Electronic Structure Investigation of the Bulk Properties of Uranium

Article pubs.acs.org/IC

Cite This: Inorg. Chem. XXXX, XXX, XXX−XXX

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

Inorg. Chem. Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 08/17/18. For personal use only.

‡

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. 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.

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 © XXXX American Chemical Society

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 Received: June 8, 2018

A

DOI: 10.1021/acs.inorgchem.8b01561 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry Table 1. Magnetic Stability of (U, Pu)O2 for Various Pu Contentsa AFM

FM

Pu content

a0 (Å)

c/a

band gap (eV)

a0 (Å)

c/a

band gap (eV)

EFM − EAFM/atom (meV)

UO2 12.5% 18.75% 25% 31.25% 37.5% 50% PuO2

5.543 5.532 5.526 5.520 5.514 5.509 5.497 5.453

0.987 0.990 0.992 0.994 0.996 0.997 1.001 1.02

2.5 1.2 1.2 1.2 1.2 1.2 1.2 2.0

5.547 5.532 5.526 5.520 5.514 5.509 5.495 5.452

0.986 0.990 0.992 0.994 0.995 0.997 1.001 1.02

2.2 0.9 0.9 0.9 0.9 0.9 0.9 1.4

+10.00 +0.01 −0.11 −0.13 −0.14 −0.03 +0.22 +3.31

a0 is the lattice constant, and EFM − EAFM/atom is the difference in total energy between FM and AFM orders.

a

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 Exc = ExGGA + EcLDA + Ecnl

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 firstprinciples 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 firstprinciples 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

(1)

is the GGA exchange energy, ELDA accounts for the local where EGGA x c correlation energy, and Enlc 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 groundstate 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 B


Article

Inorganic Chemistry

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. 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

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 12atom 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 C


Article

Inorganic Chemistry evaluate the effect of the variation of the U parameter for a Pu4+ cation on the bulk properties of (U, Pu)O2. 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 UO2, (U0.5Pu0.5)O2, and PuO2. 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 UO2 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 (U0.5Pu0.5)O2 and for PuO2 when we allow the U parameter of Pu (UPu 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)O2 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 UO2, from 1.6 to 2.3 eV for PuO2, but only from 0.9 to 1.3 eV for (U0.5Pu0.5)O2. The values calculated for UO2 compare well to the available experimental value by Schoenes,59 and the calculated values for PuO2 compare well to the experimental value by McNeilly8 for the optimal value of U. We note here that the variation is larger for both pure oxides than that of (U, Pu)O2, suggesting that, for a small variation of the U parameter on (U, Pu)O2, the electronic properties do not vary strongly. These results on the impact of the U parameter for the Pu cation show that the choice of the U values from UO2 and PuO2 is accurate enough for the electronic structure calculations of (U, Pu)O2.

Figure 2. Lattice parameter of (U, Pu)O2 as a function of Pu content: Vegard’s law. Comparison with XRD experiments.2

Table 2. Lattice Constant of UO2, (U0.5Pu0.5)O2, and PuO2 Obtained from PBE+U and vdW-optPBE+U and Comparison with XRD Experiments2 lattice constant (Å)

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.4 This behavior has been proven experimentally for the uranium−plutonium mixed oxide2,41,60−63 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.31,64 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)O2 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 (dM−O) of (U, Pu)O2, UO2, and PuO2 (see Table 3). The calculated U− O and Pu−O distances in (U, Pu)O2 are different from their

oxide

XRD

PBE+U

vdW-optPBE+U

UO2 (U0.5Pu0.5)O2 PuO2

5.47 5.434 5.398

5.543 (+1.3%) 5.497 (+1.1%) 5.453 (+1.0%)

5.559 (+1.6%) 5.513 (+1.4%) 5.469 (+1.3%)

Table 3. Interatomic Distances M−O (with M = U, Pu) in the First Coordination Shell and Comparison of Calculated Values with Experimental Data2,42 (U, Pu)O2

compound

method

dU−O (Å)

dPu−O (Å)

DFT+U

(U0.75Pu0.25)O2

PBE+U vdW-optPBE+U PBE+U vdW-optPBE+U PBE+U vdW-optPBE+U XAS XAS XAS

2.38 2.39 2.40 2.41

2.37 2.38

UO2 PuO2 expt2,42

(U0.7Pu0.3)O2 UO2 PuO2

2.35 2.37

2.36 2.37 2.35 2.33

values in the pure oxides. The addition of Pu leads to a decrease in the U−O distance in comparison to its value in UO2, and to an increase in the Pu−O distance in comparison to its value in PuO2. 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 vdWoptPBE+U functionals, yields a satisfactory description of the structural properties of (U, Pu)O2 compounds. III.2. Elastic Properties. This section presents the results on elastic properties, especially the elastic constants C11, C12, C44, and bulk modulus B0 of (U, Pu)O2. 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 UO265 has D


Article

Inorganic Chemistry Table 4. Elastic Constants C11, C12, and C44 and Bulk Modulus B0 of (U, Pu)O2a C11 (GPa)

C12 (GPa)

C44 (GPa)

B0 (GPa)

functional

UO2

PBE+Ub LDA+U Expc PBE+Ub LDA+U Exp PBE+Ub LDA+U Exp PBE+Ub LDA+U Exp

364 401d 389e 112 132d 119e 58 94d 60e 196 222d 207g

(U0.75Pu0.25)O2

(U0.5Pu0.5)O2

(U0.25Pu0.75)O2

PuO2

375 402d

365

368

375

108 151d

116

115

111

62 78d

66

68

70

197 235d 156.9−221h

199

199

199 208f 178g

a

Comparison with available experimental data. bThis study. cExp = experiments. dRef 10. eRef 66. fRef 68. gRef 67. hSee ref 69.

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

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 B 0 . 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 PuO 2 . 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.

Figure 3. Projected density of states of antiferromagnetic UO2, (U0.75Pu0.25)O2, and PuO2, respectively, for 5f orbitals and oxygen 2p orbitals. E


Article

Inorganic Chemistry Table 5. Band Gap in (U, Pu)O2 and Its Evolution as a Function of the Pu Content. band gap functional

UO2

U0.94Pu0.06O2−U0.25Pu0.75O2

U0.18Pu0.82O2

U0.06Pu0.94O2

PuO2

PBE+U* vdW-optPBE+U * LDA+U Exp

2.5 2.6 2.3a 2.1b

1.2 1.4 1.5a

1.4 1.5

1.7 1.8

2.0 2.0 1.8c 1.8d

a

Ref 10. bRef 59. cRef 68. dRef 8.

Table 6. Enthalpy of Formation of UO2, PuO2, and Pu2O3 Using Both PBE+U and vdW-opt-PBE+U Functionalsa ΔHf (eV) oxide

Exp

CALPHAD

PBE+U

vdW-optPBE+U

UO2 PuO2 Pu2O3

−11.2475 −10.9476 −17.14 ± 0.2177 (T = 1373 K)

−11.2312 −10.9913 −17.1713

−10.86 −10.10 −15.71

−11.27 −10.90 −17.11

a

Comparison is made with data from Guéneau et al.12,13 and experimental data.

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

consistent with the experimental value presented above and agrees well with other electronic structure calculations.7,65 For PuO2, a band gap of 1.8 eV has been found experimentally through the determination of the activation energy for electronic conduction.8 Although this value is widely used as reference for many theoretical studies, it has been recently challenged by McCleskey et al.9 and Scott et al.71 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.72

εgABC = xεgAC + (1 − x)εgBC − Cgx(1 − x)

(2)

where Cg 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 Cg must be equal to zero. Iliopoulos et al. have however found an increase in Cg 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 XANES42,74 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−yPuy)O2, and PuO2. The enthalpy of formation is calculated from the results of the electronic structure calculations as follows: U

Pu yO2

ΔHfMOX = E Tot1−y

met − (1 − y)E Umet − yE Pu − 2EOO2

(3) U Pu O ETot1−y y 2

where is the DFT+U total energy of (U, Pu)O2, Emet U is the calculated chemical potential of metallic uranium in the F


Article

Inorganic Chemistry met α-phase reference state, EPu is the calculated chemical potential of metallic plutonium in the α-phase reference state, and EOO2 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

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: U1 − yPu yO2

ΔHf

U

ΔHf (eV) PBE+U

vdW-optPBE+U

(U0.75Pu0.25)O2 (U0.5Pu0.5)O2 (U0.25Pu0.75)O2

−10.69 −10.51 −10.33

−11.18 −11.09 −10.99

Pu O

UO

(4)

PuO

where ΔHf 1−y y 2, ΔHf 2, and ΔHf 2 are the enthalpies of formation of the three oxides and ΔHmix 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 ΔHmix(0) = ΔHmix(1) = 0 corresponding to pure elements, it can be shown that

Table 7. Enthalpy of Formation for (U, Pu)O2 Calculated Using Both PBE+U and vdW-opt-PBE+U Functionals (U, Pu)O2 oxide

= (1 − y)ΔHfUO2 + yΔHfPuO2 + ΔHmix

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 vdWoptPBE+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%,

UinPuO2 PuinUO2 ΔHmix(y) = ΔHsol × y 2 (1 − y) + ΔHsol

× y(1 − y)2 UinPuO2 ΔHsol =−

where PuinUO2 ΔHsol =

(5)

(

∂ Δ Hmix ∂y

)

(

∂ Δ Hmix ∂y

)

and y→1

are the solution enthalpies for y→0

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 2 2 contents. The solution enthalpies ΔHUinPuO and ΔHPuinUO can sol sol 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: UinPuO2 ΔHsol = PuinUO2 ΔHsol

1 [ΔHf (1 − Δy)] and Δy 1 = [ΔHf (Δy)] Δy

(6)

In the above equations, ΔHf(Δy) is the heat of formation of (U1−yPuy)O2 calculated at y = Δy, and ΔHf(1 − Δy) is the heat of formation of (U1−yPuy)O2 calculated at y = 1 − Δy. The heats of formation are calculated according to eq 3 in a 96atom 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: G


Article

Inorganic Chemistry U

Pu yO2

ΔHmix(y) = Etot1−y

UO2 PuO2 − (1 − y)Etot − yEtot

+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.

(7)

U1−yPuyO2 UO2 PuO2 where Etot , Etot , and Etot are the total energies calculated for the three oxides. Figure 4 shows the mixing

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

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).

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

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Ibrahim Cheik Njifon: 0000-0001-6450-8944 Marjorie Bertolus: 0000-0001-6598-5312 Roland Hayn: 0000-0002-7659-0073 Notes

The authors declare no competing financial interest.

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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 H


Article

Inorganic Chemistry

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received funding from the Euratom research and training programme 2014-2018 under grant 754329.

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J


Electronic Structure Investigation of the Bulk Properties of Uranium

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