CHAPTER 3: SYMMETRY AND GROUPS, AND CRYSTAL STRUCTURES

Sarah Lambart 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. RECAP CHAP. 2

Rx/Rz C.N. Type Hexagonal or An ideal close-packing of sphere 1.0 12 Cubic for a given CN, can only be Closest Packing achieved for a specific ratio of 1.0 - 0.732 8 Cubic ionic radii between the anions and 0.732 - 0.414 6 Octahedral Tetrahedral (ex.: the cations. 0.414 - 0.225 4 4- SiO4 ) 0.225 - 0.155 3 Triangular <0.155 2 Linear 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. CONTENT CHAP. 3 (2-3 LECTURES)

„ Definitions: unit cell and lattice „ 7 Crystal systems „ 14 Bravais lattices

„ Element of symmetry 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 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 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. 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. 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. 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 inter- axial 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. CRYSTAL LATTICE IN TWO DIMENSIONS

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

„ Rules to choose a unit cell „ Smallest repeat unit

„ Highest possible symmetry (with the most 90° angles) CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS 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 inter- axial 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. CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ Each corner is „ F: Face-centered shared between 8 „ C: Side-centered cells CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ F: Face-centered „ C: Side-centered CRYSTAL LATTICE

„ Unit cell in 3 D: 4 type of unit cells: „ P: primitive

„ I: Body-centered „ Each face is shared „ F: Face-centered between 2 cells „ C: Side-centered 7 CRYSTAL SYSTEMSc c c

b a2 a b 1 P or C a R Monoclinic Hexagonal Rhombohedral Triclinic a ο ο ο α ≠ β ≠ γ α = γ = 90 ≠ β α = β = 90 γ =120 α = β = γ ≠ 90ο a ≠ b ≠ c = ≠ a ≠ b ≠ c a1 a2 c a1 = a2 = a3 a 3 c c

a2 a1 b a 2 a1 a Orthorhombic Tetragonal Isometric (or cubic) ο ο ο α = β = γ = 90 a ≠ b ≠ c α = β = γ = 90 a1 = a2 ≠ c α = β = γ = 90 a1 = a2 = a3 7 CRYSTAL SYSTEMSc c c

b a2 a b 1 P or C a R Monoclinic Hexagonal Rhombohedral Triclinic a ο ο ο α ≠ β ≠ γ α = γ = 90 ≠ β α = β = 90 γ =120 α = β = γ ≠ 90ο a ≠ b ≠ c = ≠ a ≠ b ≠ c a1 a2 c a1 = a2 = a3 a 3 c c

a2 a1 b a 2 a1 a Orthorhombic Tetragonal Isometric (or cubic) ο ο ο α = β = γ = 90 a ≠ b ≠ c α = β = γ = 90 a1 = a2 ≠ c α = β = γ = 90 a1 = a2 = a3 14 BRAVAIS LATTICES

a 3

Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES

a 3

Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES

a 3

Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES

a 3

Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES 14 BRAVAIS LATTICES

„ Example: Sphalerite (ZnS) 14 BRAVAIS LATTICES

„ Example: Sphalerite (ZnS)

„ Basis: S2- + Zn2+ 14 BRAVAIS LATTICES

„ Example: Halite (NaCl) 14 BRAVAIS LATTICES

„ Example: Halite (NaCl) SYMMETRIES “… is the hardest thing for student to understand, appreciate or visualize.” SYMMETRIES

4-fold rotational symmetry.

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. ROTATIONAL SYMMETRY

„ 1 fold rotation axis = no rotational symmetry 1 1

„ 2 fold rotation axis: identical after a rotation of 180° (360/180 = 2) 2 symbol: filled oval or A2

2

2 ROTATIONAL SYMMETRY

„ 3 fold rotation axis = identical after a

rotation of 120° (360/120 = 3) symbol: filled triangle or A 3 3 3

„ 4 fold rotation axis: identical after a rotation of 90° (360/90 = 4) 4 4 symbol: filled square or A4 ROTATIONAL SYMMETRY

„ 6 fold rotation axis = identical after a rotation of 60° (360/60 = 6)

symbol: filled hexagon or A6 6 6 IMPROPER ROTATIONAL SYMMETRY

„ 5 fold, 7 fold, 8 fold or higher: does not exist in crystals because cannot fill the space 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. MIRROR SYMMETRY MIRROR SYMMETRY MIRROR SYMMETRY CENTER OF SYMMETRY

„ A center of symmetry is an inversion through a point, symbolized with the letter "i". ROTOINVERSIONS Combinations of rotation with a center of symmetry. __

„ 1 fold rotoinversion axis = A1 center of symmetry

symbol: A1

„ 2 fold rotoinversion axis = 1) 180° rotation, 2) center of symmetry = mirror perpendicular to the axis m

symbol: A2 ROTOINVERSIONS

1) Rotation 360° 2) Inversion ROTOINVERSIONS Combinations of rotation with a center of symmetry. __

„ 1 fold rotoinversion axis = A1 center of symmetry

symbol: A1

„ 2 fold rotoinversion axis = 1) 180° rotation, 2) center of symmetry = mirror perpendicular to the axis m

symbol: A2 ROTOINVERSIONS

1) Rotation 180° 2) Inversion ROTOINVERSIONS

„ 3 fold rotoinversion axis = 1) 120° rotation, 2) center of symmetry

symbol: A3

„ 4 fold rotoinversion axis = 1) 90° rotation, 2) center of symmetry

symbol: A4 ROTOINVERSIONS 1

„ 3 fold rotoinversion axis = 1) 120° rotation, 2) center of symmetry

symbol: A3

1 „ 4 fold rotoinversion axis = 1) 90° rotation, 2) center of symmetry

symbol: A4 ROTOINVERSIONS

„ 3 fold rotoinversion axis = 1) 120° rotation, 2) center of symmetry 2 symbol: A3

„ 4 fold rotoinversion axis = 1) 90° rotation, 2) center of symmetry 2 symbol: A4 ROTOINVERSIONS

„ 6 fold rotoinversion axis = 1) 60° rotation, 2) center of symmetry = 3 fold rotation axis + 1 perpendicular mirror plan

symbol: A6

CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS COMBINATIONS OF SYMMETRICAL OPERATIONS In crystals there are 32 possible combinations of symmetry elements: the 32 Crystal Classes.

a2 a = b ≠ c a1 α = β = γ = 90°

c COMBINATIONS OF SYMMETRICAL OPERATIONS In crystals there are 32 possible combinations of symmetry elements: the 32 Crystal Classes.

a2 a = b ≠ c a1 α = β = γ = 90°

tetragonal c COMBINATIONS OF SYMMETRICAL OPERATIONS

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

„ Square-shaped top „ 4 fold rotation axis

„ A 2-fold axis that cuts diagonally through COMBINATIONS OF SYMMETRICAL OPERATIONS

„ Square-shaped top „ 4 fold rotation axis

„ A 2-fold axis that cuts diagonally through „ Mirror plan through the diagonal COMBINATIONS OF SYMMETRICAL OPERATIONS

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

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

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

„ 4-fold rotation axis: same face every 90°

„ 1 4-fold rotation axis „ 4 2 fold rotation axes „ 5 mirror plans

„ 1 center of symmetry

A4, 4A2, 5m, i ⇔ 4/m2/m2/m the ditetragonal dipyramidal class