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Recent Experiments

To progress beyond abstract studies of tilings, examples of real quasicrystals must be found which support the new paradigm. It would be particularly useful to find an example with a simple atomic decoration of the quasi-unit cell so that the correspondence is apparent. A promising system is the decagonal phase of AlNiCo , one of the most studied quasicrystals. High resolution lattice images reveal a network of overlapping decagonal clusters columns about 2 nm in diameter.[18], [19]


 
Figure 3: Superposition of a perfect decagon tiling on the high angle annular dark-field (HAADF) lattice image of water-quenched Al72Ni20Co8 obtained by the high angle annular dark field method by Saitoh et al. The overlay decagon tiling is shown separately in the following figure. Note the high degree of order in the lattice image and the near-perfect correspondence with the overlaid decagons.

A historic difficulty with AlNiCo has been disorder and superlattice effects which have confounded structural analysis. Early overlapping cluster models, such as Burkov's,[4] characterized the structure in terms of overlapping clusters with decagonal or pentagonal symmetry using overlap rules which produce random tilings. The random tiling picture seemed appropriate because of evidence of some diffuse scattering.

Recently, however, Tsai et al.[20] have found a simple decagonal phase in water-quenched Al72Ni20Co8 which exhibits no superlattice reflections and no diffuse streaks. Figure 3 shows a superposition of the high angle annular dark-field (HAADF) image for Al72Ni20Co8 obtained by Saitoh et al.[21] and a single decagon tiling. The bright spots in the HAADF image correspond to the positions of the transition metal atoms. Holding the image at an angle, one observes that the lattice image shows no detectable phason strain across nearly 15 nm. Hence, the image is isomorphic to a perfect Penrose tiling. The structure also appears to be composed of overlapping, decagonal clusters. See Figure 4. The HAADF image shows that the innermost ring of atoms inside the clusters has neither pentagonal nor decagonal symmetry. Rather, the structure breaks decagon symmetry in precisely the same sense that the overlap rules (see the superposed kite-shape decorations on decagon tiles). Using convergent beam electron diffraction, the space group was determined to be centrosymmetric P105/mmc using convergent beam electron diffraction.


 
Figure 4: An blow-up of a decagonal cluster in water-quenched Al72Ni20Co8 obtained by the high angle annular dark field method by Saitoh et al. The decagon is nearly 2 nm across. The overlay is a single decagon quasi-unit cell with decagon symmetry-breaking decoration to indicate overlap rules.


 
Figure 5: The single decagon tiling used to overlay the HAADF lattice image in Figure 3.

In Figure 3, the lattice image is superposed by a perfect single decagon tiling. The perfect tiling overlay is shown in Figure 5. The correspondence appears near-perfect across the image, with atomic decoration of each decagon and each kite-shape decoration within the decagon appearing to be identical. Jeong and Steinhardt have developed a simple calculational scheme for relating the atomic decoration of the quasi-unit cell to the stoichiometry and density.[23] Figure 6 is a candidate atomic decoration of the decagon unit cell that agrees with current measurements: the stoichiometry, Al72Ni21Co7 , and the density, 3.94 g/cm3 , lie within 1% of the measured values. The computed HREM lattice image agrees closely with the experimental image. This model is not equivalent to an atomic decoration of acute and obtuse Penrose tiles in which every acute tile is decorated equivalently and every obtuse tile is decorated identically. Hence, Al72Ni20Co8 appears to be an example for which the single decagon picture can be verified and explored in fine detail.


 
Figure 6: A candidate model for the atomic decoration of the decagonal quasi-unit cell for Al72Ni20Co8 . Large circles represent Ni (red) or Co (purple) and small circles represent Al . The structure has two distinct layers along the periodic c -axis. Solid circles represent c=0 and open circles represent c=1/2 .
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Figure 7: The computed HREM lattice image agrees closely with the experimental image; see Figure 7.
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\epsfxsize=3.3 in \epsfbox{stein6.eps}\end{center}\end{figure}


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Next: Implications Up: No Title Previous: A New Penrose Tiling

4/14/1998