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Material Sciences and Engineering, MatE271 1
Material Sciences and Engineering MatE271 1
Crystal Structure
Ashraf Bastawros
www.public.iastate.edu\~bastaw\courses\Mate271.html
Week 2
Material Sciences and Engineering MatE271 Week 2 2
- Define basic terms in crystallography
- Be able to identify 7 crystal systems and 14Bravais lattices
- To compare and contrast the structures of metal, ceramics and polymer materials
- Explain the three most important structures for metals
- Calculate atomic structure densities
Goals for this unit
Material Sciences and Engineering, MatE271 2
Material Sciences and Engineering MatE271 Week 2 3
Crystal Structure and Periodicity
� Crystalline Materials atoms are in an ordered 3-D periodic arraySingle crystal or polycrystalline solids (metals, ceramics, semiconductors, some polymers )
� Amorphous Materialsshort range order
( glasses, many polymers)� Intermediates
Material Sciences and Engineering MatE271 Week 2 4
Crystal Structure and PeriodicityLattice - a 3-D array of positions in space. A geometric framework on which to hang atoms
Unit Cell - the smallest repeat unit which defines the crystal structure
2-D 3-D
2-D
Note - In these cases, there is one atom centered on each lattice site
Material Sciences and Engineering, MatE271 3
Material Sciences and Engineering MatE271 Week 2 5
Simple Cubic (SC)
Lattice
Crystal Structure
Hang 1 atom on each lattice point
SC lattice and crystal structure
a = 2RWhere:
R = atomic radius atom a = lattice parameter
Material Sciences and Engineering MatE271 Week 2 6
Axes Labels
Lattice constant- a, b and c are lengths of edges- α, β, γ are angles (α is across from a, etc.)
a
b
c
αβγ
By convention origin - 0,0,0 z
y
x
Material Sciences and Engineering, MatE271 4
Material Sciences and Engineering MatE271 Week 2 7
Crystal Systems (7-systems)
Tetragonala=b≠c
α=β=γ=90
Cubica=b=c
α=β=γ=90
Orthorhombica≠b≠c
α=β=γ=90
Material Sciences and Engineering MatE271 Week 2 8
�Squished�tetragonal
�Pushed over�cube
Rhombohedrala=b=c
α=β=γ 90≠
Hexagonala=b≠c
α=β=90 γ=120
Crystal Systems (7-systems)
Material Sciences and Engineering, MatE271 5
Material Sciences and Engineering MatE271 Week 2 9
�Pushed over� orthorhombic(in one direction)
�Pushed over� orthorhombic(in two directions)
Crystal Systems (7-systems)
Orthorhombica≠b≠c
α=β=γ=90
Monoclinica≠b≠c
α=β=90, β ≠ 90
Triclinica≠b≠c
α ≠ β ≠ γ ≠ 90
Material Sciences and Engineering MatE271 Week 2 10
Fourteen Crystal (Bravais) Lattices
- Lattices are merely geometric constructs forming a framework of points in space on (or around) which you can position atoms
- In the simplest case, you can apply one atom centered at each lattice point�Most METALLIC materials fall into this class
- Represent the entire lattice with one unit cell
Material Sciences and Engineering, MatE271 6
Material Sciences and Engineering MatE271 Week 2 11
Cubic Systems
- Based on a Cubic unit Cell
� Simple Cubic (SC)� One atom on each corner� Coordination number of 6
� Body-centered Cubic (BCC)� One atom on each corner and one in the center� Coordination number of 8
� Face-centered Cubic (FCC)� One atom on each corner and on each face� Coordination number of 12
a=b=c, α=β=γ=90°
Material Sciences and Engineering MatE271 Week 2 12
Simple Cubic (SC)
Lattice
Crystal Structure
Hang 1 atom on each lattice point
- SC lattice and crystal structure
a = 2RWhere:
R = atomic radius atom a = lattice parameter
Material Sciences and Engineering, MatE271 7
Material Sciences and Engineering MatE271 Week 2 13
Body Centered Cubic (BCC)
- BCC lattice and crystal structure
a = 4R3
where:R = atomic radius atom a = lattice parameter
A
B
Staking order A-B-A-B
Material Sciences and Engineering MatE271 Week 2 14
Cubic Packing - BCC
a
a
√2 a √2 a
a √3a
√3a=4Ra=4R/√3
Material Sciences and Engineering, MatE271 8
Material Sciences and Engineering MatE271 Week 2 15
Face Centered Cubic (FCC)
a = 2R 2
a = 4R2
where:R = atomic radius atom a = lattice parameter
A
B
Staking order A-B-A-B
Stacking Direction[100]
Material Sciences and Engineering MatE271 Week 2 16
Cubic Packing - FCC
a
a
√2 a
4R=√2aa=2R√2
Material Sciences and Engineering, MatE271 9
Material Sciences and Engineering MatE271 Week 2 17
Atomic Packing Factor
- Fraction of solid sphere volume in a unit cell
APF = volume of atoms in unit celltotal cell volume
- Example for FCCHow many atoms are in the unit cell?What is the cell volume?
Material Sciences and Engineering MatE271 Week 2 18
Atomic Packing Factor - FCC
- There are 4 spheres in the cell
- The volume of the spheres:
4 ×43
πR3 =163
πR3
4R
Material Sciences and Engineering, MatE271 10
Material Sciences and Engineering MatE271 Week 2 19
Atomic Packing Factor - FCC
- What is the volume of the cube?� a3 (length of side
cubed)- What is that in terms
of R? (sphere radius)� a = 2R√2� (2R√2)3 = 16R3√2
Material Sciences and Engineering MatE271 Week 2 20
Atomic Packing Factor - FCC
APF = volume of atoms in unit celltotal cell volume
= Vs/Vc
163
πR 3
16R3 2 =
π3 2
= 0.74
The atomic packing factor for FCC is 0.74(74% of the space is filled)
Material Sciences and Engineering, MatE271 11
Material Sciences and Engineering MatE271 Week 2 21
HCP Crystal Structure
HCP crystal structure = HexagonalBravais lattice with 2 atoms per lattice site.
1st atom at 0,0,0 (i.e. lattice point) 2nd atom at 2/3, 1/3, 1/2
Note - 2nd atom environment different than the 1st atom
This is not a lattice point!
For �ideal� HCP onlyc = 1.633 a
a
c
0,0,0
Material Sciences and Engineering MatE271 Week 2 22
HCP Crystal Structure
This sphere is not totally inside unit cell although the partial spheres all add up so that there are two spheres per unit cell!
Material Sciences and Engineering, MatE271 12
Material Sciences and Engineering MatE271 Week 2 23
ABA vs ABC Packing
Material Sciences and Engineering MatE271 Week 2 24
FCC and HCP Compared
Material Sciences and Engineering, MatE271 13
Material Sciences and Engineering MatE271 Week 2 25
Structure of Selected Metals
Metal
Crystal Structure
Atomic Radius (nm)
Aluminum FCC 0.1431 Chromium BCC 0.1249
Cobalt HCP 0.1253 Copper FCC 0.1278
Gold FCC 0.1442 Lead FCC 0.1750
Material Sciences and Engineering MatE271 Week 2 26
Example Problem
o If you know the crystal structure, the atomic radius and the atomic weight, you can calculate the density of a particular material.
o Copper has an atomic radius 0.128 nm an FCC crystal structure and an atomic weight of 63.5 g/mol. Calculate its density.
Material Sciences and Engineering, MatE271 14
Material Sciences and Engineering MatE271 Week 2 27
Crystalline & Noncrystalline Materials
o Single crystals� repeated arrangement of atoms extends
throughout the specimen� all unit cells have the same orientation� exist in nature� can also be grown (Si)
� without external constraints, will have flat, regular faces
Material Sciences and Engineering MatE271 Week 2 28
Polycrystalline Materials
o Crystals of different� sizes� orientations� shapes
o Grain Boundaries� mismatch between
two neighboring crystals
Material Sciences and Engineering, MatE271 15
Material Sciences and Engineering MatE271 Week 2 29
Polycrystalline Materials
o Most crystalline materials are composed of many small crystals called grains
o Crystallographic directions of adjacent grains are usually random
o There is usually atomic mismatch where two grains meet - this is called a grain boundary
o Most powdered materials have a many randomly oriented grains
Material Sciences and Engineering MatE271 Week 2 30
Reading Assignment
Shackelford 2001(5th Ed)
� Read Chapter 3, pp 59-64
Read ahead to page 88, 101-110
Check class web site:
www.public.iastate.edu\~bastaw\courses\Mate271.html 2