CHAPTER 3: SYMMETRY AND GROUPS, AND CRYSTAL STRUCTURES. Sarah Lambart

Transcription

1 CHAPTER 3: SYMMETRY AND GROUPS, AND CRYSTAL STRUCTURES Sarah Lambart

2 RECAP CHAP. 2 2 different types of close packing: hcp: tetrahedral interstice (ABABA) ccp: octahedral interstice (ABCABC) Definitions: The coordination number or CN is the number of closest neighbors of opposite charge around an ion. It can range from 2 to 12 in ionic structures. These structures are called coordination polyhedron.

3 RECAP CHAP. 2 An ideal close-packing of sphere for a given CN, can only be achieved for a specific ratio of ionic radii between the anions and the cations. Rx/Rz C.N. Type Cubic Hexagonal or Cubic Closest Packing Octahedral Tetrahedral (ex.: SiO 4 4- ) Triangular < Linear

4 RECAP CHAP. 2 Pauling s rule: #1: the coodination polyhedron is defined by the ratio Rcation/Ranion #2: The Electrostatic Valency (e.v.) Principle: ev = Z/CN #3: Shared edges and faces of coordination polyhedra decreases the stability of the crystal. #4: In crystal with different cations, those of high valency and small CN tend not to share polyhedral elements #5: The principle of parsimony: The number of different sites in a crystal tends to be small.

5 CONTENT CHAP. 3 (2-3 LECTURES) Definitions: unit cell and lattice 7 Crystal systems 14 Bravais lattices Element of symmetry

6 CRYSTAL LATTICE IN TWO DIMENSIONS A crystal consists of atoms, molecules, or ions in a pattern that repeats in three dimensions. The geometry of the repeating pattern of a crystal can be described in terms of a crystal lattice, constructed by connecting equivalent points throughout the crystal. Step1: 2D crystal lattice

7 CRYSTAL LATTICE IN TWO DIMENSIONS Lattice point: A crystal lattice is constructed by connecting adjacent equivalent points (lattice points) throughout the crystal. The environment about any lattice point is identical to the environment about any other lattice point. The choice of reference lattice point is arbitrary. One choice of a reference point

8 CRYSTAL LATTICE IN TWO DIMENSIONS Lattice point: A crystal lattice is constructed by connecting adjacent equivalent points (lattice points) throughout the crystal. The basic parallelogram (parallelepiped in three dimensions) constructed by connecting lattice points defines a unit cell. A lattice constructed from the chosen point.

9 CRYSTAL LATTICE IN TWO DIMENSIONS Lattice Points and Unit Cell Because the choice of reference lattice point is arbitrary, the location of the lattice relative to the contents of the unit cell is variable. Regardless of the reference point chosen, the unit cell contains the same number of atoms with the same geometrical arrangement. The same lattice and unit cell defined from a different reference point.

10 CRYSTAL LATTICE IN TWO DIMENSIONS Unit Cell - The unit cell is the basic repeat unit from which the entire crystal can be built. - A primitive unit cell contains only one lattice point. The same lattice and unit cell defined from a different reference point.

11 CRYSTAL LATTICE IN TWO DIMENSIONS Unit cell: The unit cell of a mineral is the smallest divisible unit of a mineral that possesses the symmetry and properties of the mineral. The unit cell is defined by three axes or cell edges, termed a, b, and c and three interaxial angles alpha, beta, and gamma, such that alpha is the angle between b and c, beta between a and c, and gamma between a and b.

12 CRYSTAL LATTICE IN TWO DIMENSIONS Alternate lattice and choice of the unit cell

13 CRYSTAL LATTICE IN TWO DIMENSIONS Rules to choose a unit cell Smallest repeat unit Highest possible symmetry (with the most 90 angles)

14 CRYSTAL LATTICE IN TWO DIMENSIONS

15 CRYSTAL LATTICE IN TWO DIMENSIONS

16 CRYSTAL LATTICE IN TWO DIMENSIONS

17 CRYSTAL LATTICE IN TWO DIMENSIONS Unit cell: The unit cell of a mineral is the smallest divisible unit of a mineral that possesses the symmetry and properties of the mineral. The unit cell is defined by three axes or cell edges, termed a, b, and c and three interaxial angles alpha, beta, and gamma, such that alpha is the angle between b and c, beta between a and c, and gamma between a and b.

18 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

19 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

20 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

21 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

22 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

23 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered Each corner is shared between 8 cells

24 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

25 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

26 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered

27 CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered Each face is shared between 2 cells

28 7 CRYSTAL SYSTEMS c c c b a 2 a Triclinic α β γ a b c b a Monoclinic α = γ = 90 ο β a b c a 1 P or C Hexagonal α = β = 90 ο γ = 120 ο a 1 = a2 c R Rhombohedral α = β = γ 90 ο a 1 = a 2 = a 3 c c a 3 a 2 b a a 1 Orthorhombic Tetragonal α = β = γ = 90 ο a b c α = β = γ = 90 ο a 1 = a c 2 a 2 a 1 Isometric (or cubic) α = β = γ = 90 ο a 1 = a 2 = a 3

29 7 CRYSTAL SYSTEMS c c c b a 2 a Triclinic α β γ a b c b a Monoclinic α = γ = 90 ο β a b c a 1 P or C Hexagonal α = β = 90 ο γ = 120 ο a 1 = a2 c R Rhombohedral α = β = γ 90 ο a 1 = a 2 = a 3 c c a 3 a 2 b a a 1 Orthorhombic Tetragonal α = β = γ = 90 ο a b c α = β = γ = 90 ο a 1 = a c 2 a 2 a 1 Isometric (or cubic) α = β = γ = 90 ο a 1 = a 2 = a 3

30 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a 1 = a 2 = a 3 a 1 a 2 P F I CN = 6 CN = 12 CN = 8

31 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a 1 = a 2 = a 3 a 1 a 2 P F I CN = 6 CN = 12 CN = 8

32 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a 1 = a 2 = a 3 a 1 a 2 P F I CN = 6 CN = 12 CN = 8

33 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a 1 = a 2 = a 3 a 1 a 2 P F I CN = 6 CN = 12 CN = 8

34 14 BRAVAIS LATTICES

35 14 BRAVAIS LATTICES Example: Sphalerite (ZnS)

36 14 BRAVAIS LATTICES Example: Sphalerite (ZnS) Basis: S 2- + Zn 2+

37 14 BRAVAIS LATTICES Example: Halite (NaCl)

38 14 BRAVAIS LATTICES Example: Halite (NaCl)

39 SYMMETRIES is the hardest thing for student to understand, appreciate or visualize.

40 SYMMETRIES 4-fold rotational symmetry.

41 SYMMETRY OPERATIONS A Symmetry operation is an operation on an object that results in no change in the appearance of the object. There are 3 types of symmetry operations: rotation, reflection, and inversion.

42 ROTATIONAL SYMMETRY 1 fold rotation axis = no rotational symmetry fold rotation axis: identical after a rotation of 180 (360/180 = 2) symbol: filled oval or A

43 ROTATIONAL SYMMETRY 3 fold rotation axis = identical after a rotation of 120 (360/120 = 3) symbol: filled triangle or A fold rotation axis: identical after a rotation of 90 (360/90 = 4) symbol: filled square or A 4 4 4

44 ROTATIONAL SYMMETRY 6 fold rotation axis = identical after a rotation of 60 (360/60 = 6) symbol: filled hexagon or A 6 6 6

45 IMPROPER ROTATIONAL SYMMETRY 5 fold, 7 fold, 8 fold or higher: does not exist in crystals because cannot fill the space

46 MIRROR SYMMETRY A mirror plan is something that gives you the reflection that exactly reflects the other side: same distance, same component. The plane of the mirror is an element of symmetry referred to as a mirror plane, and is symbolized with the letter m.

47 MIRROR SYMMETRY

48 MIRROR SYMMETRY

49 MIRROR SYMMETRY

50 CENTER OF SYMMETRY A center of symmetry is an inversion through a point, symbolized with the letter "i".

51 ROTOINVERSIONS Combinations of rotation with a center of symmetry. 1 fold rotoinversion axis = center of symmetry symbol: A 1 A 1 2 fold rotoinversion axis = 1) 180 rotation, 2) center of symmetry = mirror perpendicular to the axis symbol: A 2 m

52 ROTOINVERSIONS 1) Rotation 360 2) Inversion

53 ROTOINVERSIONS Combinations of rotation with a center of symmetry. 1 fold rotoinversion axis = center of symmetry symbol: A 1 A 1 2 fold rotoinversion axis = 1) 180 rotation, 2) center of symmetry = mirror perpendicular to the axis symbol: A 2 m

54 ROTOINVERSIONS 1) Rotation 180 2) Inversion

55 ROTOINVERSIONS 3 fold rotoinversion axis = 1) 120 rotation, 2) center of symmetry symbol: A 3 4 fold rotoinversion axis = 1) 90 rotation, 2) center of symmetry symbol: A 4

56 ROTOINVERSIONS 1 3 fold rotoinversion axis = 1) 120 rotation, 2) center of symmetry symbol: A 3 4 fold rotoinversion axis = 1) 90 rotation, 2) center of symmetry symbol: A 4 1

57 ROTOINVERSIONS 3 fold rotoinversion axis = 1) 120 rotation, 2) center of symmetry symbol: A fold rotoinversion axis = 1) 90 rotation, 2) center of symmetry symbol: A 4 2

58 ROTOINVERSIONS 6 fold rotoinversion axis = 1) 60 rotation, 2) center of symmetry = 3 fold rotation axis + 1 perpendicular mirror plan symbol: A 6

59 CRYSTAL LATTICE IN TWO DIMENSIONS

60 CRYSTAL LATTICE IN TWO DIMENSIONS

61 CRYSTAL LATTICE IN TWO DIMENSIONS

62 COMBINATIONS OF SYMMETRICAL OPERATIONS In crystals there are 32 possible combinations of symmetry elements: the 32 Crystal Classes. a = b c α = β = γ = 90 a1 a2 c

63 COMBINATIONS OF SYMMETRICAL OPERATIONS In crystals there are 32 possible combinations of symmetry elements: the 32 Crystal Classes. a = b c α = β = γ = 90 a1 a2 tetragonal c

64 COMBINATIONS OF SYMMETRICAL OPERATIONS Square-shaped top 4 fold rotation axis

65 COMBINATIONS OF SYMMETRICAL OPERATIONS Square-shaped top 4 fold rotation axis A 2-fold axis that cuts diagonally through

66 COMBINATIONS OF SYMMETRICAL OPERATIONS Square-shaped top 4 fold rotation axis A 2-fold axis that cuts diagonally through Mirror plan through the diagonal

67 COMBINATIONS OF SYMMETRICAL OPERATIONS Rectangular faces: 2-fold rotation axis perpendicular to the rectangular face.

68 COMBINATIONS OF SYMMETRICAL OPERATIONS Square top + rectangular sides: mirror plan parallel to the 4-fold axis

69 COMBINATIONS OF SYMMETRICAL OPERATIONS Square top + rectangular sides: mirror plan parallel to the 4-fold axis mirror plan perpendicular to the 4- fold axis One center of symmetry (not represented)

70 COMBINATIONS OF SYMMETRICAL OPERATIONS 4-fold rotation axis: same face every 90

71 COMBINATIONS OF SYMMETRICAL OPERATIONS 4-fold rotation axis: same face every fold rotation axis 4 2 fold rotation axes 5 mirror plans 1 center of symmetry A 4, 4A 2, 5m, i 4/m2/m2/m the ditetragonal dipyramidal class