CRYSTALLINE STATE

1. CRYSTALLINE STATE

2. INTRODUCTION Primary: Ionic Covalent Metallic Van der Waals Gas Liquid Solid Octet stability Secondary: Dipole-dipole London dispersion Hydrogen

3. STATE OF MATTER GAS LIQUID SOLID • The particles are arranged in tight and regular pattern • The particles move very little • Retains its shape and volume • The particles move past one another • The particles close together • Retains its volume • The particles move rapidly • Large space between particles

4. CRYSTAL SYSTEM CLASSIFICATION OF SOLID BY ATOMIC ARRANGEMENT Atomic arrangement Order Name

5. Early crystallography Robert Hooke (1660) : canon ball Crystal must owe its regular shape to the packing of spherical particles (balls)  packed regularly, we get long-range order. NeilsSteensen(1669) : quartz crystal All crystals have the same angles between corresponding faces, regardless of their sizes  he tried to make connection between macroscopic and atomic world. If I have a regular cubic crystal, then if I divide it into smaller and smaller pieces down to an atomic dimension, will I get a cubic repeat unit?

6. Renė-Just Haūy(1781): cleavage of calcite • Common shape to all shards: rhombohedral • Mathematically proved that there are only 7 distinct space-filling volume elements 7 crystal systems

7. 3 AXES 4 AXES yz =  xz =  xy =  yz = 90 xy = yu = ux = 60 CRYSTALLOGRAPHIC AXES

8. The Seven Crystal Systems

10. (rombhohedral)

11. SPACE FILLING TILING

17. August Bravais(1848): more math • How many different ways can we put atoms into these 7 crystal systems and get distinguishable point environment? • He mathematically proved that there are 14 distinct ways to arrange points in space 14 Bravais lattices

18. The Fourteen Bravais Lattices

19. 1 2 3 Face-centered cubic Simple cubic Body-centered cubic

20. 4 5 Simple tetragonal Body-centered tetragonal

21. 6 7 Simple orthorhombic Body-centered orthorhombic

22. 8 9

23. 10 11 12

24. 13 14 Hexagonal

25. Repeat unit A point lattice

26. a, b, c z Lattice parameters , ,  b c   y O  a x A unit cell

27. Crystal structure (Atomic arrangement in 3 space) Bravais lattice (Point environment) Basis (Atomic grouping at each lattice point)

28. EXAMPLE: properties of cubic system*) C 109 C *) cubic system is the most simple most of elements in periodic table have cubic crystal structure

29. CRYSTAL STRUCTURE OF NaCl

30. CHARACTERISTIC OF CUBIC LATTICES

31. EXAMPLE: FCC 74% matter (hard sphere model) FCC 26% void

32. In crystal structure, atom touch in one certain direction and far apart along other direction. • There is correlation between atomic contact and bonding. • Bonding is related to the whole properties, e.g. mechanical strength, electrical property, and optical property. • If I look down on atom direction: high density of atoms  direction of strength; low density/population of atom  direction of weakness. • If I want to cleave a crystal, I have to understand crystallography.

34. CRYSTALLOGRAPHIC NOTATION POSITION: x, y, z, coordinate, separated by commas, no enclosure O: 0,0,0 A: 0,1,1 B: 1,0,½ z Unit cell A B O y a x

35. DIRECTION: move coordinate axes so that the line passes through origin • Define vector from O to point on the line • Choose the smallest set of integers • No commas, enclose in brackets, clear fractions z   OB 1 0 ½ [2 0 1] AO 0 -1 -1 Unit cell A O B y x

36. Denote entire family of directions by carats < > e.g. all body diagonals: <1 1 1>

37. all face diagonals: <0 1 1> all cube edges: <0 0 1>

38. MILLER INDICES For describing planes. Equation for plane: • where a, b, and c are the intercepts of the plane with the x, y, and z axes, respectively. • Let: • so that • No commas, enclose in parenthesis (h k l) • denote entirely family of planes by brace, e.g. all faces of unit cell: {0 0 1} etc.

39. (h k l)  [h k l] Intercept at  c Miller indices: (h k l) (2 1 0) (2 1 0) b Intercept at b a Parallel to z axes [2 1 0] Intercept at a/2 MILLER INDICES

40. (0 1 0) (0 2 0) Miller indices of planes in the cubic system

41. CRYSTAL SYMMETRY Many of the geometric shapes that appear in the crystalline state are some degree symmetrical. This fact can be used as a means of crystal classification. The three elements of symmetry: • Symmetry about a point (a center of symmetry) • Symmetry about a line (an axis of symmetry) • Symmetry about a plane (a plane of symmetry)

42. Symmetry about a point A crystal possesses a center of symmetry when every point on the surface of the crystal has an identical point on the opposite side of the center, equidistant from it. Example: cube 

43. Symmetry about a LINE • If a crystal is rotated 360 about any given axis, it obviously returns to its original position. • If the crystal appears to have reached its original more than once during its complete rotation, the chosen axis is an axis of symmetry.

44. Rotated 180 • Twofold rotation axis DIAD AXIS TRIAD AXIS • Rotated 120 • Threefold rotation axis AXIS OF SYMMETRY TETRAD AXIS • Rotated 90 • Fourfold rotation axis • Rotated 60 • Sixfold rotation axis HEXAD AXIS

45. The 13 axes of symmetry in a cube

46. Symmetry about a plane A plane of symmetry bisects a solid object in a such manner that one half becomes the mirror image of the other half in the given plane. A cube has 9 planes of symmetry: The 9 planes of symmetry in a cube

47. Cube (hexahedron) is a highly symmetrical body as it has 23 elements of symmetry (a center, 9 planes, and 13 axis). • Octahedron also has the same 23 elements of symmetry.

48. ELEMENTS OF SYMMETRY

49. Combination forms of cube and octahedron

50. SOLID STATE BONDING • Composed of ions • Held by electrostatic force • Eg.: NaCl IONIC COVALENT • Composed of neutral atoms • Held by covalent bonding • Eg.: diamond SOLID STATE BONDING MOLECULAR • Composed of molecules • Held by weak attractive force • Eg.: organic compounds • Comprise ordered arrays of identical cations • Held by metallic bond • Eg.: Cu, Fe METALLIC