3.012 Fund of Mat Sci: Structure – Lecture 14POINT GROUPS AND BRAVAIS LATTICESPhoto courtesy of Eric Gjerde3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Wed Nov 2• Study: Allen and Thomas from 3.1.1 to 3.1.4 and 3.2.1, 3.2.4, and 3.2.53.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. The quantization of vibrations:2. Specific heat and excitations of a Bose-Einstein ensemble3. Symmetry operations (inversion, rotation, mirror…) and elements (points, axes, planes…)4. Group theory12Enω⎛⎞=+⎜⎟⎝⎠h3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Possible symmetries in a moleculeTable of symmetry elements and their corresponding operations removed for copyright reasons.See Engel, T., and P. Reid. Physical Chemistry. Single volume ed. San Francisco, CA: Benjamin Cummings, 2005, p. 658, table 28.1.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Symmetriesof H2OFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Figure by MIT OCW.Symmetriesof H2OThe 4 symmetry operations of H2O form a group (called C2v)1. Closure: A☼B is also in G.2. Associativity: (A☼B) ☼C=A☼ (B☼C)3. Identity: I☼A=A☼I4. Inverse: A☼inv(A)=inv(A) ☼A=IMultiplication Table for Operators of the C2VGroup removed for copyright reasons.See Engel, T., and P. Reid. Physical Chemistry. Single volume ed. San Francisco, CA: Benjamin Cummings, 2005, p. 666, table 28.3.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)D2hImage of the Symmetry elements of the D2hgroup in ethene removed for copyright reasons.See Engel, T., and P. Reid. Physical Chemistry. Single volume ed. San Francisco, CA: Benjamin Cummings, 2005, p. 682, figure 28.10.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Representation of a proper rotationDiagrams of various rotational axes removed for copyright reasons.See Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999, pp. 100-101, figures 3.10 and 3.11.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Representation of D2hImage of the Symmetry elements of the D2hgroup in etheneremoved for copyright reasons.See Engel, T., and P. Reid. Physical Chemistry. Single volume ed. San Francisco, CA: Benjamin Cummings, 2005, p. 682, figure 28.10.Courtesy of Marc De Graef. Used with permission.Symmetry in three dimensions• Inversion is only meaningful in 3-dim 3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Symmetry in three dimensions• Roto-inversion (improper rotation)Diagrams of rotoinversion axes removed for copyright reasons.See Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999, p. 128, figures 3.34 and 3.35.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Symmetry in three dimensions• Roto-reflection (improper rotation)Diagrams of the operation of a threefold rotoreflection axis removed for copyright reasons. See Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999, p. 129, figure 3.36.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Representation of D3hImage of the symmetry elements of a PCL5 molecule removed for copyright reasons.See Engel, T., and P. Reid. Physical Chemistry. Single volume ed.San Francisco, CA: Benjamin Cummings, 2005, page 658, figure 28.1(b).Courtesy of Marc De Graef. Used with permission.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Translational SymmetryDiagrams of one-dimensional periodicity removed for copyright reasons.See Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999, p. 92, figure 3.1.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Primitive, multiple, and unit cellsDiagrams of primitive and nonprimitive cells removed for copyright reasons. See Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999, p. 94, figures 3.4 and 3.5.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)BravaisLatticeTriclinicMonoclinicOrthorhombicTetragonalTrigonalCubicHexagonalParametersSimple(P)VolumeCentered (I)BaseCentered (C)FaceCentered (F)a1 = a2 = a3α12 = α23 = α31a1 = a2 = a3α23 = α31 = 900α12 = 900 a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = 1200α23= α31= 900a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = α23 = α31 < 1200a3a1a24 Lattice Types7 Crystal ClassesFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Mirror and glide planesFigures of reflectional symmetry and symmetrical pattern generation removed for copyright reasons. See Allen, S. M., and E. L. Thomas. The Structure of Materials. New York, NY: J. Wiley & Sons, 1999, pp. 98-99, figures 3.7 and 3.8.Diagram of rotation axis and parallel translation removed for copyright reasons.See page 130, Figure 3.38 in Allen, S. M., and E. L. Thomas. The Structure of Materials.New York, NY: J. Wiley & Sons, 1999..Diagram of object repetitions by operation of 41, 42, and 43 screw axes. Removed for copyright reasons.See page 133, Figure 3.39 in Allen, S. M., and E. L. Thomas. The Structure of Materials.New York, NY: J. Wiley & Sons, 1999.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Combining rotations and translations()2cosmT T Tα=−3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)32 Crystallographic Point GroupsCrystal System Schoenflies SymbolHermann-Mauguin SymbolOrder of the groupLaue GroupTriclinicMonoclinicOrthorhombicTetragonalTrigonalC1C2CsC2hD2C2vD2hC4S4C4vC3C3iD3C3vD3dC6C3hC6hD6D3hD6hThTdOhC6vD2dD4hC4hD4Ci124m2/m2/mmm2mmmmmm4/m4/m22244224mm4/m mm4/m mm42m3323m612224448448888316136/m6/m mm3mTO33m66/m6226mm6/m
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