Quasi-crystals shocked the crystallographic world when they were reported in 1984. We now know that they are not a rare exception, and can be found in many alloy systems. One of the richer systems for quasi-crystals and their approximants is Al−Co−Ni. A large series of pseudo-decagonal (PD) approximants have been found. Only two of them, PD4 and PD8, have been solved by X-ray crystallography. We report here the structures of PD1, PD2, PD3 and PD5, solved from the limited information that is provided by electron diffraction patterns, unit cell dimensions and high-resolution electron microscopy images.
In 1974, Penrose  discovered that it was possible to derive a non-periodic tiling of the plane from only two tiles—far less than the 20 426 (!) tiles that Berger  had used in 1964 to construct the first non-periodic tiling. When Gardner , in 1977, presented the work by Penrose to a wider audience in Scientific American, MacKay  immediately realized that this kind of ordered but non-crystalline (by traditional definition) structures ought to exist in the real world. When Shechtman  discovered the first quasi-crystals in 1984 in an Al−Mn alloy, most other crystallographers were sceptical or even actively denying the idea of quasi-crystals . The first quasi-crystals were icosahedral with a fivefold symmetry, but later decagonal , octagonal  and other quasi-crystals were also found.
Now it is completely accepted that quasi-crystals are an important kind of structured material, even though they do not obey any of the 230 space groups with translational symmetry. Thermodynamically stable icosahedral and decagonal quasi-periodic phases are included in a large number of constitutional phase diagrams, including aluminium-transition metal (TM) alloy systems .
An adjoined group of compounds are approximants; compounds with a chemical composition very close to that of a quasi-crystal and with a diffraction pattern closely resembling that of the quasi-crystal, yet having a translational symmetry and thus being a traditional three-dimensional crystal. The Al−Co−Ni alloy system is especially rich in approximants. A whole series with almost a perfect tenfold symmetry of their electron diffraction patterns, called pseudo-decagonal (PD) have been found and described [10,11] with their spectacular electron diffraction patterns. Out of those approximants, only two, PD4  and PD8, also called the W-phase , have been solved to atomic resolution by X-ray crystallography.
Here, we present the atomic structures of four more PD approximants, based on the X-ray structure of PD4, matched with high-resolution electron microscopy images, electron diffraction patterns and unit cell dimensions from the other PD compounds.
A number of Al−Co−Ni PD approximants were synthesized and their electron diffraction patterns and high-resolution transmission electron microscopy (HRTEM) images were taken along the pseudo-tenfold axis [10,11].
The HRTEM images were analysed by the crystallographic image processing program (CRISP) , based on analysis of the Fourier transforms of HRTEM images. From the amplitudes and phases in the Fourier transforms, the symmetry can be determined. After the amplitude and phase relations of the respective plane group were applied, an inverse Fourier transform was calculated. In principle, this gives a projection of the structure, but in practice there are limitations. First, the resolution (typically approx. 3 Å) is slightly too low for atomic resolution, given that the interatomic distances in metal alloys are in the range 2.0–2.9 Å. Secondly, electron microscopy images change the appearance dramatically with focus, even though they may remain sharp. As a consequence, atoms may appear either black or white as the focus is changed slightly. If there are amorphous regions in the HRTEM image, they will contribute to a continuous white noise background between the diffraction spots in the Fourier transform. Depending on the focus value, black rings may appear (called Thon rings) in that background noise and from the radius or radii of these rings, the focus can be determined and then compensated for by crisp. Unfortunately, most of the samples here were rather clean, without any amorphous layer, making it hard to estimate the focus value.
It is well established that HRTEM images of the PD series of approximants (including twinned and disordered samples) show densely packed ‘wheels’ of about 2 nm in diameter. These wheels can be seen more or less clearly in images irrespective of the focus value. They may appear either as black or as white rings, depending on the focus. In this study, we reversed the contrast of those HRTEM images that showed white wheels, so that all images have the same contrast, with atoms being black.
The atomic structure of the wheel can safely be assumed to be identical or very similar for the whole range of PD structures. From the only compounds in the PD series that have been solved by single-crystal X-ray crystallography, PD4 and PD8, we know the atomic details of this wheel. This reduces the problem of solving the PD structures, to solving the packing of the wheels in each of the structures. For this, it is very helpful that the unit cell dimensions of all the compounds are closely related, with the a-axis being 23.2 Å or values that are multiples of 23.2 Å and τ (the golden number 1.61803…), i.e. 37.7, 60.9, 98.7 or 159.7 Å, etc. Similarly, the shortest b-axis found is 19.8 Å and the other lengths are again multiplied by τ; 32.0, 51.8 and 83.8 Å. Finally, the c-axis, along the pseudo-tenfold, is always 4.1 Å (or 8.2 Å, but the longer c-axis dimension is then just a superstructure and it is possible to treat the structures as all having a c-axis of 4.1 Å).
Several basic distances can be found in the PD4 structure (figure 1). Its b-axis is 32.0 Å. The centre-to-centre distance between two adjacent wheels that are sharing atoms on their perimeters is 19.8 Å. Two adjacent, but not intersecting, wheels are 23.2 Å apart. Finally, the angle between the centres of three intersecting wheels is 72°, a number that should not surprise anyone, considering that these structures have some relations to the fivefold symmetry.
With the above tools and background knowledge, we built models for the structures PD1, PD2, PD3c and PD5 and reinterpreted PD8 in terms of wheels. For the three largest structures in the PD series, PD6 (a=98.7 Å, b=32.0 Å), PD7 (a=159.7 Å, b=51.8 Å) and PD9 (a=60.9 Å, b=83.8 Å), we have neither electron diffraction patterns nor HRTEM images at hand. They all have unit cell dimensions that are related by τ to the solved structures. Although one can speculate on the structure from just the unit cell dimensions, we did not attempt this here. We present the five PD approximant structures, starting with the simplest, i.e. PD8 which has the smallest unit cell.
3. Structures of pseudo-decagonal compounds
The unit cell dimensions of the eight members of the PD series are given in table 1. The c-axis is always the pseudo-tenfold and either 4.1 or 8.2 Å. The a- and b-axes are always chosen such that the very strong reflection at 2.0 Å resolution is along b* and the equally strong reflection at 2.3 Å resolution is along a* (figure 3a).
PD4 is one of only two structures in the PD series that are known to atomic resolution (figure 1; TM atoms (Co and Ni) are large, Al atoms are small). It has the composition Al72.5Co18Ni7.5. It was solved by single-crystal X-ray diffraction . We shall use this structure here for deducing the other structures in the PD series. Although this structure was solved by single-crystal X-ray diffraction, there were problems of disorder. Because Co and Ni are adjacent in the periodic table, their scattering power for X-rays are so similar that it was not possible to distinguish Co from Ni, and thus they were just called TM atoms. Aluminium scatters only about half as strongly as Co and Ni, so Al could at most places be distinguished from Co/Ni, but at some positions there were indications of mixed occupancies of Al/TM.
The simplest of all the compounds in the PD series is PD8. Its chemical composition Al71.8Co21.1Ni7.1 is very close to that of PD4 (Al72.5Co18Ni7.5). Even in the absence of images and diffraction patterns, it can be solved assuming the wheel structure (figure 2). The unit cell dimensions of PD8 are only a=23.2 Å and b=19.8 Å, and so there is just room for one wheel in each unit cell. Both of these distances are found in PD4 as marked in figure 1. It is immediately concluded that the wheels in PD8 must be intersecting along the b-axis but 3.4 Å apart along a. Note that owing to the tenfold symmetry of the wheel, it is possible for a wheel to share an atom or two with two other wheels at opposite sides, but this cannot happen at a 90° angle. Thus, PD8 has the smallest possible unit cell for a PD compound having at least one full wheel.
The phase reported in the study of Sugiyama et al.  as W-(AlCoNi) is probably identical to PD8. It was reported to have unit cell dimensions a=39.668(3) Å, b=8.158(1) Å, c=23.392 Å, β=90.05(1)° and space group Cm. However, considering that the β angle is very close to 90° and that all three perpendicular electron diffraction patterns show perfect mm-symmetry, the space group may well be orthorhombic. If we use the same setting as the other members of the PD series, the short axis is c. The a-axis of PD8 (23.2 Å) would then correspond to c in the W-phase (23.392 Å). Finally, the a-axis in the W-phase (39.668 Å) is doubled compared with PD8 (19.8 Å). If the weak superstructure reflections in the W-phase are ignored (as was done for PD4), then the W-phase would have b=4.079 Å and a=19.834 Å. For the sake of completeness and as a way of evaluating our approach in solving the PD structures, we include here both results based only on metric similarities with PD4 and the X-ray structure of the W-phase, respectively.
Atoms in the high angle annular dark field (HAADF) image by Sugiyama et al.  appear as white contrast, i.e. the opposite contrast to that of which we use here, so we reproduce the HAADF image here with reversed contrast, i.e. atoms appear black, as shown in figure 2b.
The second smallest unit cell area in the PD series is that of PD2 (figures 3 and 4). a=23.2 Å is the same as that for PD8, 23.2 Å, but b is τ times larger than 19.8 Å, i.e. 32.0 Å. The length of the b-axis is identical to that in PD4, so we can expect a very similar packing of the wheels along b in PD2 and PD4.
Following the convention established by Grushko, the b*-direction in all PD compounds is chosen to be that which has a very strong reflection at 2.0 Å resolution. As a consequence, there is always a strong reflection at 2.3 Å resolution in the a*-direction (figure 3a). From this, it follows that all of the wheels are oriented as in PD4 (figure 1), with respect to the crystallographic a- and b-axes. The innermost pentagon of TM atoms points along a but not along b. On the other hand, a pair of TM atoms at the rim of the wheel is always found along the b-axis, but not along the a-axis (figure 1).
Note that all of the distances and other geometric features of PD2 can be found in PD4. The crystal structure of PD4 can be described as two pairs of the PD2 unit cells, all arranged along the a-direction. The length 32.0 Å of the b-axis is τ times that of PD8 (19.8 Å). The length of the diagonal of the PD2 unit cell is √(23.22+32.02)=39.53 Å. As there are two wheels per unit cell along that [1 −1 0] diagonal direction, they are 39.53/2=19.8 Å centre-to-centre, which is approximately 19.8 Å, given the accuracy with which the unit cell dimensions of the PD series have been given.
The almost square unit cell of PD1 has a=37.7 Å and b=39.7 Å. However, the lines of the reflections with even k are very much stronger than those with odd k (figure 5a,c). This means that PD1 has a basic structure with b=19.8 Å and only a superstructure where the alternating basic unit cells of 19.8 Å differ slightly. It is not possible at this stage to say what the differences between these alternating 19.8 Å units are.
The b-axis of PD1 (39.7 Å) is exactly twice the length of the b-axis of PD8 (19.8 Å), so it must have intersecting wheels along b. This is also observed experimentally in the HRTEM images in figure 5b,d. The a-axis of PD1 (37.7 Å) is slightly shorter than the b-axis because the wheels are packed in a zigzag fashion along a.
All of the structures in the PD series are orthorhombic and have primitive lattices, except PD3c which is centred (figure 6). The unit cell dimensions for PD3 and PD3c are both a=37.7 Å and b=51.8 Å. We have solved PD3c. The length of the a-axis is the same as that of the a-axis of PD1. Therefore, PD3c and PD1 must have the same zigzag arrangement of wheels along their a-axes (figures 5d and 7b). If there is a certain feature at the origin, then that same feature will also be found in the centre of the unit cell. This information is sufficient for specifying where the wheels are; they make up a pattern of distorted hexagons.
The largest approximant for which we have HRTEM images is PD5. Its unit cell dimensions are a=61.0 Å and b=32.0 Å. The 32.0 Å distance, already known from PD4, is due to a zigzag arrangement of wheels along the b-axis (figure 8d).
The structure of PD5 is closely related to that of PD3c; they both contain squeezed hexagons of six wheels, with exactly the same relative positions. The difference in PD3c is that the hexagons are arranged in a centred manner (cmm), parallel to the unit cell axes, whereas in PD5 the hexagons are tilted with respect to the unit cell axes. In PD5, this results in a herringbone (zigzag) pattern of hexagons, typical of pgg symmetry (marked with lines in figure 8d). Six ‘wheels’ are seen per unit cell. The similarity between PD3c and PD5 is also seen from the fact that they have the same unit cell area—1952 Å2. It should be pointed out that this structure, although very plausible given the evidence above, is still quite uncertain. The symmetry is not unambiguously clear and the HRTEM image used (figure 8c), although clear, does not show just one single crystalline area.
4. Conclusions and future work
We believe that the structure models we have presented here for four approximants in the PD series are essentially correct. We shall now try to find single crystals, large enough for single crystal electron diffraction, i.e. about 100–1000 unit cells. Then, we can collect the complete three-dimensional electron diffraction data with our newly developed rotation electron diffraction technique . Using the overall packing of the wheels as described in this article, combined with the atomic coordinates from the known PD4 structure, we expect that these structure models are close enough to allow for the refinement of each of these structures.
Many HRTEM images show defects and intergrowth of the different PD compounds . This is not surprising given the close similarities of these structures.
If the larger structures in the PD series can also be solved, they will, no doubt, be gradually closer to the Al−Co−Ni quasi-crystal structure. We expect that there could be an infinite number of PD structures, with similarly increasing unit cells. It may be possible to predict their structures in detail, using the knowledge of the packing of wheels, as shown in this article. It may be very much harder to synthesize single crystals of them (even with as few unit cells as 10–1000 as required for electron crystallography), since they will be extremely similar in chemical composition.
This project is supported by the Swedish Research Council (VR) and the Swedish Governmental Agency for Innovation Systems (VINNOVA) through the Berzelii Center EXSELENT and Göran Gustafsson Foundation. We thank Dr Markus Döblinger for kindly providing the SAED patterns and HRTEM images used in this paper, and valuable comments.
One contribution of 14 to a Theme Issue ‘Beyond crystals: the dialectic of materials and information’.
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