Inorganic Material Chemistry Apr 05, 2016 Crystal Structures Miller Index Symmetry Space Group

Crystalline Solids Crystalline solids: Solids in which atoms are arranged in an orderly repeating pattern. The orderly arrangements of atoms that produce flat surfaces cause the solids to have highly regular shapes Amorphous solids: (from the Greek words for “without form”) lack the order found in crystalline solids. At the atomic level the structures of amorphous solids are similar to the structures of liquids, but the molecules, atoms, and/or ions lack the freedom of motion they have in liquids.

Unit Cell Unit cell: A basic “building block” that repeats over and over again, in all directions in crystalline solids. It is made up of a unique arrangement of atoms and embodies the structure of the solid. Crystal lattice: The geometrical pattern of points on which the unit cells are arranged.

Primitive Lattice If we place a lattice point at each corner of a unit cell, we get a primitive lattice.

Centered Lattice If we place a lattice point at each corner of a unit cell, we get a primitive lattice. It is also possible to generate what are called centered lattices by placing additional lattice points in specific locations in the unit cell.

14 Bravais Lattice

Close Packed Structure Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them Close packing represents the most efficient use of space when packing identical spheres. 74% of the total volume is occupied by the spheres There are two ways to close pack the spheres Hexagonal close packing (hcp) Cubic close packing (ccp)

Close Packed Structure First layer top view Two kinds of hollow spaces are formed (red, yellow dots) Second layer top view The spheres of the second layer sit in half of the hollows of the first layer. Third layer top view Two possible options Directly over first layer-hcp In the red dot layer-ccp

hcp ccp Close Packed Structure First layer top view Two kinds of hollow spaces are formed (red, yellow dots) Second layer top view The spheres of the second layer sit in half of the hollows of the first layer. Third layer top view Two possible options Directly over first layer-hcp In the unmarked layer-ccp hcp ccp

Close Packed Structure B C

Crystal Structures Zinc blende vs Wurtzite Fluorite / Antifluorite Rocksalt (NaCl) / CsCl CdCl2 / CdI2 NiAs / Rutile (TiO2) Spinel (MgAl2O4) Perovskite (SrTiO3)

Close Packed Structure Zinc blende ZnS (1:1), cubic packing S2- in lattice, Zn2+ in ½ Td sites Lattice: (1/8) x 8 + (1/2) x 6 = 4 Td holes: 8 x 1/2 = 4 S Zn Wurtzite ZnS (1:1), hexagonal closed packing S2- in lattice, Zn2+ in ½ Td sites S Zn

Crystal Structures Fluorite CaF2 (1:2), cubic packing Ca2+ in lattice, F- in all Td sites (Cation framework) e.g. CeO2, ZrO2, UO2 Lattice: (1/8) x 8 + (1/2) x 6 = 4 Td holes: 8 x 1 = 8 Ca F Anti-fluorite Li2O (1:1), cubic packing O2- in lattice, Li+ in all Td sites (Anion framework) e.g. Na2O, K2O Lattice: (1/8) x 8 + (1/2) x 6 = 4 Td holes: 8 x 1 = 8 Li O

Crystal Structures Rock salt (NaCl) NaCl (1:1), cubic packing Cl- in lattice, Na+ in all Oh sites Lattice: (1/8) x 8 + (1/2) x 6 = 4 Oh holes: (1/4) x 12 + 1 = 4 CsCl CsCl (1:1) Cl- makes primitive cell, Cs+ in Oh sites Lattice: (1/8) x 8 = 1 Oh holes: 1 Cs Cl

Crystal Structures No ion CdCl2 CdI2 CdCl2 (1:2), cubic packing (ABCABC) Cl- in lattice, Cd2+ in 1/2 Oh sites Cdl2 (1:2), hexagonal closed packing (ABAB) l- in lattice, Cd2+ in 1/2 Oh sites

Crystal Structures NiAs NiAs (1:1), hexagonal closed packing (ABAB) Side view (ABAB) Top view Ni As NiAs (1:1), hexagonal closed packing (ABAB) As2- in lattice, Ni2+ in all Oh sites Rutile Ti O TiO2 (1:1), hexagonal closed packing (ABAB) O2- in lattice, Ti in 1/2 Oh sites

Crystal Structures Spinel (MgAl2O4) Mg2+ Al3+ O2- AB2X4 (e.g. MgAl2O4) O2- in ccp lattice Mg2+ in1/8 Td sites Al3+ in 1/2 Oh sites

Crystal Structures Oxygen Octaedral Perovskite (SrTiO3) Oxygen Octaedral Octaedral site B cations : Nb, Ta, Ti, Zr, Fe, Mn, …. Dodecaedral site A cations : K, Na, Ca, Sr, Ba, Pb, Bi, Y, La, … Cubic perovskite SrTiO3 ABO3 (A2+, B4+, O2-) e.g. SrTiO3, BaTiO3, LaAlO3 BO6 octahedral A site has 12 oxygen neighbors

Miller Index Three integers that determines lattice plane. (hkl) Identify the locations where the plane intercepts the x, y, z axes as the fractions of the unit cell edge lengths a, b, c. Direction: [hkl] / Plane: (hkl) Intercept at ∞ Intercept at b Intercept at 𝟏 𝟐 𝒂

Miller Index

Miller Index

Symmetry Symmetry : useful when it comes to describing shapes of regular repeating structures as it provides a way of describing similar features in different structures so they become unifying features. Point symmetry : symmetry possessed by a single object that describes the repetition of identical parts of object. Elements : mirror plane, rotation axis, center of symmetry, inversion axis Mirror plane  a two-fold rotation axis (2) passing through the centers of the top and bottom edges Inversion axis a symmetry plane (m) that relates (as a mirror does) the top to the bottom. Center of symmetry Rotation axis A four-fold improper axis implies 90º rotations followed by reflection through a mirror plane perpendicular to the axis.  a center of symmetry in its center(ī)

Symmetry Space symmetry : combining the rotation axes and the mirror planes with the characteristic translations of the crystals, new symmetry elements appear. Elements : screw axis, glide plane Screw axis : combines translation with rotation. Glide plane : combines translation with reflection. Glide plane Screw axis Glide plane. a reflection followed by a translation Twofold screw axis. a rotation followed by a translation 

Space Group Definition Space group is the symmetry group of a configuration in space usually in 3D. Details Point groups and Bravais lattice yields 73 space groups. Compound operation (Glide, screw operation) yields 157 space groups. Total: 230 space groups to describe crystalline systems

Space Group A screw axis is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number n, to describe the degree of rotation. (e.g. n=3 would mean a rotation one third of the way around the axis each time.)

Space Group a-glide plane b-glide plane c-glide plane A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b, or c, depending on which axis the glide is along.