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3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fund of Mat Sci: Structure – Lecture 16
STRUCTURE OF SOLIDS
Charge density in paraelectric and ferroelectric PbTiO3
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Mon Nov 7
• Study: Allen and Thomas 3.2.2 up to pag. 140, and 3.4
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time:
1. Symmetry operations: rotation, reflection, inversion, roto-inversion
2. Mirror+mirror=rotation3. Periodicity constrains rotations (1, 2, 3, 4, 6)
→ ten crystallographic point groups in 2d4. Bravais lattices5. International tables
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)
Compound ops. with translations:4) Glides
Illustration of glide symmetry removed for copyright reasons.See p. 99, figure 3.8 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)
Compound ops. with translations:5) Screw (in 3 dim)
Illustration of rotation and parallel translation removed for copyright reasons. See p. 130, figure 3.38 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)
Tn mτ =rr
Figure of object repetition removed for copyright reasons.See p. 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)
Space groups
• All possible combinations of point group symmetries with translations
• 230 in total• We have seen 14 (Bravais lattices)• Incorporate all possible translation with
symmetries, and add screw axis and glide planes
Bravais lattices: simple cubic
α-phase of polonium….3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.Figure by MIT OCW.
Bravais lattices: body-centered cubic
Ba, Cr, Cs, Fe, K, Li, Mo, Na, Nb, Rb, Ta, Tl, V, W…3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Bravais lattices: body-centered cubic
Figure by MIT OCW.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Primitive unit cell and conventional unit cell
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Wigner-Seitz cell
Figure by MIT OCW.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Crystal Structure = Lattice + Basis
Crystal Structure = Lattice + basis
Lattice
BasisFigure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Bravais lattices: face-centered cubic
Ag, Al, Au, Ca, Cu, Ir, Ni, Pb, Pd, Pt, Sc, Sr…
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Close-Packed Structures
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Interstitials in Close-Packed
Octahedral Tetrahedral
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Miller Indices
(001) (100) (010)
(011)(110)
(111) (111)
(101)
(111)
Figure by MIT OCW.
Diamond and Zincblend
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Perovskites
Perovskite CaTio3
Figure by MIT OCW.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Sodium Chloride (rocksalt)
Source: Wikipedia
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Cesium Chloride
Image of the structure of Cesium Chloride removed for copyright reasons.