3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES Photo courtesy of Eric Gjerde 3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fund of Mat Sci: Structure – Lecture 14
POINT GROUPS AND BRAVAIS LATTICES
Photo 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 3.2.5
3.012 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 theory
12
E nω⎛ ⎞= +⎜ ⎟⎝ ⎠
h
3.012 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 H2O
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005) Figure by MIT OCW.
Symmetriesof H2O
The 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=I
Multiplication Table for Operators of the C2V Group 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)
D2h
Image of the Symmetry elements of the D2h group 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 rotation
Diagrams 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 D2h
Image of the Symmetry elements of the D2h group 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 D3h
Image 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 Symmetry
Diagrams 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)
BravaisLattice
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
Cubic
Hexagonal
Parameters Simple(P)
VolumeCentered (I)
BaseCentered (C)
FaceCentered (F)
a1 = a2 = a3α12 = α23 = α31
a1 = a2 = a3α23 = α31 = 900
α12 = 900
a1 = a2 = a3α12 = α23 = α31 = 900
a1 = a2 = a3α12 = α23 = α31 = 900
a1 = a2 = a3α12 = 1200
α23= α31= 900
a1 = a2 = a3α12 = α23 = α31 = 900
a1 = a2 = a3α12 = α23 = α31 < 1200
a3
a1
a2
4 Lattice Types
7 C
ryst
al C
lass
es
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Mirror and glide planes
Figures 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
( )2 cosmT T T α= −
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
32 Crystallographic Point Groups
Crystal System Schoenflies Symbol
Hermann-Mauguin Symbol
Order of the group
Laue Group
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
C1
C2Cs
C2h
D2
C2v
D2h
C4S4
C4v
C3
C3iD3C3vD3dC6
C3hC6hD6
D3hD6h
Th
TdOh
C6v
D2dD4h
C4h
D4
Ci
1
2
4
m
2/m
2/m
mm2
mmm
mmm
4/m
4/m
222
4
4224mm
4/m mm
4/m mm
42m
3
323m
6
1
2
22
4
4
4
8
448
88
8
3
16
1
3
6/m
6/m mm
3m
T
O
3
3m
66/m6226mm
6/m mm6m2
23m343243mm3m
666
66
12
12121212241224242448
Hexagonal
Cubic m3
m3m
1
The Crystallographic Point Groups and the Lattice Types.
Figure by MIT OCW.