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First-principles study of surface properties of crystalline and amorphous uranium aluminides
Applied Surface Science 502 (2020) 144132
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First-principles study of surface properties of crystalline and amorphous uranium aluminides
T
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Zhi-Gang Mei , Abdellatif M. Yacout Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Surface energy Uranium aluminides First-principles calculations Amorphous phases
Surface energy is a crucial material property required for the modeling of fission gas bubble behavior in nuclear materials. However, there is no information available regarding the surface properties of uranium aluminides. In this work we systematically investigate the surface properties of uranium aluminides (UAl3 and UAl4) by firstprinciples density functional theory (DFT) calculations. A total of 13 surface orientations with a maximum Miller index (MMI) up to 3 are studied for cubic UAl3, while 19 surfaces with a MMI of 2 for orthorhombic UAl4. Using the surface energies predicted by DFT, the surface properties of equilibrium single crystal UAl3 and UAl4, including surface area weighted surface energy, dominant surface orientations and surface anisotropy, are obtained from Wulff construction. To understand gas bubble behaviors in amorphous uranium aluminides under irradiation, we study the surface properties of amorphous UAl3 and UAl4. Compared to the crystalline phases, the amorphous phases are found to have lower surface energies due to the reduced number of breaking U-Al bonds close to the surface. The currently obtained surface properties of UAl3 and UAl4 can be used for the future modeling of gas bubble behaviors in both crystalline and amorphous uranium aluminide fuels.
1. Introduction Uranium aluminides, e.g., UAl2, UAl3 and UAl4, have been extensively studied by both experiment and theory because of their scientific and technological importance [1–10]. UAlx/Al dispersion fuels (UAlx represents a mixture of UAl2, UAl3 and UAl4) are used as nuclear fuels for research reactors worldwide [11]. Meanwhile, U-Mo/Al dispersion fuels are currently under investigation as one of the candidate low enriched uranium (LEU) fuels to convert the high enriched uranium (HEU) fuels used for the European high-power research reactors [12]. Experiments of U-Mo/Al dispersion fuel show that interaction layer (IL) forms between U-Mo fuel particles and Al matrix during the fabrication of fuel plate or under neutron irradiations. By the thermal annealing of diffusion couple and nuclear fuel plate, Mazaudier et al. [13] found that the IL consists of binary U-Al compounds, i.e., UAl3 and UAl4, and several ternary U-Mo-Al compounds. In comparison, neutron irradiation tests show that the IL is mainly composed of amorphous uranium aluminide phases [14]. It is well known that the IL formed in U-Mo/Al dispersion fuel is detrimental to the fuel performance, such as reducing overall thermal conductivity and causing abnormal fuel plate swelling at high fission densities [12]. Fundamental material properties of uranium aluminides are important to the qualification of U-Mo/Al dispersion fuels. The electronic ⁎
and magnetic properties of binary uranium aluminides, i.e., UAl2, UAl3 and UAl4, have been extensively studied [2–6,9]. Recently, the thermophysical properties of binary and ternary uranium aluminides formed in U-Mo/Al dispersion fuel, including mechanical, thermodynamic and thermal transport properties, have also been studied using density functional theory (DFT) calculations [10,15]. However, the surface properties of uranium aluminides haven’t been investigated by either experiment or theory. Surface properties of a material, such as surface energy, is crucial to the description of fission gas bubble behaviors in irradiated fuels using simulation methods, such as rate theory and phase field modeling [16,17]. To this end, we systematically investigated the surface properties of uranium aluminides, including UAl3 and UAl4, using DFT calculations. At ambient conditions, UAl3 and UAl4 crystallize in the cubic and orthorhombic structures with space group of Pm3¯m and Imma, respectively. For cubic UAl3, we studied a total of 13 surface orientations with a maximum Miller index (MMI) of 3, while a total of 19 surface orientations with a MMI of 2 for orthorhombic UAl4. The surface properties of crystalline UAl3 and UAl4, including equilibrium single crystal shape, surface area weighted surface energy, dominant surface orientations and surface anisotropy, were obtained from the calculated surface energies. To understand the fission gas bubble behavior in amorphous uranium aluminides, we also studied the surface properties of amorphous UAl3 and UAl4 using amorphous
Corresponding author. E-mail address: [email protected] (Z.-G. Mei).
https://doi.org/10.1016/j.apsusc.2019.144132 Received 11 July 2019; Received in revised form 17 September 2019; Accepted 20 September 2019 Available online 17 October 2019 0169-4332/ © 2019 Published by Elsevier B.V.
Applied Surface Science 502 (2020) 144132
Z.-G. Mei and A.M. Yacout
samples generated by “melt and quench” method from ab initio molecular dynamics (AIMD) simulations. The reduced surface energies of the amorphous phases compared to the crystalline phases were discussed.
AIMD simulations were minimized through a high-precision DFT-based conjugate-gradient structural optimization calculations with a k-point sampling of 2 × 2 × 2 and a plane wave cut-off energy of 500 eV. The surface energy of the amorphous phase was obtained by averaging the surface energies calculated from three different surface orientations, i.e., [1 0 0], [0 1 0], and [0 0 1], of the amorphous phase.
2. Methodology Vienna Ab initio Simulation Package (VASP) [18,19] was utilized to perform all the DFT calculations in this work. The interaction between electrons and ions were described by the projector augmented wave method (PAW) [20], while the exchange-correlation functional was described by the generalized gradient approximation (GGA) [21]. The 6s26p65f36d17s2 of U and 3s23p1 of Al were treated as valence electrons in the DFT calculations. Ferromagnetic (FM) structures were adopted for all uranium aluminide phases. A cut-off energy of 500 eV was adopted for the plane wave basis set. The Monkhorst-Pack [22] k-point sampling for UAl3 and UAl4 were 29 × 29 × 29 and 24 × 16 × 8, respectively. Conventional DFT methods are known to poorly describe the strong correlation between f electrons in actinides. More advanced method, such as DFT + U, may provide a better description of the electronic and magnetic behaviors of uranium compounds, e.g., UN [23] and UO2 [24]. However, there is no consensus on whether methods, such as DFT + U, are necessary to correctly describe U metal and its alloys. For example, Söderlind et al. [25,26] found that conventional DFT is sufficient for U and U-Zr alloy systems. Kniznik et al. concluded that DFT + U, while essential for the strongly correlated insulating uranium oxide compounds, is at least difficult to validate against experimental data for the weakly correlated uranium aluminides [27]. Additionally, our previous studies show that the structural and mechanical properties of uranium aluminides can be well predicted using conventional DFT [10,15]. Therefore, DFT + U methodology was not considered in this work. Meanwhile, our previous study also shows that the spin-orbit coupling (SOC) has a relatively small influence on the structural and mechanical properties of uranium aluminides, such as UAl3 [10]. To further investigate the role of SOC on the surface energy, we compared the surface energy of UAl3 calculated with and without SOC. The surface energy of UAl3 (1 0 0) surface predicted with SOC is only slightly decreased by 0.03 J/m2 compared to the value of 1.50 J/m2 calculated without SOC. Due to the minor impact of SOC on the surface energy, we neglect the SOC effect in this work for computational efficiency. The structural properties of crystalline UAl3 and UAl4 have been predicted using DFT in our previous work [10]. The fully optimized lattice parameters were used to create the surface structures of UAl3 and UAl4 using periodic slab models. The surface structures of UAl3 and UAl4 were generated using the algorithm implemented in pymatgen [28] by considering all the possible terminations. Our tests show that a vacuum and slab thicknesses of 12 Å ensure that the surface energies are converged to 0.02 J/m2. The k-point samplings for all the surface 50 50 slab calculations were set as a × b × 1, where a and b represent the lattice vectors of the surface. To simulate the surface properties of amorphous UAl3 and UAl4, the “melt and quench” technique was adopted to generate the amorphous samples using AIMD simulations. A 4 × 4 × 4 supercell of UAl3 with a total number of atoms of 256 and a 4 × 3 × 1 supercell of UAl4 with a total number of atoms of 240 were used as the initial structures to create amorphous phases for UAl3 and UAl4, respectively. AIMD simulations for the liquids were run in the NVT ensemble at 4000 K for 6 ps with a time step of 2 fs with a low sample of density of 3.6 g/cm3 by increasing the supercell size from 1.0 to 1.2 times. Then the sample density was reduced to 3.5 g/cm3 by rescaling supercell size back to 1.01 times. The sample was annealed at 4000 K for 6 ps, and then quickly cooled to 300 K in 10 ps. Finally, the sample was annealed at 300 K for 8 ps. To expedite the simulations, relatively low-precision settings with Γ point only for the Brillouin zone integration and a plane wave cut-off energy of 400 eV were adopted for all AIMD simulations. Finally, the internal stresses of the amorphous samples generated from
3. Results and discussion 3.1. Stoichiometric surfaces of crystalline phases The surface energy (γhkl ) of a crystal with a facet of Miller index (h k l) can be calculated as,
γhkl =
hkl hkl Eslab − N × Ebulk , 2 × Aslab
hkl Eslab
(1)
where is the total energy of the slab, Aslab the surface area of the hkl the bulk energy per atom and N the total number of slab structure, Ebulk atoms in the slab structure. To avoid the slow convergence of surface energy with respect to slab thickness, the approach proposed by Sun et al. [29] was adopted to eliminate the Brillouin zone integration errors between the slab and the bulk, in which the bulk reference energy was calculated from a surface-oriented bulk unit cell instead of a conventional unit cell, i.e., the bulk reference energy of each individual surface orientation was calculated from the same surface model structure except without added vacuum layer and atomic relaxation but with the same k-point sampling as the surface model. For cubic UAl3 phase, there are 13 unique surface orientations with MMI up to 3. In comparison, there are 19 unique surface orientations with MMI up to 2 for orthorhombic UAl4 phase. For UAl4, there could be up to five different terminations for each surface orientation due to its low symmetry. Therefore, there are a total of 72 unique surface slab models for UAl4 by considering all the possible terminations. Figs. 1 and 2 show a few representative surface model structures of crystalline UAl3 and UAl4 after atomic relaxation studied in this work. As shown in Table 1, the (1 0 0) surface is predicted to be the most stable surface orientation for UAl3 among the 13 studied surfaces, due to its lowest surface energy. Similarly, the (1 0 0) surface is also predicted to be the most stable surface orientation of UAl4 among the 19 studied surfaces, closely followed by the (1 2 1) orientation. In comparison, the surface energy of the (1 0 0) surface orientation of UAl4 is predicted to be about 0.25 J/m2 lower than that of UAl3. Since the numbers of breaking U-Al bonds in the (1 0 0) surface layer of UAl3 and UAl4 are exactly the same, the lower surface energy of UAl4 can be ascribed to a weaker U-Al bond in UAl4 compared to that in UAl3. Therefore, the overall bonding in UAl4 is weaker than that in UAl3, consistent with our previous study of the mechanical properties of uranium aluminides [10]. To understand the cause of the surface energy difference between various surface orientations, we compared the number of breaking bonds for uranium and aluminum atoms close to the surface layers. In bulk UAl3 and UAl4, the coordination number (CN) of U-Al bond for U atom is 12 and 13, respectively. In the surface models of UAl3, the CN of U-Al for U atoms in the surface layers drops from 12 to 6–8, therefore the number of breaking U-Al bonds is 4–6. In comparison, the CN of U-Al for U atom in the surface layers of UAl4 decrease from 13 to 7–11, and the number of breaking U-Al bonds is 2–6. Overall, the surface energies of UAl3 and UAl4 are roughly proportional to the number of breaking U-Al bonds in the surface layers. For example, the number of breaking U-Al bonds in the surface layer of UAl3 (3 2 0) surface is the highest of 6, which leads to the highest surface energy of 1.89 J/m2 among the 13 studied surface orientations of UAl3. To investigate the impact of termination on the surface stability, we studied the (0 0 1) and (1 0 1) surfaces of UAl4 (two of the most stable orientations of UAl4) by considering all the possible surface 2
Applied Surface Science 502 (2020) 144132
Z.-G. Mei and A.M. Yacout
Fig. 1. Selective surface model structures of crystalline UAl3 after relaxation. Grey and cyan colors represent Al and U atoms, respectively.
comparison, the (1 2 1) surface is found to be the most dominant surface orientation for single crystal UAl4, followed by the (0 0 1), (1 0 1) and (0 1 2) surfaces. In experiment, the surface energy of a material can be estimated from the surface tension of its liquid phase. Therefore, averaged surface energy of a crystal is often used as a basis for comparisons with experimentally determined surface energy. Meanwhile, averaged surface energy of a material is commonly used to model the gas bubble behavior in a material using approaches, such as phase field method. For this reason, we also estimated the area weighted surface energy (γ¯) of crystalline UAl3 and UAl4 using the following equation,
terminations. Figs. 3 and 4 show the symmetrically unique terminations for the (0 0 1) and (1 0 1) surfaces, respectively. For both (0 0 1) and (1 0 1) surface orientations, the termination without uranium in the outermost layers (top and bottom layers) has the lowest surface energy, e.g., slab models with termination 1 shown in Figs. 3 and 4. This can be understood by the theory of bond breaking at the surface. For surface terminations with uranium atom at the top or bottom layer, the number of breaking bonds is higher than those with only aluminum atoms in the out layers.
3.2. Wulff shapes of crystalline phases
γ¯ =
To understand the geometrical shape of faceted voids for gas bubbles formed inside a material, it is important to investigate the single crystal shape under equilibrium conditions. The equilibrium crystal shapes of uranium aluminides can be obtained by the Wulff construction using the DFT predicted surface energies. The methodology developed by Miracle-Sole as implemented in pymatgen [28,30] was used to construct the equilibrium crystal shapes of UAl3 and UAl4. Fig. 5 show the equilibrium single crystal shapes of cubic UAl3 and orthorhombic UAl4 using surface energies calculated with MMI up to 3 and 2, respectively. With the constructed equilibrium crystal shape, the area fraction of each symmetrically distinct facet and the dominant surface orientation can be determined. Table 2 lists the calculated area fraction of each facet in equilibrium single crystal UAl3 and UAl4. Although the (1 0 0) surface is predicted to be the most stable surface orientation of UAl3, the most dominant surface orientation in single crystal UAl3 is found to be the (3 1 1) surface, followed by the (1 0 0) and (3 2 2) surfaces. In
∑{hkl} γhkl Ahkl ∑ Ahkl
=
A , ∑ γhkl fhkl {hkl}
(2)
A fhkl
is the area fraction of the {h k l} family in the Wulff shape, where γhkl the surface energy, and Ahkl the total area of all facets in the {h k l} family. The area weighted surface energies of crystalline UAl3 and UAl4 are predicted to be 1.59 J/m2 and 1.34 J/m2 from the calculated surface energy and area fraction as shown in Tables 1 and 2. To investigate the surface anisotropy of single crystal UAl3 and UAl4, the surface anisotropy (αγ) proposed by Tran et al. [30] was adopted in this work, as defined by the following equation,
αγ =
A ∑{hkl} (γhkl − γ¯)2fhkl
γ¯
.
(3)
αγ represents the variation of surface energies. A perfectly isotropic crystal would have αγ = 0. The surface anisotropy of UAl3 and UAl4 are predicted to be 0.034 and 0.036, respectively. Therefore, the surface of
Fig. 2. Selective surface model structures of crystalline UAl4 after relaxation. Grey and cyan colors represent Al and U atoms, respectively. 3
Applied Surface Science 502 (2020) 144132
Z.-G. Mei and A.M. Yacout
Table 1 Surface energies of stoichiometric surfaces in crystalline UAl3 and UAl4, in unit of J/m2. UAl3
(1 0 0) 1.50
(1 1 0) 1.68
(1 1 1) 1.61
(2 1 0) 1.73
(2 1 1) 1.66
(2 2 1) 1.66
(3 1 0) 1.80
(3 1 1) 1.60
(3 2 0) 1.89
(3 2 1) 1.70
(3 2 2) 1.60
(3 3 1) 1.77
(3 3 2) 1.62
UAl4
(0 0 1) 1.33 (1 2 2) 1.44
(0 1 0) 1.49 (2 0 1) 1.38
(0 1 1) 1.52 (2 1 0) 1.41
(1 0 0) 1.29 (2 1 1) 1.47
(1 0 1) 1.34 (2 1 2) 1.46
(1 1 0) 1.45 (2 2 1) 1.43
(1 1 1) 1.48
(0 1 2) 1.35
(0 2 1) 1.63
(1 0 2) 1.55
(1 1 2) 1.49
(1 2 0) 1.33
(1 2 1) 1.30
Visual Molecular Dynamics (VMD) [31]. Fig. 7 shows the calculated RDFs, g(r), of amorphous UAl3 and UAl4. The first peak positions (R) and coordination number (CN) for U-Al, Al-Al, and U-U are tabulated in Table 3. The coordination number of elements in amorphous UAl3 and UAl4 is defined as the number of atoms in the range of the first coordination sphere, which is obtained by integrating the radial distribution function (RDF) in the spherical coordination to the first minimum of the RDF. The first peak positions of U-Al, Al-Al, and U-U are very similar in amorphous UAl3 and UAl4. Compared to the crystalline phases, the distance between U and Al atoms in the amorphous phases slightly increases, while the distance between Al and Al atoms decreases slightly. The most notable change was observed for the first peak position of U-U in the amorphous phase, which decreases by 40% compared to that of the crystalline phase. Accordingly, the coordination number of U-Al in amorphous UAl3 and UAl4 reduces by about 3, while the coordination number of U-U decreases from 6 to 2. The calculated RDF and CN indicate that U atoms tend to attract another two U atoms and form small clusters in the amorphous phases of UAl3 and UAl4, which can be observed in the snap-shot of amorphous UAl3 and UAl4 in Fig. 6. With the generated amorphous samples for UAl3 and UAl4, three slab model structures were created along the [1 0 0], [0 1 0] and [0 0 1] directions to simulate the surface properties of the amorphous phases. The surface energies of amorphous UAl3 and UAl4 were obtained by averaging the surface energies calculated along these three directions of the amorphous structure. The predicted surface energies of amorphous UAl3 and UAl4 are shown in Table 4. In comparison to the crystalline phase, the surface energy of the amorphous phase is found to decrease by 30% and 25% for UAl3 and UAl4, respectively. The decreased surface energies of amorphous UAl3 and UAl4 can be understood by the reduced number of breaking U-Al bonds in the surface layers of amorphous UAl3 and UAl4 compared to the crystalline phases. With the decreased surface energy, fission gas atoms tend to form larger bubbles in the amorphous uranium aluminides compared to the crystalline phases.
Fig. 3. Surface model structures of UAl4 (0 0 1) surface with four unique terminations after relaxation. Grey and cyan colors represent Al and U atoms, respectively.
Term. 1
Term. 2
Term. 3
Term. 4
4. Conclusions The surface properties of two uranium aluminides, both crystalline and amorphous phases of UAl3 and UAl4, were systematically investigated using first-principles DFT calculations. The surface energies of the crystalline phases were predicted for facets up to MMIs of 3 and 2 for cubic UAl3 and orthorhombic UAl4, respectively. The (1 0 0) surface was determined to be the most stable surface orientations for both UAl3 and UAl4 from the calculated surface energies. The study of the (0 0 1) and (1 0 1) surfaces of UAl4 indicate that the surface termination without U atoms is more stable than those terminated with U atoms. The equilibrium crystal shapes of UAl3 and UAl4 were obtained using the Wulff construction from the DFT predicted surface energies. From the Wulff shape, the dominant facets for equilibrium single crystals of UAl3 and UAl4 are determined to be (3 1 1) and (1 2 1), respectively. The area weighed surface energies of crystalline UAl3 and UAl4 are predicted to be 1.59 J/m2 and 1.34 J/m2. Additionally, the calculated surface anisotropy indicates that the surface of UAl3 single crystal is slightly more isotropic than UAl4. To model the surface properties of amorphous phases of UAl3 and UAl4, the amorphous samples were generated using the “melt and quench” method by AIMD simulations.
Fig. 4. Surface model structures of UAl4 (1 0 1) surface with four unique terminations after relaxation. Grey and cyan colors present Al and U atoms, respectively.
UAl3 single crystal is slightly more isotropic in comparison to UAl4. 3.3. Surface properties of amorphous phases To model the surface properties of amorphous uranium aluminides, we first created amorphous samples of UAl3 and UAl4 using the “melt and quench” method by AIMD simulations. The atomic structures of the generated amorphous samples are shown in Fig. 6, with total atoms of 256 and 240 for UAl3 and UAl4, respectively. Radial distribution function (RDF) is one of the most important structural information for characterizing the local ordering of amorphous materials. The trajectories of the last 6 ps of the AIMD simulations at 300 K were used to calculate the RDFs of UAl3 and UAl4 using the plugin implemented in 4
Applied Surface Science 502 (2020) 144132
Z.-G. Mei and A.M. Yacout
Fig. 5. Equilibrium single crystal shapes of (a) cubic UAl3 and (b) orthorhombic UAl4 constructed from surface energies calculated with MMI up to 3 and 2 for UAl3 and UAl4, respectively. Table 2 Area fractions of the dominant surface orientations in equilibrium single crystals of UAl3 and UAl4. Miller indices
Area fraction (UAl3)
Miller indices
Area fraction (UAl4)
(1 0 0) (1 1 0) (3 1 1) (3 2 2) (3 3 2)
23% 10% 32% 22% 11%
(0 0 1) (0 1 1) (1 0 0) (1 0 1) (0 1 2) (1 1 2) (1 2 1)
17% 9% 6% 15% 15% 7% 29%
The local structures of the amorphous phases were analyzed using the calculated radial distribution functions. The surface energies of the amorphous phases were obtained by averaging the surface energies calculated along the [1 0 0], [0 1 0] and [0 0 1] directions. In comparison to the crystalline phases, the surface energy of the amorphous phase is found to decrease by 30% and 25% for UAl3 and UAl4, respectively, consistent with the reduced number of breaking U-Al bonds in the surface layers of the amorphous phases. With the currently obtained surface properties of uranium aluminides, the morphology of voids or gas bubbles formed in crystalline UAl3 and UAl4 due to ion or neutron irradiation can be simulated using approaches, such as phase field method. More importantly, the gas bubble-induced swelling in amorphous uranium aluminides, such as in the interaction layer of U-
Fig. 7. Radial distribution functions, g(r), of amorphous UAl3 and UAl4 calculated from AIMD trajectories simulated at 300 K.
Mo/Al dispersion fuel, can be more accurately predicted using mesoscale simulations, which is crucial to the evaluation of fuel stability under irradiation.
Fig. 6. Atomic structures of amorphous UAl3 and UAl4 generated by “melt and quench” method from AIMD simulations at 300 K. Grey and cyan color represent Al and U atoms, respectively. 5
Applied Surface Science 502 (2020) 144132
Z.-G. Mei and A.M. Yacout
Table 3 First peak position R (Å) and coordination numbers CN in amorphous and crystalline UAl3 and UAl4 evaluated from AIMD simulations at 300 K. Phases
RU-Al
RAl-Al
RU-U
CNU-Al
CNAl-Al
CNU-U
Amorphous UAl3 Crystalline UAl3 Amorphous UAl4 Crystalline UAl4
3.15 2.99 3.13 3.09
2.85 3.00 2.83 2.99
2.53 4.27 2.59 4.37
9 12 10.5 13
9 8 9.5 9.5
1.9 6 1.8 6
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Table 4 Averaged surface energy (J/m2) of crystalline and amorphous UAl3 and UAl4. Phases
Surface energy
Amorphous UAl3 Crystalline UAl3 Amorphous UAl4 Crystalline UAl4
1.12 1.59 1.00 1.34
Acknowledgements This work is sponsored by the U.S. Department of Energy, National Nuclear Security Administration (NNSA), Office of Material Management and Minimization (NA-23) Reactor Conversion Program. The efforts involving Argonne National Laboratory were supported under Contract No. DE-AC02-06CH11357 between UChicago Argonne, LLC and the U.S. Department of Energy. We gratefully acknowledge the computing resources provided on Bebop, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. References [1] W.F. Brinkman, S. Engelsberg, Spin-fluctuation contributions to the specific heat, Phys. Rev. 169 (1968) 417–431. [2] D. Aoki, N. Watanabe, Y. Inada, R. Settai, K. Sugiyama, H. Harima, T. Inoue, K. Kindo, E. Yamamoto, Yoshinori Haga, Yoshichika Ōnuki, Fermi surface properties of the enhanced Pauli Paramagnet UAl 3, J. Phys. Soc. Jpn. 69 (2000) 2609–2614. [3] A.L. Cornelius, A.J. Arko, J.L. Sarrao, J.D. Thompson, M.F. Hundley, C.H. Booth, N. Harrison, P.M. Oppeneer, Electronic properties of \(\mathrm{U}{X}_{3} (X= \mathrm{Ga},\) Al, and Sn) compounds in high magnetic fields: transport, specific heat, magnetization, and quantum oscillations, Phys. Rev. B 59 (1999) 14473–14483. [4] D. Aoki, Y. Haga, Y. Homma, Y. Shiokawa, E. Yamamoto, A. Nakamura, R. Settai, Y. Ōnuki, Magnetic and electrical properties in NpAl4 and UAl4, J. Phys. Soc. Jpn. 78 (2009) 044712. [5] H. Armbrüster, W. Franz, W. Schlabitz, F. Steglich, Transport properties, susceptibility and specific heat of UAl2, J. Phys. Colloq. 40 (1979) C4-150–C154-151. [6] S.-I. Fujimori, M. Kobata, Y. Takeda, T. Okane, Y. Saitoh, A. Fujimori, H. Yamagami, Y. Haga, E. Yamamoto, Y. Ōnuki, Electronic structures of UX3 (X=Al, Ga, and In)
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Full Length Article
First-principles study of surface properties of crystalline and amorphous uranium aluminides
T
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Zhi-Gang Mei , Abdellatif M. Yacout Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Surface energy Uranium aluminides First-principles calculations Amorphous phases
Surface energy is a crucial material property required for the modeling of fission gas bubble behavior in nuclear materials. However, there is no information available regarding the surface properties of uranium aluminides. In this work we systematically investigate the surface properties of uranium aluminides (UAl3 and UAl4) by firstprinciples density functional theory (DFT) calculations. A total of 13 surface orientations with a maximum Miller index (MMI) up to 3 are studied for cubic UAl3, while 19 surfaces with a MMI of 2 for orthorhombic UAl4. Using the surface energies predicted by DFT, the surface properties of equilibrium single crystal UAl3 and UAl4, including surface area weighted surface energy, dominant surface orientations and surface anisotropy, are obtained from Wulff construction. To understand gas bubble behaviors in amorphous uranium aluminides under irradiation, we study the surface properties of amorphous UAl3 and UAl4. Compared to the crystalline phases, the amorphous phases are found to have lower surface energies due to the reduced number of breaking U-Al bonds close to the surface. The currently obtained surface properties of UAl3 and UAl4 can be used for the future modeling of gas bubble behaviors in both crystalline and amorphous uranium aluminide fuels.
1. Introduction Uranium aluminides, e.g., UAl2, UAl3 and UAl4, have been extensively studied by both experiment and theory because of their scientific and technological importance [1–10]. UAlx/Al dispersion fuels (UAlx represents a mixture of UAl2, UAl3 and UAl4) are used as nuclear fuels for research reactors worldwide [11]. Meanwhile, U-Mo/Al dispersion fuels are currently under investigation as one of the candidate low enriched uranium (LEU) fuels to convert the high enriched uranium (HEU) fuels used for the European high-power research reactors [12]. Experiments of U-Mo/Al dispersion fuel show that interaction layer (IL) forms between U-Mo fuel particles and Al matrix during the fabrication of fuel plate or under neutron irradiations. By the thermal annealing of diffusion couple and nuclear fuel plate, Mazaudier et al. [13] found that the IL consists of binary U-Al compounds, i.e., UAl3 and UAl4, and several ternary U-Mo-Al compounds. In comparison, neutron irradiation tests show that the IL is mainly composed of amorphous uranium aluminide phases [14]. It is well known that the IL formed in U-Mo/Al dispersion fuel is detrimental to the fuel performance, such as reducing overall thermal conductivity and causing abnormal fuel plate swelling at high fission densities [12]. Fundamental material properties of uranium aluminides are important to the qualification of U-Mo/Al dispersion fuels. The electronic ⁎
and magnetic properties of binary uranium aluminides, i.e., UAl2, UAl3 and UAl4, have been extensively studied [2–6,9]. Recently, the thermophysical properties of binary and ternary uranium aluminides formed in U-Mo/Al dispersion fuel, including mechanical, thermodynamic and thermal transport properties, have also been studied using density functional theory (DFT) calculations [10,15]. However, the surface properties of uranium aluminides haven’t been investigated by either experiment or theory. Surface properties of a material, such as surface energy, is crucial to the description of fission gas bubble behaviors in irradiated fuels using simulation methods, such as rate theory and phase field modeling [16,17]. To this end, we systematically investigated the surface properties of uranium aluminides, including UAl3 and UAl4, using DFT calculations. At ambient conditions, UAl3 and UAl4 crystallize in the cubic and orthorhombic structures with space group of Pm3¯m and Imma, respectively. For cubic UAl3, we studied a total of 13 surface orientations with a maximum Miller index (MMI) of 3, while a total of 19 surface orientations with a MMI of 2 for orthorhombic UAl4. The surface properties of crystalline UAl3 and UAl4, including equilibrium single crystal shape, surface area weighted surface energy, dominant surface orientations and surface anisotropy, were obtained from the calculated surface energies. To understand the fission gas bubble behavior in amorphous uranium aluminides, we also studied the surface properties of amorphous UAl3 and UAl4 using amorphous
Corresponding author. E-mail address: [email protected] (Z.-G. Mei).
https://doi.org/10.1016/j.apsusc.2019.144132 Received 11 July 2019; Received in revised form 17 September 2019; Accepted 20 September 2019 Available online 17 October 2019 0169-4332/ © 2019 Published by Elsevier B.V.
Applied Surface Science 502 (2020) 144132
Z.-G. Mei and A.M. Yacout
samples generated by “melt and quench” method from ab initio molecular dynamics (AIMD) simulations. The reduced surface energies of the amorphous phases compared to the crystalline phases were discussed.
AIMD simulations were minimized through a high-precision DFT-based conjugate-gradient structural optimization calculations with a k-point sampling of 2 × 2 × 2 and a plane wave cut-off energy of 500 eV. The surface energy of the amorphous phase was obtained by averaging the surface energies calculated from three different surface orientations, i.e., [1 0 0], [0 1 0], and [0 0 1], of the amorphous phase.
2. Methodology Vienna Ab initio Simulation Package (VASP) [18,19] was utilized to perform all the DFT calculations in this work. The interaction between electrons and ions were described by the projector augmented wave method (PAW) [20], while the exchange-correlation functional was described by the generalized gradient approximation (GGA) [21]. The 6s26p65f36d17s2 of U and 3s23p1 of Al were treated as valence electrons in the DFT calculations. Ferromagnetic (FM) structures were adopted for all uranium aluminide phases. A cut-off energy of 500 eV was adopted for the plane wave basis set. The Monkhorst-Pack [22] k-point sampling for UAl3 and UAl4 were 29 × 29 × 29 and 24 × 16 × 8, respectively. Conventional DFT methods are known to poorly describe the strong correlation between f electrons in actinides. More advanced method, such as DFT + U, may provide a better description of the electronic and magnetic behaviors of uranium compounds, e.g., UN [23] and UO2 [24]. However, there is no consensus on whether methods, such as DFT + U, are necessary to correctly describe U metal and its alloys. For example, Söderlind et al. [25,26] found that conventional DFT is sufficient for U and U-Zr alloy systems. Kniznik et al. concluded that DFT + U, while essential for the strongly correlated insulating uranium oxide compounds, is at least difficult to validate against experimental data for the weakly correlated uranium aluminides [27]. Additionally, our previous studies show that the structural and mechanical properties of uranium aluminides can be well predicted using conventional DFT [10,15]. Therefore, DFT + U methodology was not considered in this work. Meanwhile, our previous study also shows that the spin-orbit coupling (SOC) has a relatively small influence on the structural and mechanical properties of uranium aluminides, such as UAl3 [10]. To further investigate the role of SOC on the surface energy, we compared the surface energy of UAl3 calculated with and without SOC. The surface energy of UAl3 (1 0 0) surface predicted with SOC is only slightly decreased by 0.03 J/m2 compared to the value of 1.50 J/m2 calculated without SOC. Due to the minor impact of SOC on the surface energy, we neglect the SOC effect in this work for computational efficiency. The structural properties of crystalline UAl3 and UAl4 have been predicted using DFT in our previous work [10]. The fully optimized lattice parameters were used to create the surface structures of UAl3 and UAl4 using periodic slab models. The surface structures of UAl3 and UAl4 were generated using the algorithm implemented in pymatgen [28] by considering all the possible terminations. Our tests show that a vacuum and slab thicknesses of 12 Å ensure that the surface energies are converged to 0.02 J/m2. The k-point samplings for all the surface 50 50 slab calculations were set as a × b × 1, where a and b represent the lattice vectors of the surface. To simulate the surface properties of amorphous UAl3 and UAl4, the “melt and quench” technique was adopted to generate the amorphous samples using AIMD simulations. A 4 × 4 × 4 supercell of UAl3 with a total number of atoms of 256 and a 4 × 3 × 1 supercell of UAl4 with a total number of atoms of 240 were used as the initial structures to create amorphous phases for UAl3 and UAl4, respectively. AIMD simulations for the liquids were run in the NVT ensemble at 4000 K for 6 ps with a time step of 2 fs with a low sample of density of 3.6 g/cm3 by increasing the supercell size from 1.0 to 1.2 times. Then the sample density was reduced to 3.5 g/cm3 by rescaling supercell size back to 1.01 times. The sample was annealed at 4000 K for 6 ps, and then quickly cooled to 300 K in 10 ps. Finally, the sample was annealed at 300 K for 8 ps. To expedite the simulations, relatively low-precision settings with Γ point only for the Brillouin zone integration and a plane wave cut-off energy of 400 eV were adopted for all AIMD simulations. Finally, the internal stresses of the amorphous samples generated from
3. Results and discussion 3.1. Stoichiometric surfaces of crystalline phases The surface energy (γhkl ) of a crystal with a facet of Miller index (h k l) can be calculated as,
γhkl =
hkl hkl Eslab − N × Ebulk , 2 × Aslab
hkl Eslab
(1)
where is the total energy of the slab, Aslab the surface area of the hkl the bulk energy per atom and N the total number of slab structure, Ebulk atoms in the slab structure. To avoid the slow convergence of surface energy with respect to slab thickness, the approach proposed by Sun et al. [29] was adopted to eliminate the Brillouin zone integration errors between the slab and the bulk, in which the bulk reference energy was calculated from a surface-oriented bulk unit cell instead of a conventional unit cell, i.e., the bulk reference energy of each individual surface orientation was calculated from the same surface model structure except without added vacuum layer and atomic relaxation but with the same k-point sampling as the surface model. For cubic UAl3 phase, there are 13 unique surface orientations with MMI up to 3. In comparison, there are 19 unique surface orientations with MMI up to 2 for orthorhombic UAl4 phase. For UAl4, there could be up to five different terminations for each surface orientation due to its low symmetry. Therefore, there are a total of 72 unique surface slab models for UAl4 by considering all the possible terminations. Figs. 1 and 2 show a few representative surface model structures of crystalline UAl3 and UAl4 after atomic relaxation studied in this work. As shown in Table 1, the (1 0 0) surface is predicted to be the most stable surface orientation for UAl3 among the 13 studied surfaces, due to its lowest surface energy. Similarly, the (1 0 0) surface is also predicted to be the most stable surface orientation of UAl4 among the 19 studied surfaces, closely followed by the (1 2 1) orientation. In comparison, the surface energy of the (1 0 0) surface orientation of UAl4 is predicted to be about 0.25 J/m2 lower than that of UAl3. Since the numbers of breaking U-Al bonds in the (1 0 0) surface layer of UAl3 and UAl4 are exactly the same, the lower surface energy of UAl4 can be ascribed to a weaker U-Al bond in UAl4 compared to that in UAl3. Therefore, the overall bonding in UAl4 is weaker than that in UAl3, consistent with our previous study of the mechanical properties of uranium aluminides [10]. To understand the cause of the surface energy difference between various surface orientations, we compared the number of breaking bonds for uranium and aluminum atoms close to the surface layers. In bulk UAl3 and UAl4, the coordination number (CN) of U-Al bond for U atom is 12 and 13, respectively. In the surface models of UAl3, the CN of U-Al for U atoms in the surface layers drops from 12 to 6–8, therefore the number of breaking U-Al bonds is 4–6. In comparison, the CN of U-Al for U atom in the surface layers of UAl4 decrease from 13 to 7–11, and the number of breaking U-Al bonds is 2–6. Overall, the surface energies of UAl3 and UAl4 are roughly proportional to the number of breaking U-Al bonds in the surface layers. For example, the number of breaking U-Al bonds in the surface layer of UAl3 (3 2 0) surface is the highest of 6, which leads to the highest surface energy of 1.89 J/m2 among the 13 studied surface orientations of UAl3. To investigate the impact of termination on the surface stability, we studied the (0 0 1) and (1 0 1) surfaces of UAl4 (two of the most stable orientations of UAl4) by considering all the possible surface 2
Applied Surface Science 502 (2020) 144132
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Fig. 1. Selective surface model structures of crystalline UAl3 after relaxation. Grey and cyan colors represent Al and U atoms, respectively.
comparison, the (1 2 1) surface is found to be the most dominant surface orientation for single crystal UAl4, followed by the (0 0 1), (1 0 1) and (0 1 2) surfaces. In experiment, the surface energy of a material can be estimated from the surface tension of its liquid phase. Therefore, averaged surface energy of a crystal is often used as a basis for comparisons with experimentally determined surface energy. Meanwhile, averaged surface energy of a material is commonly used to model the gas bubble behavior in a material using approaches, such as phase field method. For this reason, we also estimated the area weighted surface energy (γ¯) of crystalline UAl3 and UAl4 using the following equation,
terminations. Figs. 3 and 4 show the symmetrically unique terminations for the (0 0 1) and (1 0 1) surfaces, respectively. For both (0 0 1) and (1 0 1) surface orientations, the termination without uranium in the outermost layers (top and bottom layers) has the lowest surface energy, e.g., slab models with termination 1 shown in Figs. 3 and 4. This can be understood by the theory of bond breaking at the surface. For surface terminations with uranium atom at the top or bottom layer, the number of breaking bonds is higher than those with only aluminum atoms in the out layers.
3.2. Wulff shapes of crystalline phases
γ¯ =
To understand the geometrical shape of faceted voids for gas bubbles formed inside a material, it is important to investigate the single crystal shape under equilibrium conditions. The equilibrium crystal shapes of uranium aluminides can be obtained by the Wulff construction using the DFT predicted surface energies. The methodology developed by Miracle-Sole as implemented in pymatgen [28,30] was used to construct the equilibrium crystal shapes of UAl3 and UAl4. Fig. 5 show the equilibrium single crystal shapes of cubic UAl3 and orthorhombic UAl4 using surface energies calculated with MMI up to 3 and 2, respectively. With the constructed equilibrium crystal shape, the area fraction of each symmetrically distinct facet and the dominant surface orientation can be determined. Table 2 lists the calculated area fraction of each facet in equilibrium single crystal UAl3 and UAl4. Although the (1 0 0) surface is predicted to be the most stable surface orientation of UAl3, the most dominant surface orientation in single crystal UAl3 is found to be the (3 1 1) surface, followed by the (1 0 0) and (3 2 2) surfaces. In
∑{hkl} γhkl Ahkl ∑ Ahkl
=
A , ∑ γhkl fhkl {hkl}
(2)
A fhkl
is the area fraction of the {h k l} family in the Wulff shape, where γhkl the surface energy, and Ahkl the total area of all facets in the {h k l} family. The area weighted surface energies of crystalline UAl3 and UAl4 are predicted to be 1.59 J/m2 and 1.34 J/m2 from the calculated surface energy and area fraction as shown in Tables 1 and 2. To investigate the surface anisotropy of single crystal UAl3 and UAl4, the surface anisotropy (αγ) proposed by Tran et al. [30] was adopted in this work, as defined by the following equation,
αγ =
A ∑{hkl} (γhkl − γ¯)2fhkl
γ¯
.
(3)
αγ represents the variation of surface energies. A perfectly isotropic crystal would have αγ = 0. The surface anisotropy of UAl3 and UAl4 are predicted to be 0.034 and 0.036, respectively. Therefore, the surface of
Fig. 2. Selective surface model structures of crystalline UAl4 after relaxation. Grey and cyan colors represent Al and U atoms, respectively. 3
Applied Surface Science 502 (2020) 144132
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Table 1 Surface energies of stoichiometric surfaces in crystalline UAl3 and UAl4, in unit of J/m2. UAl3
(1 0 0) 1.50
(1 1 0) 1.68
(1 1 1) 1.61
(2 1 0) 1.73
(2 1 1) 1.66
(2 2 1) 1.66
(3 1 0) 1.80
(3 1 1) 1.60
(3 2 0) 1.89
(3 2 1) 1.70
(3 2 2) 1.60
(3 3 1) 1.77
(3 3 2) 1.62
UAl4
(0 0 1) 1.33 (1 2 2) 1.44
(0 1 0) 1.49 (2 0 1) 1.38
(0 1 1) 1.52 (2 1 0) 1.41
(1 0 0) 1.29 (2 1 1) 1.47
(1 0 1) 1.34 (2 1 2) 1.46
(1 1 0) 1.45 (2 2 1) 1.43
(1 1 1) 1.48
(0 1 2) 1.35
(0 2 1) 1.63
(1 0 2) 1.55
(1 1 2) 1.49
(1 2 0) 1.33
(1 2 1) 1.30
Visual Molecular Dynamics (VMD) [31]. Fig. 7 shows the calculated RDFs, g(r), of amorphous UAl3 and UAl4. The first peak positions (R) and coordination number (CN) for U-Al, Al-Al, and U-U are tabulated in Table 3. The coordination number of elements in amorphous UAl3 and UAl4 is defined as the number of atoms in the range of the first coordination sphere, which is obtained by integrating the radial distribution function (RDF) in the spherical coordination to the first minimum of the RDF. The first peak positions of U-Al, Al-Al, and U-U are very similar in amorphous UAl3 and UAl4. Compared to the crystalline phases, the distance between U and Al atoms in the amorphous phases slightly increases, while the distance between Al and Al atoms decreases slightly. The most notable change was observed for the first peak position of U-U in the amorphous phase, which decreases by 40% compared to that of the crystalline phase. Accordingly, the coordination number of U-Al in amorphous UAl3 and UAl4 reduces by about 3, while the coordination number of U-U decreases from 6 to 2. The calculated RDF and CN indicate that U atoms tend to attract another two U atoms and form small clusters in the amorphous phases of UAl3 and UAl4, which can be observed in the snap-shot of amorphous UAl3 and UAl4 in Fig. 6. With the generated amorphous samples for UAl3 and UAl4, three slab model structures were created along the [1 0 0], [0 1 0] and [0 0 1] directions to simulate the surface properties of the amorphous phases. The surface energies of amorphous UAl3 and UAl4 were obtained by averaging the surface energies calculated along these three directions of the amorphous structure. The predicted surface energies of amorphous UAl3 and UAl4 are shown in Table 4. In comparison to the crystalline phase, the surface energy of the amorphous phase is found to decrease by 30% and 25% for UAl3 and UAl4, respectively. The decreased surface energies of amorphous UAl3 and UAl4 can be understood by the reduced number of breaking U-Al bonds in the surface layers of amorphous UAl3 and UAl4 compared to the crystalline phases. With the decreased surface energy, fission gas atoms tend to form larger bubbles in the amorphous uranium aluminides compared to the crystalline phases.
Fig. 3. Surface model structures of UAl4 (0 0 1) surface with four unique terminations after relaxation. Grey and cyan colors represent Al and U atoms, respectively.
Term. 1
Term. 2
Term. 3
Term. 4
4. Conclusions The surface properties of two uranium aluminides, both crystalline and amorphous phases of UAl3 and UAl4, were systematically investigated using first-principles DFT calculations. The surface energies of the crystalline phases were predicted for facets up to MMIs of 3 and 2 for cubic UAl3 and orthorhombic UAl4, respectively. The (1 0 0) surface was determined to be the most stable surface orientations for both UAl3 and UAl4 from the calculated surface energies. The study of the (0 0 1) and (1 0 1) surfaces of UAl4 indicate that the surface termination without U atoms is more stable than those terminated with U atoms. The equilibrium crystal shapes of UAl3 and UAl4 were obtained using the Wulff construction from the DFT predicted surface energies. From the Wulff shape, the dominant facets for equilibrium single crystals of UAl3 and UAl4 are determined to be (3 1 1) and (1 2 1), respectively. The area weighed surface energies of crystalline UAl3 and UAl4 are predicted to be 1.59 J/m2 and 1.34 J/m2. Additionally, the calculated surface anisotropy indicates that the surface of UAl3 single crystal is slightly more isotropic than UAl4. To model the surface properties of amorphous phases of UAl3 and UAl4, the amorphous samples were generated using the “melt and quench” method by AIMD simulations.
Fig. 4. Surface model structures of UAl4 (1 0 1) surface with four unique terminations after relaxation. Grey and cyan colors present Al and U atoms, respectively.
UAl3 single crystal is slightly more isotropic in comparison to UAl4. 3.3. Surface properties of amorphous phases To model the surface properties of amorphous uranium aluminides, we first created amorphous samples of UAl3 and UAl4 using the “melt and quench” method by AIMD simulations. The atomic structures of the generated amorphous samples are shown in Fig. 6, with total atoms of 256 and 240 for UAl3 and UAl4, respectively. Radial distribution function (RDF) is one of the most important structural information for characterizing the local ordering of amorphous materials. The trajectories of the last 6 ps of the AIMD simulations at 300 K were used to calculate the RDFs of UAl3 and UAl4 using the plugin implemented in 4
Applied Surface Science 502 (2020) 144132
Z.-G. Mei and A.M. Yacout
Fig. 5. Equilibrium single crystal shapes of (a) cubic UAl3 and (b) orthorhombic UAl4 constructed from surface energies calculated with MMI up to 3 and 2 for UAl3 and UAl4, respectively. Table 2 Area fractions of the dominant surface orientations in equilibrium single crystals of UAl3 and UAl4. Miller indices
Area fraction (UAl3)
Miller indices
Area fraction (UAl4)
(1 0 0) (1 1 0) (3 1 1) (3 2 2) (3 3 2)
23% 10% 32% 22% 11%
(0 0 1) (0 1 1) (1 0 0) (1 0 1) (0 1 2) (1 1 2) (1 2 1)
17% 9% 6% 15% 15% 7% 29%
The local structures of the amorphous phases were analyzed using the calculated radial distribution functions. The surface energies of the amorphous phases were obtained by averaging the surface energies calculated along the [1 0 0], [0 1 0] and [0 0 1] directions. In comparison to the crystalline phases, the surface energy of the amorphous phase is found to decrease by 30% and 25% for UAl3 and UAl4, respectively, consistent with the reduced number of breaking U-Al bonds in the surface layers of the amorphous phases. With the currently obtained surface properties of uranium aluminides, the morphology of voids or gas bubbles formed in crystalline UAl3 and UAl4 due to ion or neutron irradiation can be simulated using approaches, such as phase field method. More importantly, the gas bubble-induced swelling in amorphous uranium aluminides, such as in the interaction layer of U-
Fig. 7. Radial distribution functions, g(r), of amorphous UAl3 and UAl4 calculated from AIMD trajectories simulated at 300 K.
Mo/Al dispersion fuel, can be more accurately predicted using mesoscale simulations, which is crucial to the evaluation of fuel stability under irradiation.
Fig. 6. Atomic structures of amorphous UAl3 and UAl4 generated by “melt and quench” method from AIMD simulations at 300 K. Grey and cyan color represent Al and U atoms, respectively. 5
Applied Surface Science 502 (2020) 144132
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Table 3 First peak position R (Å) and coordination numbers CN in amorphous and crystalline UAl3 and UAl4 evaluated from AIMD simulations at 300 K. Phases
RU-Al
RAl-Al
RU-U
CNU-Al
CNAl-Al
CNU-U
Amorphous UAl3 Crystalline UAl3 Amorphous UAl4 Crystalline UAl4
3.15 2.99 3.13 3.09
2.85 3.00 2.83 2.99
2.53 4.27 2.59 4.37
9 12 10.5 13
9 8 9.5 9.5
1.9 6 1.8 6
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Table 4 Averaged surface energy (J/m2) of crystalline and amorphous UAl3 and UAl4. Phases
Surface energy
Amorphous UAl3 Crystalline UAl3 Amorphous UAl4 Crystalline UAl4
1.12 1.59 1.00 1.34
Acknowledgements This work is sponsored by the U.S. Department of Energy, National Nuclear Security Administration (NNSA), Office of Material Management and Minimization (NA-23) Reactor Conversion Program. The efforts involving Argonne National Laboratory were supported under Contract No. DE-AC02-06CH11357 between UChicago Argonne, LLC and the U.S. Department of Energy. We gratefully acknowledge the computing resources provided on Bebop, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. References [1] W.F. Brinkman, S. Engelsberg, Spin-fluctuation contributions to the specific heat, Phys. Rev. 169 (1968) 417–431. [2] D. Aoki, N. Watanabe, Y. Inada, R. Settai, K. Sugiyama, H. Harima, T. Inoue, K. Kindo, E. Yamamoto, Yoshinori Haga, Yoshichika Ōnuki, Fermi surface properties of the enhanced Pauli Paramagnet UAl 3, J. Phys. Soc. Jpn. 69 (2000) 2609–2614. [3] A.L. Cornelius, A.J. Arko, J.L. Sarrao, J.D. Thompson, M.F. Hundley, C.H. Booth, N. Harrison, P.M. Oppeneer, Electronic properties of \(\mathrm{U}{X}_{3} (X= \mathrm{Ga},\) Al, and Sn) compounds in high magnetic fields: transport, specific heat, magnetization, and quantum oscillations, Phys. Rev. B 59 (1999) 14473–14483. [4] D. Aoki, Y. Haga, Y. Homma, Y. Shiokawa, E. Yamamoto, A. Nakamura, R. Settai, Y. Ōnuki, Magnetic and electrical properties in NpAl4 and UAl4, J. Phys. Soc. Jpn. 78 (2009) 044712. [5] H. Armbrüster, W. Franz, W. Schlabitz, F. Steglich, Transport properties, susceptibility and specific heat of UAl2, J. Phys. Colloq. 40 (1979) C4-150–C154-151. [6] S.-I. Fujimori, M. Kobata, Y. Takeda, T. Okane, Y. Saitoh, A. Fujimori, H. Yamagami, Y. Haga, E. Yamamoto, Y. Ōnuki, Electronic structures of UX3 (X=Al, Ga, and In)
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