3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci:...

22
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)

Transcript of 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci:...

Page 1: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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)

Page 2: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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

Page 3: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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

Page 4: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 5: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Symmetriesof H2O

Figure by MIT OCW.

Page 6: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005) Figure by MIT OCW.

Symmetriesof H2O

Page 7: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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)

Page 8: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 9: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 10: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 11: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

Symmetry in three dimensions

• Inversion is only meaningful in 3-dim

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Page 12: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 13: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 14: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 15: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 16: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 17: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 18: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 19: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.

Page 20: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

.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.

Page 21: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Combining rotations and translations

( )2 cosmT T T α= −

Page 22: 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS ... · 3.012 Fund of Mat Sci: Structure – Lecture 14 POINT GROUPS AND BRAVAIS LATTICES ... See Engel, T., and P. Reid.

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.