We selected irregularly shaped single crystals of Hf13.0Ni40.8Ga30.9 and Zr13.0Ni40.6Ga31.0 from their annealed samples to collect crystal X-ray diffraction data for further structural refinement. For each crystals, the analysis revealed a hexagonal unit cell and the Laue group 6/mmm with a primitive lattice. Systematic absences favored a centrosymmetric structure, yielding possible space groups P
3, P3m1, and P6/mmm. We chose the space group P6/mmm to be consistent with the one adopted by known compounds. Using direct methods, we built a structural model with 20 crystallographic sites. We assigned four positions to Hf atoms because of their large electron densities; we assigned the other positions to Ga or Ni atoms and distinguished them with the aid of site occupancy. Most sites exhibited full occupancy, but the Ga4 site only revealed an occupancy of 0.948(8). We suspect that the deficiency in electron density was influenced by nearby residuals around position (0, 0, 0), rather than by a mixed occupancy of Ga and Ni atoms. It was supported by the elongated
displacement parameters of the Ga4 site paralleling direction to residuals. Subsequent refinements reveled three residual maxima with small inter-site distances and unreasonable thermal parameters.
For improved analysis, these electron densities were refined as Ni9, Ni10 with fixed Ueq (0.01 Å2), and Ni11 with anisotropic thermal displacement. The residuals decreased significantly from 14.17 to 3.56 e/Å3 with the disordered model and yielded partial occupancies for the Ni9, Ni10 and Ni11 atoms of 0.12(1), 0.116(9), and 0.180(8), respectively. Although small electron densities remained near Ni11, further refinements with the remaining residuals were unstable and failed. All metal positions, except Ni9 and Ni10, were eventually refined anisotropically. The final R factors R1, wR2, and GOF were 0.0299, 0.0598, and 1.138, respectively, with the formula Hf13.0Ni40.8Ga30.9. The Zr analogue was processed in a similar manner to generate the formula Zr13.0Ni40.6Ga31.0 with parameters R1, wR2, and GOF of 0.0348, 0.0686, and 1.088, respectively.
6.3 Crystal Structure
The structures of Hf13.0Ni40.8Ga30.9 and Zr13.0Ni40.6Ga31.0 can be understood as comprising three fragments—CaCu5 type [159], MnCu2Al type [160] and Fe2P type [161]—viewed along the c direction. The CaCu5 fragments (Fig. 6.1) are located at the corners of the unit cell and form a tunnel-like structure with a six-fold rotation axis. The MnCu2Al fragments, which possess an inversion center, sit at the centers of the edges and the cell. The Fe2P fragments with a six-fold inversion axis are surrounded by the former two fragments, placed in the positions where x and y
are equal to ±1/4.
Fig. 6.2 and 6.3 present the detailed coordinate environments of each atomic site. The distances from the central atom are truncated at 3 Å to present reasonable bonds based on the metallic radii (rHf = 1.50 Å; rGa = 1.26 Å; rNi = 1.21 Å) [147]. The interatomic distances around Hf range from 2.72 Å to 2.97 Å (Table 2.12). Despite some Hf–(Ni, Ga) contacts being longer than the sum of their corresponding metallic radii, these distances are comparable with those found in Hf2Ga3 (dHf–
Ga = 2.96 Å) [162], Hf5Ga3 (dHf–Ga = 2.84 Å) [163] and HfNi2 (dHf–Ni = 2.86 Å) [164]. The Ga–Ga, Ni–Ni and Ni–Ga distances are truncated at a range from 2.36 Å to 2.81 Å, considered effective distances for bonding interactions. Such bond lengths are found in HfGa3 (dGa–Ga = 2.74 Å) [165], Hf2Ni7 (dNi–Ni = 2.50–2.80 Å) [166] and Ni13Ga9 (dNi–Ga = 2.35–2.67 Å) [167]. The Ni–Ga bonds were significantly shorter than the Ga–Ga and Ni–Ni contacts and were more abundant in the structure.
Fig. 6.1 Structure of Hf13.0Ni40.8Ga30.9 in a projection along the c-axis, demonstrating the forms of the CaCu5, MnCu2Al, and Fe2P phases. The red, yellow, and blue spheres represent Hf, Ni and Ga atoms, respectively.
Fig. 6.2a displays Hf-based polyhedra with a pentacapped pentagonal prism (Hf2, CN15), a pentacapped trigonal prism (Hf3, CN11) and a tricapped pentagonal prism (Hf4, CN13). The coordination environment of the atom Hf1 is a hexagonal antiprism with a shifted center distorting the Hf–Ni (2.85 Å) and Hf–Ga (3.05 Å) bond distances. Fig. 6.2b presents the stacking hexagons constructed by the Ni and Ga atoms with disordered sites randomly distributed within the framework around position (0, 0, 0). Because of the short distance of 2.44 Å in the vertical direction between the neighboring Hf atoms and the origin, these electron residuals were not assigned to a Hf atom but rather a Ni atom, considering where they reside in the same plane with the Ga hexagon.
The distance of 2.94 Å from the center hexagon to the Ga atom is significantly longer than the average Ni–Ga bond (2.47 Å), suggesting that the disordered model can be attributed to spatial considerations.
Fig. 6.2 (a) Hf-based polyhedra with distance 2.97 Å from the center. (b) Environment of disorder located around position (0, 0, 0). The colors of the spheres conform to those in Fig. 6.1.
Fig. 6.3 displays the coordination environments of Ni and Ga atoms containing 6–12 neighbors.
Some of the polyhedra adopt a trigonal prism as the main framework (Ga4 and Ni2), further capped with three atoms (Ga1, Ga6, and Ni1) or a pair of atoms (Ga2). Other notable structural units include distorted icosahedra (Ga7, Ga8, Ni4, and Ni5), cuboids (Ga3) and a pentagon capped with two pairs of atoms (Ni7). The coordination environments for the Ga5, Ni3, Ni6, and Ni8 sites are randomly constructed with neighboring atoms; they are difficult to classify.
Fig. 6.3 (a) Ga- and (b) Ni-based polyhedra with truncation distances of 2.81 and 2.73 Å, respectively. The colors of the spheres conform to those in Fig. 6.1.
The structure of Hf13.0Ni40.8Ga30.9 features three different layers along the c direction [158]; the arrangement of atoms on each layer is extended as a result of a CaCu5 structure and correlates with those of similar compounds (Fig. 6.4), namely HoNi3.4Ga1.6 and HoNi2.6Ga2.4 [168]. The structure of HoNi3.4Ga1.6 features two layers: layer 1 (L1) contains one Ho atom (Wyckoff site 1a) with two Ni and Ga mixed positions evenly separated along the diagonal (site 2c); layer 2 (L2) has Ni and Ga
mixed sites (site 3g) that form infinite triangular nets. For HoNi2.6Ga2.4, the atomic sites on layers L1 and L2 correspond to a two-fold scale of the CaCu5 structure with two sites removed from the diagonal of L1 and six sites on L2 replaced with two large atoms.
Hf13.0Ni40.8Ga30.9 is a four-fold CaCu5-related compound with a third extra layer 3 (L3). In L1, only two gallium atoms are expelled from the lattice, resulting in a puckered sheet because of the compactness of the positions within a confined cell. The substitutions on L2 occur on positions around the three-fold axis, generating open sites that allow Ga atoms to reside within. In addition, L3 is formed by rotating 180° on the blue triangular fragments of L2. As a result, both L2 and L3 are sandwiched by layer L1, forming a 3-D structure with an L1-L2-L1-L3-L1 stacking sequence.
Although we sought to discover compounds corresponding to a three-fold scale of CaCu5 composed of Ni and Ga, our many trials provided no specific results. Nevertheless, two reported compounds—LaGaBi2 [169] and La13Ga8Sb21 [170]—account for the three- and four-fold CaCu5
-related samples, for which more eliminations and substitutions are performed in both L1 and L2.
Whereas the atomic size is a very significant factor affecting the structure of intermetallic compounds, the series of isostructural phases comprising elements of different groups indicate that the valence electron concentration (vec) is another effective factor. The number of electrons per atom of Hf13.0Ni40.8Ga30.9 is 6.53 (552.7/84.7); this value is within the range 6.62–6.31 obtained from the known compounds–Y13Pd40Sn31 [138], Li13Ni40Si31, Sc12.7Ni40.7Ge31 [156], Nb(Ta)Co4Si3
[157] and Na26Cd144 [158]. The relationship between the number of valence electrons and the
structure has been investigated in RE2–xFe4Si14–y species and related intermetallic systems [171], where the framework and sequence of stacking layers differ as vec varies.
F
6.4 Calculations of electronic structure
Because the observed crystal structure is complicated, we used theoretical calculations to determine the electronic structure and bonding properties. Fig. 6.5 presents the calculated plots of the densities of states (DOS) and crystal-orbital Hamilton population (COHP). We performed these calculations using “Hf13Ni40.5Ga31” as a theoretical model, with Ni(10) and Ni(11) sites were removed and all positions designed to be fully occupied. The DOS curve does not features a gap near the Fermi level (Ef), indicative of a metallic properties. The pronounced Ni 3d orbitals are occupied from –4 eV to the Fermi level, and are mixed with contributions from Hf and Ga atoms between –4 and –2 eV. A local minimum state appeared approximately 3 eV above the Fermi level in the plot of the DOS, indicated a characteristic feature of a polar intermetallic phase, as seen in a partial DOS (Fig. 6.6) that electron transfer is revealed in the occupied Ni 4s and 3p orbitals.
Nevertheless, a large part of the Hf 4d orbitals lies below the Fermi level, indicating that the Hf atom is partial cationic and intimately involved in chemical bonding.
Fig. 6.5 Calculated densities of states (DOS) and crystal-orbital Hamiltonian-population (COHP) curves for the model compound Hf13Ni40.5Ga31. The lower horizontal line denotes the Fermi energy;
the upper line denotes the optimal level.
The COHP curves and bonding contributions from the Hf–Ni, Hf–Ga, Ni–Ga, and Ga–Ga pairs.
The integrated COHP value reveals the strong interaction of the Ni–Ga bond (1.36), more than 25%
greater than those of the Hf–Ni (1.03), Hf–Ga (1.09), and Ga–Ga (1.00) bonds. The Ni–Ni contact is relatively weak because of the anti-bonding contribution near the Fermi level. We suggest that these bonds are optimized at 3 eV above the Fermi level, revealing the local minimum state in the DOS curve. In the region between the Fermi level and the pseudogap, the interactions of the Ni–Ni, Ga–Ga, and Ni–Ga contacts appear to be non-bonding, whereas the Hf–Ni and Hf–Ga interactions are relatively weak. This observation suggests a potential phase width, through substitution of Ga by Ni, from Hf13Ni40.5Ga31 to Hf13Ni50.5Ga16, which derives from the corresponding electrons of a pseudogap. The homogenous range is, however, experimentally unattainable in the series of reactions Hf Ni Ga , because the major phase turned to HfNi Ga when x was greater than 12.
Fig. 6.6 Calculated partial densities of states (PDOS) for individual orbitals of Hf, Ni and Ga.
Chapter 7
Conclusion
In Hf–Cu–Al system, we have synthesized and characterized ternary Laves phases HfAlxCu2–x
(x = 0.2–1.0). Analysis of three crystals—HfAl1.51Cu0.49, HfAl1.12Cu0.88, and HfAl0.96Cu1.04— confirmed that the variation of the three Laves phases occurred in sequence MgCu2 → MgNi2 → MgZn2. As smaller Cu atoms gradually replaced the Al atoms, the size reduction was observed in refined volumes and cell parameters. Theoretical calculations with the coloring models suggest that the model with high number of Al–Cu contacts is more stable due to strong bonding interaction compared to Al-Al and Cu–Cu contacts.
Furthermore, We have synthesized and characterized a ternary compound Hf6Cu16Al7.58, which is isostructural with Sc11Ir4 and comprises polyhedra with coordination numbers 8–13 for Hf, Cu and Al atoms. Calculations of the electronic structure reveal a strong Cu–Al interaction that provides the greatest contribution to the structural bonding. The Al2-based octahedron exhibits a short bond from the central Al atom to the vertex Hf atom; bonding analysis indicates strong interatomic interactions from Hf–Al contacts.
In Hf–Al–Sb system, We have successfully obtained three new ternary hafnium aluminium antimonides Hf5Al3–xSbx (x = 0.70, 1.44, 2.14). The structure of HfAl2.30Sb0.70 was isostructural with
binary compound Hf5Al3 of Mn5Si3 type, which was composed of trigonal antiprism of two types stacking along the c axis, whereas HfAl1.56Sb1.44 and HfAl0.86Sb2.14 adopted W5Si3 structure with condensations of tetrahedra and square antiprisms. Resistivity measurements revealed effective electrical conductivity. A bonding analysis revealed strong Hf-Sb interactions in models of both Mn5Si3 and W5Si3 types. The large proportion of Sb also favored anti-bonding characters (Hf-Sb for the Mn5Si3 type and Sb-Sb for the W5Si3 type), which rationalizes the variation of structure and the site preference of metal atoms.
Lastly, in Hf–Ni–Ga system, the structure of two compounds Hf13.0Ni40.8Ga30.9 and Zr13.0Ni40.6Ga31.0 are characterized, which comprise polyhedra with 12–15 connected neighbors for Hf and Zr atoms and 6–12 coordination numbers for Ni and Ga atoms; these structures can be regarded as extended forms of the CaCu5 structure with reduction in numbers of atoms and valence electron concentration. According to calculations of the band structure, these phases are metallic and partially polar, with electron transfer revealed in the Ni 4s and 3p orbitals, and strengthened by the strong contributions of Ni–Ga bonds. In addition, the title structures has been synthesized with elements ranging from groups 1 to 5 (i.e., Li, Na, Y, Sc, Zr, Hf, Nb, Ta), but no isostructural compounds comprising elements from group 2. Accordingly, the possible permutations of Mg and Ca elements, via the information of valence electron concentration, are predicted; the corresponding experiments are in progress.
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Appendix
In our study, ternary crystallization diagram is utilized as a tool to organizes and illustrates stoichiometric ratios for all exploratory synthesis. Each vertex of diagram indicates a full component element (A, B or C). The line between two vertexes is concentration line of one binary system (AmBn); similarly, the lines inside diagram between two points on edges is concentration line of one ternary system [Am(B1–δCδ)n, or (A1–δCδ)mCn]. Blue dots on the ternary phases diagrams indicate the ratios of elements used for exploratory synthesis of new intermetallic compound. Red dots indicate known binary or ternary compounds.
(a) Hf–Al–Ge system (b) Hf–Al–Sb system
(c) Hf–Cu–Al system (d) Hf–Cu–Ga system
(e) Hf–Cu–In system (f) Hf–Cu–Sb system
(g) Hf–Cu–Si system (h) Hf–Cu–Sn system
(i) Hf–Ni–Ga system (j) Hf–Ag–Al system
(k) Hf–Ag–Ga system (l) Hf–Ni–Al system