3.012 Fund of Mat Sci: Structure Lecture 15 TILES, TILES, TILES, TILES, TILES, TILES Photo courtesy of Chris Applegate.
Homework for Fri Nov 4 Study: Allen and Thomas from 3.1.1 to 3.1.4 and 3.2.1, 3.2.4 (just read), and 3.2.5 (only crystal systems, Bravais lattices, unit cells)
Last time: 1. Symmetry operations forming a group, and symmetry operations as matrices 2. Molecular symmetries: rotation, inversion, rotoinversion 3. Examples of C 2v (water), D 2h (ethene) 4. Basic ideas about 3d, crystals, and lattices
Why do we do this? Constraints on physical properties Crystal structures from spectroscopies Captures universal properties
From graphite to nanotubes Photos of apples arranged to resemble atoms in graphite and carbon nanotubes removed for copyright reasons. Source: Wikipedia
Point group symmetries in 3 dim: 1) Rotations (axis: diad, triad...) Diagrams of various rotational symmetries removed for copyright reasons. See pages 100-101, Figures 3.10 and 3.11, in Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999. Figure by MIT OCW. International notation: 1, 2, 3, 4, 6 Schoenflies: C i
Point group symmetries in 3 dim: 2) Reflections (mirror), Diagrams of reflectional symmetry removed for copyright reasons. See p. 98, figure 3.7 in Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999. International: m
Point group symmetries in 3 dim: 3) Inversion, rotoinversion, rotoreflection Diagrams of rotoinversion axes removed for copyright reasons. See p. 128, figures 3.34 and 3.35, in Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999. 2-Fold Gyroid Twisted Apical Bipyramid 4 - Figure by MIT OCW. Rotoinversion: Rotoreflection: (Inversion: ) 1 1, 2, 3... 1, %%% 2,3...
Two mirror planes rotation axis Image of reflectional symmetry and rotation removed for copyright reasons. See p. 104, figure 3.14 in Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999.
Translational Symmetry 1 Dim 3 Dim t 1 a 2 Dim Double t 1 t 2 t 2 Triple Single t 1 t 2 t 1 Cl - Cu + Figure by MIT OCW.
Rotations compatible with translations Images removed for copyright reasons. See p. 102, figures 3.12 and 3.13, in Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999. mt = T 2( T cosα )
Ten crystallographic point groups in 2d Illustrations of the ten crystallographic point groups removed for copyright reasons. See p. 106, figure 3.18, in Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999.
Bravais Lattices Infinite array of points with an arrangement and orientation that appears exactly the same regardless of the point from which the array is viewed. r r r r R = la+ mb + nc l,m and n integers r r r a, b, c primitive lattice vectors 14 Bravais lattices exist in 3 dimensions (1848) M. L. Frankenheimer in 1842 thought they were 15. So, so naïve
Bravais Lattices b a Figure by MIT OCW.
4 Lattice Types Bravais Lattice Parameters Simple (P) Volume Centered (I) Base Centered (C) Face Centered (F) Triclinic a 1 = a 2 = a 3 α 12 = α 23 = α 31 Monoclinic a 1 = a 2 = a 3 α 23 = α 31 = 90 0 α 12 = 90 0 7 Crystal Classes Orthorhombic Tetragonal a 1 = a 2 = a 3 α 12 = α 23 = α 31 = 90 0 a 1 = a 2 = a 3 α 12 = α 23 = α 31 = 90 0 Trigonal a 1 = a 2 = a 3 α 12 = α 23 = α 31 < 120 0 Cubic a 1 = a 2 = a 3 α 12 = α 23 = α 31 = 90 0 Hexagonal a 1 = a 2 = a 3 α 12 = 120 0 α 23 = α 31 = 90 0 a 3 a 2 a 1 Figure by MIT OCW.
32 crystallographic point groups in 3d The Crystallographic Point Groups and the Lattice Types. Crystal System Schoenflies Symbol Hermann-Mauguin Symbol Order of the group Laue Group Triclinic C 1 1 C i 1 1 2 1 Monoclinic C 2 C s 2 m 2 2 2/m Orthorhombic C 2h D 2 2/m 222 4 4 mmm C 2v mm2 4 Tetragonal D 2h C 4 S 4 C 4h mmm 4 4 4/m 8 4 4 8 4/m Trigonal Hexagonal Cubic D 4 C 4v D 2d D 4h C 3 C 3i D 3 C 3v D 3d C 6 C 3h C 6h D 6 C 6v D 3h D 6h T T h O T d O h 422 4mm 42m 4/m mm 3 3 32 3m 3m 6 6 6/m 622 6mm 6m2 6/m mm 23 m3 432 43m m3m 8 8 8 16 3 6 6 6 12 6 6 12 12 12 12 24 12 24 24 24 48 4/m mm 3 3m 6/m 6/m mm m3 m3m Figure by MIT OCW.
Schoenflies notation A nx, where A:{C,D,T,O,S} n:{,2,3,4,6} ( means no symbol} x:{,s,i,h,v,d}
Reading the International Tables Figure removed for copyright reasons. See p. 157, figure 3.59 in Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999.