1. Chapter 1 Crystal Structures

2. Two Categories of Solid State Materials Crystalline: quartz, diamond….. Amorphous: glass, polymer…..

3. Ice crystals

4. crylstals

7. Lattice Points, Lattice and Unit Cell How to define lattice points, lattice and unit cell?

8. LATTICE LATTICE = An infinite array of points in space, in which each point has identical surroundings to all others. CRYSTAL STRUCTURE = The periodic arrangement of atoms in the crystal. It can be described by associating with each lattice point a group of atoms called the MOTIF (BASIS)

10. Notes for lattice points Don't mix up atoms with lattice points Lattice points are infinitesimal points in space Atoms are physical objects Lattice Points do not necessarily lie at the centre of atoms

11. An example of 2D lattice

12. An example of 3D lattice

13. Unit cell A repeat unit (or motif) of the regular arrangements of a crystal is defined as the smallest repeating unit which shows the full symmetry of the crystal structure Unit cell A repeat unit (or motif) of the regular arrangements of a crystal is defined as the smallest repeating unit which shows the full symmetry of the crystal structure

14. More than one ways

15. How to assign a unit cell

16. A cubic unit cell

17. 3 cubic unit cells

18. Crystal system Crystal system is governed by unit cell shape and symmetry

19. The Interconversion of Trigonal Lattices t1t1 t2t2 t1t1 t2t2 γ=120° 兩正三角 柱合併體

20. The seven crystal systems

21. Symmetry Space group Space group = point group + translation point group Space group point group

22. Definition of symmetry elements ------------------------------------------------------------- Elements of symmetry ------------------------------------------------ Symbol Description Symmetry operations --------------------------------------------------------------------- E Identity No change  Plane of symmetry Reflection through the plane i Center of symmetry Inversion through the center C n Axis of symmetry Rotation about the axis by (360/n) o S n Rotation-reflection Rotation about the axis by (360/n) o axis of symmetry followed by reflection through the plane perpendicular to the axis ---------------------------------------------------------------------

23. Center of symmetry, i

24. Rotation operation, C n

25. Plane reflection, 

26. Matrix representation of symmetry operators

27. Symmetry operation

28. Symmetry elements

29. space group = point group + translation Symmetry elements Screw axes2 1 (//a), 2 1 (//b), 4 1 (//c) 4 2 (//c), 3 1 (//c) etc Glide planes c-glide ( ┴ b), n-glide, d-glide etc

30. 2 1 screw axis // b-axis

31. Glide plane

32. Where are glide planes?

33. Examples for 2D symmetry http://www.clarku.edu/~djoyce/wallpaper/seventeen.html

34. Examples of 2D symmetry

35. General positions of Group 14 (P 2 1 /c) [unique axis b] 1x,y,z identity 2-x,y+1/2,-z+1/2Screw axis 3-x,-y,-zi 4x,-y+1/2,z+1/2Glide plane

36. Multiplicity, Wyckoff Letter, Site Symmetry 4e1(x,y,z) (-x, ½ +y,½ -z) (-x,-y,-z) (x,½ -y, ½ +z) 2d ī (½, 0, ½) (½, ½, 0) 2c ī (0, 0, ½) (0, ½, 0) 2b ī (½, 0, 0) (½, ½, ½) 2a ī (0, 0, 0) (0, ½, ½)

37. General positions of Group 15 (C 2/c) [unique axis b] 1x,y,zidentity 2-x,y,-z+1/22-fold rotation 3-x,-y,-zinversion 4x,-y,z+1/2c-glide 5x+1/2,y+1/2,zidentity + c-center 6-x+1/2,y+1/2,-z+1/22 + c-center 7-x+1/2,-y+1/2,-zi + c-center 8x+1/2,-y+1/2,z+1/2c-glide + c-center

38. P21/c in international table A

39. P21/c in international table B

40. C n and 

41. Relation between cubic and tetragonal unit cell

42. Lattice : the manner of repetition of atoms, ions or molecules in a crystal by an array of points

43. Types of lattice Primitive lattice (P) - the lattice point only at corner Face centred lattice (F) - contains additional lattice points in the center of each face Side centred lattice (C) - contains extra lattice points on only one pair of opposite faces Body centred lattice (I) - contains lattice points at the corner of a cubic unit cell and body center

44. Examples of F, C, and I lattices

45. 14 Possible Bravais lattices : combination of four types of lattice and seven crystal systems

46. How to index crystal planes?

47. Lattice planes and Miller indices

48. Lattice planes

49. Miller indices

50. Assignment of Miller indices to a set of planes 1. Identify that plane which is adjacent to the one that passes through the origin. 2. Find the intersection of this plane on the three axes of the cell and write these intersections as fractions of the cell edges. 3. Take reciprocals of these fractions. Example: fig. 10 (b) of previous page cut the x axis at a/2, the y axis at band the z axis at c/3; the reciprocals are therefore, 1/2, 1, 1/3; Miller index is ( 2 1 3 ) #

51. Examples of Miller indices

52. Miller Index and other indices (1 1 1), (2 1 0) {1 0 0} : (1 0 0), (0 1 0), (0 0 1) …. [2 1 0], [-3 2 3] : [1 0 0], [0 1 0], [0 0 1]

53. 考古題 Assign the Miller indices for the crystal faces

54. Descriptions of crystal structures The close packing approach The space-filling polyhedron approach

55. Materials can be described as close packed Metal- ccp, hcp and bcc Alloy- CuAu (ccp), Cu(ccp), Au(ccp) Ionic structures - NaCl Covalent network structures (diamond) Molecular structures

56. Close packed layer

57. A NON-CLOSE-PACKED structure

58. Close packed

59. Two cp layers

60. P = sphere, O = octahedral hole, T+ / T- = tetrahedral holes

61. Three close packed layers in ccp sequence

63. ccp

64. ABCABC.... repeat gives Cubic Close-Packing (CCP) Unit cell showing the full symmetry of the arrangement is Face-Centered Cubic Cubic: a = b =c,  =  =  = 90° 4 atoms in the unit cell: (0, 0, 0) (0, 1 / 2, 1 / 2 ) ( 1 / 2, 0, 1 / 2 ) ( 1 / 2, 1 / 2, 0)

66. hcp

67. ABABAB.... repeat gives Hexagonal Close-Packing (HCP) Unit cell showing the full symmetry of the arrangement is Hexagonal Hexagonal: a = b, c = 1.63a,  =  = 90°,  = 120° 2 atoms in the unit cell: (0, 0, 0) ( 2 / 3, 1 / 3, 1 / 2 )

69. Coordination number in hcp and ccp structures

70. hcp

71. Face centred cubic unit cell of a ccp arrangement of spheres

72. Hexagonal unit cell of a hcp arrangement of spheres

73. Unit cell dimensions for a face centred unit cell

75. Density of metal

76. Tetrahedral sites

77. Covalent network structures of silicates

78. C 60 and Al 2 Br 6

80. The space-filling approach Corners and edges sharing

81. Example of edge-sharing

83. Example of corner-sharing

84. Corner- sharing of silicates