3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Structure – Lecture 16STRUCTURE OF SOLIDSCharge density in paraelectric and ferroelectric PbTiO33.012 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 3.43.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Symmetry operations: rotation, reflection, inversion, roto-inversion2. Mirror+mirror=rotation3. Periodicity constrains rotations (1, 2, 3, 4, 6) → ten crystallographic point groups in 2d4. Bravais lattices5. International tables3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)BravaisLatticeTriclinicMonoclinicOrthorhombicTetragonalTrigonalCubicHexagonalParametersSimple(P)VolumeCentered (I)BaseCentered (C)FaceCentered (F)a1 = a2 = a3α12 = α23 = α31a1 = a2 = a3α23 = α31 = 900α12 = 900 a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = 1200α23= α31= 900a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = α23 = α31 < 1200a3a1a24 Lattice Types7 Crystal ClassesFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Compound ops. with translations:4) GlidesIllustration 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)Figure of object repetition 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.Tnmτ=&GG3.012 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 planesBravais lattices: simple cubicα-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 cubicBa, 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 cubicFigure 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 cellFigure 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 + BasisCrystal Structure = Lattice + basisLatticeBasisFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Bravais lattices: face-centered cubicAg, 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 StructuresFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Interstitials in Close-Packed Octahedral TetrahedralFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Interstitials in Close-Packed (001) (100) (010)(011)(110)(111) (111)(101)(111)Figure by MIT OCW.Diamond and Zincblend3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)PerovskitesFigure by MIT OCW.Perovskite CaTio3Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Sodium Chloride (rocksalt)Source: Wikipedia3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Cesium ChlorideImage of the structure of Cesium Chloride removed for copyright
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