Quasicrystals

Intermediate structural state now referred to as a quasicrystal. The novel structural concepts generated to describe quasicrystals have improved our understanding of the nature of traditional crystals and glasses.

Fig.4.26. Electron diffraction pattern of a rapidly cooled Al₆Mn alloy showing fivefold symmetry; that is, the pattern is identical with each rotation of 360^o/5, or 72^o, about its center. Such symmetry is impossible in traditional crystallography.

The Schechtman discovery that led to the idea of quasicrystals was an electron diffraction pattern of a micrometer-size crystallite of a rapidly cooled Al₆Mn alloy exhibiting five-fold symmetry (Fig.4.26).

(a) (b)

Fig.4.27 (a) Skinny and fat rhombuses can be repeated in (b) a two-dimensional stacking to produce a space-filled pattern with fivefold symmetry. This Penrose tiling provides a schematic explanation for the diffraction pattern of Figure 4.26.

(a) (b)

Fig.4.28. (a) The relation of the skinny rhombus of Figure 4.27 to the geometry of a regular pentagon. (b) The acute angle in the skinny rhombus is (1/5) π and, in the fat rhombus, is (2/5)π. These angles assure the fivefold symmetry of the Penrose tiling.

1) The skinny rhombus is directly related to the key dimensions of the pentagon, namely, the edge and the diagonal.

Fig.4.29. The Penrose tiling of Figure 4-27b decorated with pentagons to illustrate the fivefold symmetry of the overall pattern. Note that there is orientational order (all pentagon bases are parallel), but the lack of regular spacing of pentagons corresponds to a noncrystalline structure.

2) Note that the ratio of the length of the pentagon’s diagonal to its edge is Φ, an important irrational number equal to (√5 +1)/2 = 1.618. The number Φ is sometimes called the golden ratio because of its fundamental role in numerous shapes in the natural world, as well as in the proportions of much of the architecture of the ancient world.

3) The main role of the golden ratio in Fig. 4.28 is also shown by the fact that the ratio of the number of fat rhombuses to the number of skinny rhombuses is Φ.

4) The direct relation of the rhombuses to fivefold symmetry indicates that the acute angle in the skinny rhombus is (1/5)π and in the fat rhombus is (2/5) π.

5) The 3-dimensional Penrose tiling produces the fivefold symmetry of the 3-dimensional icosahedrons.

Fig.4.30. A theoretical diffraction pattern for a three-dimensional Penrose tiling directly matching the experimental pattern of Figure 4.26.

The icosahedron composed of 20 identical equilateral triangular faces, has a fivefold symmetry, where 5 triangles join at a vertex, as well as threefold symmetry and twofold symmetry.

Fig. 4.31. Three views of an icosahedron showing (a) fivefold symmetry,

(b) threefold symmetry, and (c) twofold symmetry.

All 3 symmetries have been seen in the diffraction patterns of Al₆Mn and other quasicrystalline materials. They referred to as icosahedra phases. To date, numerous alloy systems have exhibited quasicrystalline structure: Al – Li-Cu; Al-Co-Cu; Al-Co-Ni.

Дата добавления: 2015-08-05