Chapter 1 Crystal Structures

1. Chapter 1Crystal 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

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 • is governed by unit cell shape and symmetry

19. t1 t2 t2 t1 γ=120° The Interconversion of Trigonal Lattices 兩正三角柱合併體

20. The seven crystal systems

21. Symmetry Space group = point group + translation

22. Definition of symmetry elements ------------------------------------------------------------- Elements of symmetry ------------------------------------------------ Symbol Description Symmetry operations --------------------------------------------------------------------- EIdentity No change s Plane of symmetry Reflection through the plane i Center of symmetry Inversion through the center Cn Axis of symmetry Rotation about the axis by (360/n)o SnRotation-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, Cn

25. Plane reflection , 

26. Matrix representation of symmetry operators

27. Symmetry operation

28. Symmetry elements C3

29. space group = point group + translation Symmetry elements

30. 21 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 21/c) [unique axis b]

36. Multiplicity, Wyckoff Letter, Site Symmetry

37. General positions of Group 15 (C 2/c) [unique axis b]

38. P21/c in international table A

39. P21/c in international table B

40. Cn 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 planes1. 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 pagecut 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 ) #