Ab-initio study of the stability of the D8m-Nb5Sn2Ga and D8m-Ta5SnGa2 compounds

Journal of Alloys and Compounds 625 (2015) 57–63

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Ab-initio study of the stability of the D8m-Nb5Sn2Ga and D8m-Ta5SnGa2 compounds Catherine Colinet a,⇑, Jean-Claude Tedenac b a b

Science et Ingénierie des Matériaux et Procédés, Grenoble INP, UJF, CNRS, 38402 Saint Martin d’Hères Cedex, France Institut de Chimie Moléculaire et des Matériaux I.C.G., UMR-CNRS 5253, Université Montpellier II, Place E. Bataillon, 34095 Montpellier Cedex 5, France

a r t i c l e

i n f o

Article history: Received 24 August 2014 Received in revised form 15 October 2014 Accepted 26 October 2014 Available online 12 November 2014 Keywords: Nb5Sn2Ga Ta5SnGa2 Enthalpies of formation Lattice parameters Ab initio calculations

a b s t r a c t First principles calculations have been performed in the T–Sn–Ga (T = V, Nb, Ta) systems along the section xT = 0.625. The enthalpies of formation of the binary and ternary D8m, D81, and D88 structures have been calculated. In the V–Sn–Ga system, no ternary structure is stable in the section. In the Nb–Sn–Ga system, the ternary compound D8m-Nb5Sn2Ga is stable. In the Ta–Sn–Ga system, a combination of the ab-initio calculations and Gibbs energy calculations using the sublattice model allows the show that the phase D8m-Ta5(Sn,Ga)2Ga with a mixed occupancy of the 8h sites of the structure by Ga and Sn atoms is stable at high temperature due to the configurational entropy. These results are in agreement with the experimental determinations previously published in the literature. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The ternary D8m structure (Nb5Sn2Si prototype, tI32, space group I4/mcm) is found in several systems of the type T–X–X0 where T is a transition metal of the beginning of the series (Ti and V columns), X and X0 p elements (Al, Ga, In, Si, Ge, Sn, Pb, Sb. Bi). Kleinke [1], Tanaka et al. [2], Voznyak et al. [3] emphasized the arrangement of the p elements in the D8m structure is controlled by size differences between X and X0 elements. Applying a self-component flux method where tin acts as a solvent, Shihido et al. [4,5] and Ye et al. [6] synthesized successfully single crystals of the ternary compounds Nb5Sn2Ga and Ta5SnGa2. The crystal structures of these ternary compounds belong to the tetragonal system (space group I4/mcm, N°140, tI32). The lattice parameters were obtained [7,8] and are reported in Table 1. The main structural difference between the two compounds appears in the site occupancies: in the case of the Nb5Sn2Ga compound, the Ga atoms occupy the 4a Wyckoff positions and the Sn atoms the 8h Wyckoff positions [7] while in the case of the Ta5SnGa2 compound the Ga atoms occupy the 4a sites and the remaining Ga and Sn atoms share equally and statistically the 8h sites [8]. In the following we will quote this phase as D8m-Ta5(Sn,Ga)2Ga. In the case of the V–Sn–Ga system, no stable D8m ternary compound appears, instead the V2Sn2Ga compound [9] is synthesized in the tin flux method (see Table 2). ⇑ Corresponding author. Tel.: +33 (0) 4 76 82 67 47; fax: +33 (0) 4 76 82 67 45. E-mail address: [email protected] (C. Colinet). http://dx.doi.org/10.1016/j.jallcom.2014.10.155 0925-8388/Ó 2014 Elsevier B.V. All rights reserved.

The aim of the present work is to combine first principles calculations of formation enthalpies and Gibbs energy calculations in order to understand the stability at high temperature of D8m-Nb5Sn2Ga and D8m-Ta5(Sn,Ga)2Ga and to show that the D8m-V5Sn2Ga is not stable. The unit cell of D8m-T5Sn2Ga contains 32 atoms that are situated in a layered arrangement along c axis. The layers z = 1/4 and z = 3/4 are occupied by Ga (4a) and T (4b) atoms while the layers z = 0 and z = 1/2 are occupied by the T (16k) and Sn (8h) atoms. Fig. 1 shows the D8m structure in the case of Nb5 Sn2Ga compound. The structure is build up of two different columns of atoms running parallel to the tetragonal c axis. One column consists of Ga (4a) centered square antiprisms (GaNb8), which share square faces (see Fig. 2a). The other column consists of Nb (4b) centered Sn tetraedra (NbSn4) which share edges (see Fig. 2b). These two columns are interconnected by numerous Nb–Sn and Nb–Nb bonds. The Nb (4b) as well as the Ga (4a) atoms form parallel linear chains with short interatomic distances c/2. 2. Computational details The DFT calculations have been carried out using the Vienna ab initio simulation package (VASP) [10,11]. The calculations made use of potentials constructed by the projector-augmented waves (PAW) technique [12,13]. For the GGA exchange–correlation function, the Perdew–Burke–Ernzerhof parameterization (PBE) was applied [14]. In the PAW potentials used in the present work, the semi-core 3s and 3p electrons of V are considered as valence

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Table 1 Experimental [7,8] and calculated values of the lattice parameters and internal coordinates of the D8m-Nb5Sn2Ga, D8m-Ta5SnGa2 compounds, and the solid solution D8m-Ta5SnGa2 with mixed occupancy of the 8h sites by Ga and Sn atoms. Compound D8m-Nb5Sn2Ga

Exp. Ab-initio

D8m-Ta5Sn2Ga Sol.sol.D8m-Ta5SnGa2

Ab-initio Exp. Ab-initio

a (Å)

c (Å)

x(8h)

x(16k)

y(16k)

10.586 10.6914 10.6354 10.354 10.4522

5.177 5.1688 5.2158 5.1795 5.1819

0.168 0.16878 0.1688 0.1678 0.1665

0.0724 0.07115 0.0710 0.0734 0.0678

0.2204 0.21769 0.2188 0.2208 0.2236

Table 2 Enthalpies of formation of the perfectly ordered compounds in the D8m structure. The formation enthalpies (kJ/mol of atoms) are referred to: A2-T, A5-Sn, and A11-Ga.

a

Compound

DfH

Compound

Df H

V5Ga3a Nb5Ga3a Ta5Ga3a

19.13 32.89 24.81

V5Sn2Ga Nb5Sn2Gaa Ta5Sn2Ga

5.70 24.58 9.79

Stable compounds.

5d4 6 s1 electronic configuration). Concerning the p elements, the 3d, 4s and 4p electrons are used as valence electrons for Ga (3d10 4s2 4p1 electronic-configuration) while the 4d, 5s and 5p electrons are used as valence electrons for Sn (4d10 5s2 5p2 electronicconfiguration). All calculations are performed with a plane-wave cutoff energy of 350 eV. The structures are fully relaxed with respect to the volume, the shape and the cell-internal coordinates. For the Brillouin-zone integration, the Methfessel–Paxton [15] technique with a modest smearing of the one-electron levels (0.2 eV) is used. Care is taken that for each structure a sufficient number of k points for the Brillouin-zone integration is chosen. A Gamma centered k-point grid was used for the hexagonal structures while a Monkhorst–Pack [16] grid was chosen for the other structures. With the chosen plane-wave cutoff and k-point sampling, the reported enthalpies of formation are estimated to be converged with a precision better than ±0.5 kJ/mol of atoms. The energy of formation per atom, DfE(T5/8Sn2/8Ga1/8), is obtained from:

5 min 2 min 1 min Df EðT5=8 Sn2=8 Ga1=8 Þ ¼ Emin T5=8 Sn2=8 Ga1=8  EA2-T  EA5-Sn  EA11-Ga 8 8 8 Fig. 1. D8m-T5Sn2Ga unit cell (T = V, Nb, or Ta). Yellow circles: Ga atoms on 4a Wyckoff positions, orange circles: Sn atoms on 8h, light green circles: T atoms on 4b, dark green circles: T atoms on 16k. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

electrons (3s2 3p6 3d4 4s1 electronic configuration), the semi-core 4s and 4p electrons of Nb are considered as valence electrons (4s2 4p6 4d4 5s1 electronic configuration) while for Ta only the semi-core p electrons are considered as valence electrons (5p6

ð1Þ

The Emin are the minimum total energy of the compound (expressed per atom) and of the elements: body centered cubic A2-V, A2-Nb, A2-Ta, body-centered tetragonal A5-Sn and orthorhombic A11-Ga obtained after full relaxation, this implies a zero pressure. In the case of Sn, the ground state obtained in the ab initio calculations is the diamond A4 structure. We kept however A5-Sn as reference state because it is the one used in the enthalpy of formation databases. At T = 0 K and p = 0 Pa, the formation enthalpy, DfH(T5/8 Sn2/8Ga1/8), equals the calculated formation energy, when the

Fig. 2. (a) Column of face-shared square Ga (4a) centered antiprisms (GaNb8). The Ga (4a) atoms form linear chains parallel to c axis. (b) Column of edge-shared Nb (4b) centered Sn tetraedra. (NbSn4). The Nb (4b) atoms form linear chains parallel to c axis.

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zero-vibration contribution is ignored, since it is much smaller than the formation energy. Additionally to the D8m structures, we investigated the binary D88 (hP16, P63/mcm, N°193, Mn5Si3-type) and D81 (tI32, I4/mcm, N°140, Cr5B3-type) structures, and also the ternary D8l structures (tI32, I4/mcm, N°140, Mo5SiB2-type). These structures are often in competition with the D8m structure in alloys of early transition metals (columns Ti, V, Cr) and p elements. Calculations of enthalpies of mixing in solid solutions have also been performed in the present work. In the cases of D8m and D8l structures, the conventional unit cells were used. In order to determine the enthalpies of mixing along the sections, the atoms are randomly placed on the sublattice where the mixing of the atoms is studied. Successive relaxations are performed: volume, shape, and positions. Care is taken that the cells remain tetragonal.

Dmix HD8m



Sn

The aim of the present ab initio study is also to determine the domain of stability of the D8m phase in the section T5Sn3–T5Ga3. At non-zero temperature, it is necessary to take into account the configurational entropy due to the mixing of the Ga and Sn atoms on site 4a and 8h. For this purpose, the Gibbs energy of the solution is described with the compound energy formalism [17,18]. The following sublattice model: (T)5:(Ga,Sn)2:(Ga,Sn) is used The two transition metal sublattices, sites 4b and sites 16k, are gathered. The two last sublattices represent 8h and 4a sites with the corresponding multiplicities indicated in subscripts. Assuming a random distribution of Ga and Sn on sublattices 4a and 8h, the Gibbs expression of the D8m phase is: D8m m m GD8 ¼ ySn ySn Df HD8 m ðTÞ T5=8 Sn3=8 þ ySn yGa Df HT5=8 Sn1=8 Ga2=8 ð4aÞ ð8hÞ

ð4aÞ ð8hÞ

ð8hÞ ð4aÞ

Ga

¼ Df HD8m





T5=8 Sn2=8 Sny4a Gay4a Sn

1=8



Ga

1=8

D8m D8m 4a y4a Sn Df HT5=8 Sn3=8  yGa Df H T5=8 Sn2=8 Ga1=8

ð5Þ The interaction parameter

m LD8 Ta:Sn:Sn;Ga



  Dmix H T5=8 Sn2=8 Sny4a Gay4a Sn

Ga

1=8



is obtained from:

4a 4a m ¼ LD8 Ta:Sn:Sn;Ga ySn yGa

ð6Þ

In the following, we have considered that L is a constant interaction parameter. Along the other sections, a same procedure was applied as well as in the case of the D8l phase. The minimization of the Gibbs energy has been performed using the SGTE lattice stabilities of the elements [20]. Finally, the thermodynamic data and the phase diagrams along the section T5Sn3–T5Ga3 have been computed.

ð4aÞ ð8hÞ

D8m m þySn yGa Df HD8 T5=8 Sn2=8 Ga1=8 þ yGa yGa Df HT5=8 Ga3=8 h   ð4aÞ ð4aÞ ð4aÞ ð4aÞ þRT 18 ySn ln ySn þ yGa ln yGa  i ð8hÞ ð8hÞ ð8hÞ ð8hÞ m þ ex GD8 þ 28 ySn ln ySn þ yGa ln yGa m

ð2Þ

ðsÞ

R is the gas constant and yi the site fractions of the element i (Sn or Ga) on the site s (4a or 8h). The Gibbs energy in Eq. (2) is referred to A2-T, A5-Sn, and A11-Ga like the enthalpies of formation of the end members. The last term of Eq. (2) is an excess Gibbs energy which is written in the form of the Redlich–Kister polynomial [19]. In the case of the D8m phase, this term is: m GD8 m

The mixing enthalpy of Ga and Sn atoms along section (i), for example, has been obtained from the formation enthalpies of the solid solutions and the perfectly ordered compounds T5Sn3 and T5Sn2Ga by:

T5=8 Sn2=8 Sny4a Gay4a

3. Thermodynamic modeling

ex

(ii) T5Sn2Ga–T5Ga3 in which sites 4a are completely occupied by Sn and the remaining Sn and Ga atoms placed randomly in sites 8h. (iii) T5Sn3–T5SnGa2 in which sites 4a are completely occupied by Sn, while the remaining Sn and Ga atoms are randomly placed in sites 8h. (iv) T5SnGa2–T5Ga3 in which sites 8h are completely occupied by Ga, while the remaining Sn and Ga atoms are randomly placed in sites 4a.

ð8hÞ ð4aÞ ð4aÞ

ð8hÞ ð4aÞ ð4aÞ

0;D8m m ¼ ySn ySn yGa L0;D8 T:Sn:Sn;Ga þ yGa ySn yGa LT:Ga:Sn;Ga ð4aÞ ð8hÞ ð8hÞ m þyGa ySn yGa L0;D8 T:Sn;Ga:Ga

þ

ð4aÞ ð8hÞ ð8hÞ m ySn ySn yGa L0;D8 T:Sn;Ga:Sn

ð3Þ

The L are interaction parameters which can depend of composition among the relation:

  ð4aÞ ð4aÞ 0;D8m 1;D8m m LD8 T:Sn:Sn;Ga ¼ LT:Sn:Sn;Ga þ LT:Sn:Sn;Ga ySn  yGa  2 ð4aÞ ð4aÞ m þ L2;D8 þ þ T:Sn:Sn;Ga ySn  yGa

ð4Þ

m in the case for example of the LD8 T:Sn:Sn;Ga interaction parameter. The excess Gibbs energy includes interactions within sublattices and non-configurational terms such as vibrational and electronic contributions. In the following, the two latter contributions have been neglected. In order to estimate the interactions between Ga and Sn atoms on the 4a and 8h sublattices, we have calculated the enthalpies of formation of partially disordered compounds along the four sections:

(i) T5Sn3–T5Sn2Ga in which sites 8h are completely occupied by Sn, and the remaining Sn and Ga atoms placed randomly in sites 4a.

4. Results and discussion First of all, we investigated the ground state in each binary T–Ga and T–Sn for the xT = 5/8 composition of the transition metal. In the binaries (V, Nb, Ta)–Sn, no compound is stable for this composition, instead an equilibrium between A15-T3Sn (cP8, Pm()n, Cr3Si-type) and oF48-TSn2 (oF48, Fddd, Mg2Cu-type) is observed. In the (V, Nb, Ta)–Ga binaries, the ab-initio calculations show that the D88 structure is less stable than the D8m structure for xT = 5/ 8. In the case of the Nb5Ga3 compound, the Pearson’s Handbook [21] indicates a stabilization of the D88 structure by oxygen. It is probably also the case for the V5Ga3 and Ta5Ga3 compounds. The D81 structure is indicated as stable in the case of the Ta5Ga3 compound [21]. We found the D8m structure more stable than the D8l one with a difference of 0.7 kJ/mol of atoms. The enthalpies of formation at T = 0 K of the binary and ternary compounds along the T5Sn3–T5Ga3 sections are displayed in Fig. 3a–c. In the V–Sn–Ga system, the ternary D8m-V5Sn2Ga compound is not stable while the ternary compound D8m-Nb5Sn2Ga is stable in the Nb–Sn–Ga system. In the Ta–Sn–Ga system, neither the ternary D8m-Ta5Sn2Ga nor the D8l-Ta5 Ga2Sn compounds are stable. The lattice parameters as well as the internal parameters of the D8m-Nb5Sn2Ga compound are reported in Table 1. One may observe a good agreement with the experimental values. The environment of each atom in D8m-Nb5Sn2Ga is presented in Table 3. In the metal-rich compounds, valence electrons remain available for the formation of homoatomic bonds between the transition metal atoms Nb, here, one short (2.584 Å) and several intermediate Nb– Nb bonds (2.999–3.463 Å). The latter are typical lengths for Nb–Nb bonds (2.863 Å in A2-Nb, between 2.655 and 3.252 Å in A15-Nb3Sn). The Nb–Ga distances of 2.769 Å are almost equivalent to the sums of Slater’s covalent radii [22] (rNb = 1.45 Å, rGa = 1.30 Å,

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V5Sn3

Δ fH (kJ/mol of atoms)

(a)

V5Ga3

D8l

10 5

D8l-V5Sn2Ga

Ga (4a)

D88

Ga (4a) Sn (8h) Nb (4b) Nb (16k)

D8m

0 D8m-V5Sn2Ga

A15-V3Sn+ oF48-VSn2

D8l-V5Ga2Sn

D88

-15

D8l

-20

D8m

0.0

0.2

0.4

0.6

0.8

1.0

xGa / (xGa+xSn) Nb5Ga3

Nb5Sn3

Δ fH (kJ/mol of atoms)

(b)

-5

-10 -15 -20

D88 D8l-Nb5Sn2Ga

D8l D8m

D8m-Nb5Ga2Sn

A15-Nb3Sn+ oF48-NbSn2

D88

-25

D8m-Nb5Sn2Ga

D8l-Nb5Ga2Sn

-30

D8l

-35

D8m

0.0

0.2

0.4

0.6

0.8

1.0

xGa / (xGa+xSn) Ta5Ga3

Ta5Sn3

(c) Δ fH (kJ/mol of atoms)

10

D88 D8l

0

-10

2

2

Sn (8h)

Nb (4b)

Nb (16k)

2 2 2

8 8 8 7

2.584 Å

2.769 Å

4 4

3.565 Å 2.860 Å 2.829–3.040 Å

2.860 Å 2.584 Å 3.370 Å

2.769 Å 2.829–3.040 Å 3.370 Å 2.999–3.463 Å

D8m-V5Ga2Sn

-5 -10

Table 3 Bond lengths (<3.6 Å) in the ternary D8m-Nb5Sn2Ga compound.

D8l-Ta5Sn2Ga

D8m A15-Ta3Sn+ oF48-TaSn2

D8m-Ta5Ga2Sn D8m-Ta5Sn2Ga

D88

D8l-Ta5Ga2Sn

-20

D8l D8m

0.0

0.2

0.4

0.6

0.8

1.0

xGa / (xGa+xSn) Fig. 3. Enthalpies of formation of perfectly ordered D8m, D8l, and D88 compounds along the sections: (a) V5Sn3–V5Ga3, (b) Nb5Sn3–Nb5Ga3, and (c) Ta5Sn3–Ta5Ga3. Red line: ground state line at T = 0 K, blue circles: compounds in the D8m structure, dark pink circles: compounds in the D8l structure, green hexagons: compounds in the D88 structure. The dashed lines indicate the sections along which the disordered solid solutions are built. The formation enthalpies (kJ/mol of atoms) are referred to A2-T (T = V, Nb, Ta), A5-Sn, and A11-Ga. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

rSn = 1.45 Å) indicating strong bonding interactions. The Nb–Sn distances range between 2.829 and 3.040 Å. These values can be compared to 2.90 Å sum of the Nb and Sn Slater’s atomic radii [22]. Again, it is an indication of strong bonding between Nb and Sn

atoms. The Ga–Ga bond lengths (2.584 Å) are almost equivalent to the sum of Slater’s covalent radii while the Nb–Nb short bonds are smaller. The Sn–Sn bonds (3.565 Å) are much larger than those in A5-Sn (3.115 Å). The site preferences of Ga and Sn in D8m-Nb5GaSn2 are mostly ruled by size differences, and to a less extent by the different preferences to form homoatomic bonds as pointed out recently [23]. Since the Sn atoms are significantly larger than Ga atoms, there is minor incorporation of Sn in the Ga chains. If we consider now the structure of the unstable D8m-Nb5Ga2Sn compound, the calculated value of the interatomic distance along the chains is 2.654 Å which is unreasonable too short for Sn–Sn bonds. At high temperature, due to the thermal disorder, some mixing of Sn and Ga atoms on the 4a and 8h sublattices can occur. In order to calculate the Gibbs energies at high temperature with the sublattice model [17,18], we have calculated the formation enthalpies of compounds where Sn and Ga are randomly mixed either on the 4a or on the 8h Wyckoff positions. The results obtained in the three systems are displayed on Fig. 4a–c. Various compositions and configurations for a same composition have been investigated. In the whole, the enthalpy of mixing (cf. Eq. (3)) is near zero in the three systems. Therefore we have assumed in the following that all interaction terms L(T:Ga,Sn:Sn), L(T:Ga:Ga,Sn), L(T:Sn:Ga,Sn), and L(T:Ga,Sn:Ga) are equal to zero (cf Eq. (5)). The Gibbs energy of the D8m and D8l phases have been calculated at T = 1273 K. The choice of this temperature is motivated by the fact that the compounds were elaborated at high temperature, then slowly cooled to 1273 K, and finally quenched at room temperature [4–6]. We assumed that the state at 1273 K is retained at room temperature after the quenching. The Gibbs energy curves are displayed in Fig. 5 with A2-T, A5-Sn and A11-Ga at 1273 K as reference states. Tables 4–6 gather the values used in the phase diagram calculations. The ground state line is indicated by a red full line in Fig. 5. An important stabilization of the solid solutions due to the entropy of configuration is observed. However, in the case of the V–Sn–Ga system, the D8m solid solution extents only in a very small composition range from D8m-V5Ga3. In the case of the Nb–Sn–Ga system, the D8m solid solution is stable in a large composition range from a composition near Nb5Sn2Ga (calculated composition: Nb5Sn2.1Ga0.9) to Nb5Ga3. The site occupancies of Ga and Sn on positions 4a and 8h have been computed. They are 4a 8h 8h y4a Ga ¼ 0:8153 ySn ¼ 0:1847 and yGa ¼ 0:0424 ySn ¼ 0:9578. Even at high temperature, the Ga atoms occupy preferentially the 4a sites while the Sn atoms occupy preferentially the 8h sites. In the case of the Ta–Sn–Ga system, the phase diagram calculation shows that the D8m solid solution is stable between a composition near Ti5SnGa2 (calculated composition Ta5Ga2.1Sn0.9 and Ta5Ga3. The site occupancies of Ga and Sn on positions 4a and 8h 4a have been computed. They are y4a Ga ¼ 0:9955 ySn ¼ 0:0045 and 8h y8h ¼ 0:5523 y ¼ 0:4487. The values of the formation enthalpy Ga Sn and Gibbs energy at 1273 K are 18.02 and 19.88 kJ/mol of atoms respectively. The D81 solid solution is never stable however the difference of the Gibbs energies of the D8m and D8l solid solutions is low in the Ga rich side. The compound Ti5SnGa2 was synthesized by the self-flux method using tin as a solvent [5,6]. Our results are in good agreement with these experimental determinations (see Table 1 for comparison of lattice parameters and internal coordinates).

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V5Ga3

V5Sn3

(a)

5

0 D8m-V5Ga2Sn

-5 D8m-V5Sn2Ga

-10

V5Ga3

V5Sn3

ΔfH (kJ/mol of atoms)

Δ fH (kJ/mol of atoms)

5

D8m

D8m

0 -5

D8m-V5Sn2Ga

A15-V3Sn+ oF48-VSn2

-10 A15-V3Sn+ A15-V3Ga+ oF48-VSn2

-15

-15

(a)

D8m

-20

-20 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

Nb5Sn3

(b)

-15

D8m-Nb5Ga2Sn

-20

D8m-Nb5Sn2Ga

-30

0.2

0.4

0.6

-20

-25

(b)

A15-Nb3Sn+ oF48-NbSn2

A15-Nb3Sn+ oF48-NbSn2+ D8m-Nb5Sn2Ga

-30

D8m-Nb5Sn2Ga

D8m solid solution D8m

0.8

1.0

0.0

0.2

xGa / (xGa+xSn)

Δ fG (kJ/mol of atoms)

(c) D8l-Ta5Sn2Ga

0 D8m-Ta5Ga2Sn D8m-Ta5Sn2Ga

0.2

0.4

0.6

0.8

1.0

Ta5Ga3

(c)

D8l D8m

0

-10

A15-Ta3Sn+ oF48-TaSn2

D8m-Ta5Sn2Ga D8l-Ta5Ga2Sn

A15-Ta3Sn+ oF48-TaSn2+ D8m-Ta5(Sn,Ga)2Ga

D8l

D8l D8m

0.0

10

-20

D8l-Ta5Ga2Sn

-20

0.6

Ta5Sn3

Ta5Ga3

D8l

-10

0.4

xGa / (xGa+xSn)

Ta5Sn3

D8m

1.0

Nb5Ga3

D8m

-35

0.0

ΔfH (kJ/mol of atoms)

0.8

D8m

-35

10

0.6

Nb5Sn3

Δ fG (kJ/mol of atoms)

Δ fH (kJ/mol of atoms)

Nb5Ga3

D8m

-25

0.4

D8m

xGa / (xGa+xSn)

xGa / (xGa+xSn)

-15

D8m-V5Ga3+ A15-V3Ga+ oF48-VSn2

0.8

1.0

xGa / (xGa+xSn) Fig. 4. Enthalpies of formation of disordered D8m (and D8l, in the case of Ta) compounds along the sections: (a) V5Sn3–V5Ga3, (b) Nb5Sn3–Nb5Ga3, (c) Ta5Sn3– Ta5Ga3. Blue circles: ordered D8m compounds, blue crosshairs: disordered D8m solid solutions. Dark pink circles: ordered D8l compounds, dark pink crosshairs: disordered D8l solid solutions. The formation enthalpies (kJ/mol of atoms) are referred to A2-T (T = V, Nb, Ta), A5-Sn, and A11-Ga. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

As conclusion, our calculations confirm the experimental results obtained by Shihido et al. [4,5] and Ye et al. [6]. No ternary compound possessing the D8m structure is stable in the V–Sn–Ga system. A D8m compound Nb5Sn2Ga is stable in the Sn rich domain

D8m-solid solution

0.0

0.2

0.4

0.6

D8m

0.8

1.0

xGa / (xGa+xSn) Fig. 5. Gibbs energies of formation at T = 1273 K along the sections: (a) V5Sn3– V5Ga3, (b) Nb5Sn3–Nb5Ga3, (c) Ta5Sn3–Ta5Ga3. Red lines: equilibrium state. Dashed blue line: D8m solid solution at 1273 K. In the case of the Ta5Sn3–Ta5Ga3 section, the results obtained for the D8l structure are also reported. Dashed dark pink line: D8l solid solution at 1273 K. The formation enthalpies and Gibbs energies (kJ/mol of atoms) are referred to A2-T (V or Nb or Ta), A5-Sn, and A11-Ga. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

of the Nb–Sn–Ga system. A phase Ta5(Sn,Ga)2Ga possessing the D8m structure is stabilized at high temperature by configurational entropy due to the mixing of Ga and Sn atoms on the 8h sites in the Ta–Sn–Ga system.

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C. Colinet, J.-C. Tedenac / Journal of Alloys and Compounds 625 (2015) 57–63 Table 4 Modeling of the phases along the V5Sn3–V5Ga3 section. Phase

Description

Gibbs energy (J/mol of atoms))

D8m

(V)5/8(Ga,Sn)2/8(Ga,Sn)1/8

0;A2 m G0;D8 þ 3=8 G0;A5 Sn þ 5420 V:Sn:Sn ¼ 5=8 GV 0;A2 0;A11 m G0;D8 þ 3=8 GGa  19130 V:Ga:Ga ¼ 5=8 GV 0;A2 0;A5 m G0;D8 þ 2=8 GSn þ 1=8G0;A11 þ 5700 Ga V:Ga:Sn ¼ 5=8 GV 0;A2 m þ 2=8G0;A11 þ 1=8 G0;A5 G0;D8 Ga Sn  5300 V:Sn:Ga ¼ 5=8 GV 0;D8m m L0;D8 V:Sn;Ga:Sn ¼ 0; LV:Sn;Ga:Ga ¼ 0, 0;D8m m L0;D8 V:Ga:Ga;Sn ¼ 0; LV:Sn:Ga;Sn ¼ 0

A15-V3Sn

V3/4Sn1/4

0;A2 0;A5 G0;A15 þ 1=4 GSn  7620 V:Sn ¼ 3=4 GV

oF48-VSn2

V1/3Sn2/3

0;A2 G0;oF48 þ 2=3 G0;A5 V:Sn ¼ 1=3 GV Sn  7850

Table 5 Modeling of the phases along the Nb5Sn3–Nb5Ga3 section. Phase

Description

Gibbs energy (J/mol of atoms)

D8m

(Nb)5/8(Ga,Sn)2/8(Ga,Sn)1/8

m G0;D8 ¼ 5=8 G0;A2 þ 3=8 G0;A5 Sn  14880 Nb Nb:Sn:Sn m ¼ 5=8 G0;A2 þ 3=8 G0;A11  32890 G0;D8 Ga Nb Nb:Ga:Ga

0;A2 0;A11 m ¼ 5=8 GNb þ 2=8 G0;A5  24580 G0;D8 Sn þ 1=8 GGa Nb:Ga:Sn 0;A2 m ¼ 5=8 GNb þ 2=8G0;A11 þ 1=8 G0;A5 G0;D8 Ga Sn  19140 Nb:Sn:Ga m m ¼ 0; L0;D8 ¼ 0; L0;D8 Nb:Sn;Ga:Sn Nb:Sn;Ga:Ga m m ¼ 0; L0;D8 ¼0 L0;D8 Nb:Ga:Ga;Sn Nb:Sn:Ga;Sn

A15-Nb3Sn

Nb3/4Sn1/4

G0;A15 ¼ 3=4 G0;A2 þ 1=4 G0;A5 Sn  17700 Nb:Sn Nb

oF48-NbSn2

Nb1/3 Sn2/3

G0;oF48 ¼ 1=3 G0;A2 þ 2=3 G0;A5 Sn  15610 Nb:Sn Nb

Table 6 Modeling of the phases along the Ta5Sn3–Ta5Ga3 section. Phase

Description

Gibbs energy (J/mol of atoms)

D8m

(Ta)5/8(Ga,Sn)2/8(Ga,Sn)1/8

0;A2 0;A5 m G0;D8 Ta:Sn:Sn ¼ 5=8 GTa þ 3=8 GSn þ 1820 0;A2 0;A11 m G0;D8  24810 Ta:Ga:Ga ¼ 5=8 GTa þ 3=8 GGa 0;A2 0;A5 0;A11 m G0;D8  9790 Ta:Ga:Sn ¼ 5=8 GTa þ 2=8 GSn þ 1=8 GGa 0;A2 0;A11 0;A5 m G0;D8 þ 1=8 GSn  8240 Ta:Sn:Ga ¼ 5=8 GTa þ 2=8 GGa 0;D8m m L0;D8 Ta:Sn;Ga:Sn ¼ 0; LTa:Sn;Ga:Ga ¼ 0 0;D8m m L0;D8 Ta:Ga:Ga;Sn ¼ 0; LTa:Sn:Ga;Sn ¼ 0

D81

(Ta)5/8(Ga,Sn)2/8(Ga,Sn)1/8

0;A2 0;A5 1 G0;D8 Ta:Sn:Sn ¼ 5=8 GTa þ 3=8 GSn þ 9180 0;A2 0;A11 1 G0;D8  24110 Ta:Ga:Ga ¼ 5=8 GTa þ 3=8 GGa 0;A2 0;A5 0;A11 1 þ 2940 G0;D8 Ta:Ga:Sn ¼ 5=8GTa þ 2=8 GSn þ 1=8 GGa 0;A2 0;A11 0;A5 1 G0;D8 þ 1=8 GSn  17300 Ta:Sn:Ga ¼ 5=8 GTa þ 2=8GGa 0;D81 1 L0;D8 Ta:Sn;Ga:Sn ¼ 0; LTa:Sn;Ga:Ga ¼ 0, 0;D81 1 L0;D8 Ta:Ga:Ga;Sn ¼ 0; LTa:Sn:Ga;Sn ¼ 0

A15-Ta3Sn

Ta3/4Sn1/4

0;A15 0;A5 GTa:Sn ¼ 3=4 G0;A2 Ta þ 1=4 GSn  4720

oF48-TaSn2

Ta1/3Sn2/3

0;oF48 0;A5 GTa:Sn ¼ 1=3 G0;A2 Ta þ 2=3 GSn  4670

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