Chapter 3: The Structure of Crystalline Solids
ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
(for now, focus on metals)
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation?
Instructor: Eng. Tamer Haddad
Chapter 3 - 1
Energy and Packing
• Non dense, random packing
Energy
typical neighbor
bond length
typical neighbor
bond energy
• Dense, ordered packing
r
Energy
typical neighbor
bond length
r
typical neighbor
bond energy
Dense, ordered packed structures tend to have lower
energies.
Instructor: Eng. Tamer Haddad
Chapter 3 - 2
Materials and Packing
Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers
crystalline SiO2
Adapted from Fig. 3.22(a),
Callister 7e.
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline
Si
Oxygen
noncrystalline SiO2
Adapted from Fig. 3.22(b),
Callister 7e.
Instructor: Eng. Tamer Haddad
Chapter 3 - 3
Crystal Structure:• The manner in which the atoms, ions or molecules are
spatially arranged.
• There is a large number of different crystal structures all
having long range atomic order.
Atomic Hard Sphere Model
When describing crystalline
structures, atoms of solid
spheres and well defined
diameters
Lattice : 3D array of spheres centers
Instructor: Eng. Tamer Haddad
Chapter 3 - 4
Section 3.3 – Unit Cells
Unit cell:
It is often convenient to subdivide the structure
into small repeat entities called unit cell.
smallest repetitive volume which contains the
complete lattice pattern of a crystal.
• The unit cell is the basic structural unit or building block of
the crystal structure and defines the crystal structure by virtue
of its geometry and the atom positions within.
Instructor: Eng. Tamer Haddad
Chapter 3 - 5
Section 3.4 – Metallic Crystal Structures
• How can we stack metal atoms to minimize
empty space?
2-dimensions
vs.
Instructor: Eng. Tamer Haddad
Chapter 3 - 6
Metallic Crystal Structures
• Tend to be densely packed.
• Reasons for dense packing:
- Typically, only one element is present, so all atomic
radii are the same.
- Metallic bonding is not directional.
- Nearest neighbor distances tend to be small in
order to lower bond energy.
- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures...
Instructor: Eng. Tamer Haddad
Chapter 3 - 7
Face Centered Cubic Structure (FCC)
• Atoms touch each other along face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
ex: Aluminum, Copper, Silver, and Gold.
Adapted from Fig. 3.1, Callister 7e.
Instructor: Eng. Tamer Haddad
Chapter 3 - 8
Face Centered Cubic Structure (FCC)
a: cube edge length
R: atomic radius
2a
Unit cell edge length for facecentered Cubic
a = 2R√2
a
Adapted from
Fig. 3.1(a),
Callister 7e.
• Each corner atom is shared among eight unit cells,
whereas a face-centered atom belongs to only two..
4 atoms/unit cell may be assigned to a given unit cell: 6 face x 1/2 + 8
Instructor: Eng. Tamer Haddad
corners x 1/8
Chapter 3 - 9
Two Important Characteristics of a crystal structure:
(I) Coordination Number
Coordination number:
For metals, each atom has the same number of nearestneighbor or touching atoms, which is the coordination
number.
• For FCC, Coordination # = 12
Instructor: Eng. Tamer Haddad
Chapter 3 - 10
(II) Atomic Packing Factor (APF)
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
• APF for a face-centered cubic structure = 0.74
maximum achievable APF
2a
a
atoms
volume
4
3
p ( 2a/4)
4
unit cell
atom
3
APF =
volume
3
a
unit cell
Instructor: Eng. Tamer Haddad
Chapter 3 - 11
Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
ex: Chromium, iron and tungsten.
• Coordination # = 8
Adapted from Fig. 3.2,
Callister 7e.
2 atoms/unit cell: 1 center + 8 corners x 1/8
Instructor: Eng. Tamer Haddad
Chapter 3 - 12
Atomic Packing Factor: BCC
• APF for a body-centered cubic structure = 0.68
3a
a
2a
Adapted from
Fig. 3.2(a), Callister 7e.
R
a
Close-packed directions:
length = 4R = 3 a
atoms
volume
4
3
p ( 3a/4)
2
unit cell
atom
3
APF =
volume
3
a
unit cell
Instructor: Eng. Tamer Haddad
Chapter 3 - 13
Hexagonal Close-Packed Structure
(HCP)
• Not all metals have unit cells with cubic symmetry; hexagonal
unit cell are another common metallic crystal structure.
ex: Cadmium, magnesium, titanium, and zinc.
Instructor: Eng. Tamer Haddad
Chapter 3 - 14
Hexagonal Close-Packed Structure
(HCP)
6 atoms/unit cell =
one-sixth of each of the 12 top and bottom face corner
atoms, one-half of each of the 2 center face atoms, and all 3
mid plane interior atoms.
• Coordination # = 12
• APF = 0.74
• c /a = 1.633 with deviations for some HCP metals.
Instructor: Eng. Tamer Haddad
Chapter 3 - 15
Theoretical Density Computations, r
Density = r =
r =
where
Mass of Atoms in Unit Cell
Total Volume of Unit Cell
nA
VC NA
n = number of atoms/unit cell
A = atomic weight
VC = Volume of unit cell = a3 for cubic
NA = Avogadro’s number
= 6.023 x 1023 atoms/mol
Instructor: Eng. Tamer Haddad
Chapter 3 - 16
Theoretical Density, r
• Ex: Cr (BCC)
A = 52.00 g/mol
R = 0.125 nm
n=2
R
atoms
unit cell
r=
volume
unit cell
a = 4R/ 3 = 0.2887 nm
a
2 52.00
a3 6.023 x 1023
g
mol
rtheoretical = 7.18 g/cm3
ractual
= 7.19 g/cm3
atoms
mol
Instructor: Eng. Tamer Haddad
Chapter 3 - 17
Densities of Material Classes
In general
rmetals > rceramics > rpolymers
30
Why?
Metals have...
Ceramics have...
• less dense packing
• often lighter elements
Polymers have...
r (g/cm3 )
• close-packing
(metallic bonding)
• often large atomic masses
• low packing density
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
Metals/
Alloys
20
Platinum
Gold, W
Tantalum
10
Silver, Mo
Cu,Ni
Steels
Tin, Zinc
5
4
3
2
Titanium
Aluminum
Magnesium
Graphite/
Ceramics/
Semicond
Composites/
fibers
Polymers
Based on data in Table B1, Callister
*GFRE, CFRE, & AFRE are Glass,
Carbon, & Aramid Fiber-Reinforced
Epoxy composites (values based on
60% volume fraction of aligned fibers
in an epoxy matrix).
Zirconia
Al oxide
Diamond
Si nitride
Glass -soda
Concrete
Silicon
Graphite
1
PTFE
Silicone
PVC
PET
PC
HDPE, PS
PP, LDPE
0.5
0.4
0.3
Glass fibers
GFRE*
Carbon fibers
CFRE*
Aramid fibers
AFRE*
Wood
Data from Table B1, Callister 7e.
Instructor: Eng. Tamer Haddad
Chapter 3 - 18
Section 3.6 – Polymorphism/Allotropy
• For metals and non-metals, two or more distinct crystal
structures for the same material (polymorphism/allotropy)
due to temperature and pressure)
iron system
liquid
carbon
diamond, graphite
High
Pressures
Ambient
Conditions
BCC
1538ºC
-Fe
FCC
1394ºC
-Fe
912ºC
BCC
Instructor: Eng. Tamer Haddad
-Fe
Chapter 3 - 19
Crystalline and Non-crystalline Materials
 Single Crystals
For a crystalline solid, when the periodic and repeated
arrangement of atoms is perfect or extends throughout the
entirety of the specimen without interruption,
Instructor: Eng. Tamer Haddad
Chapter 3 - 20
 Polycrystalline Crystals
Most crystalline solids are composed of a collection of many
small crystals or grains.
Ex. Solidification of polycrystalline specimen.
Grain boundary
Instructor: Eng. Tamer Haddad
Chapter 3 - 21
SUMMARY
• Atoms may assemble into crystalline or
amorphous structures.
• Common metallic crystal structures are FCC, BCC, and
HCP. Coordination number and atomic packing factor
are the same for both FCC and HCP crystal structures.
• We can predict the density of a material, provided we
know the atomic weight, atomic radius, and crystal
geometry (e.g., FCC, BCC, HCP).
• Materials can be single crystals or polycrystalline.
Instructor: Eng. Tamer Haddad
Chapter 3 - 22
Question 3-10
Some hypothetical metal has the simple cubic crystal structure
shown in Figure 3.23. If its atomic weight is 74.5 g/mol and the
atomic radius is 0.145 nm, compute its density.
Instructor: Eng. Tamer Haddad
Chapter 3 - 23