Learn · Crystals, Symmetry and Tensors
Lecture 29 of 93Week 4
Trace of a rotation matrix
Prof. Rajesh Prasad · IIT Delhi
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- Crystals, Symmetry and Tensors, IIT Delhi (at NPTEL)
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All the lectures
- What is Crystal?
- Lattice and Motif
- Lattice Translation
- Unit Cell
- Voronoi Cell
- Unit cell, Basis, Fractional Coordinates
- Miller indices for direction
- Miller indices for planes
- Line of intersection between two planes
- Plane normal of a plane for cubic crystal
- Weiss Zone Law
- Application of Weiss Zone Law
- Coordinate transformation
- Reciprocal Basis
- Reciprocal lattice vector (G*_hkl) is perpendicular to plane (hkl)
- Length of G*_hkl is reciprocal of interplanar spacing (d_hkl)
- Metric tensor
- Review
- Some properties of metric tensor in real and reciprocal space
- Transformation of direction and planes
- Volume of a crystallographic unit cell
- Orthographic projection of crystals
- Stereographic projection-I
- Stereographic projection-II
- Definition of symmetry
- What is a group?
- Matrix Representation of symmetry operation-I
- Matrix Representation of symmetry operation-II
- Trace of a rotation matrix
- Coordinate transformation of a matrix representing symmetry operation
- Chirality
- Types of symmetry operation
- Relation between cube and tetrahedron
- 2-fold roto-inversion is a mirror: Proof using matrix method
- Invariance of a trace of symmetry operation during basis transformation
- Crystallographic restriction theorem
- Geometric proof of crystallographic restriction theorem
- Quasicrystals
- Affine transformation and Seitz Operator
- Frieze group-I
- Frieze group-II
- 2D point group-I
- 2D point group-II
- The angle between mirrors is half the angle of rotation
- Different kinds of 2D lattices
- 10- point groups associated with 5 lattices
- Plane group-I
- Plane group-II
- 3D Point Groups I : Introduction
- 3D Point Groups II : Proper Monoaxial Groups
- 3D Point Groups III : Dihedral Groups
- 3D Point Groups IV : Proper cubic point groups
- 3D Point Groups V : Enantiomorphic Point Groups
- 3D Point Groups VI : Improper Monoaxial Point Groups
- 3D Point Groups VII : Combination of a mirror passing through an n-fold axis
- 3D Point Groups VIII : Combination of a mirror perpendicular to an n-fold axis
- 3D Point Groups IX : Combination of a mirror perpendicular and parallel to an n-fold axis
- 3D Point Groups X : Mirror parallel to a rotoinversion axis
- 3D Point Groups XI : Improper Cubic Point Groups
- 3D Point Groups XII : Classification of Point Groups into 7 Crystal Systems
- 3D Point Groups XIII : Chirality Revisited
- 3D Point Groups XIV : Centrosymmetric Point Groups
- 3D Point Groups XV : Laue Classes
- 3D Point Groups XVI : Group-Subgroup Relations
- 3D Point Groups XVII : Polar Groups
- 3D Point Groups XVIII : Holohedral or Holosymmetric class
- 3D Space Groups I : Introduction
- 3D Space Groups II : Trigonal / Rhombohedral / Hexagonal confusion
- 3D Space Groups III : Bravais Lattices in Trigonal Crystal System
- 3D Space Groups IV : Screw Axes and Screw Operations
- 3D Space Groups V : Glides in 3D
- 3D Space Groups VI : Symmorphic and Nonsymmorphic Space Groups
- 3D Space Groups VII : 73 Symmorphic Space Groups
- 3D Space Groups VIII : Interpretation of International Table Page Part -1 : Headline
- 3D Space Groups IX : Interpretation of International Table Page Part -2 : Symmetry Diagrams
- 3D Space Groups X : Interpretation of International Table Page Part -3 : General Position Diagram
- 3D Space Groups XI : Interpretation of International Table Page Part -4 : Origin
- 3D Space Groups XII : Interpretation of International Table Page Part -5 : Asymmetric Unit
- 3D Space Groups XIII : Interpretation of International Table Page Part -6 : Symmetry Operations
- 3D Space Groups XIV : Interpretation of International Table Page Part -7 : Generators Selected
- 3D Space Groups XV : Interpretation of International Table Page Part -8 : Positions
- 3D Space Groups XVI : Interpretation of International Table Page Part -9 : General Position Coordinate Triplet
- Representation of Physical Properties of Crystals by Tensors
- Index Notation
- Coordinate Transformation of Vectors
- Some properties of Coordinate Transformation matrix between two orthonormal bases
- Coordinate Transformation for a second rank tensor
- Coordinate Transformation of tensors of different rank
- Magnitude of a second rank tensor property in a given direction
- Neumann's Principle
- Physical properties represented by second rank tensors are CENTROSYMMETRIC
- Effect of symmetry on the components of tensor representing a physical property
- Cubic crystals are isotropic for second rank tensor properties