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 a a1 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 a a1 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 m = mirror perpendicular to the axis 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 m = mirror perpendicular to the axis 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.