Phason halfplane

Fig. 15. Phason halfplanes in the Eg-lattice. Offset along the [001] direction due to the insertion of (a) a single and (b) two phason hal lanes.

Comparing the numbers of (001) hexagon planes in the left and right parts of Fig. 15(b), one sees that two phason planes can be understood as a compacted combination of three hexagon planes. Since the thickness of the part containing the two phason halfplanes is smaller by Ids, the thickness of a phason plane, d, can be calculated as 2d — 3dh 2ds, where dh = c is the thickness of a hexagon plane. With ds — c/2t, we obtain d = cx /2 (see Appendix), which, with c = 1.256 nm for E6-Al-Pd-Mn, takes a value of 1.644 nm. 

Fig. 18 depicts a metadislocation in Eg-Al-Pd-Mn. The matrix structure entirely consists of flattened hexagons in alternating orientation. In this example, the six associated phason halfplanes stretch out to the right-hand side. 

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. 

Because metadislocations are connected to a number of associated phason halfplanes, their Bnrgers vector cannot be determined by means of a regular Burgers circuit [40]. The phason planes are not an element of the ideal structure of the Eg-phase, and hence a comparison circuit for the metadislocation in ideal Eg cannot be performed. The same holds for the metadislocation in E2s, since the associated slab of Ee is not present in the ideal E28-strnctnre. 

Fig. 22. (a) Basis vectors in the (010) plane, (b) Burgers vector determination by a circuit around the tile representing the core of a metadislocation with six associated phason halfplanes. 

Note N, number of associated phason halfplanes b. Burgers vector length in terms of the c-lattice constant and actual value in the in sg- and 28-lattice. 

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. 

The numbers of associated phason halfplanes in the series take twice the values of the Fibonacci numbers 1, 2, 3, 5, and 8 (see Appendix). 

The sequences of associated phason halfplanes and Burgers vector lengths are opposed with an increasing number of phason halfplanes, the Burgers vector lengths are decreasing. 

Experimentally, one finds that the different members of the metadislocation series are not evenly distributed but their number distribution shows a broad maximum. In deformed and undeformed e-Al-Pd-Mn, most metadislocations found have six associated phason halfplanes. Metadislocations with ten halfplanes are observed almost as often, while those with four phason halfplanes are found considerably less frequently. Metadislocations with 16 phason halfplanes are even less frequently found and species with two halfplanes were only observed in very few occasions during an extensive number of experimental investigations. The frequency of occurrence of different metadislocation types considering their energy is discussed in Section 4.5.3. 

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. 27. Metadislocation with six associated phason halfplanes constructed by projection from a five-dimensional hyperspace (courtesy of M. Engel).

Fig. 29. Metadislocation loops in Es-Al-Pd-Mn sharing six and ten associated phason halfplanes.

The loop nature of metadislocations can also be seen in images along the [010] direction. Fig. 29 is a TEM image of Se-Al-Pd-Mn. In the center of the image, two metadislocations are seen, which share six associated phason halfplanes. Obviously, the two metadislocations terminating the phason planes are segments of a loop, which has a habit plane close to (001). In the lower right corner, a smaller example of a metadislocation loop with ten associated phason planes is seen. 

Fig. 30. (a) Metadislocation with six associated phason halfplanes represented by an alternative phason-line tihng (hatched area), (b) Glide step by one c-lattice constant, (c) Climb step by one n-lattice constant. Initial and final tile positions are shown in black and gray, respectively. 

Fig. 34 shows a more complex case of splitting into three metadislocations. A metadislocation with ten phason halfplanes (1) is connected with two metadislocations, each with six phason halfplanes (2 and 3). Two phason halfplanes leave the metadislocations at the right-hand side (4 and 5). Two additional phason planes seen in the micrograph (6 and 7) are not connected to the metadislocation arrangement. The net Burgers vector of the arrangement is (-0.113-2 X 0.183) nm = -0.480 nm, which is equal to the Burgers vector of a... 

Fig. 33. Two metadislocations associated with ten (1) and six phason halfplanes (2), which are mutually connected by six phason planes, (a) TEM micograph. (b) Schematic representation in terms of triangles and lines, representing metadislocations and phason planes, respectively. The numbers in the triangles denote the number of associated phason halfplanes, the arrows indicate the lengths and directions of the Burgers vectors, (c) Corresponding schematic representation of a metadislocation with four associated...

Let us consider a (100) phase boundary between Eg and 28 phases. The lattice constants differ by a factor 3- -t and the corresponding (100) planes are congruent over small distances of maximum 4c along the [001] direction [Fig. 36(a)]. In Section 4.2, we have seen that the phason hal lanes associated to a metadislocation can be regarded as a slab of inserted phase of a closely related structure. For example, for a metadislocation in the sg-structure, the associated phason halfplanes consist of a slab of E28-structure in an sg-matrix [Fig. 21(b)]. 

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