Ab-initio study of C and O impurities in uranium nitride

Journal of Nuclear Materials 478 (2016) 112e118

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Ab-initio study of C and O impurities in uranium nitride €r Olsson* Denise Adorno Lopes, Antoine Claisse, Pa KTH-Royal Institute of Technology, AlbaNova University Center, Roslagstullsbacken 21, SE-106 91 Stockholm, Sweden

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 March 2016 Received in revised form 24 May 2016 Accepted 2 June 2016 Available online 4 June 2016

Uranium nitride (UN) has been considered a potential fuel for Generation IV (GEN-IV) nuclear reactors as well as a possible new fuel for Light Water Reactors (LWR), which would permit an extension of the fuel residence time in the reactor. Carbon and oxygen impurities play a key role in the UN microstructure, influencing important parameters such as creep, swelling, gas release under irradiation, compatibility with structural steel and coolants, and thermal stability. In this work, a systematic study of the electronic structure of UN containing C and O impurities using first-principles calculations by the Density Functional Theory (DFT) method is presented. In order to describe accurately the localized U 5f electrons, the DFT þ U formalism was adopted. Moreover, to avoid convergence toward metastable states, the Occupation Matrix Control (OMC) methodology was applied. The incorporation of C and O in the N-vacancy is found to be energetically favorable. In addition, only for O, the incorporation in the interstitial position is energetically possible, showing some degree of solubility for this element in this site. The binding energies show that the pairs (CeNvac) and (OeNvac) interact much further than the other defects, which indicate the possible occurrence of vacancy drag phenomena and clustering of these impurities in grain boundaries, dislocations and free surfaces. The migration energy of an impurity by single N-vacancy show that C and O employ different paths during diffusion. Oxygen migration requires significantly lower energy than carbon. This fact is due to flexibility in the UeO chemical bonds, which bend during the diffusion forming a pseudo UO2 coordination. On the other hand, C and N have a directional and inflexible chemical bond with uranium; always requiring the octahedral coordination. These findings provide detailed insight into how these impurities behave in the UN matrix, and can be of great interest for assisting the development of this new nuclear fuel for nextgeneration reactors. © 2016 Elsevier B.V. All rights reserved.

1. Introduction UN is considered as a potential fuel for Generation IV nuclear reactors (GEN-IV) [1,2]. It is an attractive fuel option due to the combination of high fissile nuclide density, high thermal conductivity, and its high melting point [3]. Additionally, it has been considered as a new fuel option for commercial Light Water Reactors, allowing a prolongation of the fuel residence time in the reactor, which enhances the economy of electricity production by the existing fleet [4]. However, UN powder is highly pyrophoric, which makes the fabrication process expensive due to the requirement of an inert atmosphere. Nevertheless, due to the historical focus on UO2 fuel, there is a

* Corresponding author. E-mail address: [email protected] (P. Olsson). http://dx.doi.org/10.1016/j.jnucmat.2016.06.008 0022-3115/© 2016 Elsevier B.V. All rights reserved.

significant lack of data about systems involving UN fuel. To pave the way for qualifying a UN fuel, a number of important issues must be addressed and better understood, e.g. creep, swelling, gas release under irradiation, compatibility with structural steel and coolants, and thermal stability. Previous experimental investigations of UN show that the level of C and O impurities can influence these parameters [5]. It was noted that in the concentration range of 0.1e0.15 wt% the presence of these impurity elements strongly increased the swelling. In higher concentrations (0.3e0.45%), carbonization of the inner surface of the cladding during the irradiation has been reported [6,7]. The mutual presence of C and O is hypothesized to be a catalyst for cladding carbonization [4], but an exhaustive explanation for this phenomenon is still missing. As a consequence, the C and O mass specification for the fuel was established to not exceed 0.15% for reliable operation of fuel elements [8], which significantly increases the difficulty in the development and establishment of a fabrication process.

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Moreover, there are also uncertainties regarding the influence of C and O on the fundamental properties of UN, such as the lattice parameter. Various experimental studies have reported that C impurities linearly increase the UN lattice parameter [9e11]. On the other hand, Muromura et al. [11] conclude that O does not influence this parameter (at least in the same range of concentrations as C, around 0.0e0.095 wt%). This can suggest that the UN structure can effectively accommodate O without building up significant (local) strain, but until now, there is no theoretical model to prove or disprove this hypothesis. In order to understand such phenomena, an analysis at the atomic-scale perspective can provide critical answers. Presently, Density Functional Theory (DFT), including the Hubbard U formalism (DFT þ U) [12e14] can be applied to correctly model the highly-localized and correlated electrons in UN. A review of existing literature reveals the existence of only a few theoretical investigations for the UN system [15e21], though none of them address, properly, the influence of C and O impurities. There have been previous studies for the effect of O, however, the calculations were not performed as correlated material [22e25] but, rather, with the standard general gradient approximation GGA. An extension using the DFT þ U is thus highly motivated. In this paper, a detailed investigation of the electronic behavior of C and O in the UN structure using the DFT þ U formalism is presented. Incorporation energies are discussed on the basis of the thermodynamic preferences of the system. Calculations of binding energies are presented as an attempt to identify possible clustering trends. Furthermore, impurity transport is addressed in order to identify the possible diffusion mechanisms and evaluate the different behaviors of these elements in a dynamic system. The aim of this work is to contribute to the understanding of C and O behavior in UN, in a general effort of explaining some of the experimental observations described in the literature. The results presented here are also important for the development of the fabrication technology and fuel performance analysis of UN fuels.

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Spin-orbit coupling and non-collinear magnetic ordering effects were ignored in order to simplify the calculations. The structural optimization was carried first allow the volume change which result in a slightly orthorhombic structure with lattice parameters: a ¼ 4.9005 Å, b ¼ 4.9722 Å and c ¼ 5.0352 Å. The higher distortion is observed in the z-direction, aligned with the spins moments, in agreement with the experimental observation (tetragonal structure with c/a ¼ 0.99935 at 4 K in experiments) [39]. The defects were then introduced, allowing for atomic relaxation with all symmetries removed but restraining the cell shape and the volume. The internal structural parameters were relaxed until the total energy was converged to at most 105 eV. The nudged elastic band (NEB) method [40,41] was employed for the evaluation of the migration barriers using three intermediate images. The cases simulated were vacancy and interstitial diffusion. In this way, the minimum energy path between two stable configurations could be determined.

2. Methodology The DFT calculations were performed using the scalar relativistic projector augmented wave (PAW) and the Perdew-BurkeErnzerhof (PBE) generalized gradient approximation [26,27], as implemented in the Vienna Ab Initio Simulation Package (VASP) [28e30]. In order to describe accurately the strong, on-site Coulomb repulsion among the localized U 5f-electrons, the generalized gradient approximation þ U formalism for the exchangecorrelation term was adopted. Furthermore, the occupation matrix control (OMC) scheme, developed by Dorado et al. [31e33], was used to avoid convergence toward metastable states (for detailed information see ref. [34]). The U and J parameters were specified as 2 eV and 0.1 eV, respectively, since these values give rise to lattice parameter and magnetic properties in good agreement with experiments [15,35]. Here these parameters were adopted for all cases. In this way the application of the U and J values is an approximation, since the appropriate values may change slightly as a function of local non-stoichiometry. In light of the slight changes a small variation of U and J would induce, this approximation is considered reasonable. The valence electrons explicitly treated in the calculations were 6s26p66d25f27s2 for U; 2s22p2 for C; 2s22p3 for N; and 2s22p4 for O. The plane-wave cutoff was set to 600 eV. The simulations were performed using 64-atom and 216-atom supercells, for which 4  4  4 and 2  2  2 Monkhorst-Pack meshes [36] were used, respectively. The localized spins on the uranium ions were assumed to be ordered in an antiferromagnetic (AFM) single layering since this is the experimental configuration for UN below 53 K [37,38].

Fig. 1. UN unit cell illustrating the simulated defects; (a) N vacancy, (b) impurity in N position, (c) impurities in two nearest neighbor N sites, (d) impurity in interstitial position, and (e) impurity in U position.

Table 1 Incorporation energies (eV) of carbon and oxygen impurities in UN computed with the GGA þ U formalism and OMC scheme. The references states used are the chemical potential of graphite, N2 (g), and O2 (g). Negative values indicate the energetically favorable incorporation. Defect

Formation energy (eV)

Incorporation energy (eV)

Nvac CN ON CNeON 1 nn Same spin Opposed spin CNeCN 1 nn Same spin Opposed spin ONeON 1 nn Same spin Opposed spin Nint Cint Oint NU CU OU

2.27

e 3.22 5.90

a

Relaxed to another position.

8.90 8.89 6.25 6.49 11.68 11.78 1.69 1.45 2.71 Not stablea Not stablea Not stablea

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Fig. 2. Superposed frames of atomic relaxation for carbon (left) and oxygen (right) incorporation in U-site of UN structure showing their different behaviors. Light blue spheres represent uranium ions, dark blue spheres represent N ions, and grey and red represent C and O ions respectively.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

this convention, a negative value means a more stable state. The incorporation energies obtained using Eq. (1) are presented in Table 1. The formation of N-vacancies (Fig. 1a) in UN is an endothermic process (with N2(g) as the reference state) and, consequently UN should only exhibit limited deviation from perfect stoichiometry and only at relatively high temperatures. According to these calculations the incorporation energy for C and O in the N position (Fig. 1b) is negative, which means that they are even more stable than nitrogen in this site, in concordance with experimental evidence [4]. The O and C incorporation energies are quite negative, compared to the N vacancy formation energy. That is, even with a formation of the N vacancy (necessary for placing therein O and C atoms) the process it is still energetically favorable. The most stable configurations observed are the simultaneous incorporation of ONeON, CNeON and CNeCN in dimers, two nearest neighbor N sites (Fig. 1c), indicating a clustering trend of these elements in UN. Additionally, oxygen is observed to be stable in the interstitial

3. Results and discussion 3.1. Incorporation energy All defects simulated are illustrated in the UN unity cell presented in Fig. 1. The energy balance for the incorporation of C and O atoms in UN structure can be characterized by the incorporation energy EI as follow:

EI ¼ EUNðC;OÞ  EUNðNvac Þ  Eref

(1)

where EUN(C,O) is the total energy of the final state with an impurity incorporated; EUN(Nvac) is the total energy of the system with a Nvacancy where the impurity was inserted, and Eref is the total energy of the incorporated atom in its reference system. In the present work the reference states for calculations were the chemical potentials of C, N and O, in Graphite, N2 (g) and O2 (g) states [42]. In

Np Up Ud Uf

Density of states (arb. units)

UN

N p (int) Np Up Ud Uf

Density of states (arb. units)

UN(Nint)

-7

-6

-5

-4

-3

-2

E-EF (eV)

Cp Np Up Ud Uf

UN (Cint)

-1

0

1

2

-7

-6

-5

UN (ON)

Cp Up Ud Uf

UN(CN)

-4

-3

-2

E-EF (eV)

Op Up Ud Uf

Op Np Up Ud Uf

UN(Oint)

-1

0

1

2

-7

-6

-5

-4

-3

-2

-1

E-EF (eV)

Fig. 3. Projected density of states of U, N and impurities atoms in UN, UN(CN), UN(ON), UN(Nint), UN(Cint) and N(Oint) simulated structures.

0

1

2

D.A. Lopes et al. / Journal of Nuclear Materials 478 (2016) 112e118

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Fig. 4. Atomic distances of C and O impurities from the 1 nn uranium ions after relaxation.

position (Fig. 1d), which indicates that part of the O solubility in UN lattice report in literature [43,44] is due to the occupancy of this sublattice, in contradiction with the common proposed formula UN1xOx. No experimental evidence of this phenomenon is available inliterature due to the experimental difficulty in distinguishing oxygen from N or the interstitial lattice. However, Kotomin et al. [22] previously reported similar trends using standard DFT calculations. U-site occupation (Fig. 1e) by any of the lighter elements exhibits dynamically instability. However, this effect revealed differences in C and O behavior in the UN structure. During the relaxation, C moved toward the nearest N position (Fig. 2), forming a CeN dimer. In contrast, O moved to the interstitial position. This

Binding energy (eV)

0.3

CN+ Nvac same spin

ON+ Nvac same spin

CN+ON same spin

CN+ Nvac opposite spin

ON+ Nvac opposite spin

CN+ON opposite spin

0.2

0.1

0.0

-0.1

CN + Nvac 1 NN

ON + Nvac 2 NN

3 NN

4 NN

1 NN

CN + ON 2 NN

CN+ CN same spin

0.3

Binding energy (eV)

result aligns with the incorporation energies obtained in which the interstitial position is energetically favorable for O. It is thus evident that the apparently atypical stability of O in the interstitial position is a real trend. In order to investigate the origin of such different behaviors, the projected density of states (DOS) is analyzed. Fig. 3 show the DOS for pure UN, UN(CN), UN(ON), UN(Cint) and UN(Oint) cases. From the density of states, it is possible to observe that the 2p orbital of C, N and O has the trend to occupy lower energies as the atomic number increases. A majority of densities of the 2p electrons in C are localized in the vicinity of 2 eV. The 2p electrons in N are spread in a range between 2 eV and 5.5. The majority of the 2p electrons in oxygen are localized below 5.5 eV but, additionally,

3 NN

4 NN

1 NN

2 NN

ON+ ON same spin ON+ ON opposite spin

CN+ CN opposite spin

0.2

0.1

0.0

-0.1

CN + CN 1 NN

ON + ON 2 NN

3 NN

4 NN

1 NN

2 NN

3 NN

4 NN

Fig. 5. Calculated binding energies (eV) for C and O impurities in UN. Positive values stand for attractive interactions.

3 NN

4 NN

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D.A. Lopes et al. / Journal of Nuclear Materials 478 (2016) 112e118 3.5

Vacancy 3.0

N same spin N oppo. spin

Interstitial

C same spin C oppo. spin

2.5

N C O

O same spin O oppo. spin

Em (eV)

2.0 1.5 1.0 0.5 0.0

Fig. 6. The migration barriers of C, N and O for vacancy-assisted migration (left) and for direct-interstitial diffusion (right).

there is a small population occupying 3.5 eV. Consequently, O is more strongly bound with the U electrons occupying the energy range below 5 eV. For all cases, the U 5f and 6d electrons are strongly hybridized with the 2p orbitals. For the interstitial position, it is possible to notice that C presents a very localized chemical bond with U and a low electronic population in the N energy range, creating, in this case, an increase in the population around the Fermi level. On the other hand, O in an interstitial site shows very strong bonding with Uranium (at deep energies) and an increase in the electronic population in the N energy range. In this case, there is no increase in the Fermi level population, which makes this configuration energetically stable. Therefore, it can be concluded that C only bonds in the UN structure with U, competing for U 5f 6d electrons with N. On the other hand, oxygen can partially bond with N, without influence the EF population.

3.2. Atomic distances Fig. 4 shows the atomic distances for 1nn uranium neighbors after incorporation and relaxation of C and O. The results show that C and O increase the intrinsic atomic distances, with a larger increment for O. It is also possible to observe that O creates a homogeneous distortion in the crystal lattice, expanded by the same amount in the three directions of the orthorhombic structure. On the other hand, C creates a distortion more localized in only two directions, indicating the prevalence of a directional chemical bond. Thus, it is expected that inclusion of any of these impurities will increase the lattice parameter of the crystal. This is in apparent disagreement with the experimental behavior reported by Muromura et al. [11]. One possible explanation could be, for instance, that clustering trends which are

predicted here were not taken into consideration in the experimental analysis. 3.3. Impurity interactions The binding energy between two defects in a crystalline structure can be calculated using the following equation.

Ebind ¼ ðED1 þED2 Þ  ðED þEBulk Þ

(2)

Here ED1 and ED2 are the total energies of the supercell with either defect introduced to the UN lattice independently. ED is the total energy of UN with two defects introduced into the material at a close distance to one other. EBulk represents the reference energy of the bulk UN. In this formulation, a positive binding energy indicates an attractive force between the defects. The calculated binding energies are show in Fig. 5. Due to the crystal symmetry for the distances 2 nn and 4 nn there are no available positions with opposite spin. From the results, it is evident that all impurities present a strong binding energy in the 1 nn configuration. In general, all the interactions are stronger when the two defects are in planes with opposed spins. The pair CNeON is an exception, showing interactions only in the cases where the impurities are in same spin planes. In the 1 nn distance the interactions can be ordered from the strongest to the weakest as following; (CNeNvac), (CNeCN), (CNeON) and (ONeNvac) with the same intensity, and (ONeON). The observed trend of the binding energies are considerably lower for 2 nn distance (especially for ON þ Nvac and ON þ ON, where there is no more attraction effect) is a consequence of the fact that in this configuration there is an uranium atom between the two defects, which strongly disturbs the binding between them. C and O present significant binding energy with the vacancy up

Fig. 7. Superposed frames of initial, intermediate, and final states for C (left) and O (right) diffusion assisted by an N vacancy. Light blue spheres represent Uranium ions, dark blue spheres represent N ions, and grey and red represent C and O ions respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

D.A. Lopes et al. / Journal of Nuclear Materials 478 (2016) 112e118

Fig. 8. C (right) and O (left) chemical bonds in the saddle point in the vacancy-assisted diffusion process. Light blue spheres represent Uranium ions, dark blue spheres represent N ions, and grey and red represent C and O ions respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

to 3 nn and 4 nn, respectively. Generally speaking, this attractive binding energy results in a preferential interchange of the vacancy with the N atoms around C and O. This result suggests that C and O can be carried by vacancies towards vacancy sinks, i. e. dislocations lines, grain boundaries, and free surfaces. This mechanism can cause C and O segregation on such sinks in a UN fuel pellet. This result, coupled with the incorporation energy, suggests that in these vacancy sinks the system will prefer to form pairs of OeO, CeO and CeC respectively. The carbon accumulated in these free surfaces can be the trigger for the cladding carbonization observed under in irradiation. 3.4. Migration barriers The migration energy barriers for vacancy-assisted and directinterstitial diffusion were calculated for carbon, oxygen and nitrogen. The obtained results are presented in Fig. 6. According to the calculations, O is the most mobile impurity driven by vacancy assisted diffusion in the UN structure once its migrating barrier is the lowest. The C impurity has the highest barrier for the vacancy assisted migration; 0.46 eV higher the N self-diffusion. It is also possible to notice that in the

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antiferromagnetic model C and N prefer diffusion between planes with different spin, but this is not relevant for the O diffusion. The relatively low barrier observed for the O vacancy-assisted diffusion is explained by the path followed in the diffusion process (Fig. 7). Carbon and nitrogen follow direct paths to the vacancy position pushing the uranium neighbors away. On the contrary, oxygen moves in the direction of the interstitial site reducing the force necessary to bypass the uranium neighbors. An analysis of the chemical bonds reveals that C forms a very directional chemical bonding with the nearest U atoms, trying to rebuild the octahedral coordination. Oxygen can bend its chemical bond, in a way to decrease the energy necessary for the diffusion process. Fig. 8 shows the chemical bonds of the saddle point configurations for C and O diffusion. It is possible to observe for O that two atomic distances stabilize in 2.38 Å, which is the same as that observed in the UO2 structure [45], forming in this way a pseudo UO2 coordination. In the C case the majority of the atomic distances are close to 3.0 Å, that is the normal distance in the octahedral coordination. The direct interstitial diffusion of O has a migration barrier approximately 0.15 eV lower than that of N self-diffusion, suggesting that this diffusion mechanism is significant in UN. The low energy barrier observed for the C direct-interstitial diffusion is again a consequence of the path used for the diffusion process (Fig. 9). Carbon shows an instability when crossing the crystalline plane to the next interstitial position, consequently this atom moves toward the closest N position, sharing the site with the nitrogen, in similarity to what occurred to the incorporation in uranium site. This low energy barrier is not especially significant due to the fact that the initial configuration is not energetically stable. It does entail, however, that any carbon found in interstitial sites, either from ballistic events or from processing, will rapidly diffuse and bind to vacancies or sinks, depending on the conditions. The high mobility of O in the UN structure, coupled with the strong binding energy between O and vacancies suggest that they should bind rapidly. The dominant mechanism for oxygen diffusion will, in most conditions, be vacancy-mediated, and can be more rapid than N self-diffusion. 4. Conclusions The incorporation of C and O impurities in N, U and interstitial lattice sites of uranium mononitride was investigated by means of the GGA þ U approximation in the framework of density functional theory, coupled with the occupation matrix control (OMC) scheme. The computed incorporation energies showed that C and O are energetically favorable in the N site, and revealed that O is also

Fig. 9. Superposed frames of initial, intermediate, and final states for C (left) and O (right) in the directed interstitial diffusion process.

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stable in the interstitial lattice position. C and O interact strongly with vacancies up to 3 nn and 4 nn respectively, suggesting that these impurities can be dragged by vacancies and cluster at grain boundaries, dislocations or surfaces (vacancy sinks). For the vacancy-assisted mechanism, O has a lower energy barrier for diffusion in the UN structure than N and C have. At the saddle point, O seemingly forms a pseudo UO2 coordination. This process decreases the energy necessary to bypass the uranium neighbors. On the other hand, C has very directional and inflexible chemical bonds with uranium. During the diffusion, this element rebuilds the octahedral coordination, creating chemical bonds with the uranium neighbors, and requiring much more energy in this process. For the direct interstitial diffusion C was found to have a low energy barrier, revealing that this element, if perturbed, should rapidly diffuse to a Nvac position. The direct interstitial diffusion of O presents an energy 0.15 eV lower the N self-diffusion and, thus, can be significant for UN fuel. Acknowledgements The computations were supported by the Swedish National Infrastructure. Financial support from SKB is acknowledged. Postdoctoral scholarship from CNPq e Brazil. References [1] Hj Matzke, Science of Advanced LMFBR Fuel, 1986. North Holland, Amsterdam. [2] P.D. Wilson (Ed.), The Nuclear Fuel Cycle, University Press, Oxford, 1996. [3] S.L. Hayes, J.K. Thomas, K.L. Peddicord, J. Nucl. Mater. 171 (1990) 289e299. [4] J. Zakova, J. Wallenius, Fuel residence time BWRs nitride fuel, Ann. Nucl. Energy 47 (2012) 182. [5] B.D. Rogozkin, N.M. Stepennova, A.A. Proshkin, At. Energy 95 (2003) 208e220. [6] A.G. Vakhtin, V.D. Dmitriev, S.N. Ermolaev, et al., Experience in operating fuel elements in a nitride core of a BR-10 reactor, in: Report at a SovieteFrench Seminar, Obninsk, 1992. [7] V.D. Dmitriev, L.I. Moseev, A.G. Vahtin, et al., Capacity for work mononitride fuel in the reactor BR-10 with burning until 8.2% h.a., in: 3rd Conference on Reactor Materials, Dimitrovgrad, Russia, 1993. [8] B.D. Rogozkin, N.M. Stepennova, YuYe Fedorov, M.G. Shishkov, F.N. Kryukov, S.V. Kuzmin, O.N. Nikitin, A.V. Belyaeva, L.M Zabudko, J. Nucl. Mater 440 (2013) 445e456. [9] E.H.P. Cordfunke, J. Nucl. Mater. 56 (1975) 319. [10] V.J. Tennery, E.S. Bomer, J. Am. Ceram. Sot 54 (1971) 247. [11] T. Muromura, H. Tagawa, Lattice parameter of uranium mononitride, J. Nucl. Mater. 79 (1979) 264. [12] V.I. Anisimov, J. Zaanen, O.K. Andersen, Phys. Rev. B 44 (1991) 943.

[13] V. Anisimov, I. Solovyev, M. Korotin, M. Czy_zyk, G. Sawatzky, Phys. Rev. B 48 (23) (1993) 16929. [14] I. Solovyev, P. Dederichs, V. Anisimov, Phys. Rev. B 50 (23) (1994) 16861. [15] D. Gryaznov, E. Heifets, E. Kotomin, Phys. Chem. Chem. Phys. 14 (13) (2012) 4482e4490. [16] M. Klipfel, V.D. Marcello, A. Schubert, J. van de Laar, P.V. Uelen, J. Nucl. Mater. 442 (2013) 253e261. [17] R.A. Evarestov, M.V. Losev, A.I. Panin, N.S. Mosyagin, A.V. Titov, Phys. Stat. Sol.(b) 245 (2008) 114e122. [18] W. Sun, I. Di Marco, P. Korzhavyi, A LDAþU and LDAþ dmft study of uranium mononitride: from nonmagnetic to paramagnetic and ferromagnetic, in: MRS Proceedings, vol. 1683, Cambridge Univ Press, 2014. [19] D. Bocharov, D. Grayaznov, Yu F. Zhukovskii, E.A. Kotomin, Surf. Sci. 605 (2011) 396e400. [20] E. Kotomin, R. Grimes, Y. Mastrikov, N. Ashley, J. Phys. Condens. Matter 19 (10) (2007) 106e208. [21] A.Yu Kuksin, S.V. Starikov, D.E. Smirnova, V.I. Tseplyaev, J. Alloy Compd. 658 (2016) 385e394. [22] E.A. Kotomin, YuA. Mastrikov, J. Nucl. Mater. 377 (2008) 492e495. [23] R.A. Evarestov, A.V. Bandura, M. Losev, E. Kotomin, Y.F. Zhukovskii, D. Bocharov, J. Comput. Chem. 29 (13) (2008) 2079e2087. [24] Yu F. Zhukovskii, D. Bocharov, J. Nucl. Mater. 393 (2009) 504e507. [25] D. Bocharov, D. Gryaznov, YuF. Zhukovskii, E.A. Kotomin, J. Nucl. Mater. 435 (2013) 102e106. [26] J.P. Perdew, K. Burke, Y. Wang, Phys. Rev. B 54 (1996) 16533. [27] G. Kresse, D. Joubert, Phys. Rev. B 59 (3) (1999) 1758. [28] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [29] G. Kresse, J. Hafner, J. Phys. Condens. Matter 6 (1994) 8245. [30] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [31] B. Dorado, B. Amadon, M. Freyss, M. Bertolus, M. Phys. Rev. B 79 (2009) 235125. [32] G. Brillant, A. Pasturel, Phys. Rev. B 77 (2008) 184110. [33] B. Dorado, G. Jomard, M. Freyss, M. Bertolus, Phys. Rev. B 82 (2010) 035114. [34] A. Claisse, M. Klipfel, P. Van-Uffelen, N. Lindbom, M. Freyss, P. Olsson, GGAþU Study of Uranium Mononitride: a Comparison of the U-ramping and Occupation Matrix Schemes and Incorporation Energies of Fission Products. Submitted to Journal of Nuclear Materials, 2016. [35] Y. Lu, B.-T. Wang, R.-W. Li, H.-L. Shi, P. Zhang, J. Nucl. Mater 410 (1) (2011) 46e51. [36] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (12) (1976) 5188. [37] N. Curry, Proc. Phys. Soc. 86 (1965) 1193e1198. [38] C.F. van Doorn, P. de V. du Plessis, J. Low. Temp. Phys. 28 (3/4) (1977) 391e400. [39] J.A.C. Marples, J. Phys. Chem. Solids 31 (1970) 2431e2439. [40] G. Mills, H. Jonsson, G.K. Schenter, Surf. Sci. 324 (1995) 305. [41] H. Jonsson, G. Mills, K.W. Jacobsen, in: B.J. Berne, G. Ciccotti, D.F. Coker (Eds.), Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, 1998. [42] G. Aylward, T. Findlay, S.I. Chemical Data, second ed., John Wiley &Sons, 1974. [43] N.A. Javed, J. Less-Common Met. 29 (1972) 155e159. [44] D. Yu Lyubimov, A.V. Androsov, G.S. Bulatov, K.N. Gedgovd, Radiokhimiya 56 (2014) 423e426. [45] B. Wasserstein, Cube-edges uraninites as a criterion age, Nat. Lond. 168 (1951) 380e381.