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)
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⎞
⎛
E = hω ⎜ n + ⎟
2⎠
⎝
1. The quantization of vibrations:
2. Specific heat and excitations of a
Bose-Einstein ensemble
3. Symmetry operations (inversion,
rotation, mirror…) and elements
(points, axes, planes…)
4. Group theory
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Possible symmetries in a molecule
Table 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)
Symmetries
of H2O
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.
Symmetries
of H2O
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.
The 4 symmetry operations of H2O
form a group (called C2v)
1.
2.
3.
4.
Multiplication Table for Operators of the C2V Group
removed for copyright reasons.
Closure: A☼B is also in G.
Associativity: (A☼B) ☼C=A☼ (B☼C)
Identity: I☼A=A☼I
See Engel, T., and P. Reid. Physical Chemistry. Single volume ed.
Inverse: A☼inv(A)=inv(A) ☼A=I San Francisco, CA: Benjamin Cummings, 2005, p. 666, table 28.3.
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 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.
Courtesy of Marc De Graef. Used with permission.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Symmetry in three dimensions
• Inversion is only meaningful in 3-dim
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 cells
Diagrams 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)
7 Crystal Classes
4 Lattice Types
Bravais
Lattice
Parameters
Triclinic
a1 = a2 = a3
α12 = α23 = α31
Monoclinic
a1 = a2 = a3
α23 = α31 = 900
α12 = 900
Orthorhombic
a1 = a2 = a3
α12 = α23 = α31 = 900
Tetragonal
a1 = a2 = a3
α12 = α23 = α31 = 900
Trigonal
a1 = a2 = a3
α12 = α23 = α31 < 1200
Cubic
a1 = a2 = a3
α12 = α23 = α31 = 900
Hexagonal
a1 = a2 = a3
α12 = 1200
α23= α31= 900
Simple
(P)
a3
Volume
Centered (I)
Base
Centered (C)
Face
Centered (F)
a2
a1
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.
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.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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.
.
Combining rotations and translations
mT = T − 2 (T cos α )
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The Crystallographic Point Groups and the Lattice Types.
Crystal System
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
Hexagonal
Cubic
Schoenflies
Symbol
Hermann-Mauguin
Symbol
Order of the
group
C1
1
1
Ci
1
2
C2
Cs
2
m
2
2
C2h
2/m
4
D2
222
4
C2v
mm2
4
D2h
mmm
8
C4
S4
4
4
4
4
C4h
4/m
8
D4
C4v
422
4mm
8
8
D2d
D4h
42m
4/m mm
8
16
C3
3
3
C3i
3
6
D3
C3v
32
3m
6
6
D3d
C6
3m
12
6
6
C3h
C6h
D6
C6v
6
6/m
622
6mm
6
12
12
12
D3h
D6h
T
Th
O
Td
Oh
6m2
6/m mm
23
m3
432
43m
m3m
12
24
12
24
24
24
48
32 Crystallographic
Point Groups
Laue Group
1
2/m
mmm
4/m
4/m mm
3
3m
6/m
6/m mm
m3
m3m
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)