1. (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=7 E7E396BF006E209&playnext_from=PL&index=1 Fall 2005, lectures given by Professor Bernhardt Wuensch References for the first four parts: (2) Ref. “Elementary crystallography”, Martin J. Buerger, 1963 (out of print, available in Physics Library) (3) International Tables for Crystallography (International Unions for Crystallography) V. A, B, C, … http://it.iucr.org/Ab/contents/

2. crystallography X-ray crystallography Optical crystallography (polarized light) Geometrical crystallography (symmetry theory) crystallography CrystalMapping or geometry

3. Basic Symmetry (Two hours)

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

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

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

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

8. 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

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

10. In 3-D, one more operation x y zR L Inversion Rotation 1D: Translation

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

12. Add another translation vector X Already covered by andare non-colinear. 2D space lattice. exist O colinear.  :(p integer) not a new translation vector

13. Lattice: frame work of a periodic crystalline structure (same environment for every point)                              

14. There are many ways to choose a cell with the same area. In 2D lattices: Define the area uniquely associated with a lattice point. Unit cell Array of lattice pointscell conjugate translations

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 Cartesian coordinate Rational direction integer  Use lattice net to describe is much easier! Extended to 3D  In general 2D u, v, w: integer

17.  Notation for rational planes:  2D case – line: line equation At1At1 Bt2Bt2  3D case – plane: plane equation At1At1 Bt2Bt2 Ct3Ct3 convert to integers Rational intercept plane (h k l) Equation of intercept plane x y x y z

18.  How many planes are there?  2D: A  B lines At1At1 Bt2Bt2 At1At1 Bt2Bt2 A = 2, B = 3A = 2, B = 2

19. 1 st plane 2 nd plane 3 rd plane 1/l 1/k 1/h x y z n th plane n = ABC At1At1 Bt2Bt2 Ct3Ct3  3D: ABC planes ABC pq r Common factor number of planes =

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

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 are basic translation vectors of the cell