Quasiperiodic structures can be described geometrically either as decorated tilings or as coverings (for a general review see Ref.1, for instance). While tilings are based on two or more unit-tiles, coverings can cover the plane (space) by partially overlapping copies of a single structural repeat-unit (quasi-unit-cell2-6). In the case of decagonal quasicrystals, the so-far most abundant quasi-unit-cell is based on the Gummelt decagon7-10. Such a quasi-unit-cell, when decorated with atoms (atomic cluster), is the counterpart to a unit cell of a periodic structure. It allows a more physical description of a quasicrystal structure than the structurally fully-equivalent tiling-based models do6,11,12.
In the following, we present a quasiperiodic structure that can be described by a covering based on the Andritz decagon, a novel kind of quasi-unit-cell for a decagonal phase. This fundamental decagonal unit, with a diameter of approximately 2 nm, consists of four subunits, three flattened hexagons and one bow-tie (D3H+1BT, for short).
A decagonal quasicrystal (DQC) with a composition of Al74Cr15Fe11 was identified by transmission electron microscopy (TEM). Figure 1 shows selected-area electron diffraction (SAED) patterns of Al74Cr15Fe11 along the tenfold zone axis (a) and along two typical twofold zone axes perpendicular to it (b, c). Some characteristic features of the diffraction pattern of a DQC, such as its scaling symmetry, are visualized by pentagons of varying size in Fig. 1(a). Note that the spots marked by yellow circles in the SAED pattern are much weaker than the other spots of the largest pentagon, analogously to that of Al-Cr-Fe-Si DQC5, but quite different from other typical Al-based DQC systems such as Al-Ni-Co13,14, where only strong diffraction spots are found in the corresponding positions. The translation period of the Al74Cr15Fe11 DQC, determined from the two twofold SAED patterns, is ≈ 1.23 nm, comparable to that of decagonal Al-Mn-Pd15. Consequently, the structure has a translation period of six quasiperiodic atomic layers along the periodic tenfold axis. Every other reciprocal lattice layer is systematically extinct in Fig. 1(c) indicating the existence of a c-glide plane. Thus, the 5D symmetry group of this DQC should be P105/mmc like for decagonal Al-Mn-Pd15.
The high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) image, which corresponds to a projection of the structure along the tenfold direction, is depicted in Figs. 2(a, b). One should keep in mind that such a projection turns the higher-dimensional 105 screw axis into a simple 10 rotation axis and the c glide plane into a m mirror plane, but still in higher dimensions.
Connecting related dots of the image yields a Hexagon-Bowtie (HB) tiling (Fig. 2a). This tiling can be described by a covering as well (Fig. 2b). The covering cluster is a decagon of approximately 2 nm diameter, partitioned by three flattened hexagon tiles and one bowtie tile, called Andritz decagon (D3H+1BT, for short). An example for a covering created by copies of the Andritz decagon is shown in the tilings encyclopedia16. The underlying HB tiling is a substitutional tiling, which scales by even powers of τ. Similar as the Gummelt covering it also features an underlying Fibonacci pentagrid (see Fig. 2d).
The D3H+1BT decagons are superposed onto the HAADF-STEM image in Fig. 2(a) and are filled in light-blue for clear display in Fig. 2(b). Note that the vertices of the D3H+1BT decagons are themselves decorated with smaller decagons (see also Fig. 3b), but not all are exactly the same. For the vertices of D3H+1BT to jointly form the thin waist of nearby bowtie (BT) tile, their atomic decorations are not the same as that of the other vertices decorated with the smaller decagons. Nearby D3H+1BT tiles are connected through both tiling and one "H" covering, to fill some small areas without gaps, but with some gaps for the whole plane, as marked by purple BT and S (star) tiles. These gaps could be considered as defects of quasiperiodic package by the quasi-unit-cell, analogous to the line defects of traditional crystals, for example dislocations. The percentage of area of gaps is 6.1％ among the whole filled areas in Fig. 2(b) and can be totally eliminated through the action of phason flipping, for example those in Fig. 2(c) and SFig. 1 in the supporting material.
The purple BT gap in the left column of Fig. 2c-1 is eliminated by being included in a quasi-unit-cell of D3H+1BT decagon (outlined red) in the right column, after the change of H1 tile in the left column to the H1′ tile in the right column. Actually, the change from H1 to H1′ tile is simply realized through the phason flipping by changing only one vertex (namely, from black spot to the red spot, as marked by a red arrow in the middle), which was observed experimentally through in situ high resolution TEM observations17,18. Consequently, the defect of purple BT in the left column is mended and one more quasi-unit-cell is accordingly added, as shown by the red D3H+1BT in the right column. Sometimes, the phason flipping is somewhat more complex than that in Fig. 2c-1. For example, in order to eliminate defect of purple BT in Fig. 2c-2 in the left column and to also maintain the quasiperiodic repeating of the nearby quasi-unit-cell of D3H+1BT tiles without gaps, the change of two vertexes of tiles is needed, and resulting in the change of H1 and H2 tile in the left column to the H1′ and H2′ tile in the right column. Occasionally, the change of three atomic positions is needed (see in SFig. 1a in the supporting materials). For the defect of S tiles in Fig. 2c-3 in the left column, two atomic positions are changed and create two more D3H+1BT tiles (also see another example in SFig. 1b in the supporting materials), rather than one more D3H+1BT tile for eliminating BT-type defects mentioned above.
Figure 2d is an idealized quasiperiodic covering based on D3H+1BT decagons without gaps, derived from Fig. 2b. The gaps of purple BT and S tiles patched in the quasi-unit-cell matrix in Fig. 2b are eliminated through the action of phason flipping discussed above.
The different allowed arrangements of Gummelt decagons in a strictly quasiperiodic covering (Penrose tiling) are compared to the experimentally observed ones of the Andritz decagons in Fig. 3(a). The Andritz decagon (D3H+1BT) in Fig. 3b is generated by linking the centers of the ten smaller rings of white dots (with a diameter of ~ 0.47 nm) with tenfold symmetry. Four more of these small rings located inside the ≈ 2 nm decagon, form the three shuttle-like H tiles and one BT tile with an edge length of ≈ 0.62 nm. Therefore, the D3H+1BT has just mirror symmetry, similar to the Gummelt's decagon. However, while the reflection plane runs through corners in the case of the Gummelt decagon, it is perpendicular to decagon edges in the case of the Andritz decagon.
In contrast, the D3H+1BT are linked to their neighbors by either overlapping H tiles or sharing one edge (Fig. 3b). The distance between the centers of adjacent D3H+1BT decagons with overlapping H tiles amounts to S = 1.18 nm, and to L = τ S = 1.91 nm when the D3H+1BT decagons are sharing edges. For the linkage of more D3H+1BT decagons, for example three, four, five and more D3H+1BT tiles see SFig. 2 in the Supporting materials. The simple connection rules in Fig. 3(b) allow to cover the whole plane without gaps, implying the D3H+1BT can act as a repeating quasi-unit-cell generating a quasiperiodic long-distance order16.
We now analyze the local features and the long-distance quasiperiodic ordering of the quasi-unit-cell of D3H+1BT tiles in Fig. 2d by linking their centers with solid green lines in Fig. 4a. The somewhat disordered quasiperiodic tiling superimposed on Fig. 2d is formed by linking the centers of two nearby D3H+1BT tiles covered by one H tile, with a 1.18 nm edge length of tiles. The structural blocks include regular pentagon (P), large decagon with a tangential circle diameter of 3.63 nm, as well some irregular shapes such fat hexagon (HF), banana-like tile (BLT), and concave decagon (DC). Among them, 19 pentagons are found, with the positions one-to-one corresponding to the 19 groups of five H tiles with a fivefold symmetry in Fig. 2d, for example one position marked by one red star in the upper-left corner in Fig. 4a. Irregular polygons were mostly found in imperfect DQCs19-22, implying the quasiperiodic ordering in Fig. 4a is not perfect. Interestingly, we find that some of D3H+1BT decagons (for example those highlighted in thick blue edges) tend to locate locally at the vertexes of a larger decagon with a tangential circle diameter of 5.88 nm, as those marked by dotted purple decagons. These D3H+1BT tiles share one edge with neighbors and distribute according to tenfold symmetry around one center of D3H+1BT tile, analogous to the experimental tenfold symmetric distribution of ≈ 2 nm decagons in decagonal Al59Cr21Fe10Si1023. Furthermore, we note that there are 116 decagons in Fig. 4a distributed along 10 directions differing 36°. Fig. 4b schematically show the directions of decagons, where the BT tile in each decagon is filled by colors to guide the eye for the orientations of decagons. Among them, every two of 180° oriented decagons are in a pair and colored in the same. We count the decagons along each direction and summarized in the histogram in Fig. 4c, where the maximum, minimum, and averaged number of decagons is 16, 8 and 11.1± 2.1, respectively. The difference of the counts of decagons along each direction also implies the imperfect quasiperiodic tiling.
The finding of a quasi-unit-cell different from those based on the Gummelt decagon shows that quasiperiodic order is possible based on a variety of fundamental structural subunits. But both, the Gummelt as well as the Andritz covering have in common that their structural subunits are arranged along quasi-lattice planes with traces forming Fibonacci pentagrids. These planes may be important for the evolution of quasiperiodic long-range order (see Refs.24,25).