1. Chapter 3
Structures of
Metals and Ceramics
• How do atoms assemble into solid structures?
• How do the structures of ceramic
materials differ from those of metals?
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation?
2. ENERGY AND PACKING
Now, bonding energy is not only between two atoms, its from many atoms.
Dense, regular-packed structures tend to have lower energy.
• Non dense, random packing Energy
typical neighbor
bond length
typical neighbor r
bond energy
average
• Dense, regular packing Energy
typical neighbor
bond length
typical neighbor r
bond energy
3. Building 3D ‘ordered’ array of atoms for Dummies
(i) Construct lattice
(ii) Filling the lattice
with atoms or
molecules or group
of atoms/molecules
You could choose many
number of different unit
cells for the same building
process.
4. 7 Crystal Systems
&
14 Crystal Lattices
Any crystalline structure (3D ordered array of atoms/molecules)
must fall into one of the systems and one of the crystal lattices.
6. METALLIC CRYSTALS
• tend to be densely packed.
• have several reasons for dense packing:
-Typically, only one element is present, so all atomic
radii are the same.
-Metallic bonding is non-directional.
-Nearest neighbor distances tend to be small in
order to lower bond energy.
• have the simplest crystal structures.
We will look at three such structures...
8. SIMPLE CUBIC (SC) STRUCTURE
• Unit cell (Bravais lattice): Simple cubic
• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # (CN) = 6
(# of nearest neighboring atoms)
1/8
CN is the one way to tell
how much the structure is packed with atoms.
9. Here’s the better way to tell about packing.
ATOMIC PACKING FACTOR (APF)
Volume of atoms* in unit cell
APF =
Volume of unit cell
*assume hard spheres
Close-packed direction:
a= 2R
a volume
R=0.5a atoms atom
4
unit cell 1 π (0.5a)3
3
APF = = 0.52
There are 8 of 1/8 atoms. 3
a volume
1 atom/unit cell
unit cell
10. FACE-CENTERED CUBIC (FCC) Structure
• Unit cell (Bravais lattice): FCC
• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
• γ-Fe, Al, Ni, Cu, Ag, Pt, and Au
• Coordination # = 12
Grey and red atoms are same.
11. ATOMIC PACKING FACTOR: FCC Structure
Close-packed directions:
length = 4R
= 2a
Unit cell contains:
6 x 1/2 + 8 x 1/8
a = 4 atoms/unit cell
atoms volume
4 3
unit cell 4 π ( 2a/4)
3 atom
APF = = 0.74
3 volume
a
unit cell
12. BODY-CENTERED CUBIC (BCC) Structure
• Unit cell (Bravais lattice): BCC
• Close packed directions are cube diagonals.
• α-Fe, Cr, Mo, W, and V
• Coordination # = 8
13. ATOMIC PACKING FACTOR: BCC
Close-packed directions:
length = 4R
= 3a
Unit cell contains:
1 + 8 x 1/8
R = 2 atoms/unit cell
a
atoms volume
4 3
unit cell 2 π ( 3a/4)
3 atom
APF =
3 volume = 0.68
a
unit cell
14. Summary (Metal Cubic System + HCP)
Unit Cell
Name of Structure (Bravais lattice) CN APF
SC SC 6 0.52
FCC FCC 12 0.74
BCC BCC 8 0.68
HCP hexagonal 12 0.74
Next slide
15. HEXAGONAL CLOSE-PACKED (HCP) STRUCTURE
• Unit cell (Bravais lattice): Hexagonal
• ABAB... Stacking Sequence
• 3D Projection
A sites • 2D Projection
B sites Top layer
A sites Middle layer
Bottom layer
• Coordination # = 12
• APF = 0.74
• Be, Mg, α-Ti, Zn, and Zr Unit cell: 1/3 of it
16. Closed Packed Planes (metals)
FCC – ABCABC HCP – ABABAB
A
B B
C
A
B B B
C C
B B
A
A sites
B
B sites C
C sites
17. THEORETICAL DENSITY, ρ
# atoms/unit cell Atomic weight (g/mol)
ρ= nA
Volume/unit cell VcNA Avogadro's number
(cm3/unit cell) (6.023 x 10 23 atoms/mol)
Example: Copper
Data from Table inside front cover of texbook
• crystal structure = FCC: 4 atoms/unit cell
• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)
• atomic radius R = 0.128 nm (1 nm = 10-7cm)
Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3
Result: theoretical ρCu = 8.89 g/cm3
18. Before we study crystal structure of ceramics,
We need to learn crystallographic notations
19. Crystallographic Points,
Directions, and Planes
Points (Example - cubic system)
No parenthesis !
No comma ! In fact, we’ll only deal with
cubic in this course.
a, b, c : lattice constant
q r s : multiple or fraction of Point Coordinates?
lattice constant
27. Linear and Planar Densities
FCC crystal structure (metal)
Closed packed direction
Closed packed plane
LD = # of atoms centered on direction vector/length of direction vector
PD = # of atoms centered on a plane/area of plane
28. Closed Packed Planes (metals)
FCC – (111) : ABCABC
A
B B
C
A
B B B
C C
B B
A
A sites
B
B sites C
C sites
29. Closed Packed Planes (metals)
HCP – (0001): ABABAB
A
B B
C
A
B B B
C C
B B Unit cell:
hexagonal
31. CERAMIC CRYSTALS
• Bonding:
--Mostly ionic, some covalent.
--% ionic character increases with difference in
electronegativity.
• Large vs small ionic bond character:
H
2.1
CaF2: large He
-
Li Be C F Ne
1.0 1.5 SiC: small 2.5 4.0 -
Na Mg Si Cl Ar
0.9 1.2 1.8 3.0 -
K Ca Ti Cr Fe Ni Zn As Br Kr
0.8 1.0 1.5 1.6 1.8 1.8 1.8 2.0 2.8 -
Rb Sr I Xe
0.8 1.0 2.5 -
Cs Ba At Rn
0.7 0.9 2.2 -
Fr Ra
0.7 0.9 Table of Electronegativities
32. IONIC BONDING & STRUCTURE
• Charge Neutrality: F-
--Net charge in the CaF2: Ca2+ +
cation anions
structure should
be zero. F-
--General form: AmXp # of atoms
m, p determined by charge neutrality
• Rcation/Ranion (Ratio of ionic radii) ⇒ determines CN (next slide)
--maximize the # of nearest oppositely charged neighbors
(while maintaining charge neutrality and stability)
- - - - - -
+ + +
- - - - - -
unstable stable stable
33. COORDINATION # AND IONIC RADII
Q: How many anions
can you arrange around a cation?
rcation
• Coordination # increases with r
anion
rcation ZnS
Coord #
ranion (zincblende)
< .155 2
.155-.225 3 NaCl
(sodium
.225-.414 4 chloride)
.414-.732 6
CsCl
(cesium
.732-1.0 8 chloride)
34.
35. Different crystal structures
with the same Bravais lattice (unit cell)
FCC Bravais lattice (Metal vs. Ionic Material)
• Structure of NaCl • Structure of FCC metals
Bravais lattice: FCC Bravais lattice: FCC
Coordination #: 6 Coordination #: 12
36. APF (or Ionic pakcing factor (IPF))
metals vs ionic material
• Structure of NaCl • Structure of FCC metals
Bravais lattice: FCC Bravais lattice: FCC
Coordination #: 6 Coordination #: 12
Note the difference
in closed-packed
direction.
a = 2r Na+ + 2rCl- a = 2r√2
38. Allotropes & Polymorphs
Different stable (or metastable)
Allotropes of carbon crystal structures of the same
compounds
Graphite
Different stable (and metastable)
crystal structures of single element
Diamond
Fullerene (C60)
Carbon nanotube
39. Crystalline vs. Amorphous
Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers crystalline SiO2
Si Oxygen
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline noncrystalline SiO2
41. POLYCRYSTALS
• Most engineering materials are polycrystals.
1 mm
• Nb-Hf-W plate with an electron beam weld.
• Each "grain" is a single crystal.
• If crystals are randomly oriented,
overall component properties are not directional.
• Crystal sizes typ. range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).
42. SINGLE VS POLYCRYSTALS
• Single Crystals E (diagonal) = 273 GPa
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
E (edge) = 125 GPa
• Polycrystals
-Properties may/may not 200 µm
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
43. X-ray Diffraction to determine Crystal Structure
X-ray Beams 1 & 2 have to be in phase
to be diffracted. Detector
Source
(next slide)
variables
spacing
Extra distance travelled by wave 2
between
planes
• Incoming X-rays diffract from crystal planes.
44. n: order of reflection
Bragg’s law
n λ = 2 d sin θ
Extra distance travelled by beam 2 have to be an integer
multiple of λ.
• Bragg’s law is a necessary but not sufficient condition for diffraction.
45. θ-2θ scan
θ
Typically X-ray
source and detector
X-ray
are both rotating. Detector
source
If sample S is
polycrystalline,
X-ray data will
resemble the date below.
46. DENSITIES OF MATERIAL CLASSES
Graphite/
ρmetals ρceramics ρpolymers Metals/
Ceramics/ Polymers
Composites/
Alloys fibers
Semicond
30
Why? Based on data in Table B1, Callister
20 Platinum *GFRE, CFRE, & AFRE are Glass,
Metals have... Gold, W
Tantalum Carbon, & Aramid Fiber-Reinforced
• close-packing Epoxy composites (values based on
60% volume fraction of aligned fibers
(metallic bonding) 10 Silver, Mo
Cu,Ni
in an epoxy matrix).
ρ (g/cm3)
Steels
• large atomic mass Tin, Zinc
Zirconia
Ceramics have... 5
Titanium
4 Al oxide
• less dense packing Diamond
Si nitride
3
(covalent bonding) Aluminum Glass-soda
Concrete
Glass fibers
Silicon PTFE
• often lighter elements 2
Magnesium Graphite
GFRE*
Carbon fibers
Silicone CFRE*
Polymers have... PVC
PET
Aramid fibers
AFRE*
PC
• poor packing 1 HDPE, PS
PP, LDPE
(often amorphous)
• lighter elements (C,H,O) 0.5 Wood
Composites have... 0.4
0.3
• intermediate values
47. SUMMARY
• Atoms may assemble into crystalline or
amorphous 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).
• 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.
