Phason plane

Related phases and phase transitions in terms of phason planes... 

Phason lines and phason planes can assume a dual function in s-type phases. On the one hand, they are structural defects, for instance in the phases Sg-Al-Pd-Mn and -Al-Pd-Mn. On the other hand, phason lines and phason planes can arrange regularly, forming a related e-type phase with larger c-lattice constant, and hence become elements of a new ideal structure. Phason lines and phason planes are pivotally connected to metadislocation formation and movement, as well as in phase transitions and formation of e-phases. 

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. 12. Phason planes in Se-Al-Pd-Mn. (a) High-resolution HAADF image, (b) Corresponding tiling...

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

Fig. 14. TEM micrographs along the [010] direction of tilted phason planes, (a) Under low-resolution bright-held conditions, (b) At higher magnification.

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. 

The offset ds introduced by a phason plane can be interpreted as the vertical component of the phason-plane strain field. Since the phason plane turns hexagon columns into their alternate orientation, its strain field has in addition a lateral component corresponding to a translation of half a lattice constant along the [10 0] direction. The total phason-plane strain field can then be described by the displacement vector 7 =1/2(1 0 - 1/r ), which was experimentally verified by means of fringe-contrast analysis in TEM [37]. 

Related phases and phase transitions in terms of phason planes In Section 2.2, we have introduced the fact that the ss-phase is the basic structure of a family of related phases, the s-phase family. Fig. 4 shows examples of corresponding hexagon tilings. The S28-phase [Fig. 4(a)] is represented by a tiling which consists of hexagons and banana pentagons. The latter are thus structural elements of the S28-phase, while they are defects in the ss-phase. 

The S28-phase can be considered as a structure consisting of a periodic stacking of phason planes. Other s-phases consist of phason planes stacked with different periodicities, that is, with different amounts of hexagon rows between the phason planes. Generally, between the members of the s-phase family, we can discriminate between phases with and without phason lines as structural elements. The phases Se and fall into the first category and the phases sie, S22, and E28 into the second one. This concept can be expanded to include related monoclinic phases [31]. 

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. 25. Hexagon-column arrangement in the Eg-tiling with the presence of (a) a metadislocation core, (b) a single phason plane, and (c) a combined arrangement of the latter (see text).

Engel and Trebin demonstrated [30] that all s-phases (referred to as H-phases in their paper) can be constructed by means of a projection formalism on the basis of a three-dimensional hyperspace. The result of the projection is a two-dimensional tiling in the (010) plane. These authors were able to reproduce the lattices of all E-phases and their structural defects of phasonic type, that is, phason lines and phason planes. 

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(c) shows a climb step (movement along [100] to the left) of a metadislocation by one a-lattice constant. Again, the initial and final tiles are drawn in black and gray lines, respectively. For a climb step, a much smaller number of vertex jumps is necessary. The number of necessary vertex jumps is now limited to the core region and it is now finite, no matter how far the associated phason planes are extended. Since each vertex jump physically represents a number of local atomic movements (Section 3.1.2), climb motion obviously is connected with much less atomic rearrangement than glide motion. 

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...

Fig. 35. Complex metadislocation network formed via the mutual interconnection by phason planes. The type and Burgers vector orientation of the metadislocations are indicated.

In the following section, it is shown that metadislocations exist is a wide range of CMAs other than s-phases. To start with, we discuss metadislocations in monoclinic -phases. These are closely related to the orthorhombic s-phases, and so are their metadislocations. In Sections 6.2 and 6.3, we proceed to structures more distantly related, for which the existence of metadislocations was theoretically predicted [46]. We show that metadislocations indeed exist in these systems, albeit in a different form than expected. In particular, the associated defects are not phason planes but different types of planar fault, which leads to a more general view of the characteristic features of metadislocations. 

Note that here the term phason plane is nsed in a more general sense than introduced in Section 3.1. The term is now used to refer to any lineup of phason lines that, to the eye, forms a continuous planar arrangement. 

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