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Introduction to Crystals Symmetry

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Introduction to Crystals Symmetry Introduction to crystals symmetry Symmetry in 3D 5/1/2013 L. Viciu| AC II | Symmetry in 3D 1 Outlook • Symmetry elements in 3D: rotoinversion, screw axes • Combining symmetry elements with the lattice • Space group symbols/representation • Wyckoff positions • Crystallographic conventions • Symmetry in crystal systems 5/1/2013 L. Viciu| AC II | Symmetry in 3D 2 Definitions - a remainder • Unit cell: The smallest volume that can generate the entire crystal structure only by means of translation in three dimensions. • Lattice: A rule of translation. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 3 Bravais Basis/ Crystal + lattice Motif structure 1. single atom: Au, Al, Cu, Pt 1. FCC 2. molecule: solidCH4 2. FCC + - + 3. ion pairs: Na /Cl 3. Rock salt 4. atom pairs: Carbon, Si, Ge 4. diamond •A crystal system is described only in terms of the unit cell geometry, i.e. cubic, tetragonal, etc •A crystal structure is described by both the geometry of, and atomic arrangements within, the unit cell, i.e. face centered cubic, body centered5/1/2013 cubic, etc. L. Viciu| AC II | Symmetry in 3D 4 The Bravais lattices 5/1/2013 L. Viciu| AC II | Symmetry in 3D 5 From 2D to 3D • Bravais lattices may be seen as build up layers of the five plane lattices: cubic and tetragonal stacking of square lattice layers orthorhombic P and I stacking of rectangular layers orthorhombic C and F stacking of rectangular centered layers rhombohedral stacking of hexagonal layers hexagonal stacking of hexagonal layers monoclinic stacking of oblique layers triclinic stacking of oblique layers 5/1/2013 L. Viciu| AC II | Symmetry in 3D 6 Combining symmetry elements For two dimensions – 5 lattices – 10 point groups – 17 plane groups For three dimensions – 14 Bravais lattices – 32 point groups – 230 space groups 5/1/2013 L. Viciu| AC II | Symmetry in 3D 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 5/1/2013 L. Viciu| AC II | Symmetry in 3D 8 Symmetry elements in 3D • In three dimensions we have additional symmetry operators: 1. inversion center = symmetry center 2. inversion axis = rotoinversion or improper rotation axis 3. screw axes • In addition, the mirror line becomes a mirror plane, and the glides become glide planes. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 9 1. The center of symmetry/inversion center Symbol: 1 “one bar”; Graphical: • The symmetry center: the intersection point of a mirror plane and a 2-fold axis Inversion center (x,y,z) (x, y, z) • It is always present in one of the situations: a rotation axis with even multiplicity and a if an inversion axis with odd multiplicity reflection to it is present is present P3 P2/m 5/1/2013 L. Viciu| AC II | Symmetry in 3D 10 • An inversion center requires a 2-fold axis and a mirror plane to it. • Therefore, an even rotation with a reflection perpendicular to it will give an inversion center. A A= inversion, 1 2 1 i 1rotationby1802 reflection3 3 inversion *The order of 1 has the coordinates (x y z) the operation is not important 2 has the coordinates ( x y z ) 3 has the coordinates ( x y z ) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 11 • The point groups that contain an inversion center are called Laue groups. • If you take away the translational part of the space group symmetry and add an inversion center, you end up with the Laue group. • The Laue groups are used in diffraction: describe the symmetry of the diffraction pattern. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 12 5/1/2013 L. Viciu| AC II | Symmetry in 3D 13 Centrosymmetric structures • If an inversion center is present then the structure is centrosymmetric • In a centrosymmetric structure, the unit cell is chosen so that the origin lays on the inversion center 82% of inorganic crystals are centrosymmetric because Inversion center leads to equal forces in opposing directions favoring stability George M. Sheldrick. • In a non-centrosymmetric space group, the origin is chosen at a highest symmetry point. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 14 32 point groups 11 centrosymmetric point group 20 non-centrosymmetric point groups: + piezoelectrics 432 class* *432 class has a polar axis but no piezoelectricity 10 point groups 10 point groups with unique polar with no unique axis: polar axis Pyroelectric + ferroelectrics • Polar direction = a crystal direction that is not related by symmetry to the opposed direction 5/1/2013 L. Viciu| AC II | Symmetry in 3D 15 Space groups and enantiomorphous molecules Enantiomorphous molecules crystallise in space groups without inversion centre , i, and without reflection planes, m. Altogether there are 11 Point groups possible for enantiomorphous molecules. (“Biological Point Groups”) Enantiomorphous crystals of tartaric acid (monoclinic structure, space group P21) The most common chiral space groups are P212121, P21, P1 and C2221. 15%5/1/2013 of all crystals are enantiomorphicL. Viciu| AC II | Symmetry and in 3D potentially optically active16 Klockmanns Lehrbuch der Mineralogie, 16. Auflage, Enke VerlagStuttgart 1978 2. Axes of inversion/rotoinversion • Rotation + center of symmetry = inversion axis, n , pronounced n bar • It is also called improper rotation axes (Schoenflies symbols, Sn) • Two objects related by an operation of inversion axis are enantiomorphous 2 m Symbol/graphic 1. inversion 2. 180ᵒ 1 axis atom 2 m 3 3. 4. 4 6 m 5/1/2013 L. Viciu| AC II | Symmetry in 3D 17 if 3 31, 4 41 Therefore 4 bar is a distinct operation Spinoid (like a squashed Td) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 18 3. Screw axes: nm Rotation + Translation in axial direction The rotation is combined with a translation in a defined crystallographic direction by a defined step (half a unit cell for twofold screw axes, a third of a unit cell for threefold screw axes, etc counter clockwise rotation with 360ᵒ/n + Translation with m/n (m<n) c. Hammond, the basics of crystallography and diffraction Srew axis nm 5/1/2013 L. Viciu| AC II | Symmetry in 3D 19 3. Screw axes: nm Rotation + Translation in axial direction: 21: ½ displacement on c 1 2 31: /3 displacemnt on c; 32: /3 displacement on c; 2 41: ¼ displacemnt on c; 42 /4 = ½ displacement on c; 43: ¾ displacement on c 1 2 1 3 61: /6 displacement on c; 62: /6= /3 displacement on c; 63: /6= ½ displacement on c, etc 5/1/2013 NotL. shownViciu| AC II | Symmetry in 3D 20 (Fig. 140) Inorganic structural chemistry, U. Mueller Screw axis of order 3: 31; 32 31 32 rotation by a 3-fold axis and rotation by a 3-fold axis and then translation by 1/3 then translation by 2/3 T T is translation periodicity Red color is used only to Is translation component = 1/3 show the objects obtained by translation periodicity 5/1/2013 L. Viciu| AC II | Symmetry in 3D 21 Screw axis of order 3: 31; 32 31 32 rotation by a 3-fold axis and rotation by a 3-fold axis and then translation by 1/3 then translation by 2/3 T 3 gives a clockwise spiral 31 gives an anticlockwise spiral 2 5/1/2013 L. Viciu| AC II | Symmetry in 3D 22 41 4 43 2 T T T 4 gives an 1 4 gives a anticlockwise spiral 3 clockwise spiral Color is used only to show the objects obtained by translation periodicity; otherwise the color represent the same 5/1/2013 L. Viciu| AC II | Symmetry motifin 3D 23 42 42 gives two motifs at every level and they are 180 away from each other 42 gives two spirals - one left handed (anticlockwise) and one right handed (clockwise) – in the same pattern T 5/1/2013 L. Viciu| AC II | Symmetry in 3D 24 Quartz symmetry View along c axis - quartz (hexagonal) 573C - quartz (trigonal) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 25 P6221 and P6421 P3121 and P3221 Glide planes in Space Groups • In 2D the gliding (translation) is in the direction of the dashes: g1 g2 2 3 1 • In 3D we can look at a glide from 4 different directions: Down from the top (1) Edge on and to the gliding direction (2) Edge on and to the gliding direction (3) Edge on and in between normal to and parallel to the gliding direction (4) (1) (3) * * - gliding direction 5/1/2013 (2)L. Viciu| AC II | Symmetry in 3D 26 (4) (1) Looking down from top of a glide use or or a glide b glide n glide = diagonal glide b b a a a = ½ of a axis = ½ of b axis = ½ (a+b) axis Ex: n-glide b a c glide is represented in plane as well and the translation is ½ of c axis 5/1/2013 L. Viciu| AC II | Symmetry in 3D 27 (2) Looking normal to the gliding direction use a dashed line b Ex: b glide a (3) Looking in the direction of gliding use a dotted line b Ex: b glide a (4) Looking not parallel not perpendicular to the direction of gliding use a mixed line b Ex: b glide a 5/1/2013 L. Viciu| AC II | Symmetry in 3D 28 d-glide = diamond glide • Found in centered cells only (not in primitive lattices) • It originates from the diamond structure where the diamond glide has been first observed. • The translation is ½ of the ½(a+b) axis ½ (a+b) a+b ½ (½ (a+b)) 5/1/2013 L.
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