3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Structure – Lecture 20SYMMETRIES AND TENSORSPhotograph removed for copyright reasons.Einstein explaining the Einstein convention3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Mon Nov 28• Study: 3.3 Allen-Thomas (Symmetry constraints) • Read all of Chapter 1 Allen-Thomas3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Atoms as spherical scatterers2. Huygens construction → Laue condition3. Ewald construction4. Debye-Scherrer experiments3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Scalars, vectors, tensors3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Scalars, vectors, tensors11111221332211222 2333311322333jEEEjEEEjEEEσσσσσ σσσ σ=++=+ +=+ +3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Einstein’s convention3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Transformation of a vector3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Transformation of a vector3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Orthogonal Matrices3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Transformation of a tensor3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Transformation law for products of coordinates3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Neumann’s principle• the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Symmetry constraints• Determine the crystallographic point group• Choose a generator group (set of symmetry operation which fully generates the complete point group symmetry)• Transform all components of a tensor by each of the symmetry elements• Impose Neumann’s principle that a tensor component and its transformed remain identical for a symmetry operation3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Symmetry constraints3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Symmetry constraints3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Scalar, vector, tensor properties• Mass (0), polarization (1), strain (2)YUndeformedXDeformedZXPP'uRPP'R'u = R' - RoZYFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Physical properties and their relation to symmetry• Density (mass, from a certain volume)• Pyroelectricity (polarization from temperature)• Conductivity (current, from electric field)• Piezoelectricity (polarization, from stress)• Stiffness (strain, from stress)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Curie’s Principle• a crystal under an external influence will exhibit only those symmetry elements that are common to both the crystal and the perturbin
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