1. Symmetry in two-dimension  2D unit cell Periodicity in 3-dim. – smallest repeated unit  unit cell

2. Symmetry in two-dimension

6. Symmetry elements

7. Periodicity in 3-dimetions ---Smallest repeated Unit --- Unit cell Symmetry elements, symbols, matrix representation ： Basic symmetry elements ☆ proper rotation C n → n; symbol in ‘point group → space group’ e.g. 4 z To suit inside a repeated unit in the space 1, 2, 3, 4, 5, 6 fold matrix representation

8. ☆ mirror planes  v,  h,  d --- m (a, b, c, d, n) ☆ center of symmetry ☆ translation along edges of the cell by fractions of the edge length ☆ improper rot. s →

9. ☆ screw axis rot. + tr. ☆ glide planes m + tr along a, b, c, diagonal,  a, b, c, n, d ☆ translation along edges of the cell by fractions of the edge length + tr along c  

10. Derived Symmetry within the lattice Unit cell or crystal lattice formed by 3-non-planar vectors Limitation of symmetry by periodicity t t   n  t t

11. Lattice Centering --- Pure translational rot:+ translation

12. Lattice Centering – pure translational P P  I  A A  B B  C C  F  R R  1P 2I, A, B, C 3R 4 F

13. Lattice centering Crystal system Min. sym. Max. sym.Cell parameters PTriclinic1 a  b  c,       90  P, IMonoclinic2 a  b  c,     90  P, I, F, BOrthorhombic222 a  b  c,       90  P, ITetragonal4 a  b  c,       90  PHexagonal6 a  b  c,     90 ,   120  PRPR Trigonal Rhombohedral 3333 3m a  b  c,     90 ,   120 , V a’  b’  c’,  ’   ’   ’  90 , V’ = 1/3V P, I, FCubic23 a  b  c,       90  Unit cell classifications

14. System and Point Group Position in Point-group Symbol Stereographic representation PrimarySecondaryTertiary Triclinic Only one symbol which denotes all directions in the crystal. Monoclinic The symbol gives the nature of the unique diad axis (rotation and/or inversion). 1 st setting: z-axis unique (001) 2 nd setting: y-axis unique (010) Orthorhombic Diad (rotation and/or inversion) along x- axis (100) Diad (rotation and/or inversion) along y- axis (010) Diad (rotation and/or inversion) along z-axis (001) Tetragonal Tetrad (rotation and/or inversion) along z-axis (001) Diads (rotation and/or inversion) along x- and y-axes (100) or (010) Diads (rotation and/or inversion) along [110] and [1 0] axes (110) (1 0) Trigonal and hexagonal Triad or hexad (rotation and/or inversion) along x- axis (001) Diads (rotation and/or inversion) along x-, y- and u- axes (100) ____ Diads (rotation and/or inversion) normal to x-, y-, u-axes in the plane(001);  (100) …… CubicDiads or tetrads (rotation and/or inversion) along (100) axes (100) Triads (rotation and/or inversion) along (111) axes (111) Diads (rotation and/or inversion) along (110) axes (110) xx yy zz xx Triclinic yy zz yy zz xx Monoclinic 1 st setting 2 nd setting yy zz xx Orthorhombic yy zz xx Tetragonal yy zz xx Trigonal and hexagonal uu yy zz xx Cubic Other of Positions in the Symbols of the Three-dimensional Point Groups as applied to Lattices Poles of directions for primary position Secondary ditto Tertiary ditto

15. Symmetry Operations and Space Groups a b c    a b c    a b c    a b c a b c a b c a b c The 14 Bravais lattices

16. c a b c a b c a b 120  c a b c a b c a b 續上頁 a1a1 b1b1 c1c1 a b c or P3m1

17. Laue symmetry unique part of sphere Triclinic 1/2 Monoclinic 2/m 1/4 Orthorhombic (m m m) 1/8 Tetragonal 1/8 1/16 Hexagonal 1/12 1/24 Trigonal 1/12 1/12 1/6 Cubic 1/24 1/48

18. Space Group Group Definition 1. a i  a j = a k where a k must be an element in the group 2. must have an identity element, I, so that a i  I = a i 3. The inverse of every element must also be an element in the group 4. associative law (a i a j ) a k = a i (a j a k )

19. 32 Point Gourps C n : rot. 1, 2, 3, 4, 6 (S n ) inverse rot. C nh : rot. + m  m C nv rot. + m  3m D n 3 rot. 222, 32(2), 422, 622, 23, 432  5 5 3 1 6

20. D nh rot. +m  + m  m m 2 4 m m 6 m m 9 3

22. m

23. (7)(32) (230)

24. ( 7) (32) (230) Crystal system point group space groups

27. Space group P2 1 /c origin shift basic sym elements rot. tr. tr. 

28. after origin shift to (0 0 0) after origin shift of (0, ¼, ¼) 

29. P21/cP21/c Monoclinic P12 1 /c1 Patterson symmetry P12/m1 2/m UNIQUE AXIS b, CELL CHOICE 1

30. P21/cP21/c

31. Space group Pnc2 Fig Completed worksheet

32. =  n  basic sym derived sym

33. P6mm 6mm Hexagonal P6mm Patterson symmetry P6mm

34. P6mm

38. P  C F  I [ det ]  2 P T  R ( Trigonal  Rhombohedral Cell ) [ det ]  3 R (0, 0, 0) ; (2/3, 1/3, 1/3) ; (1/3, 2/3, 2/3) R (0, 0, 0) ; (1/3, 2/3, 1/3) ; (2/3, 1/3, 2/3) [ det ]  3

40. P21/cP21/c Monoclinic 2/m UNIQUE AXIS b, DIFFERENT CELL CHOICE 1 P12 1 /c1 UNIQUE AXIS b, CELL CHOICE 1

41. P12 1 /n1 UNIQUE AXIS b, CELL CHOICE 2

42. P12 1 /a1 UNIQUE AXIS b, CELL CHOICE 3

44. Inverse  Inverse   transpose  direct space reciprocal space

45. Cell Transformation Cell 1 Cell 2 (a, b, c) (h. k. l) (x, y, z) ; (a*, b*, c*) ; (u, v, w) u  x, v  y, w  z where u, v, w integer

46. Transformation between a1a1 a2a2 c1c1 c2c2 a, b, c h, k, l reverse transpose a*, b*, c* x, y, z u, v, w reverse

47. P I P F F I Trigonal TS rhombohedral cell trigonal cell obverse (positive) reverse (negative)

48. Trigonal lattices a hex  a R  b R b hex  b R  c R c hex  a R  b R  c R a hex  b R  c R b hex  c R  a R c hex  a R  b R  c R or a hex  c R  a R b hex  a R  b R c hex  a R  b R  c R As for the hexagonal cell, in the conventional trigonal cell the threefold axis is chosen parallel to c, with a  b, unrestricted c,     90 , and   120 . Centred cells are easily amenable to the conventional P trigonal cell. Because of the presence of a treefold axis some lattices can exist which may be described via a P cell of rhombohedral shape, with unit vectors a R, b R, c R such that a R  b R  c R,  R   R   R, and the three fold axis along the a R  b R  c R direction. Such lattices may also be described by three hexagonal cells with basis vectors a hex, b hex, c hex defined according to These hexagonal cells are said to be in obverse setting. Three further triple hexagonal cells, said to be in reverse setting, can be obtained by changing a hex and b hex to  a hex and  b hex. The hexagonal cells in obverse setting have centring point at (0, 0, 0), (2/3, 1/3, 1/3), (1/3, 2/3, 2/3) While for reverse setting centring points are at (0, 0, 0), (1/3, 2/3, 1/3), (2/3, 1/3, 2/3) It is worth nothing that a rhombohedral description of a hexagonal P lattice is always possible. Six triple rhombohedral cells with basis vectors a’ R, b’ R

49. Trigonal T Obverse (positive) Reverse (negative) Rhombohedral Cell Trigonal Cell

52. In direct space: ☆ x, y, z fractional coordinates In reciprocal space: ☆ h, k, l plane, miller indices