Basics of Crystal Structures ABAB / ABCABC stacking 7 Crystal system Miller index Space group (Bravais lattice, symmetry operation)

1. ABAB / ABCABC packing

1. ABAB / ABCABC packing

1. ABAB / ABCABC packing

2. Seven crystal system

3. 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. Miller index indicates the plane that cross over the lattice point of (a/h, b/k, c/l). Direction: [hkl] / Plane: (hkl) Intercept at ∞ Bad Good Intercept at b Intercept at 𝟏 𝟐 𝒂

3. Miller Index c b a d010 d020 (010) (020) : d-spacing formula for orthorhombic 1 𝑑0202 = 0 𝑎2 + 4 𝑏2 + 0 𝑐2 ⇔ 𝑑0202= 𝑏 2 2 ⇔ 𝑑020= 𝑏 2

3. Miller Index

3. Miller Index

4. 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 Examples

4. Space group: Bravais lattice 14 Bravais lattices

4. Space group: Screw axis (symmetry operation) 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.)

4. Space group: Glide plane (symmetry operation) 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.

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

S2- in lattice, Zn2+ in ½ Td sites Lattice: (1/8) x 8 + (1/2) x 6 = 4 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

Ca2+ in lattice, F- in all Td sites (Cation framework) 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

Cl- in lattice, Na+ in all Oh sites e.g. Na2O, K2O Rock salt (NaCl) Na Cl NaCl (1:1), cubic packing Cl- in lattice, Na+ in all Oh sites e.g. Na2O, K2O 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

CdCl2 (1:2), cubic packing (ABCABC) CdI2 Cd Cl Cd l 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

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 (1:2), hexagonal closed packing (ABAB) O2- in lattice, Ti in 1/2 Oh sites

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

Octaedral site B cations : Nb, Ta, Ti, Zr, Fe, Mn, …. Perovskite (SrTiO3) Oxygen Octaedra 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 octahedra A site has 12 oxygen neighbors