Tiling representation

Fig. 4. Tiling representation of (a) the E2g-phase, (b) the Ei -phase, and (c) the -phase. (d) Phase boundary (arrow) between the -phase and the E -phase.

Fig. 9. Tiling representation of the successive steps of phason-line movement along the [001] direction...

Fig. 11. TEM micrographs and superposed tiling representation of phason lines with different numbers of kinks along the [010] directioiL (a) No kink, (b) One kink, (c) Two kinks.

Fig. 12(a) is a high-resolution HAADF image of a phason plane in eg-Al-Pd-Mn and Fig. 12(b) shows the corresponding tiling representation. Since each individual phason line connects two-neighboring hexagon columns with their alternately oriented version (Section 3.1), the phason plane is a (001) mirror. Accordingly, phason planes can be considered as twin boundaries or inversion boundaries in the Eg-structure. 

Deviations of a phason plane from the (001) orientation can occur due to the mobility of the individual phason lines along the [0 01] direction. Fig. 13 depicts a tiling representation of a tilted phason plane. The [001] displacement of the phason lines leads to an increase of the fault area and, if the mutual displacement is larger than one c-lattice constant, to the occurrence of local (100) faults (thick hne). Hence, ahgnment of the phason plane along (001) minimizes the total fault energy and is accordingly considered as the ideal orientation. 

Fig. 13. Tiling representation of a tilted phason plane. Locally, (100) faults occur (thick tine).

Fig. 20. Tiling representation of the metadislocation in Fig. 1(a). The metadislocation core is represented by a dark-gray tile. Six phason halfplanes are terminated at the right-hand side of the core.

The tiling representation shown includes non-essential features of the metadislocation, and in this sense it is held at a low degree of abstraction. Fig. 21(a) is a tiling representation of the metadislocation in the S28-structure at a higher degree of abstraction. The curvature of the phason planes is neglected but the essential features of the metadislocation, that is, the core tile, the six inserted phason halfplanes, and the surrounding S28-structure are present. 

Fig. 21. Tiling representation at a higher degree of abstraction, (a) Metadislocation in the E28-structure.

Tiling representations for the metadislocations with two, fonr, and ten associated phason halfplanes in Se-Al-Pd-Mn are displayed in Figs 24(a)-24(c), respectively. Each metadislocation core is represented by a characteristic tile, the area of which increases with increasing number of associated phason hal lanes. 

Fig. 24. Tiling representation of metadislocations with (a) ten-, (b) four-, and (c) two associated phason...

We have seen in the previous sections that metadislocations are associated with a certain number of phason halfplanes. The latter are required to accommodate the core into the crystal lattice. In the following, we describe two different ways to construct metadislocations in terms of a tiling representation. This wiU enable the reader to comprehend the relation between the number of associated phason halfplanes and the Burgers vector length, as well as further characteristic properties of the individual metadislocations within the series. 

Fig. 36. (10 0) phase boundaries between the Ee-stmcture and the 828-structure, (a) Tiling representation of a hypothetical direct boundary, (b) Construction by a periodic stack of identical metadislocations. 

Fig. 38. Metadislocation in Al72.oPd22,8Fe5,2 associated with three planar defects in a -host structure, (a) TEM micrograph, (b) Corresponding idealized tiling representation.

Fig. 41. Tiling representation of metadislocations with five associated phason halfplanes, (a) In the S -structure. (b) In a region with a high density of phason planes.

The tiling representation of the associated phason halfplanes in Fig. 41 is different. While in Fig. 41(a) the respective area corresponds to a slab of monoclinic (1, —1) phase, the area in Fig. 41(b) corresponds to the orthorhombic Sie phase. This is merely a matter of representation and does not change the construction principle since phason lines can freely move along the [001] direction via local atomic jumps (Section 3.1.2). 

Fig. 42. (a) Projection of the unit cell of T-Al-Mn-Pd along the [010] direction and corresponding tiling description, (b) Tiling representation of the R-phase. (c) Phason lines (gray) forming (001) planes (broken lines), (d) Metadislocation with six associated phason halfplanes in the T-phase. The dark-gray... 

Fig. 44(a) shows a HAADF micrograph of a metadislocation in T-Al-Mn-Pd. The planar defect at the right-hand side, that is the slab of R-phase, is visible between the dashed lines. The metadislocation core is indicated by a polygon, which directly corresponds to the predicted polygon representing a metadislocation core [cf. Fig. 42(d)]. In addition three phason defects are visible at the left-hand side of the metadislocation core. Fig. 44(b) shows a full tiling representation of the defect. 

Fig. 44. Metadislocation in T-Al-Mn-Pd. (a) High-resolution HAADF micrograph. Dashed lines indicate the location of the slab of R-phase. The metadislocation core and three phason lines are highlighted, (b) Tiling representation of the defect...

Fig. 48. Tiling representation of experimentally observed metadislocations in (a) orthorhombic a-phases, (b) monoclinic s-phases, (c) AI13M4- and (d) T-phases. The Burgers vectors are indicated by arrows...

Fig. 49. Tiling representation of various possible metadislocations arrangements in different host structures, (a, b) In the T-phase. (c, d) In the R-phase. (e, f) In the sg-phase. (g, h) In the -phase. The Burgers vectors are indicated by arrows (not to scale).

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