Post on 11-Aug-2020
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fund of Mat Sci: Structure – Lecture 20
SYMMETRIES AND TENSORS
Einstein explaining the Einstein convention
Photograph removed for copyright reasons.
3.012 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-Thomas
3.012 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 experiments
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Scalars, vectors, tensors
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Scalars, vectors, tensors
1 11 1 12 2 13 3
2 21 1 22 2 23 3
3 31 1 32 2 33 3
j E E Ej E E Ej E E E
σ σ σσ σ σσ σ σ
= + += + += + +
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Einstein’s convention
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Transformation of a vector
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Transformation of a vector
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Orthogonal Matrices
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Transformation of a tensor
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Transformation law for products of coordinates
3.012 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 crystal
3.012 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 operation
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Symmetry constraints
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Symmetry constraints
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Scalar, vector, tensor properties
• Mass (0), polarization (1), strain (2)
Figure by MIT OCW.
Y
Und
efor
med
X
DeformedZ
X
P P'u
R
PP'
R'
u = R' - R
o
Z
Y
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 influence