New building blocks in the 2/1 crystalline approximant of a Bergman-type icosahedral quasicrystal

Abstract

The refined x-ray crystal structure of the phase Mg₂₇Al_10.7(2)Zn_47.3(2) (Pa3) establishes it as the new 2/1 Bergman-type approximant of the icosahedral quasicrystal. The primitive cubic lattice consists of condensed triacontahedral and novel prolate rhombohedral (PR) clusters. Each triacontahedron encapsulates the traditional, multiply endohedral Bergman-type clusters, and each PR encapsulates an Al₂ dimer. This phase exhibits the same long-range order as recently established for the Tsai-type Sc–Mg–Zn 2/1 approximant crystal, with substantial geometric and atomic distribution differences between the two only in the short range orders. This common feature suggests that Bergman- and Tsai-type quasicrystals may be more similar than earlier conceived. Factors germane to the formation of, and the differences between, Bergman- vs. Tsai-type 1/1 and 2/1 approximate structures are considered, including notably different distributions of the more electropositive elements.

Icosahedral quasiperiodic crystals (i-QCs) are intermetallic compounds that exhibit 5-fold rotational symmetries incompatible with classic crystallography (1–3). Since the discovery of a Al–Mn icosahedral quasicrystal in 1984 (3), three families of i-QCs have been described: Mackay-type (4), Bergman-type (5), and Tsai-type (6–8). These are distinguished by their short-range orders (SROs) within the so-called 1/1 cubic approximant crystals (ACs) that have similar chemical compositions and, presumably, similar building blocks to those in the corresponding i-QCs (9). The Mackay-type SRO, quite different from our considerations here, consists of 54-atom multiply endohedral shell of, from the center out, an icosahedron, a larger icosahedron, and a 30-atom icosidodecahedron (4). In contrast, Bergman ACs are usually defined in terms of the 104-atom multiply endohedral clusters first found in the Mg₃₂(Zn,Al)₄₉ phase by Bergman et al. (5), namely, the successive icosahedron, dodecahedron, a larger icosahedron, and a buckyball-like M₆₀ polyhedron. Additional still larger polyhedra occasionally included in the Bergman description are noted below. Tsai-type clusters refer to the successive shells in MCd₆ ACs (M = Ca, Yb) (10, 11), namely a disordered tetrahedron, a dodecahedron, an icosahedron, and a 30-atom icosidodecahedron, for a total of 66 atoms. Some variations of these are assumed to exist in the MCd_5.67 icosahedral quasicrystals reported by Tsai et al. (6).

According to higher dimensional projection methods, a series of cubic ACs exist with orders (q/p) denoted by any two consecutive Fibonacci numbers, i.e., q/p = 1/1, 2/1, 3/2, 5/3 … F_n+1/F_n (1). The ratio converges to an irrational number τ = 1.618, the golden mean. The cell parameters of the various ACs are related by a_q/p = 2a₆(p + qτ)/$\sqrt{(2+\tau )}$ to a₆, the six-dimensional quasi lattice constant calculated from the QC powder patterns by Elser's method (12). Accordingly, the higher the order of an AC, the closer its structure is thought to approach that of the QC, meaning that the i-QC can be considered as the cubic AC with an infinite lattice constant.

Because of the aperiodicity of i-QCs, their structural analyses have had to use six-dimensional approaches, which treat the aperiodicity in 3D as traditional periodicity in 6D. However, these analyses are still far from complete despite the fact that some refinements have converged to very small R1 values (see ref. 13 and references therein). In addition, there are no well established routines to describe the structures of i-QCs. Thus, most i-QC models are derived from the structures of their 1/1 ACs, which only provide a minimal amount of useful information about the long range modeling of i-QCs. Therefore, knowledge of higher-order AC structures is very important to guide the quest for modeling of i-QCs.

Recently, periodic x-ray crystal structures of Tsai-type 2/1 ACs have become available for the Ca–Cd (14), Sc–Mg–Zn (15, 16), and Ca(Yb)–Au–In (unpublished results) systems. The particular Sc–Mg–Zn result evidently represents the highest degree of structural perfection reported to date for a 2/1 array, exhibiting only three partially occupied atom positions (16). This structure also makes clear that (i) the basic building blocks in this structure (and in the 1/1 AC) are 32-vertex rhombic triacontahedra (with an average edge length close to the quasi lattice constant) rather than the encapsulated Tsai-type clusters, and (ii) the newly determined primitive cubic condensation of triacontahedral clusters also generates interstitial cavities that define eight-vertex prolate rhombohedra (PRs) (cubes elongated along the diagonal) (16). Furthermore, examination of the cluster properties and linkages allow some progress toward a possible real-space iceberg-type model for the Tsai i-QCs (unpublished results).

We note that the Bergman- and the Tsai-type i-QCs appear to have several comparable macroscopic properties. For instance, both have (i) very similar x-ray powder diffraction patterns in terms of peak positions and intensities; (ii) similar valence electron counts per atom (e/a ≈ 2.0–2.2), and (iii) similar 6D quasi lattice constants (a₆ ≈ 5.0 Å). These features encouraged us to examine whether the Bergman-type 2/1 AC might also have a long-range order (LRO) similar to that in the Tsai-type 2/1 AC. The answer is of substantial importance in modeling the i-QCs because Bergman- and Tsai-type examples have been generally assumed to be basically different in structure.

We have therefore focused on the well known Mg–Al–Zn system because both the 1/1 and 2/1 AC phases have been studied in this system. The 1/1 AC, commonly written as Mg₃₂(Al,Zn)₄₉, has been extensively reexamined by both Lee and Miller (17) and Sun et al. (18), updating the results of Bergman et al. (5). Furthermore, an additional Mg₂₀ dodecahedron with a larger radial distance of 7.72 Å together with an Mg₁₂ icosahedron at 8.34 Å that are sometimes included in the description of Mg₃₂(Al,Zn)₄₉ (Fig. 1a in ref. 18) also generate an additional triacontahedron. Thus, it appears that the Bergman 1/1 AC structure can evidently also be described in terms of body-centered-cubic packing of triacontahedral clusters that share rhombic faces and oblate rhombohedra (ORs). Similar constructions have also been described in the Bergman-type Li₃CuAl₅ 1/1 AC (19) and, with simple cubic packing, in our Tsai-type Sc–Mg–Zn 2/1 AC (16).

Fig. 1.

Polyhedral view of the unit cell of the Mg₂₇Al₁₁Zn₄₇ 2/1 AC (Pa3) with condensed triacontahedral clusters (dark green) and prolate rhombohedra (PRs, gray). Note the 3-fold axis along one diagonal of the cell. Mg atoms are denoted by red solid spheres, and Zn or Zn/Al are denoted by green. The same legend applies to Figs. 2–4.

To the best of our knowledge, the present 2/1 AC phase was first observed by Takeuchi et al. (20). Their DSC study of the nominal Mg₁₀Al₃Zn₇ i-QC (Mg₅₀Al₁₅Zn₃₅ normalized) revealed an endothermic transition from the i-QC to a 2/1 AC on heating above 400°C. Takeuchi and Mizutani (21) later found that the Mg₄₃Al₁₅Zn₄₂ i-QC was also transformed into single phase 2/1 AC by annealing at 360°C for 1 h. (The foregoing compositional differences presumably result from an appreciable homogeneity range in the same i-QC system.) However, no structural data were reported from either study. The first single crystal study of a 2/1 AC phase by Spiekermann and Kreiner^† in 1997 indicated that the Mg₄₂Al₁₄Zn₄₄ 2/1 AC (refined composition) exhibited a high degree of occupational disorder, with evident Zn/Al admixtures on all 19 crystallographic sites of this type. They also concluded from their neutron powder diffraction analyses that there was no hint of Al/Mg mixing, but no further details or data on the phase were provided. A much more detailed synthetic and structural description for a Mg_47.9Al_14.2Zn_37.9 (refined composition) 2/1 AC was provided by Sugiyama et al. (22). However, the structure was described only in terms of four-shell Bergman-type clusters, not of the larger triacontahedra. Moreover, no atom positional parameters were reported. Therefore, we have reproduced the 2/1 AC phase and determined its single crystal structure, from which we find old wine in a new bottle!

In contrast to the literature, we will show that the title Mg–Al–Zn 2/1 AC exhibits primitive cubic packing of triacontahedral clusters (rather than of just Bergman clusters) as well as interstitial prolate rhombohedra (PRs), units that have never been reported in the 2/1 system. These findings shed light on the structural modeling of the Bergman-type i-QCs. Moreover, we discover that the Bergman- and Tsai-type 2/1 ACs exhibit exactly the same LRO of triacontahedra and differ only in their SRO and the distribution of atom types. Some apparent factors in the formation and nature of Bergman- vs. Tsai-type AC structures are also considered.

Results and Discussion

The title compound, Mg₂₇Al_10.7(2)Zn_47.3(2), crystallizes in space group Pa3 with a = 23.0349 (7) Å. The best way to describe a cubic structure with such a large unit cell parameter is to discuss the SRO and LRO of the building blocks separately.

LRO of Building Blocks.

We find that the LRO of the crystal structure is primitive cubic packing of condensed triacontahedral clusters and PRs rather than of the smaller Bergman-type clusters that define the interior SRO of the former. This packing of triacontahedra automatically generates the PRs. Fig. 1 shows a polyhedral view of the unit cell, which contains eight condensed or interpenetrating triacontahedra (dark green) centered at the special Wyckoff positions 8c (0.346, 0.346, 0.346) together with four (= 1/8 × 8 + 1/2 × 6) PRs (gray) at the 4a (0, 0, 0) positions, i.e., at the cell origin and face centers. Significantly, both general LRO features are geometrically the same as that found recently for the Tsai-type 2/1 AC, Sc_11.18(9)Mg_2.5(1)Zn_73.6(2) (16).

Fig. 2 details the information-rich environment of a triacontahedron, including the linkages among triacontahedra and PRs. As can be seen, each triacontahedron is bounded by 13 (= 4 × 3 + 1) like neighbors and four PRs. Of the 13, six (black) members share rhombic faces through which pass the center-to-center connections, whereas lines to the other seven (blue) pass through the 3-fold vertices. The latter pairs instead share ORs (see Fig. 5, which is published as supporting information on the PNAS web site). Of the four PRs in Fig. 2, that on the proper 3-fold axis shares a vertex (Mg9) with the adjacent triacontahedron, whereas the other three (on the closest face centers of the unit cell) share rhombic faces.

Fig. 2.

Environment of triacontahedral clusters in the Mg₂₇Al₁₁Zn₄₇ 2/1 AC, showing the linkages among triacontahedra and PRs. For clarity, only the central triacontrahedron on the proper 3-fold axis of the unit cell is shown, with the others represented by spheres at their centers. Of the 13 like neighbors, six (black) share rhombic faces through which pass the center-to-center lines, whereas the other seven (blue) lie on lines that pass through 3-fold vertices and share ORs (see Fig. 5).

SRO Within a Triacontahedral Cluster.

For convenience, the SRO description is primarily in terms of clusters built of successive concentric shells, each with characteristic atomic populations and approximate geometries. A bonding description is not attempted; rather, lines between some atom pairs are to aid the visualization of geometries more than to denote bonds, especially within the shells, whereas the distances cited are at best only semiquantitative measures of bond populations.

Each triacontahedron consists of five concentric polyhedral shells, as shown in Fig. 3. The innermost shell is an empty Zn₁₂ icosahedron (green) generated from three Zn1, three Zn3, three Zn5, and three Zn9 atoms, with their radial distances falling within 2.49–2.60 Å. The second shell consists of only Mg atoms (one Mg1, three Mg2, three Mg4, three Mg6, three Mg7, three Mg8, one Mg10 and three Mg11) in a Mg₂₀ pentagonal dodecahedron (red polyhedron in Fig. 3a) with radial distances in the range of 4.50–4.64 Å. Each Mg has three surface bonds and also bonds to atoms that define triangular face of the dual icosahedra in both the first and third shells (these intercluster bonds are not marked). The latter, Zn₉Al₃, contains three Zn/Al13, three Zn15, three Zn/Al16, and three Zn/Al19 atoms in a larger icosahedral shell with radial distances over 5.00–5.20 Å. Atoms in this shell are presumably not directly bonded to each other, but they are bonded radially to a Zn atom in the first icosahedral shell and also to the atoms defining the pentagonal faces on both the second (dodecahedral) and the fourth shells (not shown). Alternatively, the second and third shells may be combined to generate a mixed triacontahedron (the innermost cluster shown in Fig. 3b), as has also been described for Li₃CuAl₅ (19), with an average edge length of ≈3.16 Å. An illogical consequence of this, however, is that the atoms in these two shells have the opposite polarities, as discussed later. The fourth shell, a distorted Zn_46.1Al_12.5 buckyball-like sphere, is generated from six Zn/Al2, six Zn4, six Zn6, six Zn7, three Zn/Al8, six Zn10, three Zn11a (54% occupied), six Zn12, six Zn/Al14, six Zn/Al17, and six Zn/Al18, with radial distances over 6.74–7.29 Å. Each atom in this shell presumably forms three surface bonds, plus three types of intershell bonds (not shown) to atoms in both the inner second and third shells and in the outer triacontahedron (the fifth shell).

Fig. 3.

Hierarchy of five multiply endohedral shells of a triacontahedral cluster in the Mg₂₇Al₁₁Zn₄₇ 2/1 AC. These are, radially, a Zn₁₂ icosahedron (green), an Mg₂₀ dodecahedron (red), and an Al₃Zn₉ icosahedron (green, not interconnected) (a), and the subsequent Zn_46.1Al_12.5 buckyball (green) and an Mg₃₂ triacontahedron (red, dashed connections) (b). Note that the Mg₂₀ dodecahedron and Zn₉Al₃ icosahedron shown in a can be combined to generate a small mixed triacontahedron, as shown by the innermost polyhedron in b. For clarity, most intershell bonds are not drawn. Numbers mark the atoms as listed in Table 3, which is published as supporting information on the PNAS web site.

The first four shells define a typical Bergman cluster, as explained in the Introduction. However, if the Bergman clusters are considered as the basic building blocks, as in the literature (22), many other so-called “glue atoms” that do not belong to the Bergman cluster will be left. A much more meaningful assignment of these is, as before (16, 19, 23), as parts of an augmented fifth triacontahedral shell that has radial distances in the range of 7.61–8.37 Å. This shell consists of 32 skeletal Mg atoms (12 Mg3, nine Mg5, three Mg7, three Mg8, four Mg9, and one Mg10), together with six 46%-occupied Zn11b sites that lie near the Mg9–Mg5 edges (not shown in Fig. 3b for clarity). Each skeletal Mg atom in the shared fifth shell caps a face of the fourth shell and is radially positioned above an atom in the inner mixed triacontahedral shell as well (Fig. 3b). Significantly, the average edge length of outmost triacontahedral shell, 5.14 Å, is ≈τ (≈5.14/3.16) times that of in the inner triacontahedral shell, suggesting an inflation–deflation property. This value (5.14 Å) is also very close to the experimental quasi lattice constant of the Mg–Al–Zn i-QC (5.15 Å) (24), suggesting that triacontahedral clusters might also be very closely related to the quasi unit cell of corresponding i-QC; this appears to be another advantage for identifying triacontahedral clusters as the fundamental building blocks.

Note that the highest symmetry for the five-shell cluster unit ending with the triacontahedron is .3., with the unique 3-fold rotational axis through Mg10–Mg10 (Fig. 3b). In comparison, the outer triacontahedron in the 1/1 AC Mg₃₂(Al,Zn)₄₉ has a higher m3. symmetry (18). However, triacontahedra in both in fact exhibit additional pseudo 2-, 3-, and notably, 5-fold axes, in association with their dodecahedral and icosahedral parents.

Compared with the incomplete report of a refined 2/1 AC structure for a quite different composition, Mg_47.9Al_14.2Zn_37.9 (22), the contents of the present first and fourth shells differ in Zn/Al and Zn/Mg proportions, respectively. In the present structure, the innermost shell is evidently made up of only Zn atoms, and the fourth shell is free of Mg. These differences probably occur because the Mg–Al–Zn 2/1 AC is not a line compound, but a solid solution. Such a situation is apparent for the 1/1 AC, Mg₃₂(Al,Zn)₄₉, for which a much wider range of Zn (≈14.1–51.5%) has been reported (17, 18).

SRO Within PR.

Fig. 4 shows the structure of the new PR. Each PR is directly bound to eight triacontahedral clusters, six that surround the proper 3-fold axis and share Mg9–Mg5–Mg5–Mg5 rhombic faces, and two on the extended long diagonal that share Mg9 vertices. Each PR is centered by Al–Al dimer [d(Al–Al) = 2.30 Å], which lies on the 3-fold axis. The new dimer atoms would be “glue” atoms (not belonging to any building block), were Bergman-clusters to be considered as the only building blocks; this is perhaps the reason why the PRs or their contents have never been noted before. The same applies to the quite analogous Sc–Sc dimer in the Sc–Mg–Zn 2/1 structure (16).

Fig. 4.

Atomic arrangement of the interstitial PR (red) in the Mg₂₇Al₁₁Zn₄₇ 2/1 AC, which is centered by an Al–Al dimer (yellow) on its 3-fold axis. Fractional Zn11b atoms (green) lie near the Mg5–Mg9 edges. Each PR is bounded by eight triacontahedral clusters, six that share rhombic faces and two on the 3-fold axis that share the Mg9 vertex atoms (see text).

Unlike the Tsai-type 2/1 AC in which each edge of the PR is centered by a Zn atom, only the six Mg9–Mg5 edges are off-centered by fractional [46(3)%] Zn11b sites. The latter correlate with fractional Zn11a atoms that are members of neighboring buckyball shells. These are the only positionally disordered atoms in the 2/1 structure, amounting to only ≈3.5% of the total. Although Zn11a and Zn11b generate hexagonal rings perpendicular to the proper 3-fold axis (Fig. 6, which is published as supporting information on the PNAS web site), their half occupancies and small separations (1.59 Å) suggest that they instead define 3-fold disordered atoms. Configurational disorder is also found in the structure of the Tsai-type Sc–Mg–Zn 2/1 AC (16), but, interestingly, that 3-fold disorder occurs for the innermost, symmetry-breaking tetrahedron, not in the PR.

What is the importance of the present 2/1 AC? Because a 2/1 AC is structurally closer to i-QC than 1/1 AC, the novel PRs found in this study appear to have an important role in structural modeling of Mg–Al–Zn i-QC, and the previous Bergman-type i-QC model proposed on the basis of the 1/1 AC Li₃CuAl₅ (23) should be revised. Actually, the occurrence of PRs and ORs in the 2/1 AC reminds us that its structure can be alternatively described by a packing of 3D distorted Penrose tiles (25), because each triacontahedron can also be decomposed to 10 PRs and 10 ORs. Of course, the atoms on these tiles must distort somewhat from ideal positions in the corresponding i-QC. Nevertheless, a Penrose tiling model of the Mg–Al–Zn i-QC requires only some shifts of the atoms in the 2/1 AC. However, a major problem for this description is that PRs or ORs are not unique in their atomic decorations. We are attempting to develop a better, straightforward route to assemble an i-QC model on the basis of this structure and the recent 2/1 AC structure in a Tsai-type system (16); however, this is obviously beyond the scope of this paper.

Chemistry and Structural Stabilities.

As demonstrated above, the Bergman Mg–Al–Zn and the Tsai Sc–Mg–Zn 2/1 ACs have the same LRO arrangements and symmetries at the unit cell level and above in terms of the triacontahedral clusters and PRs, with differences only in SROs. Therefore, they appear to be the different products from a same template, which is defined in the Pa3 group by fixing the centers of the triacontahedra at the Wyckoff 8c (0.346, 0.346, 0.346) and of the PRs at the 4a (0, 0, 0) positions. This finding raises the question: what factors determine the distribution between Bergman- and the Tsai-type SRO? In fact, enumeration of the SRO contents reveals that there are many more disparities than there are analogies.

The shell contents in the four 1/1 and 2/1 Tsai and Bergman ACs are compared in Table 1. Their quite similar valence electron counts per atom (e/a) values (2.09–2.13) are achieved in rather different ways. Although Mg is evidently a small but essential ingredient in the Tsai members, the Mg contents of the Bergman ACs are markedly higher (by 28 atom % or more). Mg is also the most basic component in the latter in a chemical sense. At the same time, the more acidic Zn is ≈30% less, but this is compensated by admixture of ≈11% of the still more acidic Al. These acid–base categories are parallels of Lewis ideas widely used in the comparative chemistry of, say, aqueous and solid oxo chemistry of the metals, but generalized here to intermetallics. Thus, acidic Zn in the Tsai phases or the like Al plus Zn in the Bergman prototypes can be considered the better oxidizing agents, the more electronegative atoms, and the better anionic network formers, whereas the Sc or Mg components, respectively, are better respective reducing agents and therefore act more like cations. In other words, there are clear charge transfer and polarity differences and preferences in these phases that are useful for further categorization and consideration.

Table 1.

Comparison of shell contents within triacontahedra and PRs in the 1/1 and 2/1 ACs in the Tsai-type Sc–Mg–Zn and in the Bergman-type Mg–Al–Zn systems

Shell, polyhedron	Tsai		Polyhedron	Bergman
	Sc14.4Mg0.9Zn84.8(1/1)	Sc12.7Mg3.5Zn83.8(2/1)		Mg39.3Al9.2Zn51.5(1/1)*	Mg31.8Al12.6Zn55.6(2/1)
1 Tetrahedron	Zn4†	Zn2.9†	Icosahedron	Zn12	Zn12
2 Dodecahedron	Zn20	Zn19.4	Dodecahedron	Mg20	Mg20
3 Icosahedron	Sc12	Sc9.6Mg2.4	Icosahedron	Zn9.2Al2.8	Zn9Al3
4 Icosidodecahedron	Zn29.2Mg0.8	Zn29.3Mg0.7	Buckyball	Mg12Zn36.9Al11.1	Zn46.1Al12.5
5 Triacontahedron	Zn32(Zn60‡)	Zn32(Zn60‡)	Triacontahedron	Mg32	Mg32 (Zn1.4‡)
PR content		Sc2			Al2
Reference	16	16		18	This work

Dominant cation sites are shown in bold.

*The result for the composition closest to the 2/1 AC.

^†Disordered tetrahedron in 1/1 and fractional tetrahedron in 2/1 AC.

^‡Additional decorations at or near the center of the edges.

Despite the very close geometric correspondences at the unit cell level within the 1/1 (bcc) and 2/1 (primitive cubic) version of each AC family, the distributions of shell contents (SROs) within each are strikingly different (Table 1). The most basic Sc is uniquely localized in only the third icosahedral shell in both Tsai ACs, whereas among the three constituents in the Bergman types, the most basic Mg constitutes all of second dodecahedral shell and is the predominant component in the fifth triacontahedral shells. Hence, the LROs described at the cell level and above involve condensation of triacontahedra built of the more cationic Mg shells in Bergman phases, but evidently of only the more acidic Zn in the Tsai types. [This condensation in detail involves some atoms in the penultimate (fourth) shells as well.] The Bergman shells are in places also atom-richer and larger, namely, a full icosahedron in shell 1 (vs. a disordered tetrahedra) and a 60-atom buckeyball in shell 4 (compared with a 30-atom icosidodecahedron in the Tsai version).

All of these features must arise from dominant electronic (bonding) factors. As we know, a stable phase is one that minimizes the total energy. In polar intermetallics, it is useful to describe relative stabilities in terms of both local site and bond energies (26). In a given structure, the site energy terms are related to the atomic decorations of the structure (i.e., the atom coloring or site preferences), which can be understood to include Madelung energies and to serve to maximize efficient filling of space. In the particular 1/1 Bergman Mg–Al–Zn phase, Lee and Miller have shown that the site energy terms are clearly maximized by placing Mg cations only in the inner dodecahedral and outmost triacontahedral shells (i.e., on the sites with the lowest electron populations as judged by the relative Mulliken populations) (17, 26). This is clearly an important factor in the compound's stoichiometry, making the overall Mg ≈32% in this case. On the contrary, judging from a more limited number of Tsai-type clusters, a lower proportion of higher charged or more basic cations (≈14%) preferably occupy only the third icosahedral shell. We have also theoretically verified these relative distribution via LMTO-ASA calculations of relative charge densities on all-Zn 1/1 AC models of both Bergman and Tsai types (unpublished results), which in fact simply support what is already manifest in the structures. (Mulliken electronegativities do not give same differentiations, particularly for Zn vs. Al.)

Empirically, the bond energies are more germane regarding the overall atomic network structure for a given valence electron count (vec) (26) as well as the overall stoichiometry, although at present this is at best a qualitative statement. In these cases, the Hume-Rothery rules (27) presumably play an important role in structure and composition control as seen in e/a. According to many experimental results, an e/a of ≈2.1–2.25 is more favorable for Bergman-types (21, 28), whereas e/a ≈2.00–2.15 seems more appropriate for Tsai-types (7, 29, 30).

At present, perhaps one of the simpler indicators or essentials in a Bergman- vs. a Tsai-type differentiation appears to be whether there are d orbitals on the cations to contribute to the formation of pseudogap (31). At least in 1/1 Tsai-type structures, these mix with the s and p orbitals of the anions around the Fermi energy to greatly enhance the depth of the pseudogap, as seen in the Sc–Mg–Zn (15), Sc–Mg–Cu–Ga (32), and Yb–Cd systems (33). In contrast, Bergman-type phases appear to occur only as more free-electron-like systems, without noticeable contributions from d orbitals (26, 31). Of course, there are many more factors to this differentiation than simple d-orbital features. Other great differences between the two AC families, even the structures themselves, remain unaddressed (Table 1).

Conclusions

A single crystal structure determination reveals that Mg₂₇Al_10.7(2)Zn_47.3(2), a Bergman-type 2/1 AC, crystallizes in space group Pa3. The basic building blocks at the unit cell level are recognized as the triacontahedral clusters and PRs, rather than just the Bergman-type clusters within. Importantly, the LRO motif in this structure is primitive cubic packing of condensed triacontahedral clusters centered at Wyckoff 8c and of PRs at 4a positions, remarkably the same template that applies to Tsai-type 2/1 ACs. In this packing, the first coordination sphere of a triacontahedron includes 13 like neighbors and four PRs. These results shed light on the structural modeling of both Bergman- and Tsai-type i-QCs. On the other hand, very substantial differences in the general SRO constructions and atom distributions within Bergman- vs. Tsai-type ACs are also noted and discussed.

Materials and Methods

Syntheses.

To obtain the title 2/1 AC phase, mixtures of as-received Al wire (99.9%, Alfa), Mg turnings, and Zn shot (both >99.99%, Alfa) with compositions Al₁₅Mg_xZn_85-x (x = 42, 43, 44) were loaded in tantalum tubing and in turn sealed within an evacuated SiO₂ jacket. Samples were fused at 670°C for 3 days, quenched in water, and annealed at 360°C for 15 days. Phase analyses revealed that x = 43 sample had an x-ray pattern (Fig. 7, which is published as supporting information on the PNAS web site) very similar to that reported in literature (21), and this was well fit subsequently by the pattern calculated according to single crystal data (below). The results suggest that the phase width is relatively small with respect to Mg content, but possible Zn:Al variations were not explored.

Powder X-Ray Diffraction.

Phase analyses by x-ray powder diffraction was performed on a Huber 670 Guinier powder camera equipped with an area detector and Cu Ka₁ radiation (λ = 1.540598 Å). The fine powders were homogeneously dispersed on a flat mylar film with the aid of petrolatum grease. The step length was set at 0.005°, and the exposure time was 0.5 h.

SEM-EDX Analyses.

Several pieces of the crushed x = 43 sample were visually selected for semiquantitative elemental composition analysis with the aid of energy-dispersive x-ray spectroscopy (EDX) as described (15). A sample SEM image (Fig. 8, which is published as supporting information on the PNAS web site) showed that the product contained two phases, and the EDX data suggested that the major (>90%) phase was the target 2/1 AC. (The impurity was an unknown Mg-richer phase.)

X-Ray Single Crystal Diffraction.

Selected single crystals were mounted on a Bruker APEX Platform CCD diffractometer equipped with graphite-monochromatized Mo Ka radiation. The intensity data were obtained over one hemisphere with an acquisition time of 60 s per frame. Data integration, Lorentz polarization, and absorption corrections were done by the SAINT and SADABS subprograms in the SMART software package (34). Structure determination and refinements (on F²) were performed with the SHELXTL subprogram. The assignment of space group Pa3 (no. 205) was made on the basis of the Laue symmetry and the systematic absences analysis. The cell parameter was refined from ≈999 reflections with I > 20σ(I). Three data sets were collected from different crystals, but all gave virtually the same results, i.e., with all positional parameters within 3σ.

Thirty-one atoms were located by direct methods. Environmental analyses revealed 19 of them had separations (≈2.49–2.75 Å) suitable for Zn–Zn bonds, 11 had distances (≈2.86–3.02 Å) for Mg–Zn separations, and one with ≈2.31 Å to itself, for Al–Al. After a few cycles of refinements on F², the R1 value converged at ≈16.1%. All peak heights in the difference Fourier map were below 6.5 e/ų and within 1.0 Å of neighboring atoms except one on a 24d position (0.042, 0.079, 0.1376), with a peak height of ≈19 e/ų and a distance of 1.57 Å to Zn11, which in turn had a large isotropic displacement parameter (0.085 Ų). This position and Zn11 generated a hexagonal ring perpendicular to the 3-fold axis of a prolate rhombohedron (Fig. 6). Therefore, these were considered as a pair of split atoms (Zn11a and Zn11b), and the sum of their occupancies was constrained to 100% in subsequent isotropic refinements, which smoothly converged at R1 ≈ 12%. These resulted in reasonable occupancies [54.0(3)% and 46.0(3)%] and normal isotropic displacement parameters (0.023 Ų). Rechecking the other isotropic displacement parameters revealed that eight Zn atoms had somewhat large values (>0.03 Ų) compared with the average for other atoms (≈0.018 Ų), suggesting that each might be mixed with Al or Mg. However, the possibility of significant Zn/Mg admixture was easily excluded according to our EDX results because, at this stage, the sum of the Wyckoff multiplicities of unambiguously assigned Mg atoms was 31.8% (= 216/680) of that of all atoms, in comparison with the EDX value (32 (1) %). Moreover, no hints of any Mg/Al mixing were found earlier in neutron diffraction analyses on both 1/1 and 2/1 AC phases in this system (17, †). Therefore, a Zn/Al admixture was refined for each with the occupancy constrained to unity, which decreased R1 remarkably to 8.2%. The final anisotropic refinements converged at R1 = 5.96%, wR2 = 14.78% for 4,987 independent [I > 2σ(I)] reflections and 268 parameters. The extreme peaks in the difference Fourier map were 2.64 and −2.36 e/ų, respectively. The refined composition is Mg₂₇Al_10.7(2)Zn_47.3(2) (or normalized, Mg_31.8Mg_12.6(2)Zn_55.6(2)), in very good agreement with the EDX data, Mg_32.1(11)Al_12.1(6)Zn_55.8(16).

Some crystallographic and structural refinement details are summarized in Table 2. All other parameters are given in Tables 3–5, which are published as supporting information on the PNAS web site. It should be noted that these crystals exhibited the characteristically (13) large number of relatively weak reflections, as indicated by R_int and R1 values for all data (Table 2). This finding is in agreement with the calculated intensities, there being only 227 strong reflections (F²/F²_max > 0.01) of 4,987 total.

Table 2.

Some crystallographic collection and refinement data for Mg₂₇Al₁₁Zn₄₇

Refined composition	Mg27Al10.7(2)Zn47.3(2)
Formula weight	4,037.20
Space group, Z	Pa3, 8
Lattice parameter, a (Å)	23.0349 (7)
Volume (Å3), dcalc. (g/cm3)	12,222.5(6), 4.388
Absorption coefficient (mm−1, Mo Ka)	18.596
Reflections collected/Rint	61,366/0.1133
Obs. indep. refl. (>2σI)/restraints/param.	4,987/0/268
Final R indices [I > 2σ(I)] (all data)	R1 = 0.0596, wR2 = 0.1478
	R1 = 0.1600, wR2 = 0.1874
Residual peak/hole (e/Å3)	2.64/−2.36

Abbreviations

i-QC

icosahedral quasiperiodic crystal

SRO

short-range order

LRO

long-range order

AC

approximant crystal

OR

oblate rhombohedron

PR

prolate rhombohedron.