Accepted Manuscript

Accepted Manuscript

Structural, Electronic and Mechanical Properties of C14-Mg₂RE (RE=Eu, Er, Tm, Yb and Lu) Laves Phases: A DFT Study

A. Zahague , B. Lagoun , A. Legris , R. Besson , L. Rabahi , D. Bradai

PII: S0577-9073(18)30473-8

DOI: https://doi.org/10.1016/j.cjph.2018.12.003

Reference: CJPH 714

To appear in: Chinese Journal of Physics Received date: 29 March 2018

Revised date: 1 December 2018 Accepted date: 3 December 2018

Please cite this article as: A. Zahague , B. Lagoun , A. Legris , R. Besson , L. Rabahi , D. Bradai , Structural, Electronic and Mechanical Properties of C14-Mg₂RE (RE=Eu, Er, Tm, Yb and Lu) Laves Phases: A DFT Study, Chinese Journal of Physics (2018), doi:

https://doi.org/10.1016/j.cjph.2018.12.003

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1 Highlights

 The structural properties of Mg2RE are in agreement with the experimental ones.

 Mg2RE compounds in the C14 phase exhibit metallic character.

 Mg₂RE compounds in the C14 phase are mechanically stable.

 The anisotropy decreases as follows: Mg₂Yb > Mg₂Tm > Mg₂Eu > Mg₂Er> Mg₂Lu.

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Structural, Electronic and Mechanical Properties of C14-Mg

RE (RE = Eu, Er, Tm, Yb and Lu) Laves Phases: A DFT Study

A. Zahague^1*, B. Lagoun², A. Legris³, R. Besson³, L. Rabahi^1,4and D. Bradai¹

1Laboratoire de Physique des Matériaux, Faculté de Physique, USTHB, BP 32 El Alia, 16111 Bab Ezzouar, Alger, Algérie

2Laboratoire de Physique des Matériaux, Université Amar Telidji de Laghouat, Algérie

3Unité Matériaux Et Transformations, UMR CNRS 8207, Université Lille 1, F-59655 Villeneuve D'Ascq, France

4Research Center in Industrial Technologies CRTI, P.O. Box 64, Cheraga, 16014, Algiers, Algeria.

Abstract

The Pseudo-Potential Density Functional Theory (PP-DFT) method is applied to investigate the structural, electronic and mechanical properties of C14-Mg2RE Laves phases, with RE being Eu, Er, Tm, Yb and Lu. The predicted cell parameter and c/a ratio of each compound are in good agreement with experimental and theoretical results. Moreover, the studied alloys exhibit a metallic character, which is attributed to the presence of Mg atoms in the C14 Laves phase. Also, the formation of C14-Mg2RE is found to be controlled by the hybridisation between Mg p states and RE- d and f states. From a mechanical property analysis, the studied alloys are found to be mechanically stable. The Mg2Lu compound exhibits higher ductility, while Mg2Er exhibits the smallest one.

Finally, Mg₂Yb is found to be more anisotropic than other phases.

Keywords: A DFT calculations, B Magnesium based Laves phases, C Density Functional Theory (DFT), D Mechanical properties, E Electronic properties.

(*) Corresponding author: [email protected] USTHB, Algiers, Algeria. Tel/Fax: +(213)21 247 344.

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3 1. Introduction

In aircraft and automobile industries, magnesium alloys have generated a great attention for practical applications [1-4] owing to their weak density, high specific strength and stiffness, excellent machinability and casting ability and recycling [5-9]. In fact, the limitation of their application is due to their low strength, poor plasticity and corrosion resistance [10-12]. Thus, to improve the properties of magnesium alloys, a great deal of effort has been dedicated. For the sake of overcoming the previously cited obstacles, small addition of alloying elements seems to be an efficient means [13-16]. It has been proved that the addition of rare earth elements (RE) is an appropriate means to develop mechanical, thermodynamic properties, as well as to enhance creep resistance of magnesium alloys at room and high temperatures [17-22]. For better clarifying the role of rare earth elements in Mg-based alloys, the mechanic, electronic and thermodynamic properties of the C14-Mg2RE Laves phases must be fundamentally known.

The phase stability and phase diagrams have been assessed for Mg-RE systems [23-26]. It was described in several studies, that magnesium formed a large range of intermetallics with RE metals like MgRE, Mg2RE (Mg2Cu-type and Mg2Zn-type), Mg3RE, Mg12RE and Mg24RE5 [27-29].

Among these systems, particularly pseudo-binary Mg2RE (Mg2Zn-type) (RE = Eu, Er, Tm, Yb and Lu) have interested several investigations [30-46]. Reckeweg et al.[41] have studied the formation and structural stability of Mg₂RE (RE = Eu and Yb) intermetallic phases. Buschow et al.[42] have investigated the magnetic and crystallographic properties of Mg₂RE (RE = Eu, Er, Tm and Yb) compounds. Pahlman et al.[43] have assessed the formation energies of Mg₂Er and Mg₂Lu compounds. However, very few theoretical studies were reported for these systems. Using first- principle calculations, Tao et al.[44] have studied the crystal structures and formation enthalpy of Mg₂Eu phase. Bian et al.[45] have studied the structural, electronic and thermodynamic properties of the Mg₂Eu phase. Finally, the lattice parameters and formation enthalpies of Mg₂Lu were calculated by Zhang et al.[46]. In summary, most of the previous researches have concentrated on the structural properties, enthalpy of formation and thermodynamic properties. In fact, very little works focused on physical properties such as elastic constants and modulus of C14-Mg2RE (RE = Eu, Er, Tm,Yb and Lu). The knowledge of such constants produces valuable information on the strength of materials, mechanical stability as well as the anisotropic character of the bonding.

Motivated by the lack of theoretical data [44-46], about physical properties of C14-Mg2RE, the present work aims at giving new insights on structural, mechanical and electronic properties of Mg2Eu, Mg2Er, Mg2Tm, Mg2Yb and Mg2Lu within the C14 Laves phase. To this purpose, the density functional theory (DFT) calculations are used. The present paper is organized as follows:

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Technical details of the DFT calculations are presented in Sec.2. The main results are presented and discussed in Sec.3 while concluding remarks are given in Sec.4.

2. Computational Details

The present study calculations were performed basing on the density functional theory (DFT) using the pseudopotential method as implemented in the Cambridge Serial Total Energy Package (CASTEP) [47, 48]. As a more appropriate approximation to process the on-site correlation of the 4f orbital for the lanthanide series elements, the DFT + U method has been adopted[49-51].The ensemble DFT (EDFT) method improved by Payne et al [52], was used based on the conviction that the use of Hubbard U (U=6) insures convergence instability in the f orbital metal system, relevant to the Rare Earth (RE) elements, during the total energy self-consistent minimization process.The generalized gradient approximation (GGA) [53,54] within the Perdew–Burke–Ernzerhof (PBE) function [55] was taken to include the exchange-correlation energy. The configurations of the valence electron, adopted in this work, consisted of Mg 2p⁶3s², Eu 4f⁷5s²5p⁶6s², Er 4f¹²5s²5p⁶6s², Tm 4f¹³5s²5p⁶6s², Yb 4f¹³5s²5p⁶6s² and Lu 4f¹⁴5p⁶5d¹6s².The cut-off energy value was selected as 500eV for the plane-wave basis set. In the first irreducible Brillouin Zone (BZ), the energy calculations were conducted with a 12×12×6 k-point mesh according to the Monkhorst–Pack scheme. Structural optimizations have been performed using the Broyden–Fletcher–Goldfarb–

Shanno (BFGS) minimization [56] until the total energy converged to less than 10^-6eV. The coordinates of internal atoms were allowed to relax until the maximum forces on unconstrained atoms converged to less than 0.001 eV/Å. The elastic constants of Laves phases Mg₂RE (RE = Eu, Er, Tm, Yb and Lu) were determined by the stress–strain method [57, 58]. The spin polarized density functional theory was used during all calculations.

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5 3. Results and Discussion

3.1. Structural Properties

The Mg2RE compounds investigated in the present work crystallize in the Hexagonal C14 structure, with the space group of P63/mmc (no. 194) and Pearson symbol of hP12. The unit cell of C14 phase consists of twelve atoms, two Mg1 atoms at 2a positions, six Mg2 atoms at 6h positions and four RE (RE = Eu, Er, Tm, Yb and Lu) atoms at 4f positions. Fig. 1 presents the unit cell of C14-Mg2RE.

As a preliminary step, several magnetic configurations were considered for the Mg2RE compounds, including the nonmagnetic, ferromagnetic and antiferromagnetic states. For the ferromagnetic configuration, all the RE atom’s magnetic moment are aligned along the z axis, whereas in the antiferromagnetic configuration, the symmetry of the 4f Wyckoff site was broken, and all the possible configurations were considered, but only the most favorable is reported in this work. The final magnetic configuration was determined from the evaluation of the formation energies. The formation energy is calculated as the difference between the C14 Laves phase total energy and the sum of the total energies of the constituents in their corresponding ground state structure. For each compound, the formation energies with the corresponding magnetic state are reported in Table 1. In the light of our knowledge, there are no available theoretical or experimental values for comparison; however, it is interesting to point out the similar order of magnitude with other Mg-based alloys [46]. From Table 1, it can be concluded that Mg2Eu and Mg2Er are more stable with the ferromagnetic state at 0K. These results are in good agreement with those of N. Bian et al.[45], and K. H. J. Buschow et al.[42] respectively. However in contrary to what was found in the reference [42], our results indicate that Mg2Tm, Mg2Yb and Mg2Lu are stable with the nonmagnetic states at 0 K. Indeed, K. H. J. Buschow et al.[42] found the ferromagnetic and paramagnetic as the most favorable states of the Mg2Tm and Mg2Yb compounds respectively, while there are no available studies devoted to the magnetic properties of the Mg2Lu one. In summary, the magnetism of the Mg2RE Laves phases is still unexplored and therefore needs to be deeply investigated.

In order to achieve the corresponding local minimum energy, both atomic positions and lattice parameters have been completely relaxed. After geometry optimization, the equilibrium ground-state properties of the systems with the most stable magnetic configuration have been predicted. The optimized atomic coordinates of Mg₂RE (RE = Eu, Er, Tm, Yb and Lu) as well as the structural parameters are shown in Tables 2 and 3, respectively together with some previous

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experimental values [31,35,39,42,60-62] and the available theoretical data [45,46] collected from the literature. Although very few theoretical data about these intermetallic compounds are tabulated, it can be seen that our predicted lattice parameters at 0 K are in good agreement with those measured experimentally by X-ray diffraction as well as with those predicted theoretically using the first- principles methods. Particularly, the cell parameters of the Mg₂Er and Mg₂Tm compounds are slightly overestimated (around 4 %) by our method. This reflects the modeling complexity of such materials which is due to the presence of RE-f states metals. In general, our predicted values remain very satisfactory, and this could be a clear sign of the reliability of the pseudopotential method applied in the present work. Moreover, for Eu, Er, Tm, Yb and Lu elements, the corresponding cell parameters of the C14-Mg2RE are presented in the Fig.2, in order to investigate effects of RE elements on the structural parameters of the system. Obviously, the lattice constants a and c increase with increasing atomic radii for alkaline earth metals as presented in Fig. 2. Compounds containing Eu and Lu rare earth, which atomic radii rEu =1.85 A and rLu =1.75 A, possess the highest and lowest cell parameters respectively, while those containing Er, Tm and Yb elements, with equal atomic radii rEr = rTm = rYb = 1.75A, possess intermediate values of cell parameters. These results can be easily explained by size effect, while the electronic effect has also an important role in the case of Lu.

Bond lengths of Mg(2)-RE (RE = Eu, Er, Tm, Yb and Lu) have been also calculated and the relevant average bond lengths are consigned in Table 2. The relation between lattice parameters and atomic number (Z) of the RE elements are shown in Fig.3. According to Fig.3 the bond lengths decrease with increasing atomic number for alkaline earth metals (Eu<Er< Tm <Yb< Lu), due to the increase of Rare Earth atoms electronegativity. If the difference of atomic sizes might be ignored, the larger bond length between Mg and RE indicates that the bonding between them is relatively weak.

From Table 3, it is clear that the bonding between Mg and Lu is the strongest among the studied bond types.

3.2. Electronic Properties of C14-Mg2RE

The Density of states (DOS) is an important theoretical factor for elucidating the nature of the bonding features in a compound [61]. After geometry optimization, the total and partial density of states at equilibrium lattice constants for C14-Mg2RE (RE = Eu, Er, Tm, Yb and Lu) crystal structures were calculated. Fig. 4 presents the total and partial densities of states (DOS) of Mg2Eu, Mg2Er, Mg2Tm, Mg2Yb and Mg2Lu phases. Zero energy in the plotted figures corresponds to the Fermi level (EF). Fig. 4, evidences that the valence and conduction bands overlap considerably and no band gap at the Fermi level EF for Mg2RE phases does exist. Therefore, the C14-Mg2RE (RE = Eu, Er, Tm, Yb and Lu) Phases exhibit metallic properties. From Fig. 4(a, b, c, d and e), it is seen that the numbers of bonding electrons N(EF) (per atom) between the Fermi level and -10 eV, of

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Mg₂Eu, Mg₂Er, Mg₂Tm, Mg₂Yb and Mg₂Lu are 2.02, 2.22, 1.76, 2.14 and 1.16 respectively. The lower the N(E_F), the better the stability of a compound [62,63]. Hence, Mg₂Lu has the best structural stability among them. The curves of PDOS of Mg₂RE (RE = Eu, Er, Tm, Yb and Lu) clearly indicate that the Fermi surface arises mainly from the contribution of Mg p states, although the RE d contribution is also important in the cases of Eu and Lu metals around the Fermi level. This means that the Mg p, Eu d and Lu d states contribute effectively to the metallic behavior of Mg₂RE (RE = Eu, Er, Tm, Yb and Lu). Above the Fermi level, the strong hybridization between the Mg p states and RE (RE = Eu, Yb and Lu) d states and between the Mg p states and RE (RE = Er and Lu) f states contributes to the peaks in TDOS, implying a little covalent bonding between Mg and RE atoms existing in Mg2RE (RE = Eu, Er, Tm, Yb and Lu) crystal structures, as it will be discussed in details in the next paragraph. The hybridization between Mg and Lu atoms is the strongest in the energy range. Furthermore, it can be noticed that the p states of alkaline earth metals have no effect on the TDOS near the EF. Therefore, it can be concluded that the Mg p states and RE (RE = Eu, Er, Tm, Yb and Lu) d and f states ensure the main contribution to the stability of C14-Mg2RE (RE = Eu, Er, Tm, Yb and Lu) phases.

In order to give more insight on the electronic properties of the C14-Mg2RE, the real space charge density distribution in the (110) plane is shown in the Fig 5. All the structures have similar charge density distributions; therefore, only one common distribution is shown in the figure. As it can be seen, a covalent bonding exists between neighboring Mg atoms, as well as between Mg and its nearest RE atoms. This is due to the relative high electrons density prevailing the considered regions.

This result is very consistent with that found by DOS analysis. Interestingly, the charge density of around 0.112 e /A³ at the Mg-Mg bond midpoint is larger than the charge density of 0.08 e /A³ at the middle of the Mg-RE bond. Therefore, the covalent bonding of Mg-Mg is stronger than that of Mg- RE. Unlikely, the electron density distribution displays mainly ionic bonding between RE atoms owing to the low density electron in this region.

3.3. Mechanical Proprieties

The response of materials to an applied macroscopic stress can be determined by the elastic constants which are related to the strength and bonding [64]. Generally, the mechanical properties of solids are characterized by a full set of elastic constants. Hereafter, the elastic properties of C14-Mg2RE (RE = Eu, Er, Tm, Yb and Lu) phases will be discussed. Mg2Eu, Mg2Er, Mg2Tm, Mg2Yb and Mg2Lu compounds belong to hexagonal systems. The number of independent elastic constants for these structures is five: C11, C12, C13, C33 and C44 and the relevant criteria of mechanical stability conditions are:

C11> 0; C11-C12> 0; C44> 0; (C11+C12) C33-2C²13> 0 (1)

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The calculated elastic constants for Mg₂Eu, Mg₂Er, Mg₂Tm, Mg₂Yb and Mg₂Lu are presented in Table 4. As can be seen, the conditions of mechanical stability are easily satisfied. It is worth noting that neither theoretical nor experimental data of elastic constants for the presently studied intermetallics are available in the literature.

The bulk (B) and shear modulus (G) of the hexagonal structure C14-Mg2RE (RE = Eu, Er, Tm, Yb and Lu) phases, were calculated as follows [65]:

(2) (3)

The Poisson’s ratio ν and Young’s modulus E of Mg2Eu, Mg2Er, Mg2Tm, Mg2Yb and Mg2Lu phases were deduced using to the following expressions [65]:

(4) (5)

The calculated elastic modulus, Poisson’s ratio and universal elastic anisotropy index are shown in Table 4. Since C₁₁ and C₃₃ reflect the resistance to linear compression along x and z directions and as showed in Fig. 6, it can be assumed that these phases are highly incompressible under uniaxial stress along x- and z-axis. C₁₂ and C₁₃ relate to the Poisson effect in the hexagonal structures. C₄₄ reflects the resistance to shearing of {010} planes along<001>direction. These results indicate that Mg₂Lu exhibits the highest resistance to shearing along these directions. Generally, the bulk modulus is considered as the measure of resistance to volume change due to applied pressure [66]. Since Mg₂Lu has the largest bulk modulus, it exhibits the strongest resistance to volume change. In addition, shear modulus is a measure of resistance to reversible deformations upon shear stress [66]. The larger the value of shear modulus, the more pronounced directional bonding exists between atoms. The present calculated results indicate that Mg2Lu has the largest shear modulus, followed by Mg2Yb, Mg2Tm, Mg2Er and Mg2Eu. Hence, the directional bonding in Mg2Lu would be much stronger than that in Mg2Yb, Mg2Tm, Mg2Er and Mg2Eu as showed in Fig 7. Furthermore, Young’s modulus provides a measure of the stiffness of solid materials. The larger the Young’s modulus, the stiffer the material.

From the calculated values, the Young’s modulus of Mg2Lu is 64.5 GPa larger than that of Mg2Yb, Mg2Tm, Mg2Er and Mg2Eu, indicating that Mg2Lu is much stiffer than Mg2Yb, Mg2Tm, Mg2Er and Mg2Eu. As indicated above, the elastic modulus of the three Mg2Lu, Mg2Yb and Mg2Tm phases is larger than those of pure Mg [67,68]. Hence, it is obvious that the mechanical properties are improved after alloying with RE elements.

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The ratio of shear modulus to bulk modulus introduced by Pugh [66] can predict the brittle and ductile behavior of materials. A high (low) G/B value is associated with brittleness (ductility). The critical breakdown value ductility from brittleness is about 0.57 [66]. In the present work, the values of Mg₂Eu, Mg₂Er, Mg₂Tm, Mg₂Yb and Mg₂Lu are 0.71757, 0.76422, 0.75163, 0.62549 and 0.618338, respectively, implying that Mg₂Eu, Mg₂Er, Mg₂Tm, Mg₂Yb and Mg₂Lu are essentially brittle as shown in Fig. 8. On the other hand, the Poisson’s ratio is used to quantify the stability of the crystal against shear, which usually ranges from -1to 0.5. The greater the Poisson’s ratio, the better the plasticity. Most of the calculated Poisson’s ratios are very close to 0.25, which means that most of the materials are predominantly under central inter-atomic forces [69]. In addition, Mg2Lu, Mg2Yb have bigger Poisson’s ratios, and hence Mg2Lu has good plasticity among the investigated binary alloys. Contrarily, for Mg2Er, the Poisson’s ratio is the smallest, corresponding to poorest plasticity.

All single crystals in practice are anisotropic, so an appropriate parameter to characterize the extentof anisotropyis required. Recently, Ranganathan and Ostoja-Starzewski [70] reviewed the existing theories related to anisotropy, and concluded that most of them lack universality because of their non-uniqueness and ignorance of the large part of the elastic stiffness tensor. Then they developed a new universal anisotropy index, A^U given by the equation:

The superscripts V and R denote Voigt and Reuss averages, respectively. The anisotropy indexes of Mg2Eu, Mg2Er, Mg2Tm, Mg2Yb and Mg2Lu were calculated and are listed in Table 4. The anisotropy decreases in the following sequence: Mg2Yb > Mg2Tm > Mg2Eu > Mg2Er> Mg2Lu.

Consequently,Mg2Ybis thought to exhibit the highest anisotropy among the studied phases.

4. Conclusion

Density functional theory calculations have been performed in order to study the structural, mechanical and electronic properties of C14- Mg₂RE (RE = Eu, Er, Tm, Yb and Lu). The total and partial densities of state were obtained for Mg₂Eu, Mg₂Er, Mg₂Tm, Mg₂Yb and Mg₂Lu intermetallic compounds; Mg₂Lu has the strongest structural stability among them. The calculated equilibrium parameters are consistent with the available experiments and theoretical values. The elastic constants of the hexagonal phases satisfy the criteria of mechanical stability. The elastic moduli of Mg2RE phases decrease as follows: Mg2Lu> Mg2Yb >Mg2Tm> Mg2Er >Mg2Eu. The Poisson’s ratio and the

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B/G values show that all the intermetallic phases are brittle. The low degree of anisotropy is observed inMg₂Lu, Mg₂Yb and has a high degree of mechanical anisotropy character associated to a reduction of ductility.

Acknowledgements

The authorsgratefully acknowledge Prof. Ibn Khaldoun LEFKAIR from the University Amar

Telidjiof Laghouat, Algeria, for providing the CASTEP software used to perform these calculations.

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16 Table captions

Table 1: Formation energies given in (eV/atom) of the C14-Mg2RE compounds with different magnetic configurations.

Non magnetic ferromagnetic antiferromagnetic

Mg2Eu 0.097 -0.130 -0.125

Mg₂Er 0.216 -0.098 -0.092

Mg₂Tm -0.205 0.093 0.094

Mg2Yb -0.273 -0.103 -0.103

Mg2Lu -0.048 -0.047 -0.047

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Table 2: Optimized Wyckoff positions of C14-Mg2RE (RE = Eu, Er, Tm, Yb and Lu) alloys with some experimental values given between brackets.

Phase Wyckoff site Unit Cell internal parameters

Mg₂Eu 4f

6h

z = 0.0602 (0.063)⁵⁹ x = 0.334 (0.334)⁵⁹

Mg2Er 4f

6h

z = 0.06165 (0.062)³⁷ x = 0.33625 (0.37)³⁷

Mg2Tm 4f

6h

z = 0.06241 (0.06193)³⁹ x = 0.33674 (0.3422)³⁹

Mg2Yb 4f

6h

z = 0.06076 (0.06016)³⁹ x =0.33649 (0.3376)³⁹

Mg2Lu 4f

6h

z = 0.06244 (0.057)⁴⁰ x = 0.34180 (0.333)⁴⁰

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Table 3Calculated lattice constants (in Å)and bond length (in Å)of C14-Mg2RE (RE = Eu, Er, Tm, Yb and Lu) phases, together with available experimental values and other theoretical data.

Phase a (A°) c (A°) d_Mg2-RE (A°)

Mg₂Eu

Present Exp.

Theo.

6.36 6.39³¹ 6.37⁴⁵

10.32 10.32³¹ 10.32⁴⁵

3.70 3.70⁶⁰ 3.84⁴⁵

Mg2Er

Present Exp.

Theo.

6.22 6.00⁴²

-

10.15 9.73⁴²

-

3.63 3.49⁵⁹

-

Mg2Tm

Present Exp.

Theo.

6.24 5.99⁴²

-

10.09 9.70⁴²

-

3.62 3.46³⁹

-

Mg2Yb

Present Exp.

Theo.

6.22 6.25³⁵

-

10.13 10.13³⁵

-

3.62 3.62³⁹

-

Mg₂Lu

Present Exp.

Theo.

6.02 5.96³⁵ 5.97⁴⁶

9.68 9.71³⁵ 9.68⁴⁶

3.47 3.44⁴⁰

-

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Table 4: The calculated elastic constants (in GPa), elastic Moduli (in GPa) of Mg2Eu, Mg2Er, Mg2Tm, Mg2Yb and Mg2Lu phases.

Phase C11 C12 C13 C33 C44 B G E ν G/B A^u

Mg2Eu 48.94 15.47 9.02 57.33 14.13 24.69 17.71 42.89 0.210 0.717 0.185 Mg2Er 50.51 17.19 7.83 62.29 17.23 25.44 19.44 46.48 0.195 0.764 0.174 Mg2Tm 51.77 18.33 8.40 65.22 17.84 26.54 19.95 47.86 0.199 0.752 0.190 Mg2Yb 57.17 16.06 19.47 74.85 15.04 32.56 20.37 50.57 0.241 0.625 0.218 Mg2Lu 75.56 27.62 20.62 89.12 20.86 41.95 25.94 64.52 0.244 0.618 0.147

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Figure Captions

Fig. 1 Crystal structure of C14-Mg2RE.The red spheres are Mg atoms, while the grey ones are RE atoms.

Fig. 2 Calculated lattice constants of Mg2RE (RE = Eu.Er.Tm.Yb and Lu) phases.

Fig. 3 Calculated bond lengths of Mg2RE (RE = Eu.Er.Tm.Yb and Lu) phases.

Fig. 4 Total and partial DOS of (a) Mg2Eu, (b) Mg2Er, (c) Mg2Tm, (d) Mg2Yb and (e) Mg2Lu. The Fermi level was set at zero energy and marked by the vertical lines.

Fig.5: Real space charge density distribution in the (110) plane of Mg2Eu.

Fig.6 Elastic constants C_ij (in GPa) of Mg₂RE (RE = Eu.Er. Tm.Yb and Lu) alloys.

Fig. 7 Elastic Modulii (E, B and G) of Mg₂RE (RE = Eu.Er. Tm. Yb and Lu) alloys.

Fig. 8 G/B andPoisson’s ratios of Mg2RE (RE = Eu.Er. Tm. Yb and Lu) alloys.

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21

Figure 1

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22

Figure 2

Eu Er Tm Yb Lu

6,00 6,15 6,30 9,75 10,00 10,25

a,c (A°)

Elements

a c

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23

Figure 3

62 64 66 68 70 72

3,40 3,45 3,50 3,55 3,60 3,65 3,70 3,75

Bo nd le ng th d Mg(2 )-RE (A°)

Atomic number

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24

-6 -4 -2 0 2 4 6 8

-8 -4 0 4 8 -3,2 -1,6 0,0 1,6 3,2 -2 0 2

-6 -3 0 3 6 9

0 18 36 54

-6 -3 0 3 6 9

0 10 20 30 40 50

Energy (eV)

Mg2Eu

Density of States (states/eV)

Eu-s Eu-p Eu-d Eu-f Mg-s Mg-p (a)

-6 -3 0 3 6 9 12 15

-14 0 14 28 -3,2 -1,6 0,0 1,6 3,2 -2 0 2

-6 -3 0 3 6 9

-20 -10 0 10 20 30

Energy(eV)

Mg2Er (b)

Density of States (states/eV)

Er-s Er-p Er-d Er-f Mg-s Mg-p

-6 -3 0 3 6 9 12 15

-8 0 8 16 -2,2 0,0 2,2 4,4 6,6 -2 0 2

-6 -3 0 3 6 9

-52 -26 0 26

-6 -3 0 3 6 9

-50 -40 -30 -20 -10 0 10 20 30 40 50

Energy(eV)

Mg2Tm

Density of States (states/eV)

Tm-s Tm-p Tm-d Tm-f Mg-s Mg-p (c)

-6 -3 0 3 6 9 12 15

-6,3 0,0 6,3 12,6 -3,2 -1,6 0,0 1,6 3,2 -2 0 2

-6 -3 0 3 6 9

-60 -30 0 30 60

-6 -3 0 3 6 9

-50 -40 -30 -20 -10 0 10 20 30 40 50

Energy(eV)

Mg2Yb (d)

Density of States (states/eV)

Yb-s Yb-p Yb-d Yb-f Mg-s

Mg-p

-6 -3 0 3 6 9 12 15 18 21

-8 -4 0 4 8 -4 -2 0 2 4 -2 0 2

Energy(eV)

Mg2Lu (e)

Density of States (states/eV)

Lu-s Lu-p Lu-d Lu-f Mg-s Mg-p

Figure 4

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Figure 5

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26

Figure 6

Eu Er Tm Yb Lu

10 20 30 40 50 60 70 80 90

Elastic Constants Cij (GPa)

Elements

C11 C12 C13 C33 C44

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Figure 7

Eu Er Tm Yb Lu

10 20 30 40 50 60 70

Elastic Modulus (GPa)

Elements

B G E

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Figure 8

Eu Er Tm Yb Lu

0,1 0,2 0,5 0,6 0,7 0,8 0,9

, G/B

Elements

 G/B