Structural, electronic, elastic, vibrational and thermodynamic properties of U3Si2: A comprehensive study using DFT

Journal of Alloys and Compounds 732 (2018) 160e166

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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom

Structural, electronic, elastic, vibrational and thermodynamic properties of U3Si2: A comprehensive study using DFT D. Chattaraj a, *, C. Majumder b a b

Product Development Section, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India Chemistry Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 May 2017 Received in revised form 20 October 2017 Accepted 21 October 2017 Available online 25 October 2017

Uranium silicide compound is a promising candidate as low enriched uranium nuclear fuel in light water reactors. Here we report a comprehensive study on structural, electronic, elastic, vibrational and thermodynamic properties of U3Si2 using plane wave based density functional theory. The electron-ion interaction and exchange-correlation energy terms have been described by projected-augmented wave method and generalized gradient approximation scheme, respectively. The relativistic corrections to the total energy have been accounted by incorporating the spin-orbit interactions in the total energy calculations. The results showed good agreement between the experimental and theoretical lattice parameters. The electronic structure of U3Si2 compound suggests significant contribution from the 5f and 3p orbitals of U and Si atoms at the Fermi energy level, respectively. The formation energy (DfH) of U3Si2 at 0 K, after zero point energy correction, have been estimated to be
37.40 kJ/mol. Elastic property calculation of U3Si2 showed mechanical stability and anisotropy at ambient pressure. In addition, the phonon calculation showed that U3Si2 is dynamically unstable. The temperature dependent thermodynamic properties of U3Si2 have also been evaluated using the phonon density of states. © 2017 Elsevier B.V. All rights reserved.

Keywords: U3Si2 alloy First principles study Chemical bonding Vibrational properties Elastic properties Thermodynamic properties

1. Introduction Since last decades, the demand of uranium silicide compounds has increased manifold as potential low enriched uranium (LEU) nuclear fuel in light water reactors (LWRs) [1]. These compounds are considered as suitable replacement for conventional nuclear fuel uranium dioxide (UO2) because of their higher thermal conductivity and heat capacity which provides operational safety in nuclear reactors [2,3]. High uranium atom density (upto 4.8 g/cm3) and low enrichment of uranium (<20 wt% 235U) make U3Si and U3Si2 compounds better choice for nuclear fuel among the U-Si binary compounds [4e8]. Compare to U3Si, the U3Si2 compound is found to be more suitable for its implementation as LEU fuel because of several reasons such as stability against growth of fission gas bubbles, low swelling, better compatibility with aluminium etc. A substantial number of work on U3Si2-Al dispersion fuels, UNU3Si2 composites suggest that U3Si2 may serve as a very good fuel material for LWRs [9e12]. In some existing research reactors, U3Si2 has been used as fuel to reduce the uranium enrichment [13,14].

* Corresponding author. E-mail address: [email protected] (D. Chattaraj). https://doi.org/10.1016/j.jallcom.2017.10.174 0925-8388/© 2017 Elsevier B.V. All rights reserved.

Several numbers of reports on experimental and theoretical studies of U3Si2 are available in literature. The synthesis of U3Si2 using different methods have been reported by several research groups [15e17]. The crystal structure, magnetic and thermal properties of U3Si2 has also been studied. The tetragonal crystal structure of U3Si2 is reported first by Zachariasen [18] and later examined by Dwight [19]. The magnetic behavior study by Remschnig et al. revealed that U3Si2 is paramagnetic in nature [20]. Miyadai et al. also reported that the U3Si2 is a Pauli paramagnetic metal at low temperature [21]. White et al. have extensively studied the thermophysical properties of U3Si2 along with other U-Si compounds [22e25]. A series of data on the enthalpy of formation of U-Si compounds have been provided by Berche et al. after performing a detailed thermodynamic study of those compounds [26]. Regarding the theoretical study, Yang et al. have investigated the physical properties of binary U-Si alloys by using density functional theory (DFT) [27]. They have reported the structures, elastic properties and Debye temperature of several U-Si compounds. Wang et al. reported the structural, elastic properties and point defect energetics of U3Si2 using generalized gradient approximation (GGA) and the Hubbart (DFTþU) approximations [28]. Noordhoek et al. reported the phase equilibria in the U-Si system from first-

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principles calculations using spin-orbit coupling and on-site Coulomb correction (GGAþU) method [29]. Due to the advancement of hardware architecture, computational studies are capable of providing reliable, useful and in-depth information of the nuclear fuels in the similar line with experimental techniques. Now a days, density functional theory (DFT) based ab-initio method become very much useful for predicting the physico-chemical behavior of metals, alloys and their compounds. Recent studies by Zhang et al. have inspired this comprehensive study to be carried out in a systematic way [30e33]. In this work, we have reported the structural, electronic, elastic, vibrational and thermodynamic properties of U3Si2 intermetallic using state of the art first principles approach based on density functional theory. The structural properties and energetics will help to understand the stability of the compounds, the electronic properties to understand the chemical bonding present between the constituent elements of those compounds, vibrational properties to understand the dynamical behavior of those compounds, elastic properties to understand the behavior of the materials under different stress-strain conditions, and the thermodynamic properties to understand the behavior and stability of the compounds at different temperatures. The structural and electronic properties of U3Si2 have been discussed with more useful informations and the effect of spin-orbit coupling was also incorporated. Along with that, this computational study on elastic, vibrational and thermodynamic properties of this compound provides in detail physico-chemical data which will be useful for experimental understanding. The necessity of fundamental understanding of U3Si2 system and lack of important physico-chemical data of this compound has inspired this comprehensive study.

2. Computational method All results reported in this work were calculated using Vienna ab initio simulation package (VASP), which implements Density Functional Theory (DFT) [34e36]. The plane wave based pseudopotential method has been used for the total energy calculations. The electron-ion interaction and the exchange correlation energy were described under the Projector-augmented wave (PAW) method and the generalized gradient approximation (GGA) of PerdeweBurkeeErnzerhof (PBE) scheme, respectively [37e39]. Allelectronic projector-augmented wave potentials were employed for the elements U and Si. The valence electronic configuration of U and Si are set to 6s26p66d15f37s2 and 3s23p2, respectively. The energy cut off for the plane wave basis set was fixed at 500 eV. Total energies for each relaxed structure using the linear tetrahedron € chl corrections were subsequently calculated in method with Blo order to eliminate any broadening-related uncertainty in the energies [40]. Ground state atomic geometries were obtained by minimizing the Hellman-Feynman forces using the conjugate gradient method [41,42]. The forces on each ion were minimized upto 5 meV/Å. In order to verify the magnetic nature of the systems studied in this work, we have performed the total energy calculations using the spin polarized version of the DFT. The k-point meshes were constructed using the Monkhorst-Pack scheme and the 9  9  9 k-point mesh was used for the primitive cell for Brillouin zone sampling [43]. The phonon frequencies of U3Si2 are calculated using PHONON program [44] employing the minimum forces on the atoms obtained after geometry optimization. A 1  1  3 supercell of U3Si2 containing total 30 atoms has been used for the phonon calculations. A small displacement of 0.02 Å has been given to the atoms present in the supercell of U3Si2 compound. The phonon dispersion curves and temperature dependent thermodynamic functions of

161

these compounds were obtained by using the calculated phonon frequencies. The temperature-dependent thermodynamic functions of a crystal, such as the internal energy (E), entropy (S), Helmholtz free energy (F) and constant volume heat capacity (CV) can be calculated from their phonon density of states as a function of phonon frequencies. In the present study, the phonon contribution to Helmholtz free energy (F), internal energy (E), entropy (S) and constant volume specific heat (CV) at temperature T are calculated within the harmonic approximation using the following formulas [45]: umax Z

F ¼ 3nNkB T 0

E ¼ 3nN

ħ 2

umax Z

  ħu gðuÞdu ln 2 sinh 2kB T 

u coth

0

umax  Z

S ¼ 3nNkB 0

ħu 2kB T



(1)

gðuÞdu

(2)

  ħu ħu ħu coth
ln 2 sinh gðuÞdu 2kB T 2kB T 2kB T (3)

umax  Z

CV ¼ 3nNkB 0

ħu 2kB T

2

 csc h2

ħu 2kB T



gðuÞdu

(4)

F and the E at zero temperature represents the zero point energy, which can be calculated from the expression as Z umax   ħu gðuÞdu, where n is the number of F0 ¼ E0 ¼ 3nN 2 0 atoms per unit cell, N is the number of unit cells, u is the phonon frequencies, umax is the maximum phonon frequency, and g(u) is Z umax the normalized phonon density of states with gðuÞdu ¼ 1. 0

The elastic properties of U3Si2 compound was calculated using an efficient stress-strain method [46] implemented in VASP code. 3. Results and discussion 3.1. Structural properties The study of electronic structure, elastic, vibrational and thermodynamic properties first require the determination of structural properties of the compound. At room temperature, U3Si2 stabilizes into tetragonal structure with space group P4/mbm. There are ten atoms in the unit cell. The unit cell of U3Si2 is shown in Fig. 1. To obtain the equilibrium structural parameters, the electronic structure of U3Si2 was optimized by varying the cell volume as well as the lattice parameters independently. The optimized lattice parameters of U3Si2 along with the experimental values are summarized in Table 1 [19,28]. It is found that the calculated lattice parameters differ by ±1% from the experimental values. A very good agreement between the calculated and experimental values of the structural parameters of U3Si2 establishes the accuracy of the present computational approach. 3.2. Energetics and electronic properties The thermodynamic stability of a compound depends on Gibbs free energy of formation which is related to the enthalpy of formation of that compound. The formation of the tetragonal U3Si2

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Fig. 1. Crystal structure of U3Si2. The red and blue balls represent U and Si atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. The total density of states (DOS) of U3Si2 with spin orbit (SO) and without spin orbit (NSO) coupling effect.

from its constituent elements can be written by the chemical reaction: 3U(s) þ 2Si(s) ¼ U3Si2(s)

(5)

The total energy of the reactants and products are calculated using plane wave based pseudo potential approach at T ¼ 0 K. Accordingly, from equ. (5), the enthalpy of formation of U3Si2 can be calculated using the following equation:

DfH (U3Si2, 0K) ¼ (Etot(U3Si2)þZPE) e 3*(Etot (U)þZPE) e 2*(Etot(Si)þ ZPE) (6)

Fig. 3. The orbital projected density of states (DOS) of U3Si2 without spin orbit (NSO) coupling effect.

contribution to the specific heat can be derived. The coefficient of the electronic specific heat (g) is calculated as





g ¼ p2 3 $k2B $nðεF Þ

(7)

=

From the total energy calculations, the enthalpy of formation of U3Si2 at 0 K is estimated to be
51.32 kJ/mol. Whereas, after the zero point energy correction and using equation (6), the formation energy modified to
37.40 kJ/mol. The calculated enthalpy of formation of U3Si2 (at 0 K) is good in agreement with the reported experimental value of
33.86 ± 0.42 kJ/mol (at 298 K) [47]. In order to describe the nature of chemical bonding in U3Si2, the total and orbital projected density of states (DOS) have been analyzed. The total DOS of U3Si2, U and Si in U3Si2 are shown in Fig. 2 with and without spin-orbit (SO) coupling interactions. Whereas the orbital projected DOS for U3Si2 along its constituent elements are shown in Fig. 3. It is clear from this Fig. 2 that the nature of U3Si2 DOS and U DOS are very similar which indicates that U is mainly contributing to the total DOS of U3Si2 at the Fermi level. However, the energy levels of Si lie much lower in the DOS spectrum. It is also seen that the DOS of U3Si2 and U spectrum with SO coupling is more delocalized compare to that without SO coupling. Now to understand the chemical interaction of U and Si in the U3Si2 alloy, we compare the split DOS of U3Si2 as shown in Fig. 3. It is seen in Fig. 3 that 5f orbital of U and 3p orbital of Si is mainly participating in the bonding of U3Si2. The total DOS at fermi level of a compound is an important parameter using which electronic

where, kB is the Boltzmann's constant and n(εF) is the density of states at Fermi level. The calculated total DOS of U3Si2 at the Fermi level is N(EF) ¼ 5.62 states per eV-f.u. From this, the coefficient of electronic specific heat g is estimated to be 13.20 mJ/mol-K2. The presence of DOS at the Fermi level of U3Si2 indicates that the compound is an electronic conductor. 3.3. Elastic properties The elastic properties define the physical and internal changes

Table 1 Calculated, experimental and theoretical lattice parameters of U3Si2. Compound

Present study

Experiment [Ref.]

Theory [Ref.]

U3Si2

a ¼ 7.24 Å, b ¼ 7.24 Å, c ¼ 3.90 Å, V0 ¼ 204.57 Å3

a ¼ 7.33 Å, b ¼ 7.33 Å, c ¼ 3.90 Å, V0 ¼ 209.56 Å3 [19]

a ¼ 7.32 Å, a ¼ 7.32 Å, c ¼ 3.90 Å, V0 ¼ 208.20 Å3 [28]

D. Chattaraj, C. Majumder / Journal of Alloys and Compounds 732 (2018) 160e166

U3Si2

This study (Theory) Reported [29] Reported [28]

C11

C33

C44

C66

C12

C13

155 149 167.26

142 139 205.31

65 63 67.49

46 46 74.09

47 49 45.63

50 48 50.34

of a solid material under the application of stress and strain [48]. The bonding characteristics, mechanical and structural stability, deformations etc. are also described by the elastic coefficients and moduli. To investigate the mechanical stability of the intermetallic U3Si2, a set of zero pressure elastic constants were calculated from the stress-stain approach. The calculated elastic constants of single crystal of U3Si2 are listed in Table 2 along with the reported values. For tetragonal symmetry, there are six independent elastic constants, that is, C11, C12, C13, C33, C44 and C66. The calculated elastic constants of U3Si2 fulfil the Born criteria for mechanical stability [49]. Therefore, it can be said that, U3Si2 is mechanically stable at ambient pressure. The higher C11 value of U3Si2 indicates that the bonding strength along [100] direction is stronger than that in other directions. C44 > C66 suggests that [100](010) shear is easier than shear [100](001). The effective elastic moduli of polycrystalline materials are usually calculated by Voigt and Reuss approximations, in which uniform strain/stress are assumed throughout the polycrystal [50,51]. Later, Hill suggested that the actual effective moduli could be approximated by the arithmetic mean of the Voigt and Reuss values, which is referred as the VoigteReusseHill (VRH) value [52]. The details of the calculations for the bulk and shear moduli are given in Refs. [53,54] and therefore not recalled here. The Young's modulus and the Poisson's ratio can be obtained from the bulk and shear moduli [54]. The Zener anisotropy factor A, Poisson ratio n and Young's modulus Y, which are important elastic parameters, are calculated in terms of the computed data using the following relations [55]:

A¼

2C44 C11
C12

1 n¼ 2

Y¼

" B
Bþ

2G 3 1G 3

(8) # (9)

9GB G þ 3B

(10)

where G ¼ (GV þ GR)/2 is the isotropic shear modulus, GV is Voigt's shear modulus corresponding to the upper bound of G values, and GR is Reuss's shear modulus corresponding to the lower bound of G values. The Zener anisotropy, bulk modulus, shear modulus, Young's modulus and Poisson's ratio have been estimated from the calculated single crystal elastic constants, and are given in Table 3. The Zener anisotropy factor (A) represents the degree of elastic anisotropy in solids. If A has value of 1, the compound is considered as completely isotropic, whereas A value smaller or greater than

Table 3 Elastic moduli (in GPa) of U3Si2 at 0 K. System U3Si2

This study (Theory) Reported [29] Reported [28]

B

G

L

Y

A

n

B/G

83 81 92.01

57 53 67.68

165 e e

139 130 163

1.20 e e

0.22 0.23 0.20

1.46 1.53 1.36

vl ¼ √

3B þ 4G 3r

vt ¼ √

" vm ¼

(11)

G

(12)

r

1 2 1 þ 3 3 v3t vl

!#
1 3 (13)

Debye temperature (qD) is calculated using the following equation [60].

qD ¼

  1 3 ħ 3n NA r vm k 4p M =

System

unity indicates the presence of elastic anisotropy in the compound. The calculated Zener anisotropy factor at 0 GPa for U3Si2 is 1.20, which indicates that the compound is entirely anisotropic. Bulk modulus which is a fundamental physical property of solids can be used to measure of the average bond strengths of atoms in the crystal [56]. The calculated bulk modulus of U3Si2 is 83 GPa which is in good agreement with the reported theoretical values [28,29]. The bulk modulus of U3Si2 is less than its constituent elements U (98.7 GPa) and Si (98.8 GPa). This indicates the lower fracture strength of U3Si2 as compared to its constituent elements. The shear modulus (G) measures the resistance of a material to size and shape change. Young's modulus measures the resistance against uniaxial tensions and is indicative of stiffer material. The addition of Si in U resulting U3Si2 lowers the shear and Young's modulus of U. This indicates the lowering of resistance towards shape and size change and stiffness of U. Poisson's ratio is the ratio of transverse contraction strain to longitudinal extension strain under a stretching force. It is related to the bonding properties of materials. Depending on the nature of bonding, Poisson's ratio varies in different materials. As for covalent materials, the value of n is small, typically n ¼ 0.1; for ionic materials, the typical value of n is 0.25; for metallic materials, n is typically 0.33 [57]. In this study, for U3Si2, the Poisson's ratio is 0.22, which indicates the presence of ionic nature in U3Si2. Pugh has proposed that the ratio between the bulk and the shear modulus,i.e., B/G, can be used to predict the brittle or ductile behavior of materials [58]. According to the Pugh condition, a high B/G value indicates a tendency for ductility. If B/G > 1.75, then ductile behavior is predicted, otherwise the material is brittle. The B/G ratio for U3Si2 is 1.46 which indicates that tetragonal phase of U3Si2 is brittle. The elastic and thermodynamic properties of solid are related through the Debye theory. Moreover, the Debye temperature (qD) is related to many physical properties such as specific heat, elastic constants, and melting point [59]. The longitudinal, transverse, mean sound velocities and Debye temperature of U3Si2 have been calculated using the following well-known relations. Longitudinal ðvl Þ, transverse ðvt Þ and Mean ðvm Þ sound velocities are expressed, respectively, as

=

Table 2 Elastic constants (in GPa) of U3Si2 at 0 K.

163

(14)

where ħ is the Planck's constant, k is Boltzman's constant, NA is Avogadro's number, n is the number of atoms per formula unit, r is the density. All calculated quantities from Eqs. (11)e(14) are listed in Table 4. The calculated value of Debye temperature of U3Si2 agrees well with the reported theoretical value of 280.92 K [28].

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Table 4 Sound velocities (in m/sec) and Debye temperature (in K) of U3Si2 at 0 K. System U3Si2

This study (Theory) Reported [28]

Vl

Vt

Vm

WD

3630 3865.04

2180 2355.32

2411 2601.53

262 280.92

3.4. Vibrational properties The study of vibrational properties of a compound is required to judge its dynamical stability/instability and determination of its spectroscopic, namely infrared and Raman modes. The phonon dispersion law between vibrational frequency u and wave vector q can be expressed as [61]:

u ¼ uj (q)

(15)

The subscript j is the branch index. Generally, a crystal lattice with n atoms per unit cell has 3n branches, three of which are acoustic modes and the remainders are optical modes. The vibrational frequency with q ¼ 0, i.e. at the centre G point of the first Brillouin zone, is called as normal mode of vibration. The normal modes of vibration play an important role for Raman scattering and infrared absorption [62]. The calculated phonon frequencies at Brillouin zone centre (G point) of U3Si2 are listed in Table 5. It is seen that Si, the light atom, has larger displacement amplitude, which corresponds to high frequency optical mode whereas, the heavy atom U, corresponds to low frequency acoustic modes. The infrared (IR) active and Raman (R) active modes at Brillouin zone centre (G point) for U3Si2 are also mentioned in Table 5. The phonon dispersion curve depicts how the phonon energy depends on the q-vectors along the high symmetry directions in the Brillouin zone. The curve can be compared with the experimental graph obtained from the neutron scattering experiments on single crystals. The phonon dispersion curve at 0 K for U3Si2 has been obtained by plotting vibrational frequencies along the high symmetry directions, as shown in Fig. 4. For dynamical stability of a compound, the phonon frequencies for all the wave vectors should be positive. However, in computational calculations, if a particular calculation gives an imaginary frequency for any of the wave vectors, the compound is treated as dynamically unstable. Since, In this

Table 5 Phonon frequencies (in THz) at gamma (G) point of U3Si2. Frequency

Multiplicity

Irr. Representation

Active/Silent Mode


1.926
1.449 0.094 0.101 1.166 1.340 2.090 2.340 2.512 2.925 3.026 3.049 3.160 8.085 8.105 8.194 8.387 8.522 8.625 8.835 11.309 11.435

1 1 2 1 2 1 1 2 1 2 1 1 2 2 1 1 2 2 1 1 1 1

A2u A1u Eu A2u Eu B1g A2g Eu A1g Eg B1u B2g Eu Eu B1u B1g Eu Eg A2u A2g A1g B2g

Infrared Silent Infrared Infrared Infrared Raman Silent Infrared Raman Raman Silent Raman Infrared Infrared Silent Raman Infrared Raman Infrared Silent Raman Raman

Fig. 4. Calculated phonon dispersion curves along symmetry lines in the Brillouin zone for U3Si2.

present study, few acoustic frequency of U3Si2 are imaginary which indicates that it is dynamically unstable. 3.5. Thermodynamic properties Many thermo-physical properties of solids, such as specific heat, thermal expansion, heat conduction, thermal diffusivity etc. are mainly guided by the phonons which are basically quanta of vibration. In this work, the thermal properties are determined in the temperature range from 0 to 1000 K within GGA approximation, where the harmonic model remains fully valid. The internal energy and Helmholtz free energy of U3Si2 have been calculated as shown in Fig. 5(a) and (b). The temperature-dependent thermodynamic properties of U3Si2, such as the heat capacities (CV) and entropy (S) are also calculated using harmonic approximation. The variation of S and CV are shown in Fig. 5(c) and (d) upto 1000 K, which is below the melting point of U3Si (1938 K). Fig. 5(b) shows that the Helmholtz free energy (F) of U3Si2 decreases gradually with increase in temperature. In contrast, the internal energy (E) and entropy (S) increases with increase in temperature as shown in Fig. 5(a) and (c). We have also calculated the heat capacity of U3Si2, which is an interesting parameter to understand the thermal behavior of solids. The temperature dependence of heat capacity (CV) of U3Si2 is shown in Fig. 5(d). It is seen that at low temperature (upto150 K), the heat capacities of U3Si2 increase rapidly with increase in temperature, followed by gradual increase upto 400 K and then slowly attain the saturation value of ~115 J/mol.K above 400 K. This is the Dulong-Petit classical limit. There may be two types of error contributing to the thermodynamic functions. First, the phonon density of states which are contributed by imaginary frequencies should not be taken into account for the thermodynamic functions. Second, for the higher accuracy of the calculations, the anharmonicity and electron-phonon coupling factors should be considered. 4. Conclusions The structural, electronic, elastic, vibrational and thermodynamic properties of U3Si2 have been comprehensively studied using the first principles method. The calculated equilibrium lattice parameters of the compound agree well with the experimental values. The enthalpy of formation of U3Si2 at 0 K has been calculated with the zero point energy contributions. From the density of states analysis it is seen that 5f and 3p orbitals of U and Si are

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165

Fig. 5. (aed). Temperature dependent thermodynamic functions of U3Si2.

contributing significantly to the Fermi level of U3Si2. The effect of spin-orbit coupling interactions in the DOS spectra of U3Si2 is also analyzed. The elastic properties of U3Si2 reveal that it is mechanically stable and anisotropic at ambient pressure. The phonon calculation indicates that U3Si2 is dynamically unstable. The internal energy, Helmholtz free energy, entropy and heat capacity (CV) of U3Si2 are calculated as a function of temperature. The thermodynamic data calculated in this study will be useful for designing LEU fuel and other applications U3Si2. Acknowledgements The authors are thankful to Dr. B.S. Tomar, Director, Radio Chemistry and Isotope Group, Bhabha Atomic Research Centre (BARC), Shri N.K. Shukla, Head, Product Development Section (PDS), BARC for their interest and helpful discussions during progress of this work. The authors acknowledge Dr. S.C. Parida, PDS and Dr. Smruti Dash, Fuel Chemistry Division, BARC for her valuable suggestions in this work. The authors are also thankful to the members of the Computer Division, BARC, for their kind cooperation during this work. References [1] Y.S. Kim, Uranium intermetallic fuels (U-Al, U-Si, U-Mo), in: R. Konings (Ed.), Comprehensive Nuclear Materials, vol. 3, Elsevier Ltd., 2012, pp. 391e420. [2] S. Ray, P. Xu, L. Hallstadius, F. Boylan, S. Johnson, TopFuel, European Nuclear Society, Zurich, Switzerland, 2015, p. 57. [3] A.T. Nelson, J.T. White, D.D. Byler, J.T. Dunwoody, J.A. Valdez, K.J. McClellan, American nuclear society, in: Summer Meeting, American Nuclear Society, Reno, Nevada, 2014, p. 987.

[4] R.C. Birtcher, L.M. Wang, Nucl. Instrum. Methods Phys. Res. Sect. B-Beam Interact. Mater. Atoms 59 (1991) 966. [5] S.C. Middleburgh, P.A. Burr, D.J.M. King, L. Edwards, G.R. Lumpkin, R.W. Grimes, J. Nucl. Mater. 466 (2015) 739e744. [6] J. Rest, J. Nucl. Mater. 240 (1997) 205e214. [7] I.J. Hastings, R.L. Stoute, J. Nucl. Mater. 37 (1970) 295e302. [8] B. Bethune, J. Nucl. Mater. 40 (1971) 205e212. [9] L.H. Ortega, B.J. Blamer, J.A. Evans, S.M. McDeavitt, J. Nucl. Mater. 471 (2016) 116e121. [10] Y.S. Kim, G.L. Hofman, J. Nucl. Mater. 410 (2011) 1e9. [11] J. Gan, D.D. Keiser Jr., B.D. Miller, J.F. Jue, A.B. Robinson, J.W. Madden, P.G. Medvedev, D.M. Wachs, J. Nucl. Mater. 419 (2011) 97e104. [12] S.J. Zinkle, K.A. Terrani, J.C. Gehin, L.J. Ott, L.L. Snead, J. Nucl. Mater. 448 (2014) 374e379. [13] Y.S. Kim, G.L. Hofman, J. Nucl. Mater. 410 (2011) 1e9. [14] K. Boning, W. Petry, J. Nucl. Mater. 383 (2009) 254e263. [15] J.M. Harp, P.A. Lessing, R.E. Hoggan, J. Nucl. Mater. 466 (2015) 728e738. [16] V.P. Sinha, G.P. Mishra, S. Pal, K.B. Khan, P.V. Hegde, G.J. Prasad, J. Nucl. Mater. 383 (2008) 196e200. [17] G.A. Alanko, D.P. Butt, J. Nucl. Mater. 451 (2014) 243e248. [18] W.H. Zachariasen, Acta Crystallogr. 2 (1949) 94e99. [19] A.E. Dwight, ANL-82e14, Argonne National Laboratory, Argonne, 1L, 1982. [20] K. Remschnig, T. Le Bihan, H. NoVel, P. Rogl, J. Solid State Chem. 97 (1992) 391e399. [21] T. Miyadai, H. Mori, T. Oguchi, Y. Tazuke, H. Amitsuka, T. Kuwai, Y. Miyako, J. Magn. Magn. Mater. 104107 (1992) 47e48. [22] J.T. White, A.T. Nelson, J.T. Dunwoody, D.D. Byler, D.J. Safarik, K.J. McClellan, J. Nucl. Mater. 464 (2015) 275e280. [23] J.T. White, A.T. Nelson, D.D. Byler, J.A. Valdez, K.J. McClellan, J. Nucl. Mater. 452 (2014) 304e310. [24] J.T. White, A.T. Nelson, D.D. Byler, D.J. Safarik, J.T. Dunwoody, K.J. McClellan, Nucl. Mater. 456 (2015) 442e448. [25] J.T. White, A.T. Nelson, J.T. Dunwoody, D.D. Byler, K.J. McClellan, J. Nucl. Mater. 471 (2016) 129e135. neau, J. Rogez, J. Nucl. Mater. 389 (2009) [26] A. Berche, C. Rado, O. Rapaud, C. Gue 101e107. [27] J. Yang, J.P. Long, L.J. Yang, D.M. Li, J. Nucl. Mater. 443 (2013) 195e199. [28] T. Wang, N. Qiu, X. Wen, Y. Tian, J. He, K. Luo, X. Zha, Y. Zhou, Q. Huang, J. Lang, S. Du, J. Nucl. Mater. 469 (2016) 194e199.

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[29] M.J. Noordhoek, T.M. Besmann, D. Andersson, S.C. Middleburgh, A. Chernatynskiy, J. Nucl. Mater. 479 (2016) 216e223. [30] X.D. Zhang, F. Wang, W. Jiang, Trans. Nonferrous Met. Soc. China 27 (2017) 148e156. [31] X.D. Zhang, W. Jiang, J. Alloys Compd. 663 (2016) 565e573. [32] X.D. Zhang, W. Jiang, Chin. Phys. B 25 (2016), 026301 (1e10). [33] X.D. Zhang, W. Jiang, Philos. Mag. 96 (2016) 320e348. [34] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251e14269. [35] G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15e50. [36] W. Kohn, L. Sham, Phys. Rev. A 140 (1965) 1133e1138. €chl, Phys. Rev. B 50 (1994) 17953e17979. [37] P.E. Blo [38] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758e1775. [39] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865e3868. €chl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223e16233. [40] P.E. Blo [41] R.P. Feynman, Phys. Rev. 56 (1939) 340e343. [42] H. Hellman, Introduction to Quantum Chemistry, Deuticke, Leipzig, 1937. [43] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188e5192. [44] K. Parlinski, Z.Q. Li, Y. Kawazoe, Phys. Rev. Lett. 78 (1997) 4063e4066. [45] C. Lee, X. Gonze, Phys. Rev. B 51 (1995) 8610e8613. [46] Y. Le Page, P. Saxe, Phys. Rev. B 65 (2002), 104104 (1e14). [47] P. Gross, C. Hayman, H. Clayton, Thermodynamics of Nuclear Materials, IAEA, Vienna, 1962, pp. 653e665. [48] J.F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, U.S.A, 1985. [49] M. Born, I math, Proc. Cambridge Philos. Soc., On the Stability of Crystal Lattices, vol. 36, 1940, pp. 160e172.

[50] W. Voigt, Lehrbuch de Kristallphysick, Terubner-verlag, Leipzig, 1928. [51] A. Reuss, Berechnung der Flieflgrenze von Mischlqistallen auf Grund der Plastizitatsbeding fiir Einkristalle, Angew. Z. Math. Mech. 9 (1929) 49e58. [52] R. Hill, The elastic behavior of a crystalline aggregate, Proc. Phys. Soc. Lond. A65 (1952) 349e354. [53] X.M. Tao, P. Jund, C. Colinet, J.C. Tedenac, Phys. Rev. B 80 (2009) 104103e104112. [54] Z.J. Wu, E.J. Zhao, H.P. Xiang, X.F. Hao, X.J. Liu, J. Meng, Phys. Rev. B 76 (2007) 054115e054129. [55] C. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press, Chicago, 1948. [56] A.A. Maradudin, E.W. Montroll, G.H. Weiss, I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, second ed., Academic Press, New York, 1971. [57] J. Haines, J.M. Leger, G. Bocquillon, Annu. Rev. Mater. Res. 31 (2001) 1e23. [58] S.F. Pugh, XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, Philos. Mag. 45 (1954) 823e843. [59] E. Screiber, O.L. Anderson, N. Soga, Elastic Constants and Their Measurement, first ed., McGraw-Hill, New York, 1973. [60] O.L. Anderson, J. Phys. Chem. Solid 24 (1963) 909e917. [61] M.A. Omar, Elementary Solid State Physics: Principles and Applications, Addison-Wesley, Massachusetts, 1975. [62] G.Y. Zhang, G.X. Lan, Lattice Vibration Spectroscopy, High Education Press, Beijing, 1991.