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Course of MIT 3.60 Symmetry, Structure and Tensor Properties of Materials (abbreviation: SST)

Course of MIT 3.60 Symmetry, Structure and Tensor Properties of Materials (abbreviation: SST). http://www.youtube.com/watch?v=vT_6DlaHcWQ&feature=PlayList&p=7E7E396BF006E209&playnext_from=PL&index=1. Fall 2005, lectures given by Professor Bernhardt Wuensch.

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Course of MIT 3.60 Symmetry, Structure and Tensor Properties of Materials (abbreviation: SST)

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  1. Course of MIT 3.60 Symmetry, Structure and Tensor Properties of Materials (abbreviation: SST) http://www.youtube.com/watch?v=vT_6DlaHcWQ&feature=PlayList&p=7E7E396BF006E209&playnext_from=PL&index=1 Fall 2005, lectures given by Professor Bernhardt Wuensch

  2. Ref. “Elementary crystallography”, Martin J. Buerger, 1963. (out of print) International Tables for X-ray Crystallography (International Unions for Crystallography) V. I, II, III, … International Tables for Crystallography (International Unions for Crystallography) V. A, B, C, …

  3. crystallography Crystal Mapping or geometry X-ray crystallography Optical crystallography (polarized light) crystallography Geometrical crystallography (symmetry theory)

  4. Basic Symmetry (Two hours)

  5. Geometrical crystallography: the study of patterns and their symmetry Example Motif Are any of these patterns the same or are there all different?

  6. : operation of translation magnitude, direction, no unique origin, like a plain vector

  7. Other symmetry? A A location of rotation axis Rotation: A angle of rotation A 2 fold rotation

  8. How about this one? m? Yes! New type of transformation! Reflection! Symbol used for reflection is m (mirror). m? No! m? Yes!

  9. Definition of Symmetry element: Symmetry element is the locus of points left unmoved ( invariant) by the operation. What we have found for 2-dimensional symmetry operations? Translation: Reflection: Rotation: in the above case

  10. y Reflection: x & pass through the origin y Rotation: x Translation: Reflection: Rotation: That is all we can do in 2D!

  11. In 3-D, one more operation Inversion z R y x L 1D: Translation Rotation

  12. Analytical symbol Individual Operation Geometrical symbol m  Rotation axis n = integer Analytical symbol Individual Operation Geometrical symbol Reflection m  Rotation n A n - gon 1 (no symmetry)

  13. exist Add X Already covered by and are non-colinear. 2D space lattice.

  14. Define the area uniquely associated with a lattice point. Unit cell Array of lattice points cell The reverse is not true! conjugate translations There are many ways to choice .

  15. Different cells with the same area. Which one to use? Rules: (1) pick the shortest translations; (2) pick that display the symmetry of the lattice.

  16. Handedness chiral-molecules chirality Rational direction  integer Cartesian coordinate Use lattice net to describe is much easier! In general 2D Extended to 3D 

  17. Miller Indices for rational plane:  2D case – line: line equation Bt2 At1  3D case – plane: plane equation Bt2 convert to integers Ct3 At1 Equation of intercept plane Rational intercept plane

  18.  How many planes are there? Bt2  2D: ABlines 3D: ABC planes Bt2 At1 y Ct3 At1 1/k 1/h x 1st plane 1/l 2nd plane 3rd plane z nth plane n = ABC

  19. r number of planes = A B C p q Common factor

  20. Crystallographic equivalent? Individual plane Example: (hkl) Symmetry related set {hkl} {100} z y x z {100} Different Symmetry related set y x

  21. Coordination of an atom in a cell: coordinate of an atom x: fraction of unit length of y: fraction of unit length of z: fraction of unit length of Where , , : basic translation vectors of the cell

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