1. Structure of crystalline solids
Materials Science
Structure of crystalline solids

2. The structure of solids
We understand how individual atoms may bond
What about real materials made of multiple atoms?
How do atoms arrange to form solid structures?
How do these structures influence a material’s density?
How do material properties vary with these different structures?

3. The structure of solids
Crystalline materials have atoms situated in a periodic array over large atomic distances
Metals and most ceramics
Some properties depend on the crystal structure of the material (e.g. density and ductility)
Amorphous materials have no long range order (non-crystalline).
Glasses – Silica
Plastics
Rapidly cooled metals (1x105 °C/s)

4. The structure of solids
Steel, PEO and Silica glass structures (left-right)

5. Polycrystalline materials
‘Nuclei’ form during solidification, each of which grows into crystals
Crystals grow and meet
These grains are separated by an amorphous grain boundary

6. Energy and packing of atoms in solids
• Dense, regular packing
• Non dense, random packing
Dense, regular-packed structures tend to have lower energy

7. Crystalline solids Lattice: 3D array of regularly spaced points
Crystalline material: atoms situated in a repeating 3D periodic array over large atomic distances
Hard sphere representation: atoms denoted by hard, touching spheres (a)
Reduced sphere representation (b)
Unit cell: basic building block unit (such as a flooring tile) that repeats in space to create the crystal structure; it is usually a parallelepiped or prizm

8. The unit cell
Smallest repeating arrangement

9. Metallic crystal structures
Repetitive patterns – unit cells (represents the symmetry of the crystal structure)
FCC – Face Centred Cubic
BCC – Body Centred Cubic
HCP – Hexagonal Close Packed
Tend to be densely packed due to non-directional bonding in metals
Simple crystal structures, for important engineering materials
NOTE: HARD SPHERE MODEL

10. Simple cubic structure (SC)
Cubic unit cell is 3D repeat unit
Rare (only Po has this structure)
Coordination number (CN)
Number of the nearest neighbors or touching atoms
Easy for civil engineers
SC has CN=6

11. Simple cubic structure (SC)
Simple Cubic has CN=6
Close-packed directions (directions along which atoms touch each other) are cube edges
How can we describe packing of these atoms?
Coordination # = 6
(# nearest neighbors)

12. Atomic packing factor Fill a box with hard spheres
Packing factor = total volume of spheres in box / volume of box
Question: what is the maximum packing factor you can expect?
In crystalline materials:
Atomic packing factor = total volume of atoms in unit cell / volume of unit cell
A value <1 (APF is the fraction of solid sphere volume in a unit cell)

13. APF for a simple cubic structure = 0.52
Atomic packing factor a
R=0.5a
close-packed directions
contains 8 x 1/8 = 1
atom/unit cell
Lattice constant
APF for a simple cubic structure = 0.52

14. Face centred cubic (fcc)
Unit cell of cubic geometry with atoms located at each corner and the centre of the cube faces
E.g. copper, aluminium, silver, gold, (ductile metals)
Coordination number ?...
APF ?...

15. Exercise: Atomic packing for fcc
APF for fcc structure ~ 0.74, CN=12

16. Slip systems Why are fcc metals ductile?
Ductility (ease of plastic deformation) is linked to crystal structure and close packed planes
Slip occurs on specific atomic planes and in specific crystallographic slip directions, i.e. slip systems
Slip planes – most densely packed planes, and in that plane the closely packed direction

17. Face centred cubic (fcc)
For fcc there are 4 close packed planes (face diagonals) and 3 close packed directions, making 12 slip systems
Ductile metals as there are many opportunities for planes to slide over each other

18. Defining crystal directions
What do we mean by a [111] plane ?
We assign x,y,z a magnitude (0 to 1) for the unit cell
This is sufficient to define a plane or a direction

19. Body centred cubic (bcc)
Unit cell of cubic geometry with atoms located at each of the corners and the cube centre
E.g. chromium, a-iron, tungsten
Experience:
Ductile-Brittle transition
Feature a fatigue limit
CN = 8
APF ?...

20. Body centred cubic (bcc)
The Atomium, André Waterkeyn, 1958
Brussels
The bcc unit cell of an iron crystal magnified 165 billion times… shiny!
0.1nm
x165 billion

21. Atomic packing factor – bcc
APF for a body-centered cubic structure ~ 0.68

22. Slip systems in bcc crystals
6 slip planes x 2 slip directions = 12 slip systems
{110} planes in the direction of

23. Hexagonal close packed (hcp)
Hexagonal unit cell
Top and bottom faces of hexagonal cell consist of 6 atoms (hexagon) with an atom in the centre
Middle plane consists of 3 atoms
E.g. cadmium, magnesium, titanium and zinc.
Least ductile metals (only 1 close packed plane)
CN=12, APF~0.74 (like fcc)

24. Slip systems in hcp crystals
1 slip plane x 3 slip directions = 3 slip systems
Most brittle of the metals, but subtle difference between fcc structure
{0001} plane

25. Comparison of crystal structures
Crystal structure CN APF CP directions
Simple Cubic (SC) cube edges
Body Centered Cubic (BCC) body diagonal
Face Centered Cubic (FCC) face diagonal
Hexagonal Close Pack (HCP) hex side

26. Other crystal structures
• Structure of Carbon
There are many structures for
Compounds
Polymers (molecular crystals)
More complex configurations
Polymorphism, etc.
Diamond
• Structure of NaCl
Graphite

27. Other crystal structures

28. Single crystal materials
When the periodic and repeated arrangement of atoms is perfect and extends throughout the entirety of the specimen
Benefits extreme technology
Electronic and optical material (Si wafers)
High performance turbine blades
Abrasive materials (synthetic diamond)
Crystal properties reveal features of the atomic arrangement
Anisotropic (directional) properties

29. Polycrystalline materials
‘Nuclei’ form during solidification, each of which grows into crystals
Crystals grow and meet
These grains are separated by an amorphous grain boundary

30. Polycrystalline materials
Most engineering materials are polycrystalline
Collections of very small crystals (grains)
Most metals, alloys and ceramics
Isotropic properties (same properties in all directions)

31. Single- vs poly-crystalline
• Single Crystals
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
200 mm
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.

32. Amorphous materials Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of:
-metals
-many ceramics
-some polymers
crystalline SiO2
Noncrystalline materials...
• atoms have no periodic packing
• occurs for:
-complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2

33. Summary
Atoms may assemble into crystalline, polycrystalline or amorphous structures
Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but properties are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.
We will next look at how defects in these structures governs the most important mechanical properties of materials: strength and toughness