the α-phase reference state, and $E_{\text{O}_2}^{\text{O}}$ is the calculated chemical potential of oxygen in the O₂ reference state. The metal reference states are calculated using the DFT+U total energy of the α-phases (orthorhombic for U and monoclinic for Pu), considering the same functionals and the same calculation parameters as for the uranium and plutonium cations in (U, Pu)O₂. As for the oxygen reference state, the same functionals and the same calculation parameters as for the oxygen ions in (U, Pu)O₂ are considered for the calculations of the O₂ molecule. The stable triplet configuration of O₂ is also taken into account.

Table 6 and 7 show the values of the enthalpy of formation calculated for pure oxides and mixed oxides with different

<table><caption>**Table 7.** Enthalpy of Formation for (U, Pu)O₂ Calculated Using Both PBE+U and vdW-opt-PBE+U Functionals</caption><thead><tr><th rowspan="2">(U, Pu)O<sub>2</sub> oxide</th><th colspan="2">ΔH<sub>f</sub> (eV)</th></tr><tr><th>PBE+U</th><th>vdW-optPBE+U</th></tr></thead><tbody><tr><td>(U<sub>0.75</sub>Pu<sub>0.25</sub>)O<sub>2</sub></td><td>-10.69</td><td>-11.18</td></tr><tr><td>(U<sub>0.5</sub>Pu<sub>0.5</sub>)O<sub>2</sub></td><td>-10.51</td><td>-11.09</td></tr><tr><td>(U<sub>0.25</sub>Pu<sub>0.75</sub>)O<sub>2</sub></td><td>-10.33</td><td>-10.99</td></tr></tbody></table>

compositions using two different functionals: the PBE+U and the van der Waals vdW-optPBE+U functionals. The results are compared with those obtained using the thermodynamic modeling method CALPHAD by Guéneau et al. on UO₂,¹² PuO₂, and Pu₂O₃.¹³ We also compare our results with experimental data at room temperature, except for that of Pu₂O₃.

For pure oxides UO₂, PuO₂, and Pu₂O₃ (Table 6), the absolute values of the enthalpies of formation calculated using PBE+U, which correspond to the energy released on the formation of these oxides, are slightly lower than those yielded by both experiments and CALPHAD thermodynamic modeling (3% for UO₂ and 8% for both PuO₂ and Pu₂O₃). By taking into account the nonlocal interactions through the van der Waals functional in the framework of vdW-optPBE+U, however, the difference between DFT+U and both experiments and CALPHAD is significantly decreased for the three oxides and becomes less than 0.4%. This means that the van der Waals dipole-dipole and dipole-quadrupole interactions are not negligible in the case of iono-covalent systems as pointed out by Moellmann et al.⁷⁸ and Zang et al.⁷⁹ The vdW-optPBE+U results compare very well with the results of the experimental characterization and CALPHAD modeling. The improvement in the formation enthalpy of these oxides can also be explained by the improvement in the description of the binding energy of the O₂ gas molecule, which improves the calculated chemical potential of oxygen in the O₂ reference state compared to the PBE+U calculations. The chemical potentials of uranium and plutonium in their metallic reference states are also probably improved since Klimeš et al.³¹ show an improvement of the energetic description of many simple compounds, including transition metals. In addition, the difference between the enthalpy of formation at 0 K and at room temperature is estimated to be less than 0.02 eV in UO₂, using experimental heat capacities. This promotes the efficiency of this method in the prediction of the enthalpy of formation in the U-Pu-O system. We have therefore investigated the enthalpies of formation of (U, Pu)O₂ compounds (Table 7), with plutonium contents of 25%,

50%, and 75%. The values obtained using the vdW-optPBE+U approach show a regular evolution from the value in UO₂ to that in PuO₂. However, the behavior expected is a non linear curve due to the enthalpy of mixing of UO₂ and PuO₂ according to the expression of the enthalpy of formation:

$$ \Delta H_{f}^{\text{U}_{1-y}\text{Pu}_{y}\text{O}_{2}} = (1 - y)\Delta H_{f}^{\text{UO}_{2}} + y\Delta H_{f}^{\text{PuO}_{2}} + \Delta H_{\text{mix}} \quad (4) $$

where $\Delta H_{f}^{\text{U}_{1-y}\text{Pu}_{y}\text{O}_{2}}$, $\Delta H_{f}^{\text{UO}_{2}}$, and $\Delta H_{f}^{\text{PuO}_{2}}$ are the enthalpies of formation of the three oxides and $\Delta H_{\text{mix}}$ is the mixing enthalpy.

The reason for the apparent linear evolution is due to the fact that the mixing enthalpy is very small (on the order of meV) compared to the enthalpy of formation, explaining why its effect cannot be captured here. The mixing enthalpy is discussed in the next section.

## V.2. Structural Stability: Calculation of the Mixing Enthalpy Using Parametric and Direct Calculations Using SQS Configurations on (U, Pu)O₂.
We present here our results on the mixing enthalpy of (U, Pu)O₂ obtained from calculations using SQS configurations and from a parametric model. The parametric model is based on the polynomial approximation of the mixing enthalpy as proposed by Sluiter et al.⁸⁰ for alloys. Assuming a third-order cubic representation and using the limiting conditions $\Delta H_{\text{mix}}(0) = \Delta H_{\text{mix}}(1) = 0$ corresponding to pure elements, it can be shown that

$$ \Delta H_{\text{mix}}(y) = \Delta H_{\text{sol}}^{\text{UninPuO}_2} \times y^2(1-y) + \Delta H_{\text{sol}}^{\text{PuinUO}_2} \times y(1-y)^2 \quad (5) $$

where $\Delta H_{\text{sol}}^{\text{UninPuO}_2} = -\left(\frac{\partial \Delta H_{\text{mix}}}{\partial y}\right)_{y \to 1}$ and $\Delta H_{\text{sol}}^{\text{PuinUO}_2} = \left(\frac{\partial \Delta H_{\text{mix}}}{\partial y}\right)_{y \to 0}$ are the solution enthalpies for

uranium in the lattice of PuO₂ and vice versa, respectively. Thus, the knowledge of the solution enthalpies is sufficient to determine the mixing enthalpy in the entire range of plutonium contents. The solution enthalpies $\Delta H_{\text{sol}}^{\text{UninPuO}_2}$ and $\Delta H_{\text{sol}}^{\text{PuinUO}_2}$ can be calculated using DFT+U by substituting one uranium atom in the PuO₂ matrix and one plutonium atom in the UO₂ matrix in the 96-atom supercell, as done in this study. Since the composition is varying discretely in the DFT calculations, the derivatives appearing in the solution enthalpies are replaced by finite differences:

$$ \begin{aligned} \Delta H_{\text{sol}}^{\text{UninPuO}_2} &= \frac{1}{\Delta y}[\Delta H_f(1 - \Delta y)] \quad \text{and} \\ \Delta H_{\text{sol}}^{\text{PuinUO}_2} &= \frac{1}{\Delta y}[\Delta H_f(\Delta y)] \end{aligned} \quad (6) $$

In the above equations, $\Delta H_f(\Delta y)$ is the heat of formation of (U₁-yPu_y)O₂ calculated at y = Δy, and $\Delta H_f(1 - Δy)$ is the heat of formation of (U₁-yPu_y)O₂ calculated at y = 1 - Δy. The heats of formation are calculated according to eq 3 in a 96-atom supercell containing 32 actinide atoms, namely, 1 uranium and 31 plutonium atoms or 1 plutonium and 31 uranium atoms. This corresponds to Δy = 0.03125.

As for SQS calculations, the mixing enthalpy is determined using direct calculations of the total energy in the entire accessible compositions of the SQS structures. This is given by the equation as follows:
$$ \Delta H_{\text{mix}}(y) = E_{\text{tot}}^{U_{1\to7}Pu_{y}O_{2}} - (1-y)E_{\text{tot}}^{UO_{2}} - yE_{\text{tot}}^{PuO_{2}} \quad (7) $$

where $E_{\text{tot}}^{U_{1\to7}Pu_{y}O_{2}}$, $E_{\text{tot}}^{UO_{2}}$, and $E_{\text{tot}}^{PuO_{2}}$ are the total energies calculated for the three oxides. Figure 4 shows the mixing

<figure><img src="image_5.png" /><figcaption>Figure 4. Mixing enthalpy of (U, Pu)O₂ in the entire range of Pu contents, using both parametric method (♦) and SQS configurations (▲). The parametric values are given by the light blue diamonds (PBE+U) and the dark blue diamonds (vdW-optPBE+U). The SQS values are given by the orange triangles (PBE+U) and dark red triangles (vdW-optPBE+U).</figcaption></figure>

enthalpy of the uranium–plutonium mixed oxides in the entire range of plutonium contents. PBE+U and vdW-optPBE+U calculations were carried out for both parametric (light blue and dark blue diamonds) and SQS (orange and dark red triangles) methods.

The mixing enthalpy of (U, Pu)O₂ is found negative in the entire range of Pu contents using both SQS and parametric calculations, with very small values, at most 9 meV, which represents 0.08% of the enthalpy of formation. This suggests that there is no demixing due to the variation of Pu content in the material. In other words, in the entire range of Pu contents, for O/(U + Pu) = 2, the actinide sublattice forms a solid solution. This has been suggested from the experimental results for (U₀.₇Pu₀.₃)O₂ by Vigier et al.⁴² and for (U, Ce)O₂, a surrogate material for (U, Pu)O₂, by Martin et al.⁸¹ The present assumption agrees with the U–Pu–O phase diagram, in which the miscibility gap is observed only for a O/(U + Pu) ratio that is different from 2. For perfect stoichiometry, only one phase is mentioned.

The mixing enthalpy calculations using the SQS configurations display a regular evolution as a function of the Pu content only for Pu contents close to 0 or 1, less than 0.125 and higher than 0.875. Between these two regions, the mixing enthalpy is scattered. However, considering the very low energies involved, we can conclude that our SQS results are close to the behavior of an ideal solid solution.

In contrast to the SQS calculations, the parametric model yields a regular behavior for the mixing enthalpy as a function of the Pu content. The global minimum is found at y = 0.625 for PBE+U calculations and y = 0.4375 for vdW-optPBE+U calculations.

Given the scatter of the SQS mixing enthalpies and the improvement of the enthalpy of formation using vdW-optPBE

+U (in Table 6), we consider that the parametric mixing enthalpy using vdW-optPBE+U is the best estimate of the (U, Pu)O₂ mixing enthalpy as a function of the Pu content.

## VI. CONCLUSION

The present study focuses on the investigation of the structural, elastic, electronic, and energetic properties of (U, Pu)O₂ as a function of the Pu content using first-principles DFT+U calculations. We have first demonstrated that a small variation of the plutonium U parameter of DFT+U does not impact significantly the properties of the uranium–plutonium mixed oxides, which justify the use of the values found in UO₂ and PuO₂ for the investigation of the (U, Pu)O₂ mixed oxides. We have calculated the lattice constant and interatomic distances and found a good agreement with XRD data. Elastic constants and bulk moduli are also calculated for UO₂, PuO₂, and (U, Pu)O₂. The values calculated for UO₂ and PuO₂ agree well with experimental data. We can therefore conclude that the values calculated for (U, Pu)O₂ with different Pu contents are an accurate prediction of the elastic properties of the mixed compounds. As to the electronic properties, a narrowing of the band gap is observed in (U, Pu)O₂ compared to UO₂ and PuO₂. Even if there is no experimental confirmation, the fact that the same behavior is observed in other alloyed semiconductors supports our prediction. This could be confirmed by experimental photoemission spectroscopy investigations on (U, Pu)O₂. Finally, for energetic properties, we predict enthalpies of formation of (U, Pu)O₂ and show that the calculated values for pure UO₂, PuO₂, and Pu₂O₃ oxides agree very well with experiments and CALPHAD calculations. Especially, the vdW-optPBE+U values are very close to experimental ones, within a precision of less than 0.4%, which highlights the effect of nonlocal van der Waals interactions on the energetic properties of actinide based materials. The calculated mixing enthalpies are negative for all Pu contents, showing that (U, Pu)O₂ is a stable solid solution for all Pu contents. Direct calculations using SQS approximation give the mixing enthalpy that is relatively close to the behavior of an ideal solution, even if these values are scattered. Better options for the investigation of the structural configurations which provide more accurate thermodynamic averages at finite temperature might be, for instance, methods based on Monte Carlo sampling.⁸²

## ACKNOWLEDGMENTS

This work was partly performed using HPC resources from GENCI (Grant x2017096008). The authors wish to express their gratitude to G. Jomard and E. Bourasseau for fruitful discussions. This research contributes to the joint programme on nuclear materials (JPNM) of the European energy research alliance (EERA) and is part of the INSPYRE project, which has received funding from the Euratom research and training programme 2014-2018 under grant 754329.

<footnote>*E-mail: Ibrahim.cheiknjifon@cea.fr.</footnote>

<footnote>*E-mail: michel.freyss@cea.fr.</footnote>
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