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Symmetry in 2D 4/24/2013 L. Viciu| AC II | Symmetry in 2D 1 Outlook • Symmetry: definitions, unit cell choice • Symmetry operations in 2D • Symmetry combinations • Plane Point groups • Plane (space) groups • Finding the plane group: examples 4/24/2013 L. Viciu| AC II | Symmetry in 2D 2 Symmetry Symmetry is the preservation of form and configuration across a point, a line, or a plane. The techniques that are used to "take a shape and match it exactly to another” are called transformations Inorganic crystals usually have the shape which reflects their internal symmetry 4/24/2013 L. Viciu| AC II | Symmetry in 2D 3 Lattice = an array of points repeating periodically in space (2D or 3D). Motif/Basis = the repeating unit of a pattern (ex. an atom, a group of atoms, a molecule etc.) Unit cell = The smallest repetitive volume of the crystal, which when stacked together with replication reproduces the whole crystal 4/24/2013 L. Viciu| AC II | Symmetry in 2D 4 Unit cell convention By convention the unit cell is chosen so that it is as small as possible while reflecting the full symmetry of the lattice (b) to (e) correct unit cell: choice of origin is arbitrary but the cells should be identical; (f) incorrect unit cell: not permissible to isolate unit cells from each other (1 and 2 are not identical)4/24/2013 L. Viciu| AC II | Symmetry in 2D 5 A. West: Solid state chemistry and its applications Some Definitions • Symmetry element: An imaginary geometric entity (line, point, plane) about which a symmetry operation takes place • Symmetry Operation: a permutation of atoms such that an object (molecule or crystal) is transformed into a state indistinguishable from the starting state • Invariant point: point that maps onto itself • Asymmetric unit: The minimum unit from which the structure can be generated by symmetry operations 4/24/2013 L. Viciu| AC II | Symmetry in 2D 6 From molecular point group to space groups Complete consideration of all symmetry elements and translation yields to the space groups benzene graphene graphite D6h or 6/mmm p6mm P63/mmc Point group Plane group = point Space group = point group symmetry + group symmetry + in plane translation in 3D translation 4/24/2013 L. Viciu| AC II | Symmetry in 2D 7 • Symmetry operations in 2D*: 1. translation 2. rotations 3. reflections 4. glide reflections • Symmetry operations in 3D: the same as in 2D + inversion center, rotoinversions and screw axes * Besides identity 4/24/2013 L. Viciu| AC II | Symmetry in 2D 8 1. Translation (“move”) Translation moves all the points in the asymmetric unit the same distance in the same direction. There are no invariant points (points that map onto themselves) under a translation. Translation has no effect on the chirality of figures in the plane. 4/24/2013 L. Viciu| AC II | Symmetry in 2D 9 2. Rotations A rotation turns all the points in the asymmetric unit around one axis, the center of rotation. The center of rotation is the only invariant point. A rotation does not change the chirality of figures. 4/24/2013 L. Viciu| AC II | Symmetry in 2D 10 Symbols for symmetry axes Drawn symbol One fold rotation axis --- (monad) two fold rotation axis (diad) Axes perpendicular to the plane Axes parallel to the plane three fold rotation axis (triad) four fold rotation axis (Tetrad) six fold rotation axis (Hexad) CRYSTALS MOLECULES 4/24/2013 L. Viciu| AC II | Symmetry in 2D 11 3. Reflections A reflection flips all points in the asymmetric unit over a line called mirror. The points along the mirror line are all invariant points A reflection changes the chirality of any figures in the asymmetric unit • Symbol: m • Representation: a solid line 4/24/2013 L. Viciu| AC II | Symmetry in 2D 12 4. Glide Reflections Glide reflection reflects the asymmetric unit across a mirror and then translates it parallel to the mirror There are no invariant points under a glide reflection. A glide plane changes the chirality of figures in the asymmetric unit. • Symbol: g • Representation: a dashed line 4/24/2013 L. Viciu| AC II | Symmetry in 2D 13 Point group symmetry • Point group = the collection of symmetry elements of an isolated shape • Point group symmetry does not consider translation! • The symmetry operations must leave every point in the lattice identical therefore the lattice symmetry is also described as the lattice point symmetry • Plane symmetry group or plane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern 4/24/2013 L. Viciu| AC II | Symmetry in 2D 14 Examples of plane symmetry in architecture 4/24/2013 L. Viciu| AC II | Symmetry in 2D 15 Crystallographic plane point groups = 10 1. 1 (one fold axis) 6. m (mirror line) 2. 2 (two fold axis) 7. 2 mm (two mirror lines and a 2-fold axis)* 3. 3 (three fold axis) 3 m (one 3-fold axis and 8. three mirror lines) 4. 4 (four fold axis) 9. 4 mm (4-fold axis and four mirror lines)* 6 mm (6-fold axis 10. and 6 mirror lines)* 5. 6 (six fold axis) 4/24/2013 L. Viciu| AC II | Symmetry in 2D 16 * Second “m” in the symbol refers to the second type of mirror line Non-periodic 2D patterns • 5-fold , 7-fold, etc. axes are not compatible with translation non-periodic two dimensional patterns Ex: Starfish Wikipedia.org 5m (five fold axis + mirror) A Penrose tiling Group of atoms or viruses can form “quasicrystals” (quasicristals = ordered structural forms that are non-periodic) Electron diffraction of a Al-Mn quasicrystal showing 5-fold symmetry by Dan Shechtman 4/24/2013 L. Viciu| AC II | Symmetry in 2D 17 http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html 4/24/2013 L. Viciu| AC II | Symmetry in 2D 18 Combining symmetry operations Ten different plane point groups : 1, 2, 3, 4, 6, m, 2 mm, 3 m, 4 mm, 6 mm Five different cell lattice types: 1. oblique(parallelogram) (a ≠ b, ≠ 90°) 2. Rectangular (a b, 90ᵒ) 3. Square (a = b, 90ᵒ) 4. Centered rectangular or diamond (a b, 90ᵒ) 5. Rhombic or hexagonal (a = b, 120ᵒ) When point group symmetries are combined with the possible lattice cells 17 plane groups. 4/24/2013 L. Viciu| AC II | Symmetry in 2D 19 1. Combining rotation with translation 1. The rotations will always be to the plane (the space in 2D) 2. An -fold rotation followed by translation to it gives another rotation of the same angle (same order), in the same sense 3. The new rotation will be located at a distance x = T/2 x cotg /2 along perpendicular bisector of T (T=cell edge translation) Ex: 2-fold rotation followed by translation (=180) 1 The second rotation will be on T in the middle at B B T 180 A x Steps: 2 3 1. 2-fold rotation through A moves the motif from 1 to 2 2. translation by T moves the motif from 2 to 3 is the motif Or 1. 2-fold rotation through B moves the motif from 1 to 3 4/24/2013 L. Viciu| AC II | Symmetry in 2D 20 2-fold axis combined with translation 1 8 4 1 4 T2 T1 T1 T1+T2 Pair of motifs: 6 7 T2 2 9 3 2 3 •2-fold rotation at 1 combined with translation T 1 gives the rotation 6 (rotation 6 is translated to 7 by T2) •2-fold rotation at 1 combined with translation T 2 gives the rotation 8 (rotation 8 is translated to 9 by T1) •2-fold rotation at 1 combined with translation T 1+T2 gives the rotation in the middle The blue, red, green and yellow marked are independent 2-fold axes: they relate different objects pair-wise in the pattern no any pair of the blue and one of the red, green or yellow 2-fold axis describe the same4/24/2013 pair -wise relationship L. Viciu| AC II | Symmetry in 2D 21 6-fold axis combined with translation 6-fold axis contains 2/6, 2/3, 2/2 rotations All the operations of a 3-fold axis combined with translation and of a 2-fold axis combined with translation will be included for a p6 plane group 4/24/2013 L. Viciu| AC II | Symmetry in 2D 22 Combination of the rotation axes with a plane lattice = translation Two fold axis Three fold axis Four non-equivalent 2-fold axes to the plane Three non-equivalent 3-fold axes to the plane (0 0; ½ ½ , ½ 0, 0 ½ ) 2 1 1 2 00, /3 /3, /3 /3) Four fold axis Six fold axis Two non-equivalent 4-fold axes to the plane; One non-equivalent 6-fold axes to the plane; One non-equivalent 2-fold axis to the plane; One non-equivalent 3-fold axis to the plane; 2 1 1 2 (00, ½ ½) and ( ½ 0, 0 ½ ) (0 0) ; ( /3 /3 , /3 /3) and ( ½ 0) 4/24/2013 L. Viciu| AC II | Symmetry in 2D Martin Buerger: An introduction to fundamental geometric features of crystals 23 2. Combining a reflection with translation A reflection combined with a translation to it is another reflection at ½ of that perpendicular translation 1. A rectangular cell 11 2 Translation3 1 2 2 1 3 * 2 *the mirror 2 is situated at ½ distance of the translation - Pair of motifs The mirror 2 is independent from 1 because the position of the objects (1 and 2) relative to the mirror in the center (2)of the cell is distinct from the position of the same objects relative to the first mirror (1) 4/24/2013 L.
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