Atomic structure of a decagonal Al-Pd-Mn phase

We present a detailed structure solution for the $16\text{−}\AA$ decagonal quasicrystal in the Al-Pd-Mn system by means of cluster decoration and ab initio energy minimization. It is based on structure models of the $\varepsilon$ and other approximant phases. The $\varepsilon$ phases can be represented as subsets of a hexagon-boat-star (HBS) tiling. The decagonal phase comprises further HBS tiles. We have constructed several fictitious HBS approximants and optimized their structures individually. All tiles are decorated by two types of atomic clusters: the pseudo-Mackay icosahedron (PMI) and the large bicapped pentagonal prism (LBPP). It turns out that, whereas the PMI clusters can be kept essentially unchanged, the LBPP clusters must be adjusted in occupancy with Al atoms depending on their positions in the various tiles. In this way we obtain cluster decorations for all tiles of the decagonal quasicrystal. The calculations were confirmed by evaluation of an effective tile Hamiltonian.

Figure 1

Idealized tiling representation of the $16\text{−}\AA$ d-QC (left panel). The tiling vertices (that we identify with the centers of the columns of PMI clusters) correspond to bright spots in the HRTEM image of the $16\text{−}\AA$ $d$ phase [6]. Right panel: Two high-temperature equilibrium tilings observed in the Al-Mn-Pd system [10], ${\varepsilon }_6/{\xi }^{\prime }$ (pure hexagons) and ${\varepsilon }_{28}/{\xi }_2^{\prime }/\mathrm{\Psi }$, which is an intergrowth of ${\varepsilon }_6/{\xi }^{\prime }$ with ${\varepsilon }_{16}/{\xi }_1^{\prime }$. The linkages' length—separation between neighboring PMI columns—is 7.6 Å.

Figure 2

PMI cluster consisting of three atomic shells: ${\mathrm{Al}}_{30}$ icosidodecahedron, ${\mathrm{Pd}}_{12}$ icosahedron, and an ${\mathrm{Al}}_9\mathrm{Mn}$ inner shell.

Figure 3

Top: LBPP cluster consisting of three atomic shells: ${\mathrm{Al}}_{32}$ icosidodecahedron-type shell, ${\mathrm{Pd}}_{10}$ pentagonal prism, and an ${\mathrm{Al}}_{11}\mathrm{PdMn}$ inner shell. Bottom: Alternative cluster description.

Figure 4

Hexagon (left), boat (middle), and star (right) tiles with small and large nodes (outlined with green and red circles, respectively).

Figure 5

Binary HBS tilings of the approximants studied in this paper. Vertices of the superimposed red tilings constitute the large nodes of the underlying HBS tilings. They indicate the positions of the PMI columns. All other vertices are small nodes and specify the positions of the LBPP cluster columns of which some are outlined by green circles. The dotted rectangles highlight the unit cells. (a) $B_2$, (b) $H_{2}S$, (c) $H{B}_3$, (d) $HB$, (e) $H_4{B}_{3}S$, (f) $H_{5}B{S}_2$, (g) $H_2$, and (h) $H_{3}S$. In (c) the blue line segment and the blue-hashed circle characterize tiling objects for an effective tile Hamiltonian as discussed in Sec. 4.

Figure 6

Binary HBS tiling of the decagonal phase shown in Fig. 1. Vertices of the superimposed red tiling are the large nodes of the underlying HBS tiling; all other vertices are the small nodes. The boat and star tiles of the HBS tiling are shown in gray, and the hexagonal tiles are white.

Figure 7

Tiles with adjusted occupancy of LBPP clusters. The colored circles emphasize the occupancy rules for the LBPP clusters located at the small nodes in the various tiles. A fully occupied LBPP cluster contains 55 atoms.

Figure 8

Tiling of the d-QC with adjusted occupancies of LBPP clusters.

Figure 9

Left panel: A slice parallel to the pseudofivefold axis through the $H_5{S}_{2}B$ approximant structure. The left part of the image shows the relaxed static structure. Al atoms are depicted in yellow, Pd atoms are depicted in red, and Mn atoms are depicted in blue. The black lines connect PMI cluster centers $(7.6\text{−}\AA$ linkages), and the blue lines outline the primitive cell. The right side shows the occupancy probability gathered over $\sim 4\text{−}\mathrm{ns}$ MD simulation time at 1200 K. Up to 0.4 occupancy probability, Al atoms are shown as gray-level white to black, above this value they are shown as yellow. The red points indicate the positions of the Pd atoms, and the blue points indicate the positions of the Mn atoms, both above 0.5 occupancy. Right panel. $\left(e\text{−}\mathrm{DOS}\right)$ comparing the approximants $B_2$ and $H_{5}B{S}_2$ with a 128-atom cubic approximant of an icosahedral phase [28].