3.012 Fund of Mat Sci: Structure – Lecture 15
TILES, TILES, TILES, TILES, TILES, TILES
Photo courtesy of Chris Applegate.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time:
1. Symmetry operations forming a group,
and symmetry operations as matrices
2. Molecular symmetries: rotation,
inversion, rotoinversion
3. Examples of C2v(water), D2h(ethene)
4. Basic ideas about 3d, crystals, and
lattices
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Why do we do this ?
• Constraints on physical properties
• Crystal structures from spectroscopies
• Captures universal properties
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
From graphite
to nanotubes
Photos of apples arranged to resemble
atoms in graphite and carbon nanotubes
removed for copyright reasons.
Source: Wikipedia
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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: Ci
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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.
1, 2, 3...
% 2,3...
% %
Rotoreflection: 1,
(Inversion: 1)
Rotoinversion:
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Translational Symmetry
3 Dim
1 Dim
t1
a
2 Dim
t2
Double
t1
t2 Triple
Single
t1
t1
Cl-
Cu+
t2
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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 cos α )
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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…
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Bravais Lattices
b
a
Figure by MIT OCW.
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.
32 crystallographic point groups in 3d
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
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)
Schoenflies notation
Anx , where
• A:{C,D,T,O,S}
• n:{□,2,3,4,6} (□ means no symbol}
• x:{□,s,i,h,v,d}
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)