Ni₃Sn₄ and FeAl₂ as vacancy variants of the W-type (“bcc”) structure

3.1 Evaluation of crystal structure data from literature

Ni₃Sn₄ develops readily by the reaction of Ni with Sn somewhat below and also above the melting temperature of Sn, i.e. at temperatures typically relevant during Sn-based soldering [26]. Only at temperatures considerably below the melting point of Sn, different forms of a plate-like intermetallic having with a stoichiometry likely corresponding to NiSn₄ develops [26], [27], [28]. While Ni₃Sn₄ is usually included as one of the major equilibrium phases in phase diagrams Ni–Sn [29], NiSn₄ is usually believed to be metastable although final proofs lack for low temperatures.

The crystal structure of Ni₃Sn₄ has first been determined by Nowotny & Schubert [30, 31], with the crystal structure having monoclinic C2/m symmetry. Jeitschko & Jaberg [32] re-determined the crystal structure using more modern methods and arrived at likely more accurate structure parameters than the previous works. Furuseth & Fjellvåg [33] published single-crystal and powder X-ray diffraction data (SCXRD and PXRD) explicitly dealing with the composition dependence of the structure parameters of Ni_3+xSn₄ with x = 0 … 0.7. That work in particular presented SXRD data from crystal specimens of compositions of Ni_3.08Sn₄ and Ni_3.39Sn₄. Table 1 lists the details for Ni_3.08Sn₄ and Figure 1(a) illustrates the crystal structure for the purpose of further discussion. It was demonstrated by the structure analyses [33] that it is the filling of the Ni(3) position which is used to arrive at the actual composition Ni_3+xSn₄. This Ni(3) position is vacant for the ideal Ni₃Sn₄ composition (Pearson symbol mS14) and is filled up with increasing Ni content. If the position is fully occupied (not attained within the homogeneity range of the monoclinic Ni₃Sn₄ phase), the composition would be Ni₄Sn₄ (Pearson symbol mS16). This fully filled up structure is known for CoGe [34] and a polymorph for FeGe [35]. In contrast, in the Ni–Sn system Ni-rich Ni_3+xSn₄ (NiSn) appears to develop a orthorhombic crystal structure related but distinct from the monoclinic one considered here [36, 37].

Figure 1:

Structure model of Ni_3.08Sn₄ as experimentally determined in Ref. [33], compare also Table 1: (a) experimentally determined arrangement of the regular atoms within a unit cell with superposed transformed positions of the underlying W-type structure visible as vertices between “bonds” along the shortest 1 2 〈 111 〉 b c c -like distances. Only the vacant transformed positions (Va(1)) are indicated additionally as small spheres. (b) Same as (a) but without regular atoms, highlighting (red) the rhombic dodecahedron around a single central vertex as typical coordination polyhedron in the W-type structure.

In the previous works the crystal structure of Ni₃Sn₄ was discussed in relation to those of β-Sn [31, 38] and Ni₃Sn₂/NiAs [31, 33]. In particular, already in Ref. [31], prominent atomic chains running along the [ 001 ] N i 3 S n 4 directions have been identified in the crystal structure. The idea that these chains might be associated with a 〈111〉_bcc direction allowed now, by detailed inspection of the crystal structure, identifying the lattice correspondence between the Ni₃Sn₄ structure to that of the W-type structure. The basis vectors of an ideal version of Ni₃Sn₄, a N i 3 S n 4 , i d , b N i 3 S n 4 , i d , and c N i 3 S n 4 , i d spanning its C-centered unit cell can be formulated as (see Eq. (2)):

The resulting unit cell has a 10-times larger volume than that of the W-type structure, implying 20 ideal positions for atoms per unit cell. For this ideal unit cell it holds

with values already suggesting similarity with the experimental lattice parameters (Table 1), e.g. taking an approximate value of a_bcc = 3 Å. This agreement is confirmed by the small of entries of the deviatoric strain tensor ε determined (see Eq. (3)):^[3]

The value of a_bcc obtained in the course of the fitting process amounts to 2.9239 Å. The entries of ε yield x_s = 0.042 according to Eq. (4). The maximum eigenvalue of ε amounts +0.034 with eigenvector approximately pointing along [ 1 1 ‾ 0 ] b c c .

Alongside with the unit-cell transformation, rational-valued fractional coordinates of the above-mentioned 20 ideal positions for the atoms are obtained in the course of the cell transformation process. Checks in view of the C2/m symmetry showed that one symmetry-equivalent set of the transformed positions could be taken as origin of the unit cell and that a change in origin was not necessary. Every experimental atomic position can now be associated with one of the ideal transformed positions with distance <0.5 Å, being the inspiration to label the transformed W-type positions. These crystallographically distinct ideal coordinates have been added to Table 1. For one of the (sets of) transformed positions, there are no close-by regular atoms. This position can be regarded as a vacancy which is labelled Va(1); see Figure 1 and Table 1. Actually, also the Ni(3) position is predominantly vacant. If it is fully vacant (composition Ni₃Sn₄) a fraction of 6/20 = 30 % of the positions of the W-type structure to remain vacant, whereas it is 4/20 = 20 % for the composition Ni₄Sn₄. Accordingly, Ni₃Sn₄ can be regarded as a vacancy variant of the W-type structure.

3.2 Observations fitting to the scheme

Table 2 lists the lattice-direction and lattice-plane indices of Ni₃Sn₄ corresponding to the densest directions and planes of the body-centered cubic W-type structure as obtained from Eqs. (5) and (6). The stereographic projection in Figure 2 depicts a part of these directions and planes for reference purposes. The strong resemblance to a cubic pole figure illustrates the small distortion with respect to the underlying cubic symmetry as implied by the small components of the deviatoric strain tensor in Eq. (9).

Figure 2:

Stereographic projection of the 〈111〉_bcc-like directions (black) and {110}_bcc-, {100}_bcc- and {211}_bcc-like plane normals (blue, red and green) of the lattice of Ni₃Sn₄ (see Table 1) and giving the [ u v w ] N i 3 S n 4 and ( h k l ) N i 3 S n 4 using the indices as listed in Table 2 (omitting the parentheses for the plane normals). Open, unindexed symbols present poles on the lower hemisphere. Greater circles pertaining to the 〈111〉_bcc-like directions (dotted for lower hemisphere) contain those { 1 1 ‾ 0 } b c c - and { 2 1 ‾ 1 ‾ } b c c -like planes forming valid slip systems with the corresponding 〈111〉_bcc-like direction. Note that the greater circle of { 020 } N i 3 S n 4 (not shown) constitutes a miror plane and the plane normal corresponds to a twofold axis.

The structure factors for X-rays calculated using the structure model by Furuseth et al. [33] in Table 1 have appreciable magnitudes for {110}_bcc- and {200}_bcc-like planes, indicating that the underlying W-type structure of the Ni₃Sn₄ structure strongly influences diffraction pattern of the latter. Higher order and {hkl}_bcc-like reflections are, however, considerably weakened by the atomic displacements from the ideal positions. It can also be noted that the {100}_bcc- and {111}_bcc-like reflections which are expected for a CsCl (B2) superstructure for the W-type, have large structure factors revealing some corresponding atomic ordering of Ni versus Sn. The strain/monoclinic distortion due to Eq. (9) leads to significantly different positions (spitting) for the different representatives of, e.g. the prominent {110}_bcc-like reflections in a PXRD pattern.

Yu et al. [39] have observed annealing twins with habit plane { 100 } N i 3 S n 4 having a misorientation corresponding to a 180° rotation around the plane’s normal, a twinning which can also be achieved by shearing along 〈 001 〉 N i 3 S n 4 . According to Table 2, the habit plane is {211}_bcc-like whereas the shear direction is 1 2 〈 1 1 ‾ 1 ‾ 〉 b c c -like. Indeed, this type of twinning corresponds to the common type of (deformation) twinning in the W-type metals with a shear direction of the parallel to 1 2 〈 1 1 ‾ 1 ‾ 〉 b c c [40]. In the same work [39], the same plane and directions of Ni₃Sn₄ have been identified to form a slip system becoming active upon pillar compression. While the most prominent slip planes for W-type metals are {110}_bcc, { 2 ‾ 11 } b c c is the second most prominent. Both allow for slip along 1 2 〈 1 1 ‾ 1 ‾ 〉 b c c [41].

As reported in the course of solidification experiments of Sn droplets on Ni₃Sn₄ crystal facets [38], primary Ni₃Sn₄ forms as faceted crystals upon solidification of Sn-rich Ni–Sn melts. These crystals grow as long rods with axis 〈 010 〉 N i 3 S n 4 corresponding to a low-index 〈110〉_bcc direction. These rods are faceted on { 100 } N i 3 S n 4 , { 20 1 ‾ } N i 3 S n 4 and { 001 } N i 3 S n 4 . While { 100 } N i 3 S n 4 is a {211}_bcc-like plane (see above), only higher order {hkl}_bcc planes not contained in Table 2 can be identified for { 20 1 ‾ } N i 3 S n 4 and { 001 } N i 3 S n 4 so that these cannot easily associated to prominent planes of the W-type structure.

4.1 Evaluation of crystal structure data from literature

A first complete atomic structure model of FeAl₂ having triclinic P1 symmetry was reported by Corby & Black using SXRD methods [42]. The atomic structure implied 18 atom per unit cell (aP18) with one having mixed occupation by Fe+Al. Bastin et al. [43] investigated the lamellar microstructure resulting from eutectoid decomposition of the high-temperature ε-phase Fe₅Al₈ consisting into CsCl-type FeAl and FeAl₂. Thereby a dual-phase “crystal” specimen was investigated by Weissenberg-camera based SXRD. The authors proposed an A-centered pseudomonoclinic (i.e. triclinic) unit cell for the FeAl₂ without referencing to Ref. [42], and without reporting or considering an atomic structure model. Without reference to Bastin et al., but to Corby & Black, Hirata et al. [44] performed transmission electron microscopy (TEM) complemented by selected area electron diffraction (SAED). They also elaborated an orientation relationship (OR) FeAl/FeAl₂. A SCXRD-based redetermination of the crystal structure was reported by Chumak et al. [45]. They reported an additional atom per unit cell (aP19) as compared to the model reported by Corby & Black [42]. This additional atom corresponds to one which was already predicted to be present by Mihalkovic & Widom [46] based on DFT calculations, when they analyzed the structure model by Corby & Black. Notably, the structure model offered by Chumak et al. [45] has centrosymmetric triclinic P 1 ‾ symmetry.

As pointed out by Scherf et al. [47], the unit cells, which have been reported in Refs. [42, 43, 45], differed but they can be related by basis transformations. In the following we will primarily work with the structure model by Chumak et al. [45], where the basis vectors and other quantities are without primes. The basis vectors of the unit cells formulated by Corby & Black [42] (primed) and Bastin et al. [43] (double primed) can be written in accordance with Ref. [47] as:

Using in particular EBSD investigations, Scherf et al. [47] reinvestigated the OR FeAl/FeAl₂ and showed that it basically agrees with those reported previously [43, 44] if the inversion symmetry of the crystal structure of FeAl₂ is taken into account; for more details, see Section 4.2.

In contrast to Ni₃Sn₄ some relation of the crystal structure of FeAl₂ with the W-type structure has been discussed previously in view of close atomic correspondences between certain lattice planes of bcc/CsCl-type FeAl and of FeAl₂. This was in particular elaborated by Hirata et al. [44] who used, however, the outdated structure model from Ref. [42]. It was also demonstrated that the electron beam in a TEM can transform FeAl₂ particle into a W-type material having the same orientation as the surrounding FeAl matrix [44]. However, an explicit three-dimensional lattice correspondence has not been proposed in these works.

After in-depth visual inspection of the structure model by Chumak et al. [45], a lattice correspondence between the crystal structure of FeAl₂ and the W-type structure has now been derived as

The correspondences for the other sets of basis vectors [42, 43] can be obtained straightforwardly combining Eq. (11) with Eq. (10).

The resulting unit cell due to Eq. (11) has a volume which is 10.5-times larger that of the W-type structure, implying 21 ideal transformed positions for atoms per unit cell. For this ideal (id) unit cell it holds

with values already suggesting similarity with the experimental lattice parameters (Table 3), e.g. taking an approximate value of a_bcc = 3 Å. This agreement is confirmed by the small absolute values of the entries of the deviatoric strain tensor ε determined using the methodology described in Section 2:

The value of a_bcc obtained in the course of the fitting process amounts 2.9077 Å. The entries of ε yield x_s = 0.037 according to Eq. (4). The maximum eigenvalue of ε amounts to +0.030 with eigenvector approximately into [001]_bcc direction. The values of x_s is quite close to that encountered for Ni₃Sn₄ (see Section 3.1), but the largest eigenvector is oriented differently.

Figure 3:

Structure model of FeAl₂ as experimentally determined in Ref. [45], compare also Table 3: (a) regular atomic positions within a unit cell with superposed transformed positions of the underlying W-type structure visible as vertices between “bonds” along the shortest 1 2 〈 111 〉 b c c -like distances. Only the positions of Al(2a,id) and Al(2a,id) are additionally indicated additionally as spheres with one single Al(2) atom, resulting on average in a vacancy. (b) Same as (a) but without regular atomic positions, highlighting (red) the rhombic dodecahedron around a single central vertex as typical coordination polyhedron in the W-type structure.

Alongside with the unit-cell transformation, rational-valued fractional coordinates of the above-mentioned 21 ideal positions for the atoms are obtained by the transformation process. Checks in view of the P 1 ‾ symmetry showed that one symmetry-equivalent set of the transformed positions could be taken as origin of the unit cell and that a change in origin was not necessary. Every experimental atomic position, except for that of Al(2), can now be associated with one of the ideal transformed positions with distance lower <0.5 Å, being the inspiration to label the transformed W-type positions. These crystallographically distinct ideal coordinates have been added to Table 3. The regular Al(2) atom assumes a position intermediate between the transformed positions Al(2a,id) (distance 0.97 Å) and Al(2b,id) (1.55 Å); see Table 3. It is these positions which make out the difference of the 19 regular Fe/Al atoms and 21 transformed positions in the unit cells. Accordingly, one obtains a vacancy fraction of 2/21 ≈ 9.5 % in the crystal structure of FeAl₂ as reported by Chumak et al. [45]. In view of all this, FeAl₂ can be regarded as a vacancy variant of the W-type structure.

4.2 Observations fitting to the scheme

Table 4 lists the lattice-direction and lattice-plane indices of FeAl₂ corresponding to the densest directions and planes of the body-centered cubic W-type structure as obtained from Eqs. (5) and (6). The stereographic projection in Figure 4 depicts a part of these directions and planes for reference purposes. The strong resemblance to a cubic pole figure illustrates the small distortion with respect to the underlying cubic symmetry as implied by the small components of the deviatoric strain tensor in Eq. (13). As in the case of Ni₃Sn₄ (see Section 3.2), also in the case of FeAl₂ the X-ray structure factors for the lattice planes corresponding to, in particular {110}_bcc are of appreciable magnitude. Thereby, the structure factors for { 221 } F e A l 2 , { 11 3 ‾ } F e A l 2 and { 114 } F e A l 2 are significantly larger than for { 1 2 ‾ 2 ‾ } F e A l 2 , { 2 1 ‾ 2 } F e A l 2 and { 03 1 ‾ } F e A l 2 . However, in contrast to Ni₃Sn₄, {100}_bcc- and {111}_bcc-like reflections indicative for an underlying B2-type ordering (as in FeAl) are absent.

Figure 4:

Stereographic projection of the 〈111〉_bcc-like directions (black) and {110}_bcc-, {100}_bcc- and {211}_bcc-like plane normals (blue, red and green) of the lattice of FeAl₂ (see Table 3) and giving the [ u v w ] F e A l 2 and ( h k l ) F e A l 2 using the indices as listed in Table 4 (omitting the parentheses for the plane normals). Open, unindexed symbols present poles on the lower hemisphere. Greater circles pertaining to the 〈111〉_bcc-like directions (dotted for lower hemisphere) contain those { 1 1 ‾ 0 } b c c - and { 2 1 ‾ 1 ‾ } b c c -like planes forming valid slip systems with the corresponding 〈111〉_bcc-like direction.

Scherf et al. [47] showed that the OR FeAl/FeAl₂ can be expressed as { 1 ‾ 01 } F e A l ∣ ∣ { 114 } F e A l 2 and 〈 111 〉 F e A l ∣ ∣ 〈 1 1 ‾ 0 〉 F e A l 2 , and, as mentioned before, that this OR is compatible with previous findings [43, 44]. Moreover, the habit plane was explicitly determined to be { 114 } F e A l 2 , where the authors already emphasized geometric similarity with {110}_bcc (as it is the case for the {110}_FeAl planes parallel to them according to the OR). Moreover, the parallel directions are the densest packed directions 〈111〉_FeAl and 〈 1 1 ‾ 0 〉 F e A l 2 , the latter one also being 〈111〉_bcc-like; see entry 〈 7 7 ‾ 0 〉 ( F e A l 2 ) (i.e.  〈 1 1 ‾ 0 〉 F e A l 2 ) in Table 4. As, furthermore, encountered by Scherf et al. [47], the former, experimentally determined OR has also the { 221 } F e A l 2 planes parallel to other {110}_FeAl-type planes, whereby already Hirata et al. [44] emphasized that also this plane had structural similarity with {110}_FeAl or {110}_bcc which was then, furthermore, emphasized by Scherf et al. [47] for { 11 3 ‾ } F e A l 2 . It is, in particular, these three (set of) planes of FeAl₂ which contain the above-mentioned densely packed 〈 1 1 ‾ 0 〉 F e A l 2 direction; see Figure 5(a). Using the OR and Table 4, it turns out that all {110}_bcc-like planes of FeAl₂ are parallel to corresponding planes of FeAl. Accordingly, the OR FeAl/FeAl₂ can be regarded (ignoring slight misfits due to the deviatoric strain experienced by the FeAl₂) as some kind of cube-on-(pseudo)cube OR. It is, however, emphasized here that the {110}_bcc-like planes of FeAl₂ not containing the 〈 1 1 ‾ 0 〉 F e A l 2 direction are not that regular; see Figure 5(b)–(d) versus Figure 5(a). This is well compatible with the larger structure factors of { 221 } F e A l 2 , { 11 3 ‾ } F e A l 2 , { 114 } F e A l 2 as compared to { 1 2 ‾ 2 ‾ } F e A l 2 , { 2 1 ‾ 2 } F e A l 2 , and { 03 1 ‾ } F e A l 2 ; see above. In any case, the cube-on-(pseudo)cube OR readily explains the transformation of FeAl₂ particles in the electron beam of a TEM into W-type material with the same orientation as the FeAl matrix observed by Hirata et al. [44]. Also the eutectoid microstructure resulting from γ-brass-type Fe₅Al₈ (itself a vacancy variant of the W-type structure) can now be formulated under the umbrella of a cube-on-cube like OR [47] involving all three phases.

Figure 5:

Crystal structure of FeAl₂ with view along different 〈111〉_bcc-type directions and indicated { 1 1 ‾ 0 } b c c -type planes located within the indicated direction (watching downstream that direction): (a) [ 1 1 ‾ 0 ] F e A l 2 depicting the { 114 } F e A l 2 , { 11 3 ‾ } F e A l 2 and { 221 } F e A l 2 planes already discussed as { 1 1 ‾ 0 } b c c -like in Refs. [44, 47], the former two also as habit planes towards FeAl in eutectoid colonies [47, 48], Moreover the { 001 } F e A l 2 plane ({211}_bcc-like) of the deformation twinning reported in [49] (b) [ 22 1 ‾ ] F e A l 2 , (c) 〈 813 〉 F e A l 2 , (d) 〈 5 2 ‾ 6 ‾ 〉 F e A l 2 . Same colors have been used as in Figure 3.

Strikingly, Li et al. [48] found an OR which can be reformulated for the present purpose as { 1 1 ‾ 0 } F e A l ∣ ∣ { 11 3 ‾ } F e A l 2 and 〈 111 〉 F e A l ∣ ∣ 〈 1 1 ‾ 0 〉 F e A l 2 , with { 11 3 ‾ } F e A l 2 identified as habit plane. While the habit plane is different, this OR, nevertheless, implies basically the same cube-on-(pseudo)cube OR as in Ref. [47]. More detailed TEM analysis of FeAl/FeAl₂ eutectoid showed supposed antiphase boundaries in FeAl₂ as well as misfit dislocations at the phase boundaries, which were unfortunately not characterized with respect to the crystallographic characteristics [49].

Also plastic deformation processes have been characterized for FeAl₂ contained in FeAl/FeAl₂ eutectoid FeAl/FeAl₂ microstructures [48], [49], [50].

Schmitt et al. [49] studied creep at elevated temperature. While FeAl appears to deform first, upon higher strains slip also appears to occur in the FeAl₂, where { 221 } 〈 1 1 ‾ 0 〉 F e A l 2 , { 11 3 ‾ } 〈 1 1 ‾ 0 〉 F e A l 2 , and { 114 } 〈 1 1 ‾ 0 〉 F e A l 2 were proposed as slip systems. While the structural analogy with { 1 1 ‾ 0 } 〈 111 〉 F e A l was emphasized based on direct geometrical analysis of the planes and directions (see also above), the described lattice correspondence (Eq. (11)) can be used to show that these slip systems can explicitly be regarded as { 1 1 ‾ 0 } 〈 111 〉 b c c -like. The authors [49] also directly observed mechanical twins misoriented with respect to the original FeAl₂ by 180° rotation along an axis perpendicular to { 001 } F e A l 2 , with this plane also being the habit plane. According to the lattice correspondence this plane is {211}_bcc-like (see also Table 4, contained as { 007 } F e A l 2 ), and the twinning observed is an analogue of the typical {211}_bcc deformation twins known for W-type metals and solid solutions [40] (also encountered as annealing twins for Ni₃Sn₄). Moreover, the authors [49] also observed sliding on the {110}_bcc-like phase interfaces FeAl/FeAl₂.

Li et al. [48, 50] investigated plasticity at room temperature on FeAl/FeAl₂ eutectoid microstructures of different interlamellar spacings upon indentation and pillar compression. They identified [50] a plethora of different slip planes by TEM techniques, which include the above-mentioned {110}_bcc-like { 221 } F e A l 2 and { 114 } F e A l 2 , but also many others planes which cannot be related with the other {110}_bcc- or {211}_bcc-like planes contained in Table 4. An evident system behind all these crystallographically distinct slip planes (beyond { 221 } F e A l 2 and { 114 } F e A l 2 ) was neither offered by Li et al. [48], nor could such a system be found by the present author using the insights derivable from the structure relation to the W-type structure. The authors [48] explicitly observed occurrence of sliding on the {110}_bcc-like phase interfaces FeAl/FeAl₂ upon pillar compression, in particular, also at elevated temperatures [50].

At the end one might return to the pseudomonoclinic character of the crystal structure of FeAl₂ proposed by Bastin et al. [43] (double primed basis vectors in Eq. (10)). A closely related unit cell was used in a PXRD work by Balenetskyy et al. [51], with a modified basis with a F e A l 2 ‴ = c F e A l 2 ″ , b F e A l 2 ‴ = − b F e A l 2 ″ and c F e A l 2 ‴ = a F e A l 2 ″ . The latter authors emphasized that the crystal structure might truly be monoclinic. Inspection of the atomic structure model due to Chumak et al. [45] published after [51], however, does not reveal hidden symmetry in the atomic structure. Indeed, via the lattice correspondence in Eq. (11) it can be shown that the monoclinic axis corresponds to ± [ 5 ‾ 59 ] b c c , not containing any symmetry, and that the ideal lattice angles α and γ calculated from the lattice correspondence deviate by more than 1.5° from 90°. Accordingly, to attain truly monoclinic symmetry, new symmetry elements have to build up within the atomic structure, which is, however, not the case in view of the available atomic structure models [42, 45].