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

2. 1. ABAB / ABCABC packing

3. 1. ABAB / ABCABC packing

4. 1. ABAB / ABCABC packing

5. 2. Seven crystal system

6. 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

7. 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 ⇔ 𝑑020= 𝑏 2

8. 3. Miller Index

9. 3. Miller Index

10. 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

11. 4. Space group: Bravais lattice
14 Bravais lattices

12. 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.)

13. 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.

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

15. 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

16. 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

17. 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 = 4
CsCl
CsCl (1:1)
Cl- makes primitive cell, Cs+ in Oh sites
Lattice: (1/8) x 8 = 1
Oh holes: 1 Cs Cl

18. 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

19. 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

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

21. 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