On the constitution and thermodynamic modelling of the system Ti–Ni–Sn

Abstract

Phase equilibria of the system Ti–Ni–Sn have been determined for the isothermal section at 950 °C based on X-ray powder diffraction (XPD) and electron probe microanalysis (EPMA) of about 60 ternary alloys in as cast and annealed state. The section is characterized by the formation of four ternary compounds labelled τ₁ to τ₄. Whereas two of the ternary compounds are found without significant homogeneity regions: τ₁-TiNiSn (half Heusler phase, MgAgAs-type), τ₃-Ti₂Ni₂Sn (U₂Pt₂Sn-type), τ₄-(Ti_1−x−ySn_xNi_y)Ni₃ with AuCu₃-type exhibits a solution range (0.22 ≤ x ≤ 0.66 and 0.22 ≥ y ≥ 0.02) and a particularly large homogeneity region is recorded for τ₂-Ti_1+yNi_2−xSn_1−y (Heusler phase, MnCu₂Al-type). Extended solid solutions starting from binary phases at 950 °C have been evaluated for Ti₅Ni_1−xSn₃ (filled Mn₅Si₃ = Ti₅Ga₄-type; 0 ≤ x ≤ 1), Ti_1−xSn_xNi₃ (TiNi₃-type; 0 ≤ x ≤ 0.27) and (Ti_1−xNi_x)_1−ySn_y (CsCl-type) reaching a maximum solubility at x = 0.53, y = 0.06). From differential thermal analysis (DTA) in alumina crucibles under argon a complete liquidus surface has been elucidated revealing congruent melting for τ₂-TiNi₂Sn at 1447 °C, but incongruent melting for τ₁-TiNiSn (pseudobinary peritectic formation: [small script l] + τ₂ ↔ τ₁ at 1180 °C), τ₃-Ti₂Ni₂Sn (peritectic formation: L + τ₂ + Ti₅NiSn₃ ↔ τ₃ at 1151 °C) and τ₄-Ti_1−xSn_xNi₃ (peritectic formation: L + TiNi₃ + (Ni) ↔ τ₄ at 1157 °C). A Schultz–Scheil diagram for the solidification behavior was constructed for the entire diagram involving 20 isothermal four-phase reactions in the ternary. For a thermodynamic CALPHAD assessment of the ternary diagram we relied on the binary boundary systems as modelled in the literature. As thermodynamic data in the ternary system were only available in the literature for the compounds TiNi₂Sn and TiNiSn, heat of formation data were supplied by our density functional theory (DFT) calculations for Ti₂Ni₂Sn, as well as for the solid solutions, which were modelled for Ti_1−xSn_xNi₃, Ti₅Ni_1−xSn₃ and (Ti_1−xNi_x)_1−ySn_y. Thermodynamic calculation was performed with the Pandat software and finally showed a reasonably good agreement for all the 20 invariant reaction isotherms involving the liquid.

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1 Introduction

The so-called “Half Heusler” (HH) compound TiNiSn, known as an n-type semiconductor since 1986,^1,2 has hitherto displayed a high potential for exceptional efficiency in thermoelectric (TE) generators converting (waste) heat into electricity. Thermoelectric research has, therefore, focused mainly on improvement of ternary intermetallic compounds based on TiNiSn, which crystallize with the non-centrosymmetric cubic MgAgAs-type structure (space group F43̄m). Beside skutterudites and clathrates, HH compounds are promising candidates for high temperature thermoelectric applications because they inherit a tuneable electronic structure, which can be modified through (i) doping/substitution on its three metal sublattices, (ii) engineering of a generally narrow band gap, and (iii) nanostructuring via ball-milling and precipitation of secondary system inherent phases as the most prominent among many other techniques. An overview on the thermoelectric properties of HH alloys, reaching up to 700 °C a ZT_max ∼ 1.0 for p-type Ti{Fe_xCo_1−x}{Sn_ySb_1−y} and a ZT_max ∼ 1.2 for n-type {Ti_1−u−vZr_uHf_v}Ni{Sn_1−wSb_w} can be found from a recent review article.³ However, the biggest disadvantage of HH alloys is their relatively high thermal conductivity, which has to be decreased in order to increase the thermoelectric performance of these materials. Large scale production and particularly nanostructuring of TE-materials by precipitation of preferably system inherent phases, however, needs a profound knowledge not only of isothermal phase relations, temperature dependent solubilities and vacancy concentrations but also of the solidification behaviour in each ternary subsystem of any multi-component TiNiSn-based alloy system. Although several papers in the literature provide (a) phase relations in the isothermal section of Ti–Ni–Sn at 800 °C (ref. 4) (see also refs. therein), (b) melting temperatures of TiNiSn (T_m = 1182 °C;⁵ T_m = 1182 °C (ref. 6)), TiNi₂Sn (T_m = 1447 °C (ref. 5)), (c) DFT heat of formation data for various binary and ternary compounds,^5,7 (d) calorimetric heat of formation data⁸ and (e) coefficient of thermal expansion α_ave = 11.3 × 10⁻⁶ K⁻¹ (40–690 °C),⁶ to our best knowledge hitherto no liquidus projection exists. For most reports in the literature the homogeneity regions of the Heusler and the half Heusler phase play no important role, but several authors gave proof to the non-stoichiometry of TiNi_2−ySn (0 < y < 0.04, 800 °C;⁴ 0 < y < 0.22, as cast⁵) and of TiNi_1+ySn (0 < y < 0.10, as cast;⁵ 0 < y < 0.06, 900 °C;⁹ −0.06 < y < 0.08, at 1100 °C (ref. 10)).

In order to shed light on the complicated synthesis procedures for single-phase TiNiSn-based alloys, the present paper is intended to provide (1) a liquidus surface for the entire Ti–Ni–Sn phase diagram i.e. precise information on the solidification paths, and (2) phase relations in an isothermal section at 950 °C. As the TiNiSn system is a subsystem of a multi component thermoelectric alloy system, for which a thermodynamic (pre-) calculation can save extensive experimental work, the present paper will furthermore provide a thermodynamic assessment of the Ti–Ni–Sn ternary. This CALPHAD-type modelling is based on existing assessments for the binary boundary systems as well as relies on experimental thermodynamic data for the ternary compounds and will be backed by DFT energies of formation wherever needed in the modelling.

2 Experimental details

2.1. Sample preparation and characterization

Pure elements in form of Ti-, Ni-rods, Sn-shot or bars with a minimum purity of 99.95 mass% from Alfa Aesar were used for the preparation of about 60 alloys with various compositions for the system Ti–Ni–Sn. First stoichiometric amounts of Ti and Ni were arc melted together under 6 N argon and then the proper amount of Sn was added. The reguli were flipped 3 times for a better homogenization. Afterwards the samples were vacuum sealed in quartz ampullae (which were backfilled at RT with 200 mbar Ar), annealed at different temperatures for 7 days (200 °C → 950 °C with a heating rate of 10°C min⁻¹) and water quenched. Some samples were further annealed in evacuated quartz ampullae (which were backfilled at RT with 200 mbar Ar) at different temperatures in Al₂O₃ crucibles for better equilibration.

For sample characterization we used Scanning Electron Microscopy (SEM), Electron Probe Microanalysis (EPMA) and X-ray Powder Diffraction (XPD). The microstructure and chemical composition of the alloys were analyzed by SEM on a Zeiss Supra 55 VP equipped with an energy dispersive X-ray (EDX) detector operated at 20 kV. Samples for EPMA were prepared by standard metallographic methods. In some cases polishing was performed under glycerine instead of water to avoid oxidation and/or hydrolysis of samples. X-ray powder diffraction profiles for all alloys were collected from a HUBER-Guinier image plate with monochromated CuK_α1-radiation. For Rietveld refinements we used the FULLPROF program,¹¹ whilst precise lattice parameters were obtained by least square methods with program STRUKTUR¹² employing pure Ge (99.9999%) as internal standard (a_Ge = 0.5657906 nm).

Single crystals of Ti₅NiSn₃ were isolated from alloy Ti₅₃Ni₁₁Sn₃₄ annealed at 1100 °C for 5 days. The crystals were inspected on an AXS-GADDS texture goniometer for quality and crystal symmetry prior to X-ray single crystal (XSC) intensity data collection on a four-circle Nonius Kappa diffractometer (CCD area detector and graphite monochromated MoK_α radiation, λ = 0.071069 nm). Orientation matrix and unit cell parameters were derived using the program DENZO (Nonius Kappa CCD, Program Package, Nonius Delft, The Netherlands). Besides psi-scans no additional absorption correction was necessary because of the rather regular crystal shape and small dimensions of the investigated specimens. The structure was solved by direct methods and refined with the SHELXS-97 and SHELXL-97 programs,¹³ respectively.

The differential thermal analyses (DTA) measurements were performed on annealed samples in a Netzsch 404 Pegasus DSC (differential scanning calorimetry) equipment in Al₂O₃-crucibles under a stream of 6 N argon and a heating rate of 5 K min⁻¹. The equipment was calibrated in the temperature range from 300 to 1400 °C against pure metal standards supplied by Netzsch to be within ±1 °C.

2.2. Thermodynamic modelling

2.2.1. First-principles calculations. The DFT calculations were carried out using the Elk v2.3.22 package¹⁴ – an all-electron full-potential linearised augmented-plane wave (FP-LAPW) code with Perdew–Burke–Enzerhoff exchange–correlation functional in generalized gradient approximation (GGA).¹⁵ The APW basis set cut-off used in the calculations was set to 190 eV, and the k-grid was equal or higher than 10 × 10 × 10 k-points depending on the structure. Prior to final total energy calculations the geometry of the initial structures (lattice vectors and atomic coordinates) was completely relaxed. The proper values of the muffin-tin radii were selected automatically at the initial stage of the calculations. In general the enthalpy of formation (ΔH in (meV per atom)) at T = 0 K for a specific compound was calculated according to the following formula:

ΔH = 10³[E_tot(Ti_aNi_bSn_c) − a(E_tot(Ti)/j) − b(E_tot(Ni)/k) − c(E_tot(Sn)/l)]/(a + b + c),

where a, b, c are the number of each type of atoms in the crystal lattice of compound used in calculations; j, k, l are the number of atoms in the crystal lattice of Ti, Ni, and Sn, respectively, used in the calculations; E_tot is the total energy of compound in eV.

2.2.2. CALPHAD modelling. For thermodynamic and phase diagram calculations as well as for optimization of thermodynamic parameters based on the CALPHAD method, the Pandat software package¹⁶ was used. Thermodynamic modelling of phases existing in this ternary system relied on the well-known Compound Energy Formalism (CEF)¹⁷ enabling us to respect the real crystallographic structure of a phase by means of a sublattice description. As fixed values we used thermodynamic data from existing thermodynamic assessments for the respective binary sub-systems as well as energy of formation data for the end-members of intermetallic phases at 0 K (data from DFT-calculations of the ground state energies for binary compounds).

For the thermodynamic description of the respective phase structures, commonly used thermodynamic models were applied with respect to reference Gibbs energies of components in given phase ϕ, which have a polynomial form:¹⁸

where A, B, C, D_n and n (typically equal to 2, 3, and −1) are constants characteristic for the particular structure of the element i in a given temperature interval in Kelvin. The reference Gibbs energy is defined relative to the molar enthalpy of the element ⁰H^SER_i at 298 K and 1 bar in its Standard Element Reference state (SER).

The Gibbs energy of a given phase like liquid and solid solution or a compound is expressed as a sum of several contributions:

where ^refG^ɸ_m is the reference level of the molar Gibbs energy of a given phase ϕ, ^idG^ɸ_m describes the ideal mixing of components and ^exG^ɸ_m is the excess Gibbs energy describing a non-ideal behavior of components due to their mutual interactions. The last external term ^extG^ɸ_m, which includes other additional contributions to the overall Gibbs energy like e.g. pressure or surface except for magnetic contributions has not been used in the thermodynamic modelling of the Ti–Ni–Sn system.

The particular terms in case of solution phases like liquid or solid solutions are described as follows:

with

L^ɸ_i,j,k = ⁰L^ɸ_i,j,kx_i + ¹L^ɸ_i,j,kx_j + ²L^ɸ_i,j,kx_k (3e)

and

ⁿL^ɸ_i,j = a_n + b_nT + c_nT ln T (3f)

where x_i, x_j and x_k are molar fractions of elements i, j and k (Ti, Ni, Sn), respectively in the given phase and L are thermodynamic interaction parameters, allowing to express the excess Gibbs energy. The expressions (3d)–(3f) introduce the generally used Redlich–Kister–Muggianu method^19,20 for evaluation of thermodynamic non-idealities in a binary and ternary phase.

Thermodynamic modelling of intermetallic phases is based on the compound energy formalism in which the particular terms are described as follows:

where y_i is the site fraction of element i in the first or second sublattice (y′_i or y′′_i) and ν_i is the stoichiometric coefficient of element i of the real or hypothetical compound and hence its stoichiometric ratio in the first or second sublattice (ν′_i or ν′′_i). The meaning of other expressions is the same as above, but in this case G_i:j indicates the reference Gibbs energy for a real or hypothetical compound i:j with respect to the standard element enthalpy at 298.15 K (⁰H^SER_i or ⁰H^SER_j, respectively).

The difference in the reference Gibbs energy for a given real or hypothetical compound i:j (⁰G_i:j) of an intermetallic phase and the Gibbs energies of the elements i, j in their Standard Element Reference states (SER) (⁰G_i, ⁰G_j) is given by the equation:

Δ⁰G_i:j = ⁰G_i:j − ν⁰_iG_i − ν⁰_jG_j = ΔH − TΔS (5)

where H is enthalpy and S is entropy.

At T = 0 K, one may write ΔH(T = 0) = ΔE(T = 0), i.e. the difference in enthalpies is equal to the difference of total energies. These total energy differences have been calculated ab initio at the equilibrium volume in the present paper. The difference in enthalpies ΔH, at finite temperature is then obtained as (Kirchhoff's law):

ΔH = ΔE + ∫ΔC_pdT (6)

where ΔC_p is the difference between heat capacity of the given intermetallic phase and that of the SER structure.

In the region without phase transformation, entropy can be expressed as:

ΔS = ∫(ΔC_p/T)dT (7)

In general, the heat capacity difference ΔC_p is temperature dependent, and in the simplest case it can be described by a linear function:

ΔC_p = a + bT (8)

Substitution of the enthalpic and entropic term with the relations (6)–(8) in eqn (5) yields after integration:

Δ⁰G_i:j = ΔE + a(1 − ln T)T − (b/2)T² (9)

This equation has been employed in the phase diagram calculations. ΔE is calculated ab initio and ΔC_p is optimized as a curve fitting parameter to the experimental phase equilibrium data.

3 Results

3.1. The binary boundary systems

Information on the three binary phase diagrams is based on the compilation by Massalski.²¹ Assessments of experimental phase diagram data and thermodynamic modelling are available from various research groups: Ni–Sn,^22,23 Ti–Sn,²⁴ and Ti–Ni.^25–27 Detailed crystallographic data on unary and binary boundary phases reported in the literature (ref. 21–43) and obtained in the current work are summarized in Table S1 (ESI†); data for ternary compounds are listed in Table 1.

Table 1 Crystallographic data of the ternary phases of the system Ti–Ni–Sn

Phase, temperature range (°C)	Space group, prototype	Lattice parameters (nm)		Comments/references
		a	c
a This work.
τ1-TiNiSn	F4̄3m	0.5927	—	Ref. 44
	MgAgAs	0.59332(6)	—	Ref. 4
		0.59349(1)	—	^a
Ti1+yNi2−xSn1−y		0.59655(4)	—	x = 0.779, y = −0.008 at 1100 °C; in equilibrium with τ2^a
τ2-Ti1+yNi2−xSn1−y	Fm3̄m	0.6097(3)	—	x = 0, y = 0, z = 0 (ref. 4)
	MnCu2Al	0.60973(4)	—	x = 0, y = 0, z = 0,^a see also Fig. 9
		0.60710(5)	—	x = 0.344, y = 0.038 at 950 °C; in equilibrium with τ1^a
		0.60503(8)	—	x = 0.008, y = 0.344 at 1100 °C; in equilibrium with TiNi^a
		0.6049(2)	—	x = −0.651, y = −0.121 at 1100 °C; in equilibrium with TiNi3 and Ni3Sn2^a
τ3-Ti2+xNi2Sn1−x	P42/mnm	0.68168(4)	0.64379(6)	x = 0 (ref. 4)
	U2Pt2Sn	0.68273(6)	0.63850(4)	x = 0.195 in equilibrium with TiNi and Ti5NiSn3 at 950 °C
τ4-(Ti1−x−ySnxNiy)Ni3	Pm3̄m	0.36316(5)		Ti14.0Ni80.5Sn5.5; x = 0.22 and y = 0.22; see also Fig. 10
	AuCu3	0.36904(7)		Ti6.2Ni77.4Sn16.4; x = 0.66 and y = 0.1
τ4′	Unknown			At ∼ Ti16–14Ni75Sn9–11
τ5	Unknown			At ∼ Ti52Ni23Sn25

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3.2. Phase equilibria in the ternary system Ti–Ni–Sn

In order to determine the phase equilibria in the ternary system, about 60 alloys were investigated in as-cast state and after annealing at 800 and 950 °C (7 days). In some cases these temperatures were not sufficient for equilibration of the samples and therefore they were additionally annealed for 7 days in the temperature range from 450 to 770 °C (for Sn contents > 40 at%) and at 1050, 1080 and 1100 °C (for alloys containing high melting compounds such as TiNi₃, TiNi, Ti₅Sn₃ and τ₂).

Combined evaluation of EDX and XPD data in equilibrated samples defined the number of three-phase fields, which are labelled alphabetically in the figures from “a” to “w”. The temperatures of the invariant four-phase reactions have been determined with DTA on equilibrated samples. Microstructures of the as-cast samples were used to define the primary crystallization fields and reactions during solidification. The phase equilibria established in the system are presented as projections of liquidus- and solidus surfaces (Fig. 1 and 2), as a melting-crystallization diagram (Fig. 3) and in form of a Schultz–Scheil diagram (Fig. 4) for the solidification behavior in the entire system involving 20 isothermal four-phase reactions in the ternary. Phase equilibria at 950 °C are presented as an isothermal section in Fig. 5. A summary of the phases involved in invariant reactions is available from Table 2. Phase compositions of the selected alloys, which have been investigated in different states, are presented in Table S2 (ESI†) which contains links to the SEM images (Fig. 6–8) that document respective statements.

Fig. 1 Liquidus projection of Ti–Ni–Sn. The labels inside the circles denote indices for the microstructures of as-cast samples presented in Fig. 6 (open circles) and Fig. 8 (gray circles). The small triangles show compositions of eutectics observed in as-cast samples. Composition of the phases measured by EPMA is listed in Table S2 (ESI†).

Fig. 2 Solidus projection with corresponding temperatures, determined by DSC measurements. The labels inside the circles denote indexes for the microstructures of the samples annealed at 950 °C (Fig. 7, open circles) and other temperatures Fig. 8 (gray and semi-filled circles).

Fig. 3 Sub-solidus surface with superimposed mono-variant lines from the liquidus. The composition of phases involved in the invariant reactions is summarized in Table 2.

Fig. 4 Schultz–Scheil diagram for the ternary Ti–Ni–Sn system. The homogeneity regions: Ti_1−xSn_xNi₃, Ti₅Ni_xSn₃, (Ti_1−xNi_x)_1−ySn_y and Ti_1+yNi_2−xSn_1−y are noted as TiNi₃, Ti₅NiSn₃, (Ti_1−xNi_x)_1−ySn_y and τ₂.

Fig. 5 Isothermal section at 950 °C. The labels inside the circles denote indices for the microstructures of the annealed samples presented in Fig. 7.

Table 2 Comparison of experimental and calculated data characterizing the invariant equilibria in the Ti–Ni–Sn system

Reaction (exp.)	Phase	Ti	Ni	Sn	Type t °C	Reaction (calc.)	Phase	Ti	Ni	Sn	Type t °C
L + Ti2Sn ↔ Ti5Sn3(Ni) + Ti3Sn	L	62	8	30	Ur	L + Ti2Sn ↔ Ti5Sn3(Ni) + Ti3Sn	L	63.6	4.0	32.4	U
	Ti2Sn	65.0	1.0	34.0	n.d.		Ti2Sn	67.2	0.3	32.5	1510.4
	Ti5Sn3(Ni)	57.5	8	34.5			Ti5Sn3(Ni)	60.5	3.2	36.3
	Ti3Sn	74	1	25			Ti3Sn	75.1	0.0	24.9
L + τ2 ↔ Ti5Sn3(Ni) + τ1	L	42	24	34	Um	L + τ2 + Ti5Sn3(Ni) ↔ τ1	L	34.5	29.5	36.0	P
	τ2	28.4	45.3	26.3	1179		τ2	24.4	51.2	24.4	1168.5
	τ1	31.2	37.4	31.5			τ1	33.3	33.3	33.4
	Ti5Sn3(Ni)	56.4	8.6	35.0			Ti5Sn3(Ni)	56.3	9.9	33.8
L + Ti5Sn3(Ni) ↔ τ1 + Ti6Sn5	L	37	15	48	Uf	L + Ti5Sn3(Ni) ↔ τ1 + Ti6Sn5	L	34.0	26.1	39.9	U
	Ti5Sn3(Ni)	57.9	5.9	36.2	1133		Ti5Sn3(Ni)	56.4	9.7	33.9	1154.8
	τ1	33.2	33.6	33.2			τ1	33.3	33.3	33.3
	Ti6Sn5	52.7	4.5	42.8			Ti6Sn5	54.6	0.8	44.6
L + τ2 + Ti5Sn3(Ni) ↔ τ3	L	46	39	15	P1	L + τ2 ↔ Ti5Sn3(Ni) + τ3	L	40.9	39.3	19.8	U
	τ2	41.2	40	18.8	1151		τ2	24.9	51.7	23.4	1156.4
	Ti5Sn3(Ni)	55.6	11.1	33.3			Ti5Sn3(Ni)	56.1	10.2	33.7
	τ3	28.9	48.9	22.2			τ3	40.0	40.0	20.0
L + τ2 ↔ τ3 + (TiNi)	L	46	41	13	Uk	L + τ2 ↔ τ3 + (TiNi)	L	40.7	42.0	17.3	U
	τ2	41.1	49.8	9.1	1143		τ2	25.0	51.9	23.1	1151.0
	τ3	42.3	40.4	17.3			τ3	40.0	40.0	20.0
	(TiNi)	44.0	50.0	6.0			(TiNi)	31.3	48.7	20.0
L + τ2 ↔ (TiNi3) + (TiNi)	L	36	62	2	Uj	L ↔ (TiNi3) + τ2 + (TiNi)	L	31.7	60.9	7.4	E
	τ2	37.8	52	10.2	1118		τ2	25.1	52.5	22.4	1130.8
	(TiNi3)	25	74.9	0.1			(TiNi3)	25.0	75.0	0.0
	(TiNi)	41.3	53.2	5.5			(TiNi)	37.0	54.5	8.5
L ↔ τ3 + (TiNi) + Ti5Sn3(Ni)	L	49	37	14	En	L ↔ τ3 + (TiNi) + Ti5Sn3(Ni)	L	43.5	39.8	16.7	E
	τ3	43.9	40.2	15.9	1132		τ3	40.0	40.0	20.0	1139.9
	(TiNi)	47.5	49.8	2.7			(TiNi)	32.6	49.3	18.1
	Ti5Sn3(Ni)	55.6	11.1	33.3			Ti5Sn3(Ni)	56.1	10.2	33.7
L + Ti5Sn3(Ni) ↔ (TiNi) + Ti3Sn	L	58	33	9	Uo	L ↔ Ti5Sn3(Ni) + (TiNi) + Ti3Sn	L	53.2	35.1	11.7	E
	Ti5Sn3(Ni)	55.6	11.1	33.4	1107		Ti5Sn3(Ni)	56.5	9.6	33.9	1108.1
	(TiNi)	49.3	49.6	1.1			(TiNi)	38.2	51.8	10.0
	Ti3Sn	72.5	3.8	23.7			Ti3Sn	75.0	0.1	24.9
L + (TiNi3) + (Ni) ↔ τ4	L	12.0	77.0	11.0	Pw	L + (TiNi3) ↔ (Ni) + τ4	L	11.8	74.6	13.6	U
	(TiNi3)	18.7	77.9	3.4	1157		(TiNi3)	13.5	77.6	8.9	1061.8
	(Ni)	10.2	86.2	3.6			(Ni)	10.6	79.3	10.1
	τ4	13.7	80.8	5.5			τ4	13.4	72.4	14.2
L + (Ni) ↔ Ni3Sn + τ4	L	6.1	77.0	16.9	Ug	L + (Ti Ni3) ↔ (Ni) + Ni3Sn	L	8.0	74.2	17.8	E
	(Ni)	6.2	84.9	8.9	1113		(Ni)	7.5	78.9	13.6	1045.8
	Ni3Sn	0.8	75.6	23.6			Ni3Sn	1.5	73.7	24.8
	τ4	8.6	78.5	12.9			τ4	11.3	72.3	16.4
L + (TiNi3) ↔ Ni3Sn2 + τ4	L	9.0	70.0	21.0	Uv	L + (TiNi3) ↔ τ2 + τ4	L	14.8	71.1	14.1	U
	(TiNi3)	17.9	75.4	6.7	1114		TiNi3	18.6	76.2	5.2	1066.7
	Ni3Sn2	0.9	63.1	36.0			τ2	24.0	52.4	23.6
	τ4	15.9	75.7	8.4			τ4	14.5	72.1	13.4
L ↔Ni3Sn + τ4 + Ni3Sn2	L	7.0	72.0	21.0	Eh	L ↔ Ni3Sn + τ4 + Ni3Sn2	L	8.5	70.7	20.8	E
	Ni3Sn	1.2	73.3	25.5	1112		Ni3Sn	1.9	72.7	25.4	1045.5
	τ4	11.8	75.5	12.7			τ4	11.2	72.0	16.8
	Ni3Sn2	0.4	63.4	36.2			Ni3Sn2	2.9	62.6	34.5
L + τ2 ↔ (Ti Ni3) + Ni3Sn2	L	10.0	69.0	21.0	Ui	L ↔ τ2 + τ4 + Ni3Sn2	L	10.6	67.5	21.9	E
	τ2	18.9	57	24.1	1120		τ2	23.5	52.1	24.4	1044.6
	(TiNi3)	19.6	75.3	5.1			τ4	12.0	71.7	16.3
	Ni3Sn2	1.8	62.6	35.6			Ni3Sn2	4.1	62.3	33.6
L + (Ti Ni) ↔ Ti3Sn + Ti2Ni	L	67.0	29.5	3.5	Up	L ↔ (Ti Ni) + Ti3Sn + Ti2Ni	L	66.8	30.9	2.3	E
	(TiNi)	50.3	49.2	0.5	984		(TiNi)	45.3	54.4	0.3	957.5
	Ti3Sn	73.9	2.5	23.6			Ti3Sn	75.9	1.1	23.0
	Ti2Ni	65.1	33.4	1.5			Ti2Ni	66.7	33.3	0.0
L + Ti3Sn ↔ Ti2Ni + (Ti)	L	74	24	2	Uq	L ↔ Ti3Sn + Ti2Ni + (Ti)	L	68.6	29.3	2.1	E
	Ti3Sn	74.9	1.5	23.6	969		Ti3Sn	76.2	1.2	22.6	957.2
	Ti2Ni	66	33.1	0.9			Ti2Ni	66.7	33.3	0.0
	(Ti)	83.6	5.8	10.6			(Ti)	76.1	0.7	23.2
L + Ni3Sn2 ↔ τ2 + Ni3Sn4	L	1	17	82	Ub	L + Ni3Sn2 ↔ τ2 + Ni3Sn4	L	8.3	27.4	64.3	U
	Ni3Sn2	0	56.2	43.8	793		Ni3Sn2	0.1	56.2	43.7	748.0
	τ2	24.6	49.6	25.8			τ2	23.6	51.0	25.4
	Ni3Sn4	0.5	42.4	57.1			Ni3Sn4	0.0	44.0	56.0
L + Ti6Sn5 ↔ τ1 + Ti2Sn3	L	18	6	76	Ue	L + Ti6Sn5 ↔ τ1 + Ti2Sn3	L	14.7	3.5	81.8	U
	Ti6Sn5	52.8	4	43.2	753		Ti6Sn5	54.5	0.0	45.5	738.7
	τ1	33.4	33.2	33.4			τ1	33.3	33.3	33.4
	Ti2Sn3	40.3	0	59.7			Ti2Sn3	40.0	0.0	60.0
L + τ2↔ τ1 + Ni3Sn4	L	2	10	88	Uc	L + τ2↔ τ1 + Ni3Sn4	L	8.4	24.8	66.8	U
	τ2	26.8	46.3	26.9	692		τ2	23.6	51.0	25.4	716.7
	τ1	30.8	38.1	31.1			τ1	33.3	33.3	33.4
	Ni3Sn4	0	42.9	57.1			Ni3Sn4	0.0	43.8	56.2
L ↔ Ni3Sn4 + Ti2Sn3 + (Sn)	L	0.2	0.6	99.2	Ea
	Ni3Sn4	0	42.9	57.1	∼232
	Ti2Sn3	40	0	60
	(Sn)	0.8	0.8	98.4
L + τ1 ↔ Ni3Sn4 + Ti2Sn3	L	2	4	94	Ud
	τ1	32.6	34.6	32.8	232–600
	Ni3Sn4	0	42.9	57.1
	Ti2Sn3	40	0	60
[small script l] + τ2 ↔ τ1	[small script l]	37.5	25.0	37.5	pcm
	τ2	27.5	45.0	27.5	>1179
	τ1	31.0	38.0	31.0
						[small script l] + τ2 ↔ τ3	[small script l]	40.8	39.3	19.8	p
							τ2				1156.4
							τ3
[small script l] ↔ τ2 + Ni3Sn2	[small script l]	6.3	57.6	36.1	ebi	[small script l] ↔ τ2 + Ni3Sn2	[small script l]	9.7	56.8	33.5	e
	τ2	23	52	25	1201		τ2				1129.4
	Ni3Sn2	1	59.4	39.6			Ni3Sn2
[small script l] ↔ τ2 + (TiNi3)	[small script l]	24.2	64.9	10.9	eij	[small script l] ↔ τ2 + (TiNi3)	[small script l]	24.8	66.0	9.2	e
	τ2	23.6	52.7	23.7	>1180		τ2				1197.2
	TiNi3	23.8	75.4	0.8			TiNi3
[small script l] ↔ τ2 + (TiNi)	[small script l]	40.5	51	8.5	ejk	[small script l] ↔ τ2 + (TiNi)	[small script l]	35.9	51.3	12.7	e
	τ2	40	51	9	>1143		τ2				1220.8
	(TiNi)	41	51	8			(TiNi)
[small script l] ↔ τ2 + Ti5Sn3(Ni)	[small script l]	40	32	28	elm	[small script l] ↔ τ2 + Ti5Sn3(Ni)	[small script l]	37.8	34.3	27.8	e
	τ2	28	48	24	1247		τ2				1209.5
	Ti5Sn3(Ni)	55.6	11.1	33.3			Ti5Sn3(Ni)
						[small script l] ↔ (TiNi) + Ti3Sn	[small script l]	54.8	35.0	10.3	e
							(TiNi)				1111.5
							Ti3Sn
[small script l] ↔ (TiNi) + Ti5Sn3(Ni)	[small script l]	51	36	13	eno	[small script l] ↔ (TiNi) + Ti5Sn3(Ni)	[small script l]	45.9	38.8	15.3	e
	(TiNi)	48.6	49.6	1.8	>1132		(TiNi)				1144.6
	Ti5Sn3(Ni)	55.6	11.1	33.3			Ti5Sn3(Ni)
						[small script l] ↔ Ti2Ni + Ti3Sn	[small script l]	67.7	30.1	2.2	e
							Ti2Ni				957.8
							Ti3Sn

---

Fig. 6 Microstructure of selected as-cast samples. The composition of the samples is marked with respective indices inside of the open circles on Fig. 1.

Fig. 7 Microstructures of the selected samples annealed at 950 °C. Labels inside of the triangles denote the respective three phase fields (Fig. 1–5). Compositions of the phases are listed in Table S2 (ESI†). The compositions of the samples are marked with respective indices inside of the open circles on Fig. 5.

Fig. 8 Microstructures of selected as-cast and annealed samples. Labels inside of the triangles denote the respective three-phase fields (Fig. 1–5). Composition of the phases after EPMA is listed in Table S2 (ESI†). The composition of the as-cast samples is marked with respective indices inside of the gray circles on Fig. 1. The annealed samples are denoted with indices inside gray and semi-filled circles on Fig. 2.

Investigation of the alloys in as-cast state (Fig. 6 and 8) shows, that the Heusler phase (HP, τ₂-TiNi₂Sn, MnCu₂Al type) has the largest field of primary crystallization. This phase melts congruently at 1447 °C (ref. 5) and exhibits a wide homogeneity region at sub-solidus temperatures. Thus in the sample with nominal composition TiNi₂Sn, the HP crystallizes primarily (Ti_25.2Ni_49.7Sn_25.1, at.%), followed by small grains of τ₂ with composition Ti₁₅Ni₆₁Sn₂₄ (at.%) and the crystallization ends by solidification of Ni₃Sn₂ (Fig. 6a). After annealing at 950 °C (Fig. 7a), an almost single-phase sample of τ₂ was obtained with small residual amounts of Ni₃Sn₂. In order to determine details in constitution, a sample with composition Ti₁₈Ni₅₈Sn₂₄ was prepared and investigated in three states. The as-cast specimen shows primary τ₂ with a composition close to stoichiometric TiNi₂Sn (Ti_23.3Ni_52.2Sn_24.4), Ni₃Sn₂ and a eutectic structure with the composition Ti_7.5Ni_71.7Sn_20.8 (Fig. 6f). The samples annealed at 950 °C and 1100 °C (Fig. 7f and 8f) reveal the three-phase field “i”: τ₂ + Ni₂Sn₃ + TiNi₃ (Ti_1−xSn_xNi₃). However, the composition of τ₂ in the annealed specimens differs significantly: Ti_23.0Ni_52.3Sn_24.7 at 950 °C but Ti_18.9Ni_57.0Sn_24.1 at 1100 °C, respectively. The Rietveld refinement of the sample annealed at 1100 °C confirms the crystal structure of the Heusler phase with Ti/Ni substitution in the 4b site (½, ½, ½, , ). At high temperatures the homogeneity region of this phase extends further towards the binary phase TiNi. Thus, the sample Ti₃₇Ni₅₀Sn₁₃ in as-cast state defines τ₂ and TiNi with compositions Ti_29.5Ni_50.0Sn_20.6 and Ti_41.3Ni_50.7Sn_7.9, respectively. However, after annealing at 1100 °C this sample exhibits almost single-phase τ₂ with a composition in the range from Ti_35.9Ni_50.3Sn_13.8 to Ti_38.2Ni_50.6Sn_11.2. A two-phase gap between TiNi and τ₂ is defined to exist in the composition range between 6 and 9 at% Sn considering EPMA data obtained from samples Ti₃₆Ni₅₈Sn₆ and Ti₄₃Ni₄₉Sn₈ annealed at 1100 °C. Temperatures of invariant reaction that were measured on these samples by DTA were defined to be 1118 °C (U_j) and 1146 °C (U_k). Taking into account the extended homogeneity region of τ₂, a structural chemical formula for the Heusler phase may be expressed as Ti_1+yNi_2−xSn_1−y. The homogeneity region of this phase at 1100 °C extends towards binary Ni–Sn, τ₁ and TiNi reaching compositions Ti₁₉Ni₅₇Sn₂₄, Ti₂₇Ni₄₆Sn₂₇ and Ti₄₁Ni₅₀Sn₉, respectively. Details on atom site preference in the crystal structure of τ₂ will be discussed in Section 3.3.

In contrast to the full Heusler phase (τ₂), the half Heusler phase (HH, τ₁, MgAgAs-type) has a much lower thermodynamic stability, resulting in a smaller primary crystallization field. τ₁ forms incongruently (Fig. 8k) by the invariant peritectic reaction l + τ₂ ↔ τ₁ at 1180 °C. τ₁ shows a homogeneity region that tends toward τ₂. Thus, the sample Ti₂₉Sn₄₂Ni₂₉ after annealing at 1100 °C (Fig. 8n) reveals two compositions for the two Heusler phases (τ₁-Ti_30.8Sn_38.1Ni_31.1 and τ₂-Ti_26.8Sn_46.2Ni_26.9 in at%) and Ni₃Sn₄, which forms from liquid during quenching. The crystallization of as-cast samples with tin contents outside the section Ni₃Sn₂–τ₂–τ₁–Ti₅Sn₃ finishes with the formation of a tin rich liquid (Fig. 6b, c, o, 8a, b, d, g and k). Consequently, all samples from this region, that were annealed at 800 and 950 °C, were found to be in equilibrium with the Sn-rich liquid (see fields labelled “s”, “t”, “u” in Fig. 5). Such a behavior suggests a cascade of transition type reactions (U) with temperatures decreasing from this section to the Sn-rich corner of the diagram. In order to produce equilibrium samples for the determination of these reaction temperatures, the samples were annealed in the temperature range from 450 to 770 °C. Comparing the microstructures of the as-cast alloys with those annealed at different temperatures (Fig. 7a–c, 8c, e and h) and considering the temperatures determined by DTA, four invariant reactions were established: U_b, U_c, U_d, U_e.

Besides of the Heusler phase (τ₂), the solid solution based on binary Ti₅Sn₃ (Ti₅Ni_xSn₃) also plays a dominant role in the formation of phase equilibria in the ternary system. Although this phase forms incongruently in the binary system and crystallizes in a narrow field from the liquid, the primary field of crystallization in the ternary extends up to 40 at% Ni (Fig. 1 and 3). The solubility of nickel in Ti₅Sn₃ reaches a maximum extent at Ti₅NiSn₃, significantly increasing the thermodynamic stability of this phase (see DFT calculations in Section 3.4.2.1).

As-cast samples located within the composition triangle Ti₅NiSn₃–Ti₂Ni₂Sn–TiNi and in the primary crystallization field of Ti₅NiSn₃ show similar but rather complicated microstructures. As an example Fig. 8j represents the crystallization of four phases in the as cast sample Ti₄₈Ni₃₄Sn₁₈: primary Ti₅Ni_xSn₃ (Ti_55.5Ni_11.4Sn_33.0) followed by the crystallization of τ₂ (Ti_37.8Ni_49.5Sn_12.7), τ₃-Ti₂Ni₂Sn (Ti_43.8Ni_40.1Sn_16.1), TiNi (Ti_44.8Ni_49.2Sn_6.0) and a two-phase eutectic TiNi + Ti₅Ni_xSn₃ (Ti_49.1Ni_38.0Sn_12.9).

Eutectics with similar compositions are observed in several other samples (see small triangles in Fig. 3) and they are mainly located inside the three-phase region: τ₃ + TiNi + Ti₅NiSn₃. For example Fig. 6k shows the microstructure of the as-cast sample Ti₅₃Ni₂₃Sn₂₄ with a primary crystallization of Ti₅Ni_xSn₃ (Ti_56.1Ni_10.7Sn_33.2) and a eutectic with the composition: Ti_48.8Ni_38.2Sn_13.0. As the composition of the liquid of the afore mentioned samples never crosses the tie-line TiNi − Ti₅NiSn₃ during crystallization, we can define the eutectic type reaction, E_n: L ↔ TiNi + Ti₂Ni₂Sn + Ti₅Ni_xSn₃, and the quasi-binary reaction, e_max,no: l ↔ TiNi + Ti₅NiSn₃. At higher titanium content, we already observe the crystallization of Ti₃Sn. In the sample Ti₆₀Ni₂₀Sn₂₀, Ti₃Sn with the composition Ti_72.9Ni_2.4Sn_24.7 solidifies after Ti₅NiSn₃ (Ti_56.5Ni_11.0Sn_32.5) (Fig. 6l). Besides Ti₃Sn, the microstructure shows two eutectics with very close compositions ∼ Ti₅₆Ni₃₅Sn₉ (TiNi + Ti₅NiSn₃, bright eutectic) and ∼Ti₅₇Ni₃₃Sn₁₀ (TiNi + Ti₃Sn, dark eutectic). From the location of these eutectics (see Fig. 3) the transition type reaction U_o was assigned: L + Ti₅NiSn₃ ↔ TiNi + Ti₃Sn. The crystallization of as-cast samples from this region (Fig. 7l–n), the phase composition and temperatures measured on the samples annealed at 950 °C (Fig. 6l–n), allow us to define another cascade of transition type reactions: U_o (1107 °C), U_p (984 °C) and U_q (969 °C). Phase equilibria involving Ti₂Sn (phase region “r”) were not investigated, because the equilibration occurs at temperatures exceeding the technical limits of our furnaces and DTA.

The sample with stoichiometry τ₃-Ti₂Ni₂Sn, in as-cast state (Fig. 7i), shows primary crystallization of Ti₅NiSn₃ (Ti_55.4Ni_11.7Sn_32.9), than solidifies as τ₂ with two compositions (Ti_27.7Ni_48.1Sn_24.1 and Ti_32.5Ni_48.8Sn_18.8) and subsequently we observe the crystallization of τ₃ with composition Ti_43.6Ni_40.1Sn_16.2. The crystallization is finished by solidification of TiNi (Ti_43.1Ni_49.6Sn_7.3) and the formation of a two-phase eutectic TiNi + Ti₅Ni_xSn₃ with the composition Ti_49.3Ni_37.0Sn_13.8. Such a crystallization behavior indicates a peritectic formation of this phase via the reaction P_l: L + τ₂ + Ti₅Ni_xSn₃ ↔ τ₃. The sample annealed at 800 and 950 °C was not completely equilibrated and contains four phases (τ₂, τ₃, Ti₅Ni_xSn₃ and TiNi), however, after annealing at 1050 °C (Fig. 8i), TiNi disappears and only three equilibrium phases remain: τ₂ (Ti_28.9Ni_48.8Sn_22.3), τ₃ (Ti_41.2Ni_40.0Sn_18.8) and Ti₅Ni_xSn₃ (Ti_55.7Ni_11.0Sn_33.3). The temperature of the invariant equilibrium P₁ at 1151 °C is derived from DTA on the sample annealed at 1050 °C. A similar morphology that confirms the peritectic crystallization of τ₃ is observed for the as-cast sample Ti₄₀Ni₄₆Sn₁₄ (Fig. 7h) indicating a peritectic solidification of TiNi (Ti_41.6Ni_49.9Sn_8.5) around τ₂ with compositions varying from Ti_30.7Ni_49.5Sn_19.9 to Ti_29.4Ni_49.4Sn_21.2. The solidification in this sample also ends in the two-phase eutectic TiNi + Ti₅Ni_xSn₃ with composition Ti_50.6Ni_34.5Sn_14.9. All these observations establish a transition type reaction U_k: L + τ₂ ↔ τ₃ + TiNi at 1143 °C.

Another four-phase equilibrium, that involves the Heusler phase (τ₂) and binary TiNi is documented for the sample Ti₃₇Ni₅₇Sn₆. In as-cast state (Fig. 6g), we observe primary crystallization of τ₂ with compositions Ti₃₃Ni₅₁Sn₁₆ and Ti₃₆Ni₅₃Sn₁₁ for big and small grains, and a fine eutectic with overall composition Ti₃₆Ni₆₁Sn₁₃: the eutectic consists of TiNi (Ti₄₀Ni₅₅Sn₅) and TiNi₃, and significantly coagulates after annealing at 950 °C for 10 days (Fig. 7g). As the composition of the eutectic lies outside of the three-phase triangle τ₂ + TiNi₃ + (Ti_1−xNi_x)_1−ySn_y (field “j” on Fig. 3), a transition type reaction U_j is defined: L + τ₂ ↔ TiNi₃ + (Ti_1−xNi_x)_1−ySn_y. However, the temperature of this reaction, determined by DTA is 2 °C lower than the respective reaction l ↔ TiNi₃ + TiNi (1118 °C) in the binary system.

Binary TiNi₃ has the highest melting point (1380 °C) within the Ti–Ni system and also exhibits rather extended Ti/Sn substitution (Ti_1−xSn_x)Ni₃ up to x_max = 0.27 at 1080 °C. However, at a higher tin content, the hexagonal structure of (Ti_1−xSn_x)Ni₃ undergoes a structural transformation into a cubic structure with AuCu₃-type: τ₄-(Ti_1−x−ySn_xNi_y)Ni₃. The homogeneity region of this phase extends at 1080 °C from x = 0.22 and y = 0.22 (in equilibrium with (Ti_1−xSn_x)Ni₃ and (Ni)) to x = 0.54 and y = 0.06 (in equilibrium with Ni₃Sn and (Ni)). However, the maximal solubility of tin in τ₄ at temperatures below solidus increases to 16.4 at% (x = 0.66 and y = 0.1) at 950 °C) as determined from an annealed alloy Ti₅Ni₈₀Sn₁₅ (Fig. 7d) in equilibrium with Ni₃Sn and (Ni).

The sample in as cast state shows primary crystallization of (Ni) with grains of τ₄-(Ti_1−x−ySn_xNi_y)Ni₃ solidifying around them, and both phases are embedded in a Ni₃Sn matrix (Fig. 6d). In alloy Ti₁₀Ni₇₆Sn₁₄, τ₄ crystallizes as a primary phase with composition Ti_13.0Ni_77.2Sn_9.8 and solidification finishes in a two-phase eutectic (τ₄ + Ni₃Sn) with composition Ti_5.8Ni_74.6Sn_19.6. This type of solidification indicates incongruent formation of this phase during a peritectic reaction. But our guess on the formation of this phase in a three-phase peritectic, L + (Ti_1−xSn_x)Ni₃ ↔ τ₄, was ruled out by the observation that for the as-cast alloys with nickel contents from 75 to 81 at% Ni the last portion of the liquid becomes depleted by nickel. Such a solidification behavior agrees well with a cascade of reactions, peritectic P_w: L + (Ni) + (Ti_1−xSn_x)Ni₃ ↔ τ₄ followed by a transition type reaction U_g: L + (Ni) ↔ Ti_1−xSn_x)Ni₃ + Ni₃Sn.

The temperature of the invariant reaction U_g (1113 °C) is only one degree higher than that determined for E_h: L ↔ Ni₃Sn + Ni₃Sn₂ + τ₄ (1112 °C), which is evident from the microstructure of the as-cast alloy with nominal composition Ti₇Ni₇₀Sn₂₃ (Fig. 6e): we observe primary crystallization of Ni₃Sn₂, continued by a solidification of NiSn₃ and the final liquid crystallizes in form of a eutectic with composition Ti_7.5Ni_71.7Sn_20.8. The composition of this eutectic lies outside of the tie-triangle Ni₃Sn + Ni₃Sn₂ + τ₄ determined on a sample annealed at 950 °C (Fig. 7e, Table S2 (ESI†)), but it is located inside of this three-phase field as it was established for the sample annealed at 1080 °C. It should be noted, that this invariant eutectic is much finer than the two-phase mono-variant eutectic l → (Ti_1−xSn_x)Ni₃ + Ni₃Sn₂ that is observed in the as-cast sample Ti₁₈Ni₅₈Sn₂₄ (Fig. 7f). The microstructure of the latter sample also documents a secondary crystallization of Ni₃Sn₂ after primary τ₂. Considering this and the fact that the composition of this mono-variant eutectic (Ti₁₀Ni₆₉Sn₂₁) is located slightly outside of the three-phase field (Ti_1−xSn_x)Ni₃ + Ni₃Sn₂ + τ₂ (“i” in Fig. 3), the invariant reaction U_i: L + τ₂ ↔ (Ti_1−xSn_x)Ni₃ + Ni₃Sn₂ was established. Another transition-type invariant three-phase reaction U_v: L + (Ti_1−xSn_x)Ni₃ ↔ τ₄ + Ni₃Sn₂ located between the afore mentioned reactions was established at 1114 °C. The respective three-phase field on the solidus is rather narrow and we were unable to detect the eutectic structure that would document this invariant reaction.

An additional uncertainty in the determination of phase equilibria in this region arises from the characterization of the samples by XPD that show the existence of a further novel phase with composition Ti_1−xSn_xNi₃ (0.36 ≤ x ≤ 0.48), which exists in the Ni-poor part of the solid solution of τ₄-(Ti_1−xSn_xNi_y)Ni₃. In order to determine details on the crystal structure and phase equilibria involving this phase, (labelled as τ_4′-Ti_1−xSn_xNi₃) additional investigations need to be performed.

The invariant four-phase reactions established in the system are summarized in a Schulz–Scheil diagram (Fig. 4), and in order to complete this crystallization scheme, some additional invariant three-phase reactions (e_max) are requested. Besides the above discussed quasi-binary eutectic e_max,no: l ↔ TiNi + Ti₅NiSn₃, the high melting τ₂ phase forms a set of invariant eutectics with the solid solutions of binary compounds Ti₅Sn₃ (e_lm), TiNi (e_jk), TiNi₃(e_jk) and Ni₃Sn₂ (e_bi). The microstructure of the as-cast alloy Ti₈Ni₅₇Sn₃₅ (Fig. 8m) represents a fine eutectic structure (with composition Ti_6.3Ni_57.6Sn_36.1) that forms via the invariant reaction e_bi: l ↔ τ₂ + Ni₃Sn₂. The temperature of this reaction is 1201 °C, being above the neighboring invariant reactions U_b (793 °C) and U_i (1120 °C). This provides an additional proof for this quasi-binary eutectic. The isothermal section at 950 °C (see Fig. 6) represents phase equilibria similar to those reported earlier at 800 °C (ref. 4) for exception that two additional three-phase fields involving τ₄ are present in the up-dated section. The difference in extension of the homogeneity regions for solid solutions may be interpreted as the typical shrinkage of single phase fields with decrease of temperature as well as by the general difficulties of equilibration of samples at low temperatures. In this context we need to mention difficulties in the homogenization of alloys containing the high melting compounds τ₂, Ti₅Ni_xSn₃, TiNi and (Ti_1−xSn_x)Ni₃ at 950 °C and even at sub-solidus temperatures (1050–1100 °C). Therefore thermodynamic modelling of the phase diagram is more suitable to describe the equilibrium compositions at temperatures below 1000 °C.

3.3. Crystal structures and homogeneity regions of ternary phases

With respect to the importance of crystallographic data for defining models for the thermodynamic parameters we performed a number of Rietveld refinements for τ₁-TiNiSn, τ₂-Ti_1+yNi_2−xSn_1−y, (Ti_1−xSn_x)Ni₃ and Ti₅Ni_xSn₃ (see Table S3 (ESI†)). The half Heusler phase (τ₁) shows a narrow homogeneity region that is slightly off-stoichiometric on the tin-rich side Ti_32.6Ni_34.6Sn_32.8 at 1100 °C, and Ti_32.6Ni_34.3Sn_33.1 at 950 °C) and extends up to Ti_30.9Ni_37.8Sn_31.3 in equilibrium with τ₂ at 1100 °C. Rietveld refinement for TiNiSn confirms full atom order in the structure (Table S3 (ESI†)). The Heusler phase (τ₂-Ti_1+yNi_2−xSn_1−y) exhibits the biggest homogeneity region that extends at 1100 °C to Ti₂₇Ni₄₆Sn₂₇ (x = 0.30 and y = 0.00 in equilibrium with τ₁), to Ti₄₁Ni₅₀Sn₉ (x = 0 and y = 0.64 in equilibrium with TiNi) and to Ti₁₉Ni₅₇Sn₂₄ (x = −0.65 and y = −0.12 in equilibrium with TiNi₃ and Ni₃Sn₂).

Rietveld refinements (Table S3 (ESI†)) confirm a full order for stoichiometric TiNi₂Sn, whereas Ti/Sn substitution occurs in sites 4a (0, 0, 0) and 4b (½, ½, ½), respectively for tin-poor and nickel-rich extensions of the homogeneity region. The refinement did not reveal any substantial deviation from a full occupancy of Ni in site 8c (¼, ¼, ¼). It seems that this site is only affected for compositions that extend from stoichiometric τ₂-TiNi₂Sn towards τ₁-TiNiSn. Considering the close crystallographic relation between both Heusler phases and TiNi (all are derivatives of the W-type structure, we plotted the compositional dependence of the lattice parameters for the section TiNi–TiNi₂Sn (Fig. 9). One can see that lattice parameters increase with increasing tin content in the 4a site (0, 0, 0) and the dependence shows a positive deviation from Vegard's law. Maximal lattice parameters (a ≈ 0.610 nm) are observed for compositions near stoichiometric τ₂-TiNi₂Sn and the unit cell shrinks when Sn-atoms substitute for Ti in the 4b site (½, ½, ½). Due to the small difference in lattice parameters for τ₂ and the TiNi-based solution (Ti_1−xNi_x)_1−ySn_y the XPD reflexes for these phases strongly overlap and we were unable to perform an unambiguous deconvolution of the diffraction maxima and to determine precise lattice parameters for the equilibrium compositions. However, these compounds appear as well separated individual phases on SEM images in as-cast (Fig. 6h and 8o) and annealed (Fig. 7g) alloys, allowing a reliable determination of the compositions by EPMA. A much more significant decrease of the lattice parameters from a ≈ 0.610 nm to a ≈ 0.593 nm is observed for the structural change from τ₂-TiNi₂Sn to τ₁-TiNiSn and because of this fact these two phases are well distinguishable in XPD profiles.

Fig. 9 Compositional dependence of lattice parameters for homogeneity regions for the phases: TiNi, τ₁-TiNiSn and τ₂-TiNi₂Sn. The composition of the solid solution is expressed as Ti_1+yNi_2−xSn_1−y. Extensions of the homogeneity regions at 1100 °C are marked with dashed lines. Literature (open symbols), squares;^{32,36,44–51} circles;^{1,5,9,52–56} triangles-up.⁵⁷

Compositional dependence of lattice parameters for the new compound τ₄-(Ti_1−x−ySn_xNi_y)Ni₃ with AuCu₃-type structure is shown in Fig. 10. Lattice parameters increase with increase of tin content and this dependence extrapolates to the value of a = 0.3738 nm reported for the high pressure modification of Ni₃Sn which also adopts the AuCu₃ type structure.

Fig. 10 Compositional dependence of lattice parameters for the homogeneity region of τ₄-(Ti_1−x−ySn_xNi_y)Ni₃ (0.02 ≤ x ≤ 0.22) for samples annealed at 950 °C (circles) and 1080 °C (squares). Diamonds: calculated values (this work). Triangle: isotypic high-pressure modification of Ni₃Sn.²⁸

The close structural relation between AuCu₃ and the crystal structure of Ni (Cu-type) allows a second order transformation between these structures. However, the phase separation between these structures was clearly documented by SEM images for sample Ti₅Ni₈₀Sn₁₅ annealed at 950 °C (Fig. 7d) and was confirmed by XPD showing two sets of diffraction spectra of cubic structures with primitive (τ₄) and face-centered (Ni) lattices. We note that primitive X-ray diffraction peaks from the τ₄ lattice almost vanish at low Sn contents (zero intensity for Ti_0.786Sn_0.214Ni₃) due to the fact that the electron density in both crystallographic sites becomes similar. This may lead to possible misinterpretation of XPD patterns of this phase with the fcc nickel-based solid solution.

Furthermore, we subjected the ternary solid solution Ti₅Ni_xSn₃ (Hf₅CuSn₃-type structure; i.e. a filled variant of binary Ti₅Sn₃ with the Mn₅Si₃-type) to a detailed crystallographic investigation. Due to non-equilibrium solidification of as-cast samples Ti₅Ni_xSn₃ with x = 0.9 and 1.0, EPMA line scans showed that the nickel content in this phase increases from 6.3 at% Ni at the grain centre to 10.2 at% Ni at the rim of the grains. This heterogeneity results in some broadening of the XPD reflections. Annealing at 1100 °C for 6 days provides a complete homogenization of the composition, but a comparison of XPD profiles for as-cast and annealed samples shows that the half-width of only some reflections is reduced whilst others show a clear split suggesting a structural transformation in this phase. Several single crystals were selected from the annealed sample but inspection on an AXS-GADDS texture goniometer did not reveal any significant distortion of the unit cell. One crystal specimen was characterized by means of four-circle Nonius Kappa diffractometer intensity data. Although the measured data were processed in triclinic symmetry, single crystal refinement fully complies with the symmetry and atom site distribution of the Hf₅CuSn₃-type structure (ordered Ti₅Ga₄-type; for crystallographic details see Table 3).

Table 3 X-Ray single crystal data for Ti₅Ni_0.96Sn₃ (Hf₅CuSn₃-type); space group P6₃/mcm no. 193 (redundancy <10, 7.73 ≤ 2θ ≤ 72.27°, crystal size 30 × 35 × 35 μm). Standardized with program Structure Tidy⁵⁸

Parameter	Ti5Ni0.96Sn3
EPMA, at%	Ti56.0Ni10.0Sn34.0
Refinement; at%	Ti55.8Ni10.7Sn33.5
a; c [nm]	0.81440(1); 0.555922(9)
Reflections in refinement	304Fo > 4sig(Fo) of 309
Number of variables	16
RF = ∑|F0 − Fc|/∑F0	0.0298
RInt	0.0303
wR2	0.0595
GOF	1.143
Extinction (Zachariasen)	0.0021
Ti1 in 6g (x, 0, ¼), x; occ.	0.2550(4); 1.00(1)
U11; U22	0.052(1); 0.0227(9)
U33; U23 = U13 = 0; U12 [nm^2]	0.0252(8); 0.0114(4)
Sn1 in 6g (x, 0, ¼), x; occ.	0.61161(8); 1.002(7)
U11; U22	0.0253(3); 0.0077(2)
U33; U23 = U13 = 0; U12 [nm^2]	0.0117(2); 0.0038(1)
Ti2 in 4d (1/3, 2/3, 0), occ.	0.996(8)
U11 = U22; U33	0.0098(1); 0.0063(5)
U23 = U13 = 0; U12 [nm^2]	0.0045(2)
Ni in 2b (0, 0, 0), occ.	0.96(2)
U11 = U22; U33	0.094(3); 0.066(3)
U23 = U13 = 0; U12 [nm^2]	0.047(2)
Residual density: max; min	2.31; −3.42
Principal mean square atomic	Ti1 0.0620, 0.0252, 0.0227
Displacements (U)	Sn1 0.0312, 0.0117, 0.0077
	Ti2 0.0091, 0.0091, 0.0063
	Ni 0.0945, 0.0945, 0.0660

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3.4. Thermodynamic assessment of the Ti–Ni–Sn system

3.4.1. Thermodynamic data for binary boundary systems. Thermodynamic data were taken from the most recent CALPHAD modelling of the binary systems by various research groups: Ni-Sn,²³ Ti-Sn,²⁴ and Ti–Ni.²⁶

3.4.2. Thermodynamic evaluation of the Ti–Ni–Sn system.
3.4.2.1. DFT calculations of ternary compounds TiNiSn, TiNi₂Sn, Ti₂Ni₂Sn and of the solid solutions Ti₅Ni_1−xSn₃, Ti_1−xSn_xNi₃ and (Ti_1−xNi_x)_1−ySn_y.
Ti₅Ni_1−xSn₃. The modelling of the Ti₅Ni_1−xSn₃ solid solution consisted of two steps. At the first step the ground state energy of Ti₅Sn₃ with optimised crystal structure was calculated (Table 4). The negative enthalpy of formation (Table 5) proves the possibility of its formation. At the second step the chemical composition of the compound was modified by introducing additional Ni atoms into the 2b site to fit the Ti₅NiSn₃ composition. After that the ground state energy of Ti₅NiSn₃ with optimised crystal structure (Table 4) was calculated. Comparing the enthalpy of formation of Ti₅Sn₃ and Ti₅NiSn₃ (Table 5) one can see that it is lower for Ti₅NiSn₃, which means that the filling of the 2b site with Ni atoms is energetically favourable and predicts the formation of the interstitial solid solution Ti₅Ni_1−xSn₃. For further thermodynamic calculations the ground state energy and the enthalpy of formation were calculated for the hypothetical compound Ni₅NiNi₃ (Table 5) with the same structure as Ti₅NiSn₃ and optimised crystal structure geometry (Table 4).

Table 4 Optimized crystallographic data of Ti₅Sn₃, Ti₅NiSn₃, hypothetical Ni₅NiNi₃, SnNi₃, TiNi₃, Ti_0.5Sn_0.5Ni₃, Sn_0.5Ti_0.5Ni₃ and Ti₂Sn₃ compounds derived from DFT calculations

Atom	Wyckoff	x/a	y/b	z/c
Ti5Sn3 (a = 0.8102119, c = 0.5453824 nm)
Ti1	6g	0.242235	0	0.25
Ti2	4d	1/3	2/3	0
Sn1	6g	0.608723	0	0.25
Ti5NiSn3 (a = 0.8250848, c = 0.5511669 nm)
Ti1	6g	0.256722	0	0.25
Ti2	4d	1/3	2/3	0
Sn1	6g	0.610405	0	0.25
Ni1	2b	0	0	0
Ni5NiNi3 (a = 0.6970264, c = 0.4840398 nm)
Ni1	6g	0.333297	0	0.25
Ni2	4d	1/3	2/3	0
Ni3	6g	0.666532	0	0.25
Ni4	2b	0	0	0
SnNi3 (a = 0.535448, c = 0.8647618 nm)
Sn1	2a	0	0	0
Sn2	2d	1/3	2/3	3/4
Ni1	6g	1/2	0	0
Ni2	6h	0.177094	0.354189	1/4
TiNi3 (a = 0.5137315, c = 0.8382629 nm)
Ti1	2a	0	0	0
Ti2	2d	1/3	2/3	3/4
Ni1	6g	1/2	0	0
Ni2	6h	0.171415	0.34283	1/4
Ti0.5Sn0.5Ni3 (a = 0.5237464, c = 0.8544106 nm)
Ti1	2a	0	0	0
Sn2	2d	1/3	2/3	3/4
Ni1	6g	1/2	0	0
Ni2	6h	0.171782	0.343565	1/4
Sn0.5Ti0.5Ni3 (a = 0.5248831, c = 0.8498294 nm)
Sn1	2a	0	0	0
Ti2	2d	1/3	2/3	3/4
Ni1	6g	1/2	0	0
Ni2	6h	0.176477	0.352954	1/4
Ti2Sn3 (a = 0.6013709, b = 2.0176674, c = 0.7073997 nm)
Ti1	8f	0	0.080577	0.047598
Ti2	8e	1/4	0.33657	1/4
Sn1	8f	0	0.120959	0.426385
Sn2	8f	0	0.224113	0.073991
Sn3	8e	1/4	0.479599	1/4

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Table 5 DFT values of the enthalpy of formation (ΔH) for selected compounds and their derivatives in the Ti–Ni–Sn system

Compound	ΔH (meV per atom)	References
TiNiSn	−547.083	This work
	−715	Ref. 5
	−549	Ref. 7
TiNi1.25Sn	−493.933	This work
TiNi1.5Sn	−466.162	This work
TiNi1.75Sn	−461.613	This work
TiNi2Sn	−472.872	This work
	−622	Ref. 5
Ti2Ni2Sn	−468.209	This work
	−485	Ref. 5
Ti5NiSn3	−404.251	This work
	−388	Ref. 5
Ti5Sn3	−385.435	Ref. 5
	−348	This work
Ni5NiNi3	+242.338	This work
Ti2Sn3	−354.214	This work
TiNi3	−524.463	This work
	−478	Ref. 5
TiNi3 (Cu3Au-type)	−532.018	This work
	−750	Ref. 59
	−484 (LMTO-ASA)	Ref. 60
	−488 (FP-LMTO)	Ref. 60
SnNi3	−219.215	This work
SnNi3 (Cu3Au-type)	−240.291	This work
Ti0.5Sn0.5Ni3	−341.340	This work
Ti0.5Sn0.5Ni3 (Cu3Au-type)	−341.886	This work
Sn0.5Ti0.5Ni3	−318.140	This work
TiNi	−390.862	This work
	−411	Ref. 5
	−610	Ref. 59
	−395 (LMTO-ASA)	Ref. 60
	−373 (FP-LMTO)	Ref. 60
Ti0.875Sn0.125Ni	−364.133	This work
Ti0.75Sn0.25Ni	−378.446	This work
Ti0.625Sn0.375Ni	−413.794	This work
Ti0.25Sn0.75Ni	−185.791	This work
NiSn	−15.954	This work
Ni0.75Ti0.25Sn	−106.858	This work
Ni0.5Ti0.5Sn	+24.483	This work
Ni0.25Ti0.75Sn	−141.487	This work
TiSn	−101.017	This work
TiSn0.75Ni0.25	−126.421	This work
TiSn0.5Ni0.5	−219.074	This work
TiSn0.25Ni0.75	−232.767	This work

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Ti_1−xSn_xNi₃. The substitutional (Ti by Sn) solid solution based on TiNi₃ was modelled in two ways. In the first case the Ti2 atoms in the 2d position and in the second case the Ti1 atoms in the 2a site were substituted by Sn atoms yielding the compositions Ti_0.5Sn_0.5Ni₃ and Sn_0.5Ti_0.5Ni₃, respectively. The ground state energy and the enthalpy of formation (Table 5) were calculated for SnNi₃, TiNi₃, Ti_0.5Sn_0.5Ni₃, and Sn_0.5Ti_0.5Ni₃ with a completely relaxed geometry (Table 4). The comparison of Ti_0.5Sn_0.5Ni₃ and Sn_0.5Ti_0.5Ni₃ at the same composition (x = 0.5) shows that the configuration of Ti_0.5Sn_0.5Ni₃ is more preferable due to the lower enthalpy of formation value. The analysis of the ΔH values of TiNi₃, SnNi₃, and Ti_0.5Sn_0.5Ni₃ predicts that the formation of the solid solution at T = 0 K is not possible. However, at higher temperature the formation of Ti_1−xSn_xNi₃ is still possible due to the entropy contribution.

The calculations were also carried out for the high-pressure modifications (Cu₃Au-type) of TiNi₃ and Ni₃Sn binaries and the intermediate composition Ti_0.5Sn_0.5Ni₃. For the intermediate composition the unit cell was doubled in all three directions of the basis vectors (2a × 2b × 2c) giving in total 32 atoms in the supercell and the symmetry was reduced to P1. The optimised values of the lattice parameter a for TiNi₃, Ni₃Sn, and Ti_0.5Sn_0.5Ni₃ are 0.36077234, 0.37354910, and 0.367873 nm, respectively. The slope of the a(x) dependence in the range 0 ≤ x ≤ 0.5 is higher than for the higher x values. Most likely this is due to the impact of s- and p-states of Sn atoms that could increase the covalent contribution into the system of chemical bonds.

(Ti_1−xNi_x)_1−ySn_y. For the solid solution based on binary TiNi three isotypic structures were modelled: TiNi, TiSn, and NiSn with optimized lattice parameter a = 0.302378757, 0.3430739, and 0.32247303 nm, respectively. The calculated ΔH values of the hypothetical solid solutions between TiNi, TiSn, and NiSn are collected in Table 5. For the intermediate compositions the unit cell was doubled in all three directions of the basis vectors (2a × 2b × 2c) giving in total 16 atoms in the supercell and the symmetry was reduced to P1. The composition Ti_0.5Sn_0.5Ni is omitted in Table 5 as it corresponds to the TiNi₂Sn compound.
TiNi_1+xSn. The interstitial solid solution based on the half-Heusler TiNiSn phase (Ti in 4a, Ni in 4c, Sn in 4b) was modelled by reducing the symmetry of the crystal to P1 and successively filling of four available voids (4d site) with Ni atoms. According to the calculated ΔH values (Table 5) the formation of limited solid solutions between TiNiSn and TiNi₂Sn at T = 0 K is not possible (Table 5). For Ti₂Sn₃ and Ti₂Ni₂Sn the calculated ΔH values that were used in thermodynamic calculations are collected in Table 5 and the optimised crystallographic data of the Ti₂Sn₃ compound are listed in Table 4. The calculated ΔH values for the selected compounds in Ti–Ni–Sn system using full-potential elk code (GGA-PBE) are mostly in a good agreement with those obtained with pseudopotential VASP code (GGA-PBE),⁵ except for TiNiSn and TiNi₂Sn (Table 5). This is strange, as for TiNiSn the ΔH values obtained using the same VASP code and GGA-PBE approximation by Colinet et al.⁷ are comparable with our results (Table 5). The heat of formation values for binary TiNi (Table 5) obtained with FLAPW, pseudopotential,⁵ LMTO-ASA,⁶⁰ and FP-LMTO⁶⁰ methods are comparable with experimental data collected in ref. 59, while the value calculated using the semi-empirical tight-binding method⁵⁹ significantly differs.

The calculated ΔH value for the high-pressure modification of TiNi₃ (Table 5) using the FLAPW method is comparable with those obtained by LMTO-ASA and FP-LMTO methods,⁶⁰ while for the semi-empirical tight-binding method⁵⁹ the ΔH value is significantly higher. It is interesting to note, that calculations showed higher absolute values of ΔH for the high-pressure modifications of TiNi₃, Ni₃Sn and intermediate Ti_0.5Sn_0.5Ni₃ in comparison with the normal-pressure modification.

Douglas et al. observed an increase of lattice parameter a for the TiNi_1+xSn solid solution in the range 0 ≤ x ≤ 0.1 according to the XRD data.⁵ The comparison of optimized (DFT) and experimental values of lattice parameter with a set from ref. 5 (Fig. 11) shows that the theoretical and interpolated experimental dependencies are almost parallel, while the slope of the reference dependence is more horizontal. Such deviation in slope means that the x(Ni) values corresponding to the lattice parameters in ref. 5 might be significantly lower. To explain the increase of lattice parameter we have modelled a structure that consists of four TiNiSn unit cells where one Ni atom from each of them diffused to form one full-Heusler TiNi₂Sn unit cell. This is a model of the coherent grain boundary between TiNiSn and TiNi₂Sn phases. The crystal structure geometry optimisation for this model shows that the lattice parameters of TiNiSn and TiNi₂Sn unit cells vary in the ranges 0.58693 ÷ 0.58686 and 0.60766 ÷ 0.60762 nm, respectively (Fig. 12).

Fig. 11 Concentration dependences of optimized (DFT), experimental and reference⁵ values of lattice parameter a for the TiNi_1+xSn solid solution.

Fig. 12 The initial and relaxed geometry of the hypothetical TiNiSn–TiNi₂Sn structure.

The projection of the electron localization function (Fig. 13) shows localized maxima between Ni and Sn atoms within the half-Heusler (HH) cell and somewhat lower in the full-Heusler (FH) cell. The calculations predict that the presence of the TiNi₂Sn phase strongly affects the TiNiSn lattice parameter value due to the stress on the grain boundary between Heusler and half-Heusler phases. The distribution of the total density of states (Fig. 14) for TiNiSn, TiNi_1.25Sn, TiNi₂Sn, and TiNiSn–TiNi₂Sn grain boundary confirmed that TiNiSn is a semiconductor and TiNi₂Sn is characterized by a metallic type of conductivity. In the case of TiNi_1.25Sn (MgAgAs-type) the additional Ni atoms in position 4d generate a new band inside the energy gap and eliminate it. Thus the material predicted has metallic type of conductivity similar to TiNi₂Sn. The DOS distribution at the grain boundary of TiNiSn–TiNi₂Sn significantly differs from those in pure TiNiSn and TiNi₂Sn. The main difference between TiNiSn and TiNi₂Sn is that the energy bands are strongly delocalised. Like in the case of TiNi_1.25Sn and TiNi₂Sn the electrical conductivity at the grain boundary is predicted to be metallic. The deformation of the crystal structure at the TiNiSn–TiNi₂Sn grain boundary leads to additional phonon scattering and decrease of the thermal conductivity of the material. The increased electrical conductivity and decreased thermal conductivity should have a positive effect on the value of the thermoelectric figure of merit (Z).

Fig. 13 Distribution of the electron localization function along the lattice plane in the TiNiSn–TiNi₂Sn structure.

Fig. 14 Distribution of the total density of states (per formula unit) for TiNiSn, TiNi_1.25Sn, TiNi₂Sn, and for the grain boundary TiNiSn–TiNi₂Sn. The Fermi level (EF) is at E = 0 eV.

3.4.2.2 Results of thermodynamic modelling. For thermodynamic modelling of the stability of ternary compounds TiNiSn, TiNi₂Sn and Ti₂Ni₂Sn, the DFT calculated values of the formation energy (ΔH) (Table 5) were used and the entropic term was optimized. Similarly, DFT data were used for Ti₂Sn₃. Data for Ti₅Sn₃ were taken from ref. 24 and the line solubility to Ti₅NiSn₃ was optimized. The mutual solubility of TiNi₃ and SnNi₃ was modelled on the basis of binary data from ref. 23 and 26 and optimized. Stability of TiNi was modelled as B2_BCC and BCC_A2 binary phases^25,26 with optimized solubility of Sn. Note, both phases TiSn and NiSn are in the respective binary systems. The parameters used for the calculations are presented in Table 6. The results of the thermodynamic optimisation are summarized in a liquidus surface (Fig. 15), in a series of isopleths: Ni–TiSn, Sn–TiNi, NiSn–TiSn (Fig. 16), as well as in a series of isothermal sections: 800 °C, 900 °C, 950 °C (Fig. 17), 1050 °C and at 1200 °C (Fig. 18).

Table 6 Parameters for calculation of phase equilibria in Ni–Sn–Ti system

Phase	Ref.
a This work.b Optimization.c DFT.
LIQUID/(Ni,Sn,Ti)
(0)L(LIQUID,NI,Ti) = −153 707.39 + 34.859449 × T	26
(1)L(LIQUID,NI,TI) = −81 824.755 + 25.809901 × T	26
(2)L(LIQUID,NI,TI) = −10.077897 × T	26
(0)L(LIQUID,SN,TI) = −91 598.90 − 0.9416 × T	24
(1)L(LIQUID,SN,TI) = 45 682.64 − 12.1045 × T	24
(0)L(LIQUID,NI,SN) = −104 602.87 + 197.8089 × T − 21.6959 × T × LN	23
(1)L(LIQUID,NI,SN) = −30 772.17 + 52.5528 × T − 7.56094 × T × LN(T)	23
(2)L(LIQUID,NI,SN) = 6582.31	23
Ternary parameter
(0)L(LIQUID,NI,SN,TI) = −10 000	^a^,^b
Ni3Sn2/(Ni,Va)1 (Ni,Va)1 (Ni,Sn,Ti)1
(0)G(NI3SN2,NI:NI:NI) = 31 637.299 + 355.176064 × T − 66.288 × T × LN(T) − 0.0145221 × T × 2	23
(0)G(NI3SN2,VA:NI:NI) = 21 091.532 + 236.784043 × T − 44.192 × T × LN(T) − 0.0098814 × T × 2	23
(0)G(NI3SN2,NI:VA:NI) = 21 091.532 + 236.784043 × T − 44.192 × T × LN(T) − 0.0098814 × T × 2	23
(0)G(NI3SN2,VA:VA:NI) = 10 545.766 + 118.58688 × T − 22.096 × T × LN(T) − 0.0048407 × T × 2	23
(0)G(NI3SN2,NI:NI:SN; 0) = 2 × GHSERNI + GHSERSN − 83 734.818 + 14.6888165 × T	23
(0)G(NI3SN2,VA:NI:SN) = 5000 + GHSERNI + GHSERSN	23
(0)G(NI3SN2,NI:VA:SN) = GHSERNI + GHSERSN − 52 177.98 + 10.774 × T	23
(0)G(NI3SN2,VA:VA:SN) = GHSERSN + 20 000.0	23
(0)L(NI3SN2,NI:NI,VA:SN) = −9784.4 − 12.385 × T	23
(1)L(NI3SN2,NI:NI,VA:SN) = +12 000.0	23
Ternary parameters
(0)G(NI3SN2,NI:NI:TI) = GHSERTI + 2 × GHSERNI − 76 451.5 + 3.08647 × T	^a
(0)G(NI3SN2,VA:NI:TI) = GHSERTI + GHSERNI + 123 451.5 + 3.08647 × T	^a
(0)G(NI3SN2,NI:VA:TI) = GHSERTI + GHSERNI − 67 451.5 + 3.08647 × T	^a
(0)G(NI3SN2,VA:VA:TI) = GHSERTI + 5451.5 + 3.08647 × T	^a
(0)L(NI3SN2,NI:NI:TI,SN) = −50 000	^a
Ni3Sn4/(Ni)0.25(Ni,Sn)0.25(Sn)0.5
(0)G(NI3SN4,NI:NI:SN) = −25 078.56 + 4.291 × T + 0.5 × GHSERNI + 0.5 × GHSERSN	23
(0)G(NI3SN4,NI:SN:SN) = +7613.24 + 8.749 × T + 0.25 × GHSERNI + 0.75 × GHSERSN	23
(0)L(NI3SN4,NI:NI,SN:SN) = −52 928.16	23
Ni3Sn_LT_TYPE/(Ni,Sn)0.75(Ni,Sn)0.25
(0)G(NI3SN_LT_TYPE,NI:NI) = +6300 + GHSERNI	23
(0)G(NI3SN_LT_TYPE,SN:NI) = +5000 + 0.25 × GHSERNI + 0.75 × GHSERSN	23
(0)G(NI3SN_LT_TYPE,NI:SN) = −28 408 + 7.0009 × T + 0.75 × GHSERNI + 0.25 × GHSERSN	23
(0)G(NI3SN_LT_TYPE,SN:SN) = +5000 + GHSERSN	23
FCC_A1/(Ni,Sn,Ti)1 (Va)1
(0)L(FCC_A1,NI,TI:VA) = −98 143 + 6.706 × T	26
(1)L(FCC_A1,NI,TI:VA; 1) = −62 430	26
(0)TC(FCC_A1,NI:VA) = +633	18
(0)BMAGN(FCC_A1,NI:VA) = +0.52	18
(0)TC(FCC_A1,NI,TI:VA) − 2500s	26
(1)TC(FCC_A1,NI,TI:VA) = −3000	26
(2)TC(FCC_A1,NI,TI:VA) = +1300	26
(0)BM(FCC_A1,NI,TI:VA) = 50 000	26
(0)L(FCC_A1,NI,SN:VA) = −69 507.35 + 74.5697727 × T − 8.0319551 × T × LN(T)	23
(1)L(FCC_A1,NI,SN:VA) = −12 395.19	23
(0)TC(FCC_A1,NI,SN:VA) = −6000	22
(1)TC(FCC_A1,NI,SN:VA) = 3000	22
(0)BMAGN(FCC_A1,NI,SN:VA) − 6.8002	23
(1)BMAGN(FCC_A1,NI,SN:VA) = 4.3689	23
Ternary parameter
(0)L(FCC_A1,NI,SN,TI:VA) = −170 000	^a
BCC_A2/(Ni,Ti,Sn,Va)1(Va)3
(0)L(BCC_A2,NI,TI:VA) = −97 427 + 12.112 × T	26
(1)L(BCC_A2,NI,TI:VA1) = −32 315	26
(0)TC(BCC_A2,NI,TI:VA) = −575	26
(0)G(BCC_A2,VA:VA) = 80 × T	26
(0)BMAGN(BCC_A2,NI,TI:VA) = −0.85	26
(0)L(BCC_A2,NI,SN:VA) = +2369 + 43.736 × T	23
(1)L(BCC_A2,NI,SN:VA) = −654 762.56 + 63.272352 × T	23
(2)L(BCC_A2,NI,SN:VA) = +689 895.17 − 94.74087 × T	23
(0)L(BCC_A2,SN,TI:VA) = −142 089.52 + 28.14226 × T	24
(1)L(BCC_A2,SN,TI:VA) = 42 811.467	24
Ternary parameter
(0)L(BCC_A2,NI,SN,TI:VA) = +30 000	^a
Disordered part of BCC_B2, identical with BCC_A2
A2_BCC/(Ni,Ti,Sn)1(Va)3
(0)L(A2_BCC,NI,TI:VA; 0) = −97 427 + 12.112 × T	26
(1)L(A2_BCC,NI,TI:VA) = −32 315	26
(0)TC(A2_BCC,NI,TI:VA) = −575	26
(0)BMAGN(A2_BCC,NI,TI:VA; 0) = −0.85	26
HCP_A3/(Ni,Ti,Sn)1(Va)0.5
(0)L(HCP_A3,NI,TI:VA) = −20 000	26
(0)L(HCP_A3,NI,SN:VA) = 2000	^a
(0)G(HCP_A3,SN,TI:VA) = −127 549.582 + 23.2048828 × T	24
(1)G(HCP_A3,SN,TI:VA) = 64 500.46 + 7.7566 × T	24
(2)G(HCP_A3,SN,TI:VA) = 31 287.55	24
BCC_B2/(Ni,Ti,Sn)1(Ni,Ti,Sn)1(Va)3
(0)G(BCC_B2,NI:TI:VA) = −33 193 + 10.284 × T	26
(0)G(BCC_B2,TI:NI:VA) = −33 193 + 10.284 × T	26
(0)G(BCC_B2,NI:SN:VA) = −40 000 + 11 × T	^a
(0)G(BCC_B2,SN:NI:VA) = −40 000 + 11 × T	^a
(0)G(BCC_B2,TI:SN:VA) = −40 000 + 11 × T	^a
(0)G(BCC_B2,SN:TI:VA) = +40 000 + 11 × T	^a
(0)L(BCC_B2,TI:NI,TI:VA) = +60 723.7 − 15.4024 × T	26
(0)L(BCC_B2,NI,TI:TI:VA) = +60 723.7 − 15.4024 × T	26
(0)L(BCC_B2,NI:NI,TI:VA) = −55 288.8 + 25.4416 × T	26
(0)L(BCC_B2,NI,TI:NI:VA) = −55 288.8 + 25.4416 × T	26
(2)L(BCC_B2,NI:NI,TI:VA) = +6010.11 + 3.95974 × T	26
(2)L(BCC_B2,NI,TI:NI:VA) = +6010.11 + 3.95974 × T	26
Ternary parameters
(0)L(BCC_B2,TI,SN:NI:VA) = −69 000 + 2 × T	23
(0)L(BCC_B2,NI,SN:NI:VA) = −69 000 + 2 × T	23
BCT_A5/(Ni,Sn,Ti)1
(0)L(BCT_A5,SN,TI) = 50 000	24
(0)L(BCT_A5,NI,SN) = −21 500	23
Ti3Sn/(Ti)3(Ni,Sn,Va)1
(0)G(TI3SN,TI:SN) = 3 × GHSERTI + GHSERSN − 141 133.07 + 1.1272 × T	24
(0)G(TI3SN,TI:VA) = 3 × GHSERTI + 15 000	24
Ternary parameter
(0)G(TI3SN,TI:NI; 0) = 3 × GHSERTI + GHSERNI − 45 000	^a
Ti2Sn/(Ti,Va)2(Sn,Ni,Va)1
(0)G(TI2SN,TI:SN) = 2 × GHSERTI + GHSERSN − 122 344.77 + 6.0034 × T	24
(0)G(TI2SN,TI:VA) = 2 × GHSERTI + 10 000	24
(0)G(TI2SN,VA:SN) = GHSERSN + 5000	24
(0)G(TI2SN,VA:VA) 298.15, 300 000	24
(0)L(TI2SN,TI:SN,VA) = −34 085.17	24
(0)L(TI2SN,TI,VA:SN) = −49 803.91 + 24.4710 × T	24
Ternary parameters
(0)G(TI2SN,VA:NI) = GHSERNI + 15 000	^a
(0)G(TI2SN,TI:NI) = 2 × GHSERTI + GHSERNI − 70 000 + 6.0034 × T	^a
Ti5Sn3/(Ti)5(Sn)3(Ni,Va)1
(0)G(TI5SN3,TI:SN:VA) = 5 × GHSERTI + 3 × GHSERSN − 330 180 + 5.3 × T	24
Ternary parameter
(0)G(TI5SN3,TI:SN:NI) = 5 × GHSERTI + 3 × GHSERSN + GHSERNI − 395 000 + 2.3 × T	^a
Ti6Sn5/(Ti)6(Ti,Sn,Ni)5
(0)G(TI6SN5,TI:SN) = 6 × GHSERTI + 5 × GHSERSN − 468 938.25 + 5.3729 × T	24
Ternary parameters
(0)G(TI6SN5,TI:NI) = 6 × GHSERTI + 5 × GHSERNI − 300 000.25 + 5.3729 × T	^a
(0)L(TI6SN5,TI:TI,SN,NI) = 40 000	^a
TiNi3/(Ni,Ti)1(Ni,Sn,Ti)3
(0)G(TINI3,NI:NI) = +4 × GNIHCP	25
(0)G(TINI3,NI:TI) = −157 744 + 18.6544 × T + 3 × GNIHCP + GHSERTI	26
(0)G(TINI3,TI:NI) = +157 744 − 18.6544 × T + GNIHCP + 3 × GHSERTI	26
(0)G(TINI3,TI:TI) = +4 × GHSERTI	25
(0)L(TINI3,NI:NI,TI) = +143 216 − 101.776 × T	26
(1)L(TINI3,NI:NI,TI) = +109 156 − 66.448 × T	26
(0)L(TINI3,TI:NI,TI) = +20 000	26
(0)L(TINI3,NI,TI:TI) = +60 000	26
Ternary parameters
(0)G(TINI3,TI:SN) = −100 000 + GHSERSN + 3 × GDHCTI	^a
(0)G(TINI3,NI:SN) = −80 000 + GHSERSN + 3 × GDHCNI	^a
(0)L(TINI3,NI:NI,SN,TI) = −110 000	^a
(0)L(TINI3,TI:NI,SN,TI) = −150 000	^a
Ti2Ni/(Ni,Ti)2(Ni,Ti)1
(0)G(TI2NI,TI:NI) = +3 × GTI2NI	26
(0)G(TI2NI,NI:TI) = +2 × GLAVNI + GLAVTI + 30 000 − 3 × GTI2NI	26
(0)L(TI2NI,NI,TI:NI) = +60 000	26
(0)L(TI2NI,NI:NI,TI) = +60 000	26
(0)L(TI2NI,TI:NI,TI) = +60 000	26
(0)L(TI2NI,NI,TI:TI) = +60 000	26
GTI2NI = 0.333333 × GHSERNI + 0.666667 × GHSERTI – 27 514.218 + 2.85345219 × T	26
Ti2Sn3/(TI)2(SN)3
(0)G(TI2SN3,TI:SN) = 2 × GHSERTI + 3 × GHSERSN − 170 900 + 4.886 × T	^a^,^c
Ternary compounds
NiSnTi/(Ni)1(Sn)1(Ti)1
(0)G(NISNTI,NI:SN:TI) = +GHSERNI + GHSERTI + GHSERSN − 150 455 + 14.2 × T	^a^,^c
Ni2SnTi2/(Ni)2(Sn)1(Ti)2
(0)G(NI2SNTI2,NI:SN:TI) = +2 × GHSERNI + 2 × GHSERTI + GHSERSN − 225 876 + 10.51 × T	^a^,^c
Ni2SnTi/(Ni,Sn,Ti)2(Sn,Va)1(Ti,Va)1
(0)G(NI2SNTI,NI:SN:TI) = +2 × GHSERNI + GHSERTI + GHSERSN − 176 384 + 2.83 × T	^a^,^c
(0)G(NI2SNTI,NI:SN:VA) = +2 × GHSERNI + GHSERSN + 31 200	^a
(0)G(NI2SNTI,NI:VA:TI) = +2 × GHSERNI + GHSERTI + 31 200	^a
(0)G(NI2SNTI,NI:VA:VA) = +2 × GHSERNI + 31 200	^a
(0)L(NI2SNTI,NI:SN:TI,VA) = −49 000	^a
(2)L(NI2SNTI,NI:SN:TI,VA) = −139 000	^a
(0)L(NI2SNTI,NI:SN,VA:TI) = −149 000	^a
Ti5Sn5Ni30/(Sn,Ti)1(Ni,Va)3
(0)G(TI5SN5NI30,SN:NI) = 3 × GHSERNI + GHSERSN − 131 928 + 68.0 × T	^a
(0)G(TI5SN5NI30,TI:NI) = 3 × GHSERNI + GHSERTI − 131 928 + 34.0 × T	^a
(0)L(TI5SN5NI30,SN:NI,VA) = −140 000	^a
(0)L(TI5SN5NI30,TI:NI,VA) = −100 000	^a
(0)L(TI5SN5NI30,SN,TI:NI) = −130 000	^a

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Fig. 15 System Ti–Ni–Sn; calculated liquidus projection (top) and 3D view (bottom).

Fig. 16 System Ti–Ni–Sn; calculated isopleths Ni–TiSn, Sn–TiNi and NiSn–Ti–Sn.

Fig. 17 System Ti–Ni–Sn; calculated isothermal sections at 800, 900 and 950 °C.

Fig. 18 System Ti–Ni–Sn; calculated isothermal sections at 1050 and 1200 °C.

The calculation respects well the experimental melting points T_m of the three ternary compounds: τ₁ (1169 °C calculated/versus 1179 °C experimental), τ₂ (1141 °C/1447 °C), τ₃ (1156 °C/1151 °C). Although the overall shape of the liquidus surface is close to the experimental data, several calculated reaction types differ from those experimentally described (for details see the comparison of calculated with experimental reactions given in Table 2). Thus more parameters for liquid and solids may be needed for better consistency between calculation and experiment.

4 Conclusions

From a combined activity of experimental investigation (XPD, EMPA, DTA) and CALPHAD calculation we have established the phase relations in the system Ti–Ni–Sn. The system is characterized by the formation of four ternary compounds labelled τ₁ to τ₄. A particularly large homogeneity region is recorded for τ₂-Ti_1+yNi_2−xSn_1−y (Heusler phase, MnCu₂Al-type).

Extended solid solutions starting from binary phases at 950 °C have been evaluated for Ti₅Ni_1−xSn₃ (filled Mn₅Si₃ = Ti₅Ga₄-type; 0 ≤ x ≤ 1), Ti_1−xSn_xNi₃ (TiNi₃-type; 0 ≤ x ≤ 0.27) and (Ti_1−xNi_x)_1−ySn_y (CsCl-type) reaching a maximum solubility at x = 0.53, y = 0.06). From DTA measurements in alumina crucibles under argon a complete liquidus surface has been elucidated revealing congruent melting for τ₂-TiNi₂Sn at 1447 °C, but incongruent melting for τ₁-TiNiSn (pseudobinary peritectic formation: [small script l] + τ₂ ↔ τ₁ at 1180 °C), τ₃-Ti₂Ni₂Sn (peritectic formation: L + τ₂ + Ti₅NiSn₃ ↔ τ₃ at 1151 °C) and τ₄-(Ti_1−x−ySn_xNi_y)Ni₃ (peritectic formation: L + TiNi₃ + (Ni) ↔ τ₄ at 1157 °C). A Schultz–Scheil diagram for the solidification behavior was constructed for the entire diagram involving 26 isothermal reactions in the ternary.

As for a CALPHAD assessment of the ternary diagram thermodynamic data in the ternary system were only available in the literature for the compounds TiNi₂Sn and TiNiSn, heat of formation data were supplied by our DFT calculations for Ti₂Ni₂Sn, as well as for the solid solutions, which were modelled for Ti_1−xSn_xNi₃, Ti₅Ni_1−xSn₃ and (Ti_1−xNi_x)_1−ySn_y. Thermodynamic calculation was performed with the Pandat software and finally showed a reasonably good agreement for all the 20 invariant reaction isotherms involving the liquid.

Acknowledgements

This research was supported by the Grant Agency of the Czech Republic (Project No. GA14-15576S), by the Project CEITEC-Central European Institute of Technology (CZ.1.05/1.1.00/02.0068) from the European Regional Development Fund and by the Ministry of Education of the Czech Republic (7AMB15AT002). Part of the research was funded by the Austrian FWF under project P24380. The authors are furthermore grateful to some lattice parameter evaluations made by Philipp Sauerschnig.

References

1. R. V. Skolozdra, Y. V. Stadnyk and E. E. Starodynova, Ukr. Fiz. Zh., 1986, 31(8), 1258.
2. Y. M. Goryachev, S. V. Kal'chenko, R. V. Skolozdra and Yu. V. Stadnyk, Elektron. Stroenie i Svoistva Tugoplavk. Soed., I Met. AN USSR. In-t Probl. Materialoved., Kiev, 1991, p. 81.
3. G. Schierning, R. Chavez, R. Schmechel, B. Balke, G. Rogl and P. Rogl, Transl. Mater. Res., 2015, 2, 025001.
4. V. V. Romaka, P. Rogl, L. Romaka, Y. Stadnyk, N. Melnychenko, A. Grytsiv, M. Falmbigl and N. Skryabina, J. Solid State Chem., 2013, 197, 103.
5. J. E. Douglas, C. S. Birkel, N. Verma, V. M. Miller, M.-S. Miao, G. D. Stucky, T. M. Pollock and R. Seshadri, J. Appl. Phys., 2014, 115, 043720.
6. D.-Y. Jung, K. Kurosaki, C.-E. Kim, H. Muta and S. Yamanaka, J. Alloys Compd., 2010, 489, 328.
7. C. Colinet, P. Jund and J.-C. Tédenac, Intermetallics, 2014, 46, 103.
8. M. Yin and P. Nash, Data presented at CALPHAD XLIII, Changsha. Hunan, China, June 1st to June 6th, 2014.
9. R. A. Downie, D. A. MacLaren, R. I. Smith and J. W. G. Bos, Chem. Commun., 2013, 49, 4184.
10. H. Hazama, M. Matsubara, R. Asahi and T. Takeuchi, J. Appl. Phys., 2011, 110, 063710.
11. J. Rodriguez-Carvajal, FULLPROF, a program for Rietveld refinement and pattern matching analysis, Abstract of the satellite meeting on powder diffraction of the XV congress, p. 127, Int. Union of Crystallography, Talence, France, 1990, see also Physica B., 1993, 55, 192.
12. W. Wacha, Program STRUKTUR, Master thesis, University of Technology Vienna, 1989.
13. G. M. Sheldrick, Acta Crystallogr., Sect. A: Found. Crystallogr., 1990, 46, 467.
14. ELK, Program package, http://elk.sourceforge.net/.
15. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
16. Pandat software, version 2014 2.0, CompuTherm LLC, Madison, Wisconsin, USA, 2014, http://www.computherm.com/.
17. H. L. Lukas, S. G. Fries and B. Sundman, Computational Thermodynamics, Cambridge Univ. Press, Cambridge, 2007.
18. A. T. Dinsdale, Calphad, 1991, 15, 317.
19. O. Redlich and A. Kister, Ind. Eng. Chem. Res., 1948, 40, 345.
20. Y. M. Muggianu, M. Gambino and J. P. Bros, J. Chim. Phys. Phys.-Chim. Biol., 1975, 72, 83.
21. Binary Alloy Phase Diagrams, ed. T. B. Massalski, H. Okamoto, P. R. Subramanian and L. Kacprzak, ASM International, Materials Park, Ohio, USA, 2nd edn, December 1990, xxii, pp. 3589.
22. H. S. Liu, J. Wang and Z. P. Jin, Calphad, 2004, 28, 363.
23. A. Zemanova, A. Kroupa and A. Dinsdale, Monatsh. Chem., 2012, 143, 1255.
24. F. Yin, J.-C. Tedenac and F. Gascoin, Calphad, 2007, 31(3), 370.
25. P. Bellen, K. C. Hari Kumar and P. Wollants, Z. Metallkd., 1996, 87, 12.
26. J. de Keyzer, G. Cacciamani, N. Dupin and P. Wollants, Calphad, 2009, 33, 109.
27. E. Povoden-Karadeniz, D. C. Cirstea, P. Lang, T. Wojcik and E. Kozeschnik, Calphad, 2013, 41, 128.
28. F. C. Cannon, Mater. Res. Soc. Symp. Proc., 1984, 22, 113.
29. P. Pietrokowsky and E. P. Frink, Trans. Am. Soc. Met., 1957, 49, 339.
30. H. Nowotny, H. Auer Welsbach, J. Bruss and A. Kohl, Monatsh. Chem., 1959, 90, 15.
31. J. C. Schuster, M. Naka and T. Shibayanagi, J. Alloys Compd., 2000, 305, L1.
32. K. Schubert, K. Frank and R. Gohle, Naturwissenschaften, 1963, 50, 41.
33. J. H. N. van Vucht, H. A. C. M. Bruning, H. C. Donkerloot and A. H. Gomes de Mesquita, Philips Res. Rep., 1964, 19, 407.
34. H. Kleinke, M. Waldeck and P. Gütlich, Chem. Mater., 2000, 12, 2219.
35. S. Sridharan, H. Nowotny and S. F. Wayne, Monatsh. Chem., 1983, 114, 127.
36. E. L. Semenova and Y. V. Kudryavtsev, J. Alloys Compd., 1994, 203, 165.
37. W. Buhrer, R. Gotthardt, A. Kulik, O. Mercier and F. Staub, J. Phys. F: Met. Phys., 1983, 13, L77.
38. J. L. Glimois, P. Forey, R. Guillen and J. L. Feron, J. Less-Common Met., 1987, 134, 221.
39. M. Wilkens, J. Wegst, E. Gunzel, W. Burkhardt, H. G. Meissner, W. Schutt, K. Schubert and P. Esslinger, Naturwissenschaften, 1956, 43, 248.
40. S. K. Shadangi, M. Sing, S. C. Panda and S. Bhan, Cryst. Res. Technol., 1986, 21, 867.
41. H. Fjellvag and A. Kjekshus, Acta Chem. Scand., Ser. A, 1986, 40, 23.
42. W. Jeitschko and B. Jaberg, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem., 1982, 38, 598.
43. M. K. Bhargava and K. Schubert, J. Less-Common Met., 1973, 33, 181.
44. V. A. Romaka, Y. V. Stadnyk, D. Fruchart, T. I. Dominuk, L. P. Romaka, P. Rogl and A. M. Goryn, Semiconductors, 2009, 43, 1124.
45. H. P. Stüwe and Y. Shimomura, Z. Metallkd., 1960, 51, 180.
46. G. R. Purdy and J. G. Parr, Trans. Am. Inst. Min., Metall. Pet. Eng., 1961, 221, 636.
47. P. Duwez and J. L. Taylor, Trans. Am. Inst. Min., Metall. Pet. Eng., 1950, 188, 1173.
48. D. M. Poole and W. Hume Rothery, J. Inst. Met., 1954/55, 83, 473.
49. J. L. Taylor and R. W. Floyd, J. Inst. Met., 1951/52, 80, 577–587.
50. T. V. Philip and P. A. Beck, Trans. Am. Inst. Min., Metall. Pet. Eng., 1957, 209, 1269.
51. F. J. J. van Loo, G. F. Bastin and A. J. H. Leenen, J. Less-Common Met., 1978, 57, 111.
52. A. Shinogi, M. Tanaka and K. Endo, J. Phys. Soc. Jpn., 1978, 44, 774.
53. E. A. Görlich, K. Latka, A. Szytula, D. Wagner, R. Kmiec and K. Ruebenbauer, Solid State Commun., 1978, 25, 661.
54. W. Jeitschko, Metall. Trans., 1970, 1(11), 3159.
55. J. Pierre, R. V. Skolozdra and Y. V. Stadnyk, J. Magn. Magn. Mater., 1993, 128, 93.
56. P. G. van Engen, K. H. J. Buschow and M. Erman, J. Magn. Magn. Mater., 1983, 30, 374.
57. H. Hazama, M. Matsubara, R. Asahi and T. Takeuchi, J. Appl. Phys., 2011, 110, 063710.
58. E. Parthé, L. Gelato, B. Chabot, M. Penzo, K. Cenzual and R. Gladyshevskii, TYPIX Standardized Data and Crystal Chemical Characterization of Inorganic Structure Types, Springer, Berlin, Germany, 1994.
59. P. Mikušik and Š. Pick, Solid State Commun., 1993, 86(7), 467.
60. A. Pasturel, C. Colinet, D. Nguyen Manh, A. T. Paxton and M. van Schilfgaarde, Phys. Rev. B: Condens. Matter Mater. Phys., 1995, 52(21), 15176.

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Footnote

† Electronic supplementary information (ESI) available: Tables S1–S3. See DOI: 10.1039/c5ra16074j

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