12/10/2023

CRYSTAL STRUCTURES &

ENERGY AND PACKING MATERIALS AND PACKING
PROPERTIES • Non dense, random packing Energy
Crystalline materials...
typical neighbor • atoms pack in periodic, 3D arrays
bond length
• typical of: -metals
ISSUES TO ADDRESS...
typical neighbor r
-many ceramics
• How do atoms assemble into solid structures? bond energy -some polymers crystalline SiO2
Adapted from Fig. 3.18(a),
Callister 6e.

• Dense, regular packing Energy

Si Oxygen
• How does the density of a material depend on typical neighbor Noncrystalline materials...
its structure? bond length • atoms have no periodic packing
• occurs for: -complex structures
• When do material properties vary with the typical neighbor r
-rapid cooling
bond energy
sample (i.e., part) orientation?
"Amorphous" = Noncrystalline noncrystalline SiO2
Dense, regular-packed structures tend to have Adapted from Fig. 3.18(b),
Callister 6e.
lower energy.
Chapter 3- 1 Chapter 3- 2 Chapter 3- 3

METALLIC CRYSTALS SIMPLE CUBIC STRUCTURE (SC) ATOMIC PACKING FACTOR

• tend to be densely packed. • Rare due to poor packing (only Po has this structure) Volume of atoms in unit cell*
• Close-packed directions are cube edges. APF =
Volume of unit cell
• have several reasons for dense packing:
-Typically, only one element is present, so all atomic • Coordination # = 6 *assume hard spheres
radii are the same. (# nearest neighbors)
-Metallic bonding is not directional. • APF for a simple cubic structure = 0.52
-Nearest neighbor distances tend to be small in volume
order to lower bond energy. atoms
4 atom
a unit cell 1 (0.5a)3
• have the simplest crystal structures.
3
R=0.5a APF =
We will look at three such structures...
a3 volume
close-packed directions
unit cell
contains 8 x 1/8 =
1 atom/unit cell
(Courtesy P.M. Anderson) Adapted from Fig. 3.19,
Chapter 3- 4 Chapter 3- 5 Callister 6e. Chapter 3- 6

FACE CENTERED CUBIC BODY CENTERED CUBIC

ATOMIC PACKING FACTOR: FCC
STRUCTURE (FCC) • APF for a body-centered cubic structure = 0.74 STRUCTURE (BCC)
• Close packed directions are face diagonals. • Close packed directions are cube diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded Close-packed directions: --Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing. differently only for ease of viewing.
length = 4R
= 2a
• Coordination # = 12
• Coordination # = 8
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
a
Adapted from
Fig. 3.1(a),
Callister 6e. atoms volume
4
unit cell 4 ( 2a/4)3
3 atom
APF =
volume
a3
Adapted from Fig. 3.1(a),
Callister 6e. Adapted from Fig. 3.2,
Callister 6e.
(Courtesy P.M. Anderson)
unit cell
Chapter 3- 9 Chapter 3- 10 (Courtesy P.M. Anderson) Chapter 3- 7
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HEXAGONAL CLOSE-PACKED
ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68 STRUCTURE (HCP)
Close-packed directions: • ABAB... Stacking Sequence
length = 4R • 3D Projection • 2D Projection
= 3a
Unit cell contains: A sites
Top layer
1 + 8 x 1/8
= 2 atoms/unit cell B sites
Middle layer
R
Adapted from a A sites
Fig. 3.2, Bottom layer
Callister 6e. atoms volume
Adapted from Fig. 3.3,
4 Callister 6e.

unit cell 2 ( 3a/4)3

3 atom • Coordination # = 12
APF =
volume • APF = 0.74
a3
unit cell
Chapter 3- 8 Chapter 3- 12 Chapter 3-

DEMO: HEATING AND

THEORETICAL DENSITY, 
COOLING OF AN IRON WIRE
The same atoms can
• Demonstrates "polymorphism" have more than one # atoms/unit cell Atomic weight (g/mol)

 nA
crystal structure.
Temperature, C

1536
Liquid
Volume/unit cell VcNA Avogadro's number
BCC Stable
(cm3/unit cell) (6.023 x 10 23 atoms/mol)
1391
longer
heat up
FCC Stable Example: 1. Copper
shorter! 2. Chromium
914 longer!
BCC Stable 3. Gold
cool down
Tc 768 magnet falls off
shorter
Chapter 3- Chapter 3- 22 Chapter 3- 14

Characteristics of Selected Elements at 20C

At. Weight Density Crystal Atomic radius Data from Table inside front cover of Callister (see
Element Symbol (amu) (g/cm3) Structure (nm) next slide):
Aluminum Al 26.98 2.71 FCC 0.143 Cu = 4atoms/unit cell (63.55g/mol)
Argon Ar 39.95 ------ ------ ------ • crystal structure = FCC: 4 atoms/unit cell [4(0.128nm)/√2]3 (6.023x1023 atoms/mol)
Barium Ba 137.33 3.5 BCC 0.217
Beryllium Be 9.012 1.85 HCP 0.114
• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)
Boron B 10.81 2.34 Rhomb ------ Adapted from
Table, "Charac-
• atomic radius R = 0.128 nm (1 nm = 10 -7cm)
Bromine Br 79.90 ------ ------ ------ teristics of Cu = 8.89 g/cm3
Cadmium Cd 112.41 8.65 HCP 0.149 Selected
Calcium Ca 1.55 FCC
Elements",
0.197 inside front
Vc = a 3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3
40.08
Carbon C 12.011 2.25 Hex 0.071 cover,
Callister 6e.
Cesium Cs 132.91 1.87 BCC 0.265
𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟 = /𝑡ℎ𝑒𝑜. 𝑣𝑎𝑙𝑢𝑒 − 𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒/
Chlorine Cl 35.45 ------ ------ ------
Result: theoretical Cu = 8.89 g/cm3
Chromium Cr 52.00 7.19 BCC 0.125 actual value
Cobalt Co 58.93 8.9 HCP 0.125
Copper Cu 63.55 8.94 FCC 0.128 Compare to actual: Cu = 8.94 g/cm3
Flourine F 19.00 ------ ------ ------
Gallium Ga 69.72 5.90 Ortho. 0.122
Germanium Ge 72.59 5.32 Dia. cubic 0.122
Gold Au 196.97 19.32 FCC 0.144
Helium He 4.003 ------ ------ ------
Hydrogen H 1.008 ------ ------ ------ Chapter 3- 15 Chapter 3- Chapter 3-
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DENSITIES OF MATERIAL CLASSES SUMMARY

metals ceramics polymers Metals/
Alloys
Graphite/
Ceramics/ Polymers
Semicond
Composites/
fibers
EXERCISES
Why? 30
Based on data in Table B1, Callister
• Atoms may assemble into crystalline or Estimate the densities of the following materials:
Metals have... 20 Platinum
Gold, W
Tantalum
*GFRE, CFRE, & AFRE are Glass,
Carbon, & Aramid Fiber-Reinforced
amorphous structures. 1. Aluminum
• close-packing Epoxy composites (values based on
2. Barium
(metallic bonding) 10 Silver, Mo
60% volume fraction of aligned fibers
in an epoxy matrix).
• We can predict the density of a material,
• large atomic mass
Cu,Ni
Steels provided we know the atomic weight, atomic 3. Beryllium

 (g/cm3)
Tin, Zinc
Ceramics have... 5
Zirconia
radius, and crystal geometry (e.g., FCC, 4. Cadmium
Titanium
• less dense packing 4 Al oxide
Diamond BCC, HCP). 5. Calcium
3 Si nitride
(covalent bonding) Aluminum Glass-soda
Concrete
Glass fibers 6. Cobalt
• often lighter elements 2
Magnesium
Silicon PTFE GFRE*
Carbon fibers 7. Cesium
Graphite CFRE*
Polymers have... Silicone
PVC
PET
Aramid fibers
AFRE* Calculate percentage error with the theoretical
• poor packing 1 PC

(often amorphous)
HDPE, PS
PP, LDPE density.
• lighter elements (C,H,O) 0.5 Wood
Composites have... 0.4
0.3
• intermediate values Data from Table B1, Callister 6e.
Chapter 3- 16 Chapter 3- 23 Chapter 3-