An Icosahedral Al–Cu–Fe Alloy - ACS Publications - American

Jun 21, 2017 - High-dimensional (HD) crystallography is a standard method to locate atom positions explicitly. However, in practice, it is still chall...
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Three-Dimensional Modeling of Quasicrystal Structures from X‑ray Diffraction: An Icosahedral Al−Cu−Fe Alloy Seung-Tae Hong* Department of Energy Systems Engineering, DGIST (Daegu Gyeongbuk Institute of Science & Technology), Daegu 42988, South Korea S Supporting Information *

ABSTRACT: Quasicrystals (QCs) are well-ordered but aperiodic crystals with classically forbidden symmetries (such as 5-fold). High-dimensional (HD) crystallography is a standard method to locate atom positions explicitly. However, in practice, it is still challenging because of its complexity. Here, we report a new simple approach to threedimensional (3D) atomic modeling derived from X-ray diffraction data, and apply it to the icosahedral QC Al0.63Cu0.25Fe0.12. Electron density maps were calculated directly from 3D diffraction data indexed with noninteger (fractional) numbers as measured, with proper phases; each of 25 = 32 possible phase assignments for the five strongest reflections was used for Fourier synthesis. This resulted in an initial phasing model based on chemically sensible electron density maps. The following procedure was exactly the same as that used to determine ordinary crystal structures, except that fractional indices were assigned to the reciprocal vectors relative to the three orthogonal 2-fold axes in icosahedral (Ih) symmetry to which the observed diffraction data conformed. Finally, ∼30 000 atoms were located within a sphere of a ∼48 Å radius. Structural motifs or basic repeating units with a hierarchical nature can be found. Isolated icosahedral clusters are surrounded by a concentric dodecahedron, beyond which there is a concentric truncated icosahedron. These are strikingly similar to those obtained via HD crystallography, but show very clear real-space relationships between the clusters.

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INTRODUCTION

Nevertheless, the complexity of HD crystallography in practice is apparent because full structure analyses with quantitative data such as atomic coordinates, which would allow the reproducibility of a structure model, are barely reported despite the large number (>100) of discovered QC phases.4,18 The first accurate structural determination of an icosahedral phase was given for the YbCd5.7 QC,22 seven years after its discovery.23 Because of its complexity, the information about the crystalline approximant structures played a key role in deducing the building clusters and their linkages of QCs. The approximant is a normal crystalline material with a similar or identical composition as the corresponding QC, which is wellcharacterized in most cases. The cluster building blocks and linkages are presumed to be similar between a QC and its approximant. In fact, the clusters in the crystalline approximant YbCd624 are also observed in the YbCd5.7 QC,22 as in other QC cases.19 Here, we report the crystal structure of the icosahedral QC Al0.63Cu0.25Fe0.12, and the discovery of a new approach to 3D atomic modeling derived directly from X-ray diffraction data of the QC. The motivation stemmed from a curiosity to see electron density maps (Fourier synthesis maps) calculated directly from the aperiodic diffraction data indexed with noninteger (fractional) numbers as observed in reciprocal

1

Quasicrystals (QCs) are long-range ordered but translationally aperiodic atomic structures featuring rotational diffraction symmetries of 5-, 8-, 10-, or 12-fold that are forbidden by classical crystallographic theory. Since their discovery,2 an enormous amount of effort has gone into attempts to answer the key question of where the atoms are.3 There are basically two different methods of describing a QC structure: the tiling approach and the high-dimensional (HD) approach.4 The former uses at least two different nonoverlapping unit cells,5 such as the Penrose tiling.6 Later, this was developed into the concept of a single aperiodic prototile called a “quasi-unit cell” that is decorated with atoms or clusters, similar to the unit cell for a periodic crystal; however, it is allowed to partially overlap with neighboring cells to cover the full QC structure.7−9 The HD approach, on the other hand, was originally developed to analyze diffraction data of incommensurate crystals,10 and it did not take long to recognize that a QC structure can also be obtained from a periodic structure in a higher-dimensional space by taking a three-dimensional (3D) intersection.3,11−14 For a detailed description of QC structures, i.e., to locate atom positions explicitly, the HD methods have been considered more appropriate,15 and significant progress has been made in determining the structures of diverse QCs over the past three decades.4,16−19 Several models have been described comprehensively for two-dimensional15 and icosahedral18−21 QCs. © 2017 American Chemical Society

Received: January 27, 2017 Published: June 21, 2017 7354

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space. The comparison between this work and the HD method is schematically made in Figure 1.

For an ordinary crystal, 3D periodic electron density (ρxyz) (Fourier synthesis) at a point (x, y, z) is represented by the Fourier transform of the structure factors:

hkl

(1)

where (x, y, z), V, (h, k, l), and Fhkl are dimensionless fractional coordinates, the volume of the unit cell, integer indices, and the structure factor, respectively. It can be shown that eq 1 is equivalent to the following equation (Figure S1 in the Supporting Information and the descriptions therein): ρx ′ y ′ z ′ = (1/V )

∑ h′k′l′

Fh ′ k ′ l ′e 2π i(h ′ x ′+ k ′ y ′+ l ′ z ′)

EXPERIMENTAL SECTION

Synthesis of the QC Al0.63Cu0.25Fe0.12. The icosahedral quasicrystal composition Al0.63Cu0.25Fe0.12 first reported by Tsai et al.27,28 was prepared from the corresponding elements by arc-melting followed by annealing in a sealed Nb tube at 830 °C for 17 days, and then quenching to room temperature. X-ray Diffraction Data Collection. Diffraction data from a ∼0.1 × 0.1 × 0.1 mm3 crystal, picked from the smashed product, were collected at room temperature with the aid of a Siemens (Bruker) SMART Platform diffractometer, a CCD area detector, and Mo Κα radiation (λ = 0.710 73 Å) over the reciprocal space sphere with 2θ < 55°, following a quality check with X-ray Laue and precession photos as shown in Figure S2. A total of ∼1400 reflections were measured with I > 0.002Imax. These gave a very clear view of the reciprocal space which can be completely described at this level by 31 concentric polyhedra consistent with the icosahedral point group Ih (Figure S3 and Table S1), exhibiting 10 3-fold, 15 2-fold, and 6 5-fold rotation axes. Crystal Structure Determination Process. The presented method followed the same procedure as that used to determine ordinary crystal structures, except that fractional indices were assigned to the reciprocal vectors relative to the three orthogonal 2-fold axes in icosahedral (Ih) symmetry to which the observed diffraction data conformed (Figure S3 and Table S1). To obtain an initial phasing model, an assumption was made that the QC is centrosymmetric, allowing the reflection phases to be either +1 or −1. Five stronger members of the 31 sets of independent reflections, with inverse dspacings (and multiplicities) of 0.2663 (20), 0.2925 (12), 0.4738 (12), 0.4983 (30), and 0.6874 (60) Å−1, were selected for the Fourier synthesis. In the first trial, electron density maps for volumes of 10 × 10 × 10 Å3 were calculated for each of the 25 = 32 possible phase assignments, for which careful examination showed that only one sensible model, an icosahedron of atoms in the center, was produced considering the peak shapes, interatomic distances, and strengths of the phasing. The tentative phases for the other reflection sets were then obtained via calculations of structure factors, and in turn, an electron density map was calculated by Fourier synthesis again, giving only the sharper peaks, and so on. The entire process was quite straightforward, resulting in no discernible ambiguities in the model. The electron density maps such as that in Figure S4 or electron densities for any region can be calculated using eq 2 from only the information given in Table S1 such as the fractional indices, intensities, and phases of the reflections. More than 30 000 atoms were eventually located within a sphere of a ∼46 Å radius, each position determined to about 0.1 Å (Table S3). Applying the Ih symmetry operation to each of the atom positions given in Table S3 generates the equivalent atoms in real space, which is needed to reproduce the atomic positions in the models (Figures 2−4 and Figure S5), or for calculation of intensities (Figure 5 and Figure S6). The atom types were assigned according to the calculated electron densities. The highest was either Cu or Fe, which was not distinguishable at this stage, while the smaller ones were taken to be Al. The overall stoichiometry was adjusted to the nominal composition.

Figure 1. Schematic comparison between the method in this work and the HD crystallographic method to determine an icosahedral QC.

ρxyz = (1/V ) ∑ Fhkl e 2π i(hx + ky + lz)

Article

(2)

where (x′, y′, z′) is a position in the Cartesian coordinate system (in Å), and (h′, k′, l′) is a fractional index (in Å−1) for a reflection in reciprocal space. Equation 2 does not require a definition of a unit cell, and a trivial unit volume (1 Å3) can be used since V is a constant, and can be embedded in the structure factor by adjusting the scale factor of intensity data in practice. Equivalence of the two equations implies that a structural solution (electron density maps) based on fractional indices (in Å−1 units) without defining a unit cell is exactly the same as the solution from a conventional method based on integer indices with a defined unit cell. In this work, eq 2 was experimentally applied to the 3D QC data because defining a unit cell was no longer required. Nonetheless, applying eq 2 to 3D QC data is not theoretically justified because the presumption behind the Fourier synthesis is periodicity in 3D, but periodicity does not exist in the 3D QC structure. To our knowledge, this experimental trial has never been reported before, even though a similar trial was reported with 3D Patterson maps, but was unsuccessful in explicitly providing any atomic positions.25 Interestingly, the reverse procedure (diffraction pattern calculation from an aperiodic model) has been a common practice since Mackay’s optical diffraction experiment with Penrose tiling, which was the first demonstration.26 The icosahedral Al−Cu−Fe phase was chosen for this experiment because of its stability27,28 and reported structural model. This model was first suggested in 199129 and was analyzed and refined further during the next 15 years, all using the HD crystallographic method.30−35

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RESULTS AND DISCUSSION The absence of the 3D periodicity (or unit cells) of icosahedral QCs does not allow integer indices of the diffraction data (reciprocal vectors) in 3D, while each of them can be indexed with six integers in six-dimensions (6D). The out-of-the-box idea of this work is to use the observed values (in Å−1) of the reciprocal vectors in the 3D Cartesian coordinate system, that is, to not be restricted to the idea of integer indexing, maintaining the QC (icosahedral Ih) symmetry without assuming any unit cell, and applying eq 2, with all the others following the conventional crystallographic process. A detailed comparison between this method and the conventional one is made in Table S2. Consequently, the indices consist of three fractional numbers in general. In principle, fractional indexing 7355

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structure. In principle, the models, i.e., (x′, y′, z′) values, can be chosen in any other region regardless of the inclusion of the origin, in any shape such as a box or a slab, or in any size in space without limit, as long as eq 2 is applied using (h′, k′, l′) values from Table S1. The derived 3D structural model of Al0.63Cu0.25Fe0.12 has global icosahedral symmetry, and at the smallest scale appears to consist of isolated icosahedral clusters. The spherical model of a ∼45 Å radius shown in Figure 2 has only these icosahedra identified for clarity. These are of two types: with the center site either filled (red) or empty (blue), and with center-to-vertex distances all between 2.6 and 2.9 Å. Every icosahedron is in turn surrounded by a concentric dodecahedron with a centerto-vertex distance of 3.9−4.2 Å, and these share edges or vertex atoms with neighboring dodecahedra. The central region shown in Figure 3a with a radius of ∼14 Å is a dodecahedron Figure 2. Basic spherical model of Al0.63Cu0.25Fe0.12 for a diameter of ∼90 Å as viewed along a 5-fold axis. Only icosahedral components are shown for clarity, with their centers either filled (red) or empty (blue).

can be applied to any crystalline material, e.g., the (001) vector of a cubic crystal with a cell parameter of 4 Å should be indexed as (0 0 0.25) Å−1. In this case, the information on the unit cell dimensions is not necessarily required. The resulting electron density maps following the standard crystallographic process should display a periodic building block, which is called the unit cell (mathematically proven in Figure S1 and the description therein). However, fractional indexing is not used, usually because integer indexing with predetermined unit cell dimensions is a common and much more convenient practice. In this regard, just applying eq 2 to XRD data is nothing new at all if it is applied to an ordinary crystal. The key point in this work is that eq 2 was applied to 3D QC data for the first time. The fractional indices were assigned to the reciprocal vectors relative to the three orthogonal 2-fold axes in the icosahedral symmetry. The observed diffraction data (Table S1) conformed to the icosahedral symmetry. The only assumption was that the QC is centrosymmetric, allowing the reflection phases to be either +1 or −1. This assumption may be justified by the experimental observation that even a single crystal of Al−Cu− Fe QC grows to 200 μm to form a nearly perfect dodecahedron that exhibits centrosymmetric icosahedral symmetry.27,28,36 The details of the process are described in the Experimental Section and Table S2. More than 30 000 atoms were eventually located within a sphere of a ∼48 Å radius, with each position determined to about 0.1 Å (Table S3). This is a rather rough result compared with that of ordinary crystallography because the positions were manually determined within the limitations of the map resolution, and the parameters were not refined. The structural motifs are strikingly similar to those obtained via HD crystallography,32,33,35 clearly showing real-space relationships between these. The structure is consistent with the observed X-ray diffraction pattern and the measured stoichiometry, density, and symmetry. The presented model is concentric with a limited volume containing the origin. However, it should be noted that the volume of the model can be chosen arbitrarily: It is a matter of calculation and analysis time. A spherical model was chosen simply because a spherical one describes Ih symmetry appropriately. A sphere of a ∼48 Å radius, though it could be larger, was chosen because it was just manageable to locate the ∼30 000 atoms under the circumstances, and to sufficiently provide a unique feature of the QC

Figure 3. Spherical atomic model for Al0.63Cu0.25Fe0.12 for a diameter of ∼29 Å: (a) dodecahedron of dodecahedral clusters around the central one in red is shown, where each dodecahedron surrounds an unseen icosahedron; (b) all atoms outside icosahedral components are shown. Bonds are drawn between neighboring atoms, showing the skeleton of each dodecahedron surrounding an icosahedron, and other intercluster atoms.

of 20 dodecahedra (each centered by an icosahedron) that surround a central dodecahedron. To better understand the structure, all atoms outside the icosahedral components are shown as spheres in Figure 3b, where bonds are drawn between neighboring atoms to clearly exhibit the skeleton of each dodecahedron surrounding an icosahedron as well as some of the intercluster atoms. The dodecahedra seem to be the largest basic (nonpenetrating) motifs, and the edge-sharing feature between them is clear. Beyond this, larger and different polyhedra constructed from dodecahedral clusters are found: icosahedra (12), icosidodecahedra (30), truncated icosahedra (60), etc. (Figure S5). The dodecahedral clusters in Figure 3a do not fill the 3D space completely; however, additional atoms (“glue”) found between these completely fill interdodecahedral space. These are not shown in Figure 2 or Figure 3a because of the complexity, but some of them are shown as intercluster atoms in Figure 3b or as vertices in Figure 4c−e, below. These intercluster atoms also participate in the formation of outer polyhedral shells such as truncated icosahedra, larger dodecahedra, and even larger icosahedra, forming a hierarchy of shells. Despite the complexity of the model, structural motifs or basic repeating units can be found, the hierarchical natures of which are shown in Figure 4. Beyond the smallest central icosahedron (Figure 4a) and dodecahedron (Figure 4b) there is a concentric truncated icosahedron (Figure 4c) that consists of some extra interdodecahedral atoms shared by neighboring 7356

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should be emphasized that this model was built from only 31 independent reflections, which is very small in number compared with those (up to 1000 independent reflections) obtained from state-of-the-art instruments capable of collecting very weak reflections.18 An improved model can be obtained with higher-quality data providing a higher resolution to locate atoms. The X-ray diffraction patterns viewed along the (a) 2-, (b) 3-, and (c) 5-fold axes calculated for the spherical model of >30 000 atoms were compared with observed X-ray data in Figure 5. The overall Debye−Waller factor used was 1.0 Å2.

Figure 4. Hierarchy of concentric structural motifs for Al0.63Cu0.25Fe0.12: (a) icosahedron; (b) dodecahedron; (c) part b with truncated icosahedron (60); (d) larger dodecahedron, now without direct bonding between vertices; (e) still larger icosahedron.

clusters of the same or other types. Not all of the proper vertices of Figure 4c are occupied by atoms: on average, only 40−50 out of 60 atoms, according to Ih symmetry. The next outer shells are dodecahedra (Figure 4d) and icosahedra (Figure 4e) with center-to-vertex distances of 6.4−6.8 and 7.1− 7.5 Å, respectively, with the two combined (Figure 4d,e) forming a triacontahedron. Most atoms on these two shells are vertices of neighboring polyhedral shells of all types but the truncated Figure 4c. The structural motifs Figure 4a,b and Figure 4a−e are surprisingly similar to those derived through 6D crystallography, described as a Bergman- and an extendedBergman-type cluster, respectively.32,33,35 With respect to the latter cluster, the slight difference is that there are two truncated icosahedra with atom occupancies of 22−24 each in the reported models, in contrast to only one with an atom occupancy of 40−50 in the present model. Interestingly, the total number of atoms of the cluster is similar: 112 versus 104− 115.32,33 One of the simplest periodic approximants of this QC is cubic α-(Al, Si)CuFe-phase with a = 12.3 Å and space group Pm3,̅ 37 where the structural motifs are icosahedron, bigger icosahedron, and icosidodecahedron with their centers either filled or empty. The structural motifs found in the approximant are different from, but still closely related to, the structural motifs presented in Figure 4. The atomic density of the model (27 065 atoms within a radius of 46 Å) is 0.0664 atom/Å3, or 4.24 g/cm3, if the nominal composition Al0.63Cu0.25Fe0.12 is assumed. That atomic density is in the range of those of the constituent metals or their compounds: Al (0.0602), Cu (0.0847), Fe (0.0820), Al7Cu2Fe (0.0670), and Al9Fe2 (0.0632 atom/Å3).38 A density smaller than the experimental 4.57(5) g/cm3 for Al0.63Cu0.255Fe0.11531 indicates that not all of the atoms may have been identified in the outer region of the model; it is noted that the density is calculated to be 4.62 g/cm3 with a smaller concentric model (697 atoms within a radius of 13.2 Å). It

Figure 5. Observed (left) and calculated (right) X-ray diffraction patterns along (a) 2-fold, (b) 3-fold, and (c) 5-fold axes. Each spot area is proportional to intensity.

These comparisons showed remarkable matches in intensities, recognizing that the calculated pattern utilizes roughly assigned atoms and unrefined positional, compositional, and displacement parameters. The atomic model resembles an “onion” in the sense that it has a global (icosahedral) symmetry with respect to a unique origin; however, the structure exhibits a similar chemical and structural environment everywhere. Moreover, the simulated diffraction patterns for a fragment model, an octant of the spherical model that excludes the origin, are remarkably similar to those for the whole spherical model (Figure S6), indicating that our model does not depend on the presence of an origin.

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CONCLUSIONS It is demonstrated for the first time that a Fourier synthesis of aperiodic 3D diffraction data without substantial or uncommon assumptions resulted in a meaningful aperiodic 3D structure, particular to Al0.63Cu0.25Fe0.12, that has never been anticipated. 7357

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thanks P. A. Thiel for valuable information and discussions at ISU. This research was supported by the Creative Materials Discovery Program through the National Research Foundation of Korea (NRF) and was funded by the Ministry of Science, ICT and Future Planning (2015M3D1A1069707).

At this stage, the large and unbounded model cannot be the final answer. The presented model is concentric containing the origin. However, in principle the models can be chosen in any other region regardless of the inclusion of the origin, in any shape such as a box or a slab, or in any size in space without limit. Compared to the HD method, this is at present incomparably simpler in practice (the electron densities can be calculated for any region from only the information given in Table S1), and clearer in real space even though the descriptions between the two methods must be related. Despite the brilliance of the HD crystallography, it provides a rather abstract model. It is also surprising that there is no literature that explicitly presents the atomic positions in 3D for a QC based on diffraction data. Thus, this work is the first to our knowledge. The resulting real-space atomic coordinates can thus be readily used for calculating the physical properties or studying the linkage of the clusters (all of the atomic coordinates of this model can be reproduced from only the information given in Table S3), and can even aid in building models for the HD method, instead of, or combined with, models from the approximants. The method should be general in principle, and thus should be applied to any other aperiodic materials, and it is expected to be a breakthrough in revealing their mysterious structures. However, a quantitative structural analysis using this method will require the development of facile computing methods (software) to perform Fourier syntheses, refine atomic parameters, and view structures all in 3D. This work shows only a high similarity in the hierarchy of concentric structural motifs between this work and HD methods. A further investigation should be made to analyze the relationship between the models by the two methods.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.7b00245. Figure with proof that eq 2 is equivalent to eq 1, figures of X-ray Laue and precession photos, polyhedral representations of diffraction data, a 48 Å × 48 Å calculated electron density map, polyhedral real-space models, calculated X-ray diffraction patterns, tables of measured diffraction data, comparison between this work and the conventional crystallographic procedure, and coordinates of 362 independent atoms in icosahedral symmetry (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Seung-Tae Hong: 0000-0002-5768-121X Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS Late Prof. John D. Corbett (1926−2013) should have been a coauthor of this Article because the work was performed initially under his guidance with full support when the author was working at Iowa State University (ISU). The author also 7358

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