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

2. 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.

3. 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.

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

5. 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.

6. 14 Bravais Lattice

7. 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)

8. 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

9. 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

10. Close Packed StructureB C

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

12. 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

13. 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

14. 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 = 4
CsCl
CsCl (1:1)
Cl- makes primitive cell, Cs+ in Oh sites
Lattice: (1/8) x 8 = 1
Oh holes: 1 Cs Cl

15. 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

16. 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

17. 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

18. 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

19. 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

20. Miller Index

21. Miller Index

22. 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)

23. 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 

24. 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

25. 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.)

26. 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.