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# ssp_6y - 2 Continued 2.2 Point symmetry Reflection in a...

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2 Continued

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2-4 2.2 Point symmetry Reflection in a plane (mirror symmetry): symbol m E.g., H 2 O molecule 2 mirror planes: plane of molecule and ---------- Inversion: symbol 1 Reflection in a point E.g., homonuclear diatomic molecules: 1 and m bcc lattice (center of cube) n-fold rotation axes: symbol 1, 2, 3, 4, 6 Rotation by 360°/n only n = 1, 2, 3, 4, 6 compatible with translational symmetry! (small molecule, clusters: n = 5, 7, . .. also possible) E.g., n = 6 for hexagonal plane and hexagonal lattice (Fig. 2.4) n = 3 for hcp structure! (Fig. 2.6) Rotation-inversion axes: symbol 2346 ,,, n-fold rotation with simultaneous inversion
2-5 Fig. 2.5. A 5-fold axis of symmetry cannot exist in a periodic lattice because it is not possible to fill the area of a plane with a concentrated array of pentagons. We can, however, fill all the area of a plane with just two distinct designs of “tiles“ or elementary polygons. In three dimensions this building principle leads to so-called quasicrystals (see below). Fig. 2.6. Close-packed layer of atoms with centers at points marked A. This corresponds to space-filling arrangement of triangles or hexagons made up of 6 triangles. This 2-dim. lattice has 6-fold symmetry. A second and identical layer of atoms can be placed on top of this, above and parallel to the plane of drawing, with centers over the points marked with B. There are two choices for the third layer. It can go in over A or over C. Packing sequence ABCABCABC. ... fcc structure ABABAB. ... hcp structure

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2-6 Definition of crystal structures via symmetry Crystals may be described by combination of symmetry elements * that carry the crystal into itself. Combination of point symmetry elements 7 point groups (crystal systems) Lattice + basis (has its own symmetry) 32 crystallographic point groups Crystal systems + translation operation (translation centered lattices) 14 “Bravais lattices“ (space groups) Bravais lattices + basis 230 space groups 230 space groups also encompass the following symmetry elements: screw axis: rotation + translation glide mirror plane: reflection in a plane + translation
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ssp_6y - 2 Continued 2.2 Point symmetry Reflection in a...

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