Boron-phil and boron-phob structure units in novel borides Ni 3 Zn 2 B and Ni 2 ZnB : experiment and fi rst principles calculations †

The crystal structures of two novel borides in the Ni–Zn–B system, τ5-Ni3Zn2B and τ6-Ni2ZnB, were determined by single crystal X-ray diffraction (XRSC) in combination with selected area electron diffraction in a transmission electron microscope (SAED-TEM) and electron probe microanalysis (EPMA). Both compounds crystallize in unique structure types (space group C2/m, a = 1.68942(8) nm, b = 0.26332(1) nm, c = 0.61904(3) nm, β = 111.164(2)°, RF = 0.0219 for Ni3Zn2B, and space group C2/m, a = 0.95296(7) nm, b = 0.28371(2) nm, c = 0.59989(1) nm, β = 93.009(4)°, RF = 0.0163 for Ni2ZnB). Both compounds have similar building blocks: two triangular prisms centered by boron atoms are arranged along the c-axis separated by Zn layers, which form empty octahedra connecting the boron centered polyhedra. Consistent with the (Ni+Zn)/B ratio, isolated boron atoms are found in τ5-Ni3Zn2B, while B–B pairs exist in τ6-Ni2ZnB. The crystal structure of Ni2ZnB is closely related to that of τ4-Ni3ZnB2, i.e. Ni2ZnB can be formed by removing the nearly planar nickel layer in Ni3ZnB2 and shifting the origin of the unit cell to the center of the B–B pair. The electrical resistivity and specific heat of τ5-Ni3Zn2B reveal the metallic behavior of this compound with an anomaly at low temperature, possibly arising from a Kondo-type interaction. Further analysis on the lattice contribution of the specific heat reveals similarity with τ4-Ni3ZnB2 with some indications of lattice softening in τ5-Ni3Zn2B, which could be related to the increasing metal content and the absence of B–B bonding in τ5-Ni3Zn2B. For the newly found phases, τ5-Ni3Zn2B and τ6-Ni2ZnB as well as for τ3-Ni21Zn2B20 and τ4-Ni3ZnB2 density functional theory (DFT) calculations were performed by means of the Vienna Ab initio Simulation Package (VASP). Total energies and forces were minimized in order to determine the fully relaxed structural parameters, which agree very well with experiment. Energies of formations in the range of −25.2 to −26.9 kJ mol were calculated and bulk moduli in the range of 179.7 to 248.9 GPa were derived showing hardening by increasing the B concentration. Charge transfer is discussed in terms of Bader charges resulting in electronic transfer from Zn to the system and electronic charge gain by B. Ni charge contributions vary significantly with crystallographic position depending on B located in the neighbourhood. The electronic structure is presented in terms of densities of states, band structures and contour plots revealing Ni–B and Ni–Zn bonding features.


Introduction
Additions of Zn in novel Ni-B-Zn coatings on steels (used in the automotive, aerospace, petrochemical, textile and electronics industries) were found to significantly improve crystallinity, mechanical properties, and corrosion resistance. 1,2 Little, however, is known how crystalline Ni-Zn-borides might influence long term stability. Although the first investigation of the Ni-Zn-B system in 1962 by Stadelmaier 3 provided an isothermal section at 800°C and a liquidus projection in the Ni-rich part, only the crystal structure of the τ-phase Ni 20 Zn 3 B 6 has since been studied carefully by several groups, [3][4][5][6] whilst the crystal structure of the other three compounds remained unknown: ψ-Ni 57 Zn 13 B 30 , χ-Ni 45 Zn 30 B 25 , and φ-Ni 49 Zn 31 B 20 . With the increasing interest in Ni-B-Zn coatings on steels, our group (for details see the papers by Malik et al. [7][8][9] ) has recently reinvestigated the system and identified six ternary compounds: the crystal structures were solved for Ni 12 ZnB 8−x (labeled as τ 2 , probably to be identified as ψ-Ni 57 Zn 13 B 30 of Stadelmaier 3 ) as well as for Ni 21 Zn 2 B 20 (labeled as τ 3 , new structure type) and Ni 3 ZnB 2 (labeled as τ 4 , new structure type). 7 The crystal structures of two further compounds are still unknown: Ni 48 Zn 32 B 20 (in at%, labeled as τ 5 , 8 likely similar to the φ-phase of ref. 3), and Ni 47 Zn 23 B 30 8 (in at%, labeled as τ 6 , likely similar to the χ-phase of ref. 3). Moreover, a detailed study on the homogeneity region of the τ-phase (labeled as τ 1 ) was also presented, 8 which showed both Ni/Zn and Zn/B 4 atom substitutions. Physical properties have been investigated so far only for the compounds, τ 3 -Ni 21 Zn 2 B 20 and τ 4 -Ni 3 ZnB 2 , both revealing metallic and paramagnetic behavior. 9 In this paper, we aim to complete the previous work by: (i) solving the crystal structures of τ 5 and τ 6 from single crystal X-ray analyses backed by SAED-TEM, (ii) elucidating the electronic structure and chemical bonding by DFT calculations, (iii) defining the physical properties of τ 5 and τ 6 , and accordingly (iv) revising the isothermal section at 800°C.
Besides phase stability issues, the combination of transition metal elements and main group metals in intermetalliclike borides reveals an interesting combination of structural units which are composed of transition metal boride slabs ("boron-phil" part), that are connected by bonds between the transition element and the main group metal ("boron-phob" part). [10][11][12] It is worth mentioning that due to the electron deficiency of boron in most cases, only electropositive elements such as rare earth metals can form the so called "higher borides" with three-dimensional boron frameworks. [13][14][15][16][17][18] In this work, we have particularly focused on the four ternary borides (τ 3 , τ 4 , τ 5 , and τ 6 ) which all adhere to the isopleth extending from NiB to NiZn. As bonding defines the physical properties, the DFT studies accompanying the experiment will provide thermodynamic and elastic stability and elucidate the underlying electronic structure in terms of charge transfer, density of states, band structures and charge density distribution.

Sample preparation and characterisation techniques
For sample preparation, the following materials were used: nickel rods, foils, or wires, zinc filings (all metals from Alfa Aesar, D with 99.9 mass% purity), and crystalline boron pieces (Alfa Aesar, D, 99.5 mass% B). Zn filings were prepared mechanically from zinc, which was purified by heating Zn granules in an evacuated quartz ampoule at 800°C. Due to the volatility of Zn and in order to reach equilibrium at the desired temperature, the sample preparation was conducted in two steps. First, Ni x B (x ≤ 3) master alloys were prepared by arc-melting of nickel pieces/foils and boron pieces under a Ti-gettered Ar atmosphere.
The powdered Ni x B master alloys were subsequently mixed with Zn filings and compacted in a steel die without lubricant. For samples with compositions close to the Ni-Zn binary system, finely cut Ni wires (Φ = 0.25 mm, ℓ ≤ 3 mm) were used in addition to Ni 3 B powder and Zn filings. The pellets were placed inside protective Al 2 O 3 crucibles and sealed inside separate quartz tubes under ∼250 mbar of Ar.
Heat treatments were performed in a resistance furnace at 875 and 900°C for a week, with an intermediate holding at 500°C overnight. After the first heat treatment, samples were quenched in cold water, re-powdered in a tungsten carbide mortar, compacted, sealed, and heat treated in the same way as described above. Single phase samples for physical properties measurements were prepared as indicated above. Additionally, sintering processes were performed by hotpressing the pre-compacted powder pellets in alumina powder inside a graphite die at 850°C for 30 minutes. After removing the residual alumina powder on the sample's surface, they were cut into bars with dimensions of ∼10 × 2 × 1 mm 3 with a low speed diamond saw.
X-ray powder diffraction patterns were collected after every heat treatment on a Guinier-Huber G670 image plate recording system with monochromatic CuKα 1 radiation. Microstructures and compositions were examined by light optical microscopy (LOM) and scanning electron microscopy (SEM) on a Zeiss Supra 55 VP equipped with an EDX detector. The compositions of each phase were obtained by averaging several point measurements (on grains larger than at least 5 µm) and area measurements for the solid and the solidified liquid phase, respectively. Due to the difficult identification of B spectral lines, only the Ni : Zn ratio could be measured reliably with SEM-EDX, while the boron content determinations rely solely on the single crystal refinements, or from estimation based on the weight percent measurement in EDX.
Due to the ductile nature of the metal rich borides, mechanical isolation and fragmentation significantly deformed the single crystal specimens, making them unsuitable for single crystal data collection. Consequently, the single crystal specimens were separated from the flux via acid leaching (details on the system inherent flux are given in section 3.1.). Hot diluted HCl solution was used for τ 5 , while a cold mixture of concentrated HCl and HNO 3 was used for τ 6 . In the latter, the leaching process was first performed by immersing the sample in cold concentrated HCl. After the sample's surface was passivated, drop-wise concentrated HNO 3 was added while stirring the mixture until a violent reaction was observed. The reaction was stopped immediately by diluting the mixture with distilled water. In both cases, the single crystal specimens were filtered and washed several times with distilled water. Needle-shaped crystals (length/cross section ratio <3) were mounted on glass capillaries and inspected on an AXS D8-GADDS to ensure high crystal quality, lattice parameters, and crystal symmetry.
Single crystal data collections were performed at room temperature on a Bruker APEXII diffractometer equipped with a CCD area detector and an Incoatec Microfocus Source IµS (30 W, multilayer mirror, Mo-Kα; λ = 0.071069 nm; detector distance of 3 cm; full sphere; 2°< 2θ < 70°). Orientation matrices and unit cell parameters were derived using the Bruker APEXII software suite. 19 Besides the general treatment of absorption effects using the multi-scan technique (SADABS; redundancy of integrated reflections >8), no additional absorption corrections were performed because of the rather irregular cross-section (mainly due to the attack of the acid) and practically small dimensions of the investigated specimens. The structures were solved by direct methods using program SIR-92 20 and refined with the SHELXL-97 21 program within the WinGX package. 22 The crystal structure data were subsequently standardized with program STRUCTURE-TIDY. 23 For TEM investigation, a Philips CM12 STEM transmission electron microscope (TEM) operated at 120 kV with an EDAX EDX analyser was employed to get information about the crystal symmetry and lattice parameters of selected phases. The samples for the TEM study were prepared in the form of thin lamellae (lateral dimensions about 10 × 7 μm 2 ) using a focused ion beam (FIB) technique in a TESCAN LYRA 3 XMU FEG/SEM × FIB scanning electron microscope.
Electrical resistivity measurement was performed on a bar shaped polycrystalline sample by a standard four probe AC bridge technique in a homemade equipment from 2 to 300 K. Contacts were made by spot welding thin gold wires (d = 100 µm) on the sample's surface. For better mechanical stability, the spot welded contacts were coated with silver epoxy. Specific heat was measured in a Quantum Design PPMS in the same temperature range as the electrical resistivity measurement.
Hardness was measured with two different pieces of equipment, a microhardness tester, AD Paar MHT-4, mounted on a Zeiss Axioplan optical microscope employing a load of 1 N, applying a rate of 0.1 N s −1 and a loading time of 10 s (referred to as HV0.1, "static"). For all these indentations the hardness . It should be noted that prior to hardness measurements, the samples were polished with a 0.3 μm aluminum oxide paste to produce an optical finish. A capacitance dilatometer 24 served to acquire thermal expansion coefficients in the temperature range of 4.2 to 300 K.

DFT calculations
The density functional theory (DFT) calculations for the Ni-Zn-B compounds were performed by means of the planewave based Vienna ab initio simulation package (VASP) 25,26 making use of the pseudopotential construction according to the projector-augmented-wave method. 27 For the Ni pseudopotential, the 16 valence states 3p 6 and 3d 10 were considered, whereas for Zn, the 12 valence states 3d 10 and 4s 2 and for B, the 3 valence states 2s 2 and 2p 1 were taken into account. For the exchange-correlation functional, the generalized gradient approximation according to Perdew et al. 28 was chosen. The VASP calculations were done with high precision according to the energy cutoff of 600 eV. For k space integration, the Gaussian smearing method with a value for σ of 0.1 eV was applied. Regular k space grids of (10 × 10 × 3), (7 × 22 × 8), (4 × 24 × 11), and (7 × 23 × 1) k-points for the τ 3 , τ 4 , τ 5 , and τ 6 phases were constructed according to the parameter KSPACING = 0.1 in the INCAR file of VASP. For all the calculations, all structural parameters such as lattice parameters and atomic positions are fully relaxed by minimizing atomic forces and total energy. The k space integration for deriving the density of states (DOS) was made by applying the linear tetrahedron integration 29 for the k-point grids mentioned above.
The enthalphy of formation at T = 0 K and zero pressure per mol atoms, ΔH f , was derived for each compound from differences of total DFT energies per unit cell, in which U(Ni ℓ Zn m B n ) represents the total energy of the compound and U(X), X = Ni, Zn, B, the total energies of the respective elemental ground state phases. The radii for calculating the local DOS were R Ni = 1.058, R Zn = 1.270, and R B = 0.905 Å. Atomic volumes and charges were computed by analyzing charge densities in terms of the quantum theory of atoms in molecules according to Bader et al. [30][31][32][33] His method utilizes the gradient of the density searching for surfaces of zero flux. The definition of charge transfer Δq ion , as it appears in the paper, is the difference of the self-consistent Bader charge minus the Bader charge of the superposed free atom charge densities. By that, there is no need to fall back to any ad-hoc assumptions based on free atoms or ions. More details are given in ref. 34.

Results and discussion
3.1. Formation, stability and crystal growth of τ 5 and τ 6 phases During the investigation of the ternary Ni-Zn-B system by Malik et al., 8 attempts to grow single crystals of τ 5 and τ 6 were unsuccessful due to the intergrowth of phases and a hair-like morphology inside the grains. Moreover, strong adhesion of eutectic regions to the τ 5 and τ 6 grains made it difficult to isolate proper crystals. Based on these facts in the present investigation we revised the strategy for the single crystal growth and used isothermal annealing followed by quenching. Indeed, samples with a nominal composition of Ni 47 Zn 22 B 31 (in at%) annealed at 900°C (Fig. 1a) showed large crystallites of τ 4 surrounded by a thin layer of τ 6 . Comparing this microstructure with the microstructure presented by Malik et al. 8 at 880°C suggests that τ 6 is formed via a peritectic-like reaction from τ 4 at 890 ± 10°C. Details on the phase compositions of samples prepared for the single crystal growth are given in Table I of the ESI. † Accordingly, a temperature of 875°C was chosen for the single crystal growth based on the existence of the binary Ni-Zn eutectic ℓ ↔ NiZn-ht + Ni 2 Zn 11 at 875°C, 35 which provides the liquid phase for the single crystal growth. Selected micrographs in Fig. 1b-f document the attempts to grow the single crystals of τ 5 and τ 6 . The single crystals obtained in this investigation are grown from equilibrium with the Zn-rich eutectic with a Ni/Zn ratio of 39.

Structure analyses
3.2.1. The crystal structure of τ 5 -Ni 3 Zn 2 B. Selected area electron diffraction (SAED) patterns obtained in TEM from a thin lamella tilted by a series of angles around the common axis yielded a reciprocal space indicating a body centered orthorhombic unit cell (a = 1.60, b = 0.63, and c = 0.27 nm), consistent with our earlier proposition. 8 Preliminary single crystal tests on an AXS D8-GADDS instrument seemed to confirm the body centered orthorhombic unit cell; however, careful analysis of the X-ray diffraction pattern on a four-circle diffractometer resulted in a lower symmetry, i.e. a C-centered monoclinic unit cell with a = 1.68942(8) nm, b = 0.26332(1) nm, c = 0.61904(3) nm, and β = 111.164(2)°. Although the deviation of orthorhombic gamma is only in tenths of a degree (but significant), the main argument stems from the much lower internal R-value after merging reflection intensities in the monoclinic setting. The difference in the TEM and XRSC results can be easily explained with simple geometrical relationships among the unit cells shown in Fig. 2.
Using this relationship, TEM SAED patterns obtained at various sample tilts were re-indexed on the basis of the C-centered monoclinic cell. Fig. 3 shows several examples together with simulated SAED patterns using software JEMS. 36,37 It should be emphasized that all diffraction patterns were completely indexed on the monoclinic cell without any signs for superstructure formation.
The crystal structure of τ 5 was successfully solved with direct methods in the lowest symmetry i.e. in the C-centered monoclinic space group C2, yielding five crystallographic metal positions and one boron position, all in general Wyckoff  The initial refinement in the space group C2 settled quickly at a reasonable R F 2 value of 0.035 with residual electron density of less than ±3.6 e Å −3 . The analysis of a missing symmetry employing program PLATON 39 suggested the addition of a mirror plane perpendicular to the two-fold rotation axis, which consequently results in an inversion symmetry. The structure was subsequently transformed and refined in the centrosymmetric space group C2/m. The final refinement with anisotropic ADPs for metal atoms and isotropic ADPs for boron converged to a low residual factor of R F 2 = 0.0219 and a residual electron density less than ±1.33 e Å −3 . The detailed and standardized crystallographic data of τ 5 are presented in Table 1. Interatomic distances are listed in Table II of the ESI. † Refinement of boron occupancy results in only a slight deviation from full site occupancy. Thus, the structure refinement yields a fully ordered structure with three Ni sites, two Zn sites and one B site, all in position 4i (x,0,y), which corresponds to a Wyckoff sequence of 12,i 6 . Search for isotypic compounds in Pearson's Crystal Data, 40 based on the lattice parameter and the Wyckoff sequence, gave no result: this suggests that τ 5 -Ni 3 Zn 2 B is a new structure type. A Rietveld refinement (see The Ni : Zn ratio obtained from single crystal refinement (60 : 40) deviates only slightly from the EPMA ratio of 62.5 : 37.5. From the Ni-Zn-B isothermal section at 800°C, τ 5 exhibits a small homogeneity range from Ni : Zn ratio 60.6 : 39.4 to 63.8 : 36.2 at a constant boron concentration, which is also preserved up to 875°C. Zinc is known to have poor affinity to boron in many intermetallic compounds, i.e. Zn-B contacts (<0.24 nm) are rarely found in the literature. 40 Therefore, the deviation from the stoichiometric ratio could be explained by a minor nickel substitution on one or both Zn sites. As mentioned above, due to the similarity between nickel and zinc, as well as the small amount of the substitution, crystallographic quantification of the disorder cannot be carried out reliably based on X-ray intensity data. The increasing nickel content, however, results in a practically linear decrease of the unit cell volume, consistent with Vegard's law as shown in Fig. II of the ESI. † Although nickel and zinc atoms have the same coordination number of 13 and 12, respectively (see Fig. 4), the environment of the three nickel atoms, Ni1, Ni2, and Ni3, differ one to the other: a monocapped distorted cuboctahedron, a monocapped elongated pentagonal bipyramid/tricapped pentagonal prism, and a distorted pseudo Frank-Kasper polyhedron. In contrast to the nickel atoms, both Zn atoms have the same coordination number and the coordination polyhedron, which takes the form of a distorted anti-cuboctahedron. Whilst the Zn1 atom is coordinated by 7 nickel atoms and 5 Zn atoms with Ni-Zn distances less than 0.26 nm, the distorted anti-cuboctahedron around the Zn2 atom consists of 8 nickel atoms and 4 Zn atoms with some longer Ni-Zn distances (>0.26 nm). The boron atom in this compound is surrounded by 7 nickel atoms, which form a monocapped triangular (trigonal) prism.
The crystal structure of τ 5 -Ni 3 Zn 2 B is shown in Fig. 5. It can be viewed as an arrangement of two edge sharing monocapped trigonal prisms formed by boron and nickel atoms (B[Ni 7 ]) along the c-axis, separated by corrugated rectangular sheets of Zn. Along the b-axis, infinite columns of face sharing trigonal prisms are formed with a shift of 1 2 b between adjacent columns. Such an arrangement of boron polyhedra in τ 5 is based on the Re 3 B-type, where infinite boron polyhedra are connected along the c-axis, forming layers of edge sharing bicapped trigonal metal prisms along the b-axis separated by empty metal octahedra. On the other hand, stacking the boron polyhedra in the direction of the a-axis results in similar infinite columns of face sharing trigonal prisms with a shift of 1 2 a between the columns. A similar edge sharing arrangement can also be found in Ni 3 B (Fe 3 C-type) along the b-axis. In this case, however, additional edge sharing polyhedra via one of the trigonal prism edges are observed forming a zigzag-like arrangement of the capped triangular prisms, also connecting the two layers of edge sharing trigonal prisms. It is interesting to mention here that in the binary Ni-B system, Ni 3 B crystallizes in the Fe 3 C-type instead of the Re 3 B-type, whereas the layer of empty octahedra can be considered to be related to the CsCl (ht)/CuTi(rt)-type NiZn phase in the binary Ni-Zn system. 40 The structure of τ 5 can be formed by cleaving the infinite edge sharing capped trigonal prisms in the Re 3 B structure by a corrugated rectangular zinc sheet, which also fills the space  between the layers of edge sharing capped trigonal prisms. The insertion of a Zn sheet also results in the formation of some edge and face sharing empty octahedra Ni 2 Zn 4 formed by nickel from the boron polyhedra and the zinc sheet (see Fig. 6). The building block arrangement of τ 5 is analogous to that in the neighboring compounds τ 4 -Ni 3 ZnB 2 and τ 6 -Ni 2 ZnB (see the next section).
The possibility of small interstitial atoms X (X = C, N, and O) residing inside the octahedra is ruled out due to its rather small cavity size, i.e. the shortest distances between the center of the octahedra to the nearest Ni/Zn atoms are less than 0.186 nm (for details see Table III of the ESI †). Such a distance would be within the range of Ni-X contacts in the Ni-X binaries (0.175-0.196 nm), but too short for Zn-X contacts in binary Zn-X compounds (>0.195 nm). 40 Consequently, partial filling of the octahedral void(s) would create split positions of the Zn atoms, like those in V 5 Sb 4 C 1−x . 41 Moreover, the low value of R F 2 and the residual electron density of the single crystal refinement documents that there is practically no electron density at or near the center of the octahedra.   cell obtained from TEM. Indexation of the X-ray intensity pattern on a 4-circle diffractometer, however, indicated a higher symmetry in a C-centered monoclinic cell with a = 0.95296 (7), b = 0.28371 (2), c = 0.59989(1) nm, and β = 93.009 (4)°. The triclinic and monoclinic unit cells of τ 6 can be related geometrically with simple relationships: a monoclinic = 2b triclinic × sin α triclinic , b monoclinic = c triclinic , c monoclinic = a triclinic , β monoclinic = γ triclinic (see Fig. 7). Based on the X-ray single crystal data, TEM SAED patterns could be re-indexed using the proper C-centered monoclinic unit cell, as shown on several examples in Fig. 8 together with simulated patterns. Also in this case, all diffraction patterns were completely indexed on the monoclinic cell without any signs for superstructure formation.
The analysis of the systematic extinctions in the X-ray single crystal diffraction pattern suggested C2/m as the space group with the highest symmetry possible. Structural solution employing direct methods in this space group was straightforward and resulted in three metal and one boron positions, all in site 4i (x,0,y). Refinement of the structure quickly converged to a low R F 2 value of 0.0162 with residual electron density less than ±1.03 e Å −3 . The final formula from the refinement Ni 2 ZnB indicates a fully ordered structure. The Ni : Zn ratio from EPMA (68.5 : 31.5), however, indicates a small substitution of zinc by nickel. Since there is only one Zn site in the structure, a fixed amount (5.5%) of nickel based on EPMA was introduced into the Zn site. Refinement with this configuration insignificantly lowered the R F 2 and the residual electron density to 0.0160 and ±1.01 e Å −3 , respectively. The detailed results of the single crystal refinement are presented in Table 1. The single crystal structure solution was further confirmed by Rietveld refinement of the alloy Ni 42 Zn 44 B 14 from which the single crystals were obtained (see Fig. IV of the ESI †). Table IV of the ESI † lists all the interatomic distances in τ 6 -Ni 2 ZnB, which comply with the sum of atomic radii. The coordination polyhedra obtained from program DIDO95 42 are presented in Fig. 9. The atomic environment around Ni1 and Ni2 is in the form of distorted 14-and 15-vertex Frank-Kasper polyhedra, respectively. In these coordination polyhedra, several long Ni-Ni distances (0.27 < d Ni-Ni < 0.285 nm) are observed. Despite their rather long interatomic distances, these nickel atoms are included in the coordination polyhedra due to a significant area contribution to the Dirichlet domain. In contrast, the B atom that resides inside the coordination sphere of the Zn atom is excluded from the coordination polyhedron due to the same reason described in the case of τ 5 , and thus the coordination polyhedron around the Zn atom takes the form of an anti-cuboctahedron, similar to that in τ 5 . As a consequence of the lower M/B ratio of 3 in τ 6 , direct B-B contacts (d B-B = 0.1850 nm) in the form of boron pairs are observed, 34 with the B-B distance being somewhat longer than the sum of covalent radii. The triangular prism around the boron atoms in τ 6 is capped by boron and nickel atoms. In general, all coordination polyhedra in τ 6 look almost exactly the same as those in τ 4 -Ni 3 ZnB 2 7 if the Zn-B contacts are omitted. The similarity in the unit cell, and the coordination polyhedra in τ 4 -Ni 3 ZnB 2 and τ 6 -Ni 2 ZnB, suggests that these two compounds are closely related. τ 4 can be considered as a √2 superstructure of τ 6 , particularly c(τ 4 ) = √2c(τ 6 ) (see Fig. 10). τ 4 -Ni 3 ZnB 2 is the compound with the higher boron content Fig. 7 Relationship between the triclinic (red) and C-centered monoclinic (black) unit cells of τ 6 -Ni 2 ZnB. Alternatively, the structure of τ 4 can be considered as an arrangement of boron pairs and slightly puckered metal layers of Ni at z ∼ 1 4 , Zn at z ∼ 1 2 , and a nearly planar Ni layer at z ∼ 0 along the c-axis (see Fig. 11). The boron pairs, which form the 4 membered zigzag boron chains, are formed by B1 and B2 atoms, and they are located between two nickel layers. A similar arrangement is found in τ 6 . In this case, however, as the result of its lower boron content (M/B = 3), one AlB 2 layer is missing. Consequently, the 4-membered zigzag B-chain is now replaced by a B-B pair. The missing AlB 2 layer is also reflected in the shorter c-axis of τ 6 compared to that of τ 4 . The same arrangement of five edge sharing distorted octahedra formed by nickel and zinc, which connect the AlB 2 layers, is also found in τ 6 . An alternative representation of the τ 6 structure is presented in Fig. 11. One can see an identical arrangement of metal layers and boron pairs along the c-axis. The planar nickel layer at z ∼ 0, however, is missing, while the slightly puckered nickel and zinc layers are still present in the same sequence. The possibility of filling the octahedral voids in τ 4 and τ 6 by small atoms (carbon, nitrogen, and oxygen) is excluded based on the same arguments described in the case of τ 5 (for details see also In contrast to those in τ 4 , the boron pairs in τ 6 are formed between boron atoms of the same crystallographic site. This is a consequence of the inversion symmetry operating on the B atom, i.e. the center of the boron pairs lies exactly on the inversion center. Due to its larger lattice parameters together with the different choice of the origin of the unit cell, the center of gravity of the boron pairs in τ 4 is further away from the inversion center. From the projection of τ 4 along the b-axis in Fig. 12, one can clearly see the unit cell of τ 6 within τ 4 , which can be considered as the result of the removal of the planar nickel layer at z ∼ 0 and shifting the origin of the unit cell to the center of the boron pairs.
Finally, the comparison of the crystal structures of τ 4 , τ 5 , and τ 6 reveals similar building block arrangements, i.e. boron polyhedra separated by a Zn layer or connected by empty octahedra. The main difference is that the Zn layer in τ 5 is corrugated, which can be considered as the result of shifting the rectangular sheet of Zn in the c direction. Furthermore, due to   Based on the single crystal refinements presented in this paper, it deemed necessary to correct the phase compositions of τ 5 and τ 6 , particularly the boron contents, which were previously estimated from the total weight percent measurement from EDX. By keeping the Ni/Zn ratio from the previous investigation and following the thermodynamic rules for the construction of the phase diagram, the revised isothermal section at 800°C is presented in Fig. 13. The boron contents in the homogeneity regions of τ 5 and τ 6 are assumed to be constant at 16.7 and 25 at%, respectively, which are approximately 3-5 at% lower than the values reported previously. 8

DFT calculations
Discussing charge transfer (see Fig. 14; for details see Tables VI to IX of the ESI †), it is generally observed that B gains an electronic charge, whereas Zn is becoming a positive ion. In the Zn-poor τ 3 -phase, Ni also donates an electronic charge in an appreciable amount to B, and the more B-neighbours Ni has, the more positive it gets ranging up to 0.38q (in units of the proton charge) for Ni2. There are, however, also the Ni4 and Ni5 positions with a rather small loss of the electronic charge. Zn delivers an appreciable amount of electronic charge resulting in Δq ion = 0.38q. As compared to the other three phases, the positive charge of Zn is about 0.12 to 0.14q larger. Concerning the charges of B for the different phases, their variations ranging from −0.41 to −0.34q are substantially smaller than those for Zn. However, when only the Zn rich phases τ 4 , τ 5 , and τ 6 are considered, the positive charge of Zn is rather constant varying from Δq ion = 0.24 to 0.26q. For Ni, there are much larger differences in the charge ranging from even negative values for the τ 5 phase to neutral and a positive charge of 0.34q for Ni3 in the τ 4 phase.
The changes of the Bader volume for Zn and B clearly correspond to the ionic charges, namely the negative ionic charge means a larger atomic volume and vice versa. For Ni, the volume changes fluctuate more significantly. For some Ni positions, one even finds an increased volume for a positive ion, as for example for τ 4 and τ 6 . Fig. 15 shows the density of states (DOS) of the 4 phases τ 3 to τ 6 . In addition, for the newly found τ 5 and τ 6 phases, the sand p-like DOS for B and Zn are presented in Fig. 16 The second predominant DOS feature is a large peak at about −7.5 eV below E F with a width of approximately 1.5 eV for the τ 4 , τ 5 and τ 6 phases. This peak, which is very narrow Fig. 12 Comparison of unit cells of τ 6 -Ni 2 ZnB (red) and τ 4 -Ni 3 ZnB 2 (black).  for the Zn-poor τ 3 -phase, is due to the rather localised d-like states of Zn. It should be noted that the Zn-d states have to be taken into account as valence states for a high-quality DFT calculation in order to derive reliable total energies, relaxed structural parameters and elastic properties. At the lower edge of the Zn peak, Fig. 16 shows B-s states with hybridisation patterns similar to the Zn s-and p-like DOS, which presumably are mediated by Ni states since Zn is not neighbouring B.
Between −6.5 and −5 eV, there are features of a substantially smaller but increasing DOS. In particular, B-p states arise due to hybridization with Ni-d states. Fig. 16 reveals the rising of the B p-like DOS. Above −5 eV, clearly, Ni-d states start to become important and are strongly dominant in the energy  region up to a sharp dropdown at −1 eV. In this energy region, B-p states strongly hybridize with Ni-d states. Noticeably in Fig. 16a for the τ 5 -phase, there occurs a distinct splitting of B-p states with peaks at about −4 and −2.5 eV. The splitting of the B-p states correlates with the splitting of the Ni-d states.
The B-p peak at −2.5 eV coincides with a remarkable peak of Zn-p character. The steep decrease of the Ni d-like and total DOS at −1 eV is followed by small values with a still significant Ni-d character. The admixture of B and Zn states is now rather small. Interestingly, for the τ 5 -phase the Fermi energy is located at a small peak of the DOS which is of predominantly Ni-d character. Because of this peak, the τ 5 -phase might be prone to the formation of defects (vacancies and/or anti-sites) which could shift the Fermi energy up or down, where the DOS is smaller. In fact, the experimental stoichiometry as discussed above finds small deviations from the ideal stoichiometry, for which, however, the DFT calculations were made. The values of the DOS at the Fermi energy and the corresponding specific heat constant γ as presented in Table 2 clearly indicate metallic behaviour for all phases. The values per atom are rather small and increase with increasing Zn concentration.  Table 3 summarizes DFT data on the enthalphy of formation at T = 0 K and zero pressure per mol atoms, ΔH f , which were derived for each compound from differences of the total DFT energies per unit cell (according to eqn (1)). Data are plotted in Fig. 18 as a function of the B-content within the isopleth NiZn to NiB of the Ni-Zn-B phase diagram. From the DFT derived energies of formation, it is found that τ 6 -Ni 2 ZnB (ΔH f = −27.07 kJ mol −1 ) would thermodynamically not be stable at T = 0 by 0.06 kJ mol −1 when compared with the neighbouring phases τ 5 -Ni 3 Zn 2 B (ΔH f = −26.90 kJ mol −1 ) and τ 4 -Ni 3 ZnB 2 (ΔH f = −27.35 kJ mol −1 ). It is possible that deviations of experimental samples from ideal stoichiometry might be responsible for the experimentally observed stability because of mixing entropy. Also vibrational entropies might stabilize the τ 6 -phase. The DFT data document the relative stability of the ternary with respect to the binary compounds. [43][44][45] The present calculation for the Ni-borides (see in our forthcoming paper) reveals slightly lower heat of formations for the end members Ni 3 B and NiB in relation to the thermodynamically derived values from recent CALPHAD assessments. 44,45 The bulk moduli (listed in Table 3) increase with increasing boron concentrations (from B = 179.7 for τ 6 -Ni 2 ZnB to B =   248.9 GPa for the B-rich τ 3 -Ni 21 Zn 2 B 20 phase), in accordance with the increase of the rigid B-B bonds. Fig. 19a illustrates the ( positive) electron density in the τ 5 -Ni 3 Zn 2 B phase in a plane containing Ni, Zn and B atoms orientated perpendicular to the b axis. Strikingly, triangular/tetrahedral patterns corresponding to Ni 3 -B units appear in the density. The B atom placed above the plane causes broad circular features around the Ni atoms in the plane, which widely overlap in between the Ni atoms. B in the plane is surrounded by a triangular shaped density distribution which is caused by bonding features towards the three Ni atoms below the plane of the slice. A small bridge of density is built from Zn to its neighbouring Ni atoms indicating weak bonding. Fig. 19b shows the difference of self-consistent minus superposed free atom electron densities illustrating the gain and loss in density after bonding occurs (i.e. after solving Schrödinger's equation by means of DFT). Clearly, the density gain between the in-plain B and Ni atoms is visualized. Somewhat smaller is the gain in between the in-plane Ni atoms below B out-of-plane. Green rings around Ni and Zn atoms correspond to a density loss. Fig. V of the ESI † shows the same contours as Fig. 19b but with rotated axes. Here, the empty regions (regions of negative difference charge density), from which charge was transferred, are directly visible as they surround Ni and Zn atoms. The two different types of B, namely in-and out-of-plane, are now distinguishable. In Fig. 19c for the density of the τ 6 -Ni 2 ZnB phase, similar features as for τ 5 -Ni 3 Zn 2 B are observed: broad overlapping circles for the Ni atoms situated below the sticking out B atoms. Now Ni-Zn-Ni density bridges appear indicating that Zn prefers Ni neighbours. In Fig. 19d, striking features occur for one atomic plane above the plane as shown in Fig. 19c. Now, the plane contains Ni, Zn and nearest neighbour B atoms. A σ-type bonding pattern between the B atoms with a distance of 0.186 nm is seen which is accompanied by Ni-B bonding features. Again, Ni-Zn bridges appear in the contour plot.

Physical properties
In this study, only τ 5 could be obtained in the single phase and the dense form, therefore only the physical properties of τ 5 will be presented. Attempts to prepare τ 6 resulted in multiphase samples containing τ 4 , τ 5 , and τ 6 , possibly due to the uncontrolled evaporation of Zn during hot pressing that shifted the overall composition to the narrow three-phase field τ 4 -τ 5 -τ 6 .

Electrical resistivity
The electrical resistivity of τ 5 is shown in Fig. 20. The absolute resistivity value at room temperature (∼20 µΩ cm) is rather low, and the residual resistivity ratio of 3.2 evidences a rather good sample quality. As expected from the DOS (see section 3.3.), the electrical resistivity of τ 5 shows a metallic behavior from room temperature to ∼40 K. Below 40 K, a resistivity minimum is observed, followed by a slight increase down to 2 K. The low temperature upturn is well described by a simple logarithmic temperature dependence (see the inset in Fig. 20), indicating the presence of a Kondo type interaction. The nonstoichiometry in τ 5 could be responsible for the existence of a small amount of uncompensated Ni spins, which act as magnetic impurities in an otherwise non-magnetic host. The EDX measurement of this sample shows a slightly higher Ni content, x = 0.03 for Ni 3+x Zn 2−x B. Similarly, an upturn is also observed in the specific heat (see the next section). Above 40 K, the electrical resistivity behavior could be well described by the Bloch-Grüneisen model (ρ BG ), accounting for the electron-phonon scattering. The total electrical resistivity of τ 5 can be expressed as a combination of the aforementioned scattering processes and the residual resistivity (ρ 0 ) arising from static defects and impurities: A least squares fit based on eqn (2) results in a Debye temperature of Θ D = 309 K, practically similar to that of 310 K for τ 4 . 9 The Debye temperature is significantly smaller than the value obtained from the specific heat data, which could be related to the assumptions used in the model.   21 shows the specific heat of τ 5 from room temperature down to 2 K. An anomaly is observed below 5 K. By excluding the upturn, a Sommerfeld coefficient γ = 10.7 mJ mol −1 K −2 and a β coefficient of 0.1732 mJ mol −1 K −4 could be extracted from a least-squares fit based on the equation in Fig. 21. The latter corresponds to a Debye temperature of 407 K, a value lower than that of the related compound τ 4 (Θ D = 510 K), suggesting a lattice softening in this compound due to the absence of B-B contacts. A slightly higher value of the Sommerfeld coefficient in τ 5 compared to that in τ 4 is consistent with the slightly higher DOS at the Fermi level in τ 5 . A comparison of the Sommerfeld coefficient extracted from the DFT calculation of the density of states of τ 5 (3.13 states per eV per f.u.; γ band = 7.38 mJ mol −1 K −2 ; see Table 2) with the experimental value γ exp. = 10.7 mJ mol −1 K −2 gives a hint on the electron-phonon enhancement factor λ = 0.45 (from the relationship γ exp. = (1 + λ)γ band ). A similar enhancement factor of 0.38 is also found in τ 4 (γ exp. = 7 mJ mol −1 K −2 , 9 γ band = 5.07 mJ mol −1 K −2 ). A more detailed description of the lattice contribution to the specific heat is possible in terms of the Debye and Einstein model, i.e.: An analysis based on eqn (3) requires the usage of, at least, three Einstein modes to account for the experimental data. Here six atoms in the formula unit are represented by 18 phonon branches, which consist of 3 acoustic branches accounted for by the Debye model and 15 optical branches distributed over three Einstein modes. Least-squares fits to the experimental data, with an addition of the electronic contribution, result in a Debye contribution with a frequency of 218 K and the spectral weight of 3. The Einstein contribution consists of three frequencies, 149, 275 and 601 K with the corresponding spectral weights of 1, 10 and 4, respectively. In order to compare with the Debye temperature obtained from the low temperature T 3 coefficient β yielding 407 K, the Debye frequency obtained from the fit with eqn (3) must be multiplied by ffiffiffi n 3 p (n is the number of atoms in the formula unit) yielding 396 K, which is in reasonably close agreement.
Comparing the result with that of τ 4 , 9 it is evident that the phonon frequencies in τ 5 are smaller compared to those in τ 4 (ω D = 269 K, ω E1 = 191 K, ω E2 = 348 K and ω E3 = 714 K), possibly as the result of the increasing metal content and consequently, the absence of B-B bonding in τ 5 . Similar to τ 4 , 9 the spectral weight of the high frequency mode corresponds well with the boron stoichiometry. The absence of B-B bonds could also be indicated by softening of the highest frequency Einstein mode compared to that of τ 4 where B-B bonds in the form of 4-membered zigzag B chains are observed.

Thermal expansion
The temperature dependent thermal expansion of τ 5 is shown in Fig. 22. At a first glance, the thermal expansion behaves like ordinary metals, showing only a weak temperature dependence for T < 50 K, followed by an almost linear dependency at a higher temperature. 47 From the linear behaviour at temperatures between 200 to 300 K, a linear thermal expansion coefficient (CTE) of 15.1 × 10 −6 K −1 was obtained, a value that is substantially larger than that of τ 4 (10.4 × 10 −6 K −1 ). This relationship is backed by the heat capacity data, where the difference in Debye and Einstein temperatures hint to substantial differences of the lattice stiffness, when τ 5 is compared with τ 4 .  A careful inspection of the low temperature part (below 70 K) of the thermal expansion data reveals some anomalies (see the inset in Fig. 22), in which the coefficient of thermal expansion α shows negative values between ∼25 and 45 K. Moreover, a kink in the length change (dℓ/ℓ), which is associated with a jump in the CTE at around 10 K, is also observed. The origin of those anomalies is still unknown and further detailed investigations are required to reveal the origin of these anomalies. However, in the absence of distinct magnetic contributions, the negative thermal expansion of τ 5 , from about 20 to 50 K, is presumably associated with transverse vibrational modes ("guitar string vibrations") which cause negative acoustic branches in the Grüneisen tensor (for further details, see ref. 48 and 49). Non-magnetic materials known so far for their negative expansion contain Si, GaAs, CuCl, Rb-halides, ZrMo 3 O 8 or ZrW 3 O 8 (see for instance ref. [48][49][50][51]. An analysis of the temperature dependence of the thermal expansion (dℓ/ℓ), based on the semi-classical model developed by Mukherjee, 52 allows us to extract the Debye and Einstein temperatures of τ 5 . The range of the Debye temperatures was constrained in the fitting process based on the value obtained from the specific heat, while the other parameters were allowed to vary freely. Debye and Einstein temperatures of 410 K and 111 K were obtained from the fit, respectively. The Einstein temperatures obtained from the thermal expansion fit are somewhat lower than those obtained from the specific heat data, suggesting the existence of a low lying optical mode. A similar situation was also encountered in τ 4 , where a low Einstein temperature of 151 K was obtained from the thermal expansion data, which is smaller than those obtained from specific heat data.

Hardness
Due to a well-polished surface, the Vickers hardness indentations could easily be placed inside of the grains of the as-cast samples and 2ℓ could be measured correctly (see Fig. VI of the ESI †). For the grains of τ 5 -Ni 3 Zn 2 B, the static HV0.1 = 432 ± 13, whereas for the grains of τ 6 -Ni 2 ZnB, the static hardness is slightly higher with HV0.1 = 470 ± 14. Hardness measurements were also performed on the hot-pressed samples. For hotpressed τ 5 -Ni 3 Zn 2 B HV0.1 = 468 ± 19 and for the hot-pressed τ 6 -Ni 2 ZnB HV0.1 = 475 ± 19. Both values are slightly larger than for individual grains. Dynamic hardness data are usually smaller than the static ones as they are evaluated directly during the impression, whereas in case of the static hardness, the impression can "shrink" after the load is withdrawn. The surface of the hot-pressed samples had very small pores, but big enough that the microindenter MHT4 was not applicable. For the grains of τ 5 , the dynamic hardness HV0.1 = 288, of τ 6 HV = 325.
Using a Poisson number ν = 0.29(1), as obtained from the RUS measurement of the closely related compound τ 4 -Ni 3 ZnB 2 , 9 elastic moduli can be evaluated. In case of τ 5 , E = 163, B = 136, and G = 63; for τ 6 , E = 99, B = 60, and G = 40 (all values given in GPa). Considering the softer nature of both τ 5 and τ 6 , these values compare well with those of τ 4 -Ni 3 ZnB 2 where H V = 500, E = 199 GPa, B = 158 GPa and G = 77 GPa. 9 The data for the bulk moduli show good agreement with the DFT calculation (see Table 3).
From the measured elastic moduli, average sound velocity, and thus the Debye temperatures, were calculated by Anderson's equation. 53 For τ 5 , a Debye temperature of 418 K was obtained, which agrees well with the value obtained from other measurements. For τ 6 , a smaller Debye temperature of 342 K was derived.

Conclusions
The crystal structures of the hitherto two unknown phases in the Ni-Zn-B system, τ 5 -Ni 3 Zn 2 B and τ 6 -Ni 2 ZnB, have been elucidated from a combination of X-ray single crystal study, selected area electron diffraction, and electron probe microanalysis. Both compounds crystallize in new structure types and have a similar building block construction: the crystal structures are characterized by the arrangement of Zn layers/empty octahedra, separating the boron centered polyhedra. The B-B interactions agree well with the M/B ratio described in the literature, where isolated boron atoms and B-B pairs are found in τ 5 -Ni 3 Zn 2 B and τ 6 -Ni 2 ZnB, respectively. The arrangement of boron polyhedra in τ 5 -Ni 3 Zn 2 B is related to that in Re 3 B, whereas AlB 2 fragments are found in the τ 6 -Ni 2 ZnB structure. The crystal structures of the neighboring compounds are closely related: τ 4 -Ni 3 ZnB 2 can be considered as a √2c superstructure of Ni 2 ZnB and can be derived by inserting Ni layers and shifting the origin of the unit cell. The boron concentrations of Ni 3 Zn 2 B and Ni 2 ZnB obtained from the single crystal studies allow us to revise the Ni-Zn-B isothermal section at 800°C, particularly the phase composition of τ 5 -Ni 3 Zn 2 B and τ 6 -Ni 2 ZnB. τ 5 -Ni 3 Zn 2 B shows metallic-type electrical resistivity over a wide temperature range with a Kondo type behavior at low temperature, possibly due to some uncompensated Ni-spins distributed in a non-magnetic-metallic host. Specific heat and hardness measurements of τ 5 -Ni 3 Zn 2 B confirm the softer nature of this compound due to the high metal content and the absence of B-B bonding as indicated by the lower hardness, and Debye and Einstein temperatures compared to those of τ 4 -Ni 3 ZnB 2 .
DFT calculations resulted in a very good agreement with the experimental structure data. The ground state total energies and forces were minimized for deriving the ab initio structural parameters. Hitherto unknown energies of formation were evaluated revealing a small instability of the τ 6 -Ni 2 ZnB phase. Bulk moduli were determined from volume dependent total energies showing a distinct increase for B-rich compounds reaching 248.9 GPa for the τ 3 -Ni 21 Zn 2 B 20 compound. Charge transfer was defined in terms of differences of the socalled Bader charge purely based on the densities of the compounds. As an effect of bonding, B accumulates charge whereas Zn delivers charge to the system. Ni reacts in a more complicated way by varying its charge transfer depending on the local arrangement of B atoms. The analysis of the underlying electronic structure reveals Ni-B bonding characteristic for trigonal geometrical arrangements and short B-B distances. Zn prefers Ni as neighbours as expressed by the bonding features of the corresponding density.

Conflicts of interest
There are no conflicts to declare.