Atomic structure of the quasicrystal Al₇₂Ni₂₀Co₈

Steinhardt et al. reply

The purpose of our Letter¹ was to present experimental support for the quasi-unit cell picture of quasicrystals. This model proposes that the atomic structure can be reduced to a single repeating cluster satisfying certain ‘overlap rules’ (sharing of atoms by neighbouring clusters). We proposed that the quasicrystalline phase of AlNiCo can be decomposed into a repeating decagonal atom cluster (20 Å in radius). Yan and Pennycook do not refute the quasi-unit cell concept — they also propose a repeating cluster obeying the same overlap rules. However, they propose a different atomic decoration for the repeating cluster that is ten-fold symmetric, whereas our decoration explicitly breaks ten-fold symmetry. This is important because our symmetry breaking corresponds precisely to the symmetry breaking of the overlap rules, and hence provides key evidence for the quasi-unit cell picture.

Yan and Pennycook's decoration is motivated by their impressive high-angle annular dark-field (HAADF) imaging, obtained with higher resolution than we had available. As they show, the imaging disagrees with the sites of four columns of transition metal (TM) atoms in our proposal (shown by arrows in their Fig. 1). However, we find that the problem can be resolved by a modest rearrangement of the previous decoration, switching 8 out of 100 atoms and retaining the broken ten-fold symmetry. The improved model in Fig. 1 has all the same qualitative properties as the original in ref. 1, matches the new HAADF (including Yan and Pennycook's Fig 2.) and even more recent high-resolution transmission electron microscopy (HRTEM) imaging, and has a density and stoichiometry that fits measured values to better than 2 per cent.

Figure 1: Improved decoration of the quasi-unit cell for AlNiCo compared to lattice image.

Problematic TM sites in our earlier model¹ have been removed. The figure includes atoms added by overlap of neighbour clusters; these lead to the formation of neighbour TM column pairs, as seen near the centre. Large circles represent Ni (red) or Co (purple) and small circles represent Al. Solid circles represent c=0 and open circles represent c=1/2 along the periodic c-axis.

As more data become available (for example, from X-ray diffraction), further small refinements to our current best-fit decoration may be required, but the ten-fold symmetry breaking should remain as an essential property. The broken symmetry is necessary to explain three established features of AlNiCo: the broken symmetry consistently observed in through-focus HRTEM imaging of the clusters²; the broken symmetry found within the central ring of most clusters in HAADF imaging, such as our Fig. 1 and Yan and Pennycook's Fig. 2 (ref. 2) (the very rare, more symmetric rings, as shown in their Fig. 1, can be explained as defects; ref. 2 and M. Widom, personal communication); and the apparent quasiperiodic correlation in the broken symmetry direction on moving from cluster to cluster in HAADF images (see Fig. 1 of ref. 1), as is found for a configuration of overlapping decagons.

None of the features can be explained by symmetric clusters, even if chemical disorder is introduced to randomly break the ten-fold symmetry. M. Widom and co-workers (personal communication) have completed a total-energy-based prediction of the structure of AlNiCo, making no prior assumption about the existence of repeating 20-Å clusters. Yet decagonal clusters with broken ten-fold symmetry emerge as the lowest-energy configuration with nearly identical assignments of Al and TM positions, as in our improved model.