Physical properties of materials depend on the geometry of crystals
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?
Chapter 31

• Non dense, random packing
Energy typical neighbor bond length typical neighbor bond energy
• Dense, ordered packing Energy typical neighbor bond length r typical neighbor bond energy
Dense, ordered packed structures tend to have lower energies.
Chapter 3r

Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers
LONG RANGE ORDER crystalline SiO
2
Adapted from Fig. 3.18(a),
Callister 6e.
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
" Amorphous " = Noncrystalline
SHORT RANGE ORDER noncrystalline SiO
2
Adapted from Fig. 3.18(b),
Callister 6e.
Chapter 33

• The unit cell is the smallest structural unit or building block that uniquely can describe the crystal structure. Repetition of the unit cell generates the entire crystal. By simple translation, it defines a lattice .
b a
Lattice Parameter : Repeat distance in the unit cell, one for in each dimension
Chapter 3-

a
• Units cells and lattices in 3-D:
– When translated in each lattice parameter direction, MUST fill
3-D space such that no gaps, empty spaces left.
b c
Lattice Parameter : Repeat distance in the unit cell, one for in each dimension
Chapter 3-

Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
7 crystal systems
14 crystal lattices
Fig. 3.4, Callister 7e.
a, b, and c are the lattice constants
Chapter 3 -

• How can we stack metal atoms to minimize empty space?
2-dimensions vs.
Now stack these 2-D layers to make 3-D structures
Chapter 3 -

• 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 not 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...
Remember metallic bond => non-directional, does not restrict number of nearest-neighbors ( covalent ; 8-N’ rule), allows for dense atomic packing
Chapter 3- 4

http://professor.wiley.com/CGI-BIN/LANSAWEB?PROCFUN+PROF1+PRFFN22
Chapter 3-

• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
Closed packed direction is where the atoms touch each other
• Coordination # = 6
(# nearest neighbors)
(Courtesy P.M. Anderson)
Chapter 3- 5

• APF for a simple cubic structure = 0.52
Adapted from Fig. 3.19,
Callister 6e.
Chapter 3- 6

• Close packed directions are cube diagonals.
--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
ex: Cr, W, Fe (

), Tantalum, Molybdenum
• Coordination # = 8
2 atoms/unit cell: 1 center + 8 corners x 1/8
(Courtesy P.M. Anderson) Chapter 37

• APF for a body-centered cubic structure = 0.68
R a
3 a
2 a a
Chapter 3- 8

• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
•
Coordination # = 12
Adapted from Fig. 3.1, Callister 7e.
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
(Courtesy P.M. Anderson)
Chapter 3- 9

• APF for a body-centered cubic structure = 0.74
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell a
Chapter 310

• ABCABC... Stacking Sequence
• 2D Projection
A
A
B
C
B
A sites B
C
B
C
B
B sites B B
C sites
• FCC Unit Cell
Chapter 3- 11

• ABAB... Stacking Sequence
• 3D Projection • 2D Projection
A sites
B sites
A sites
Adapted from Fig. 3.3,
Callister 6e.
• Coordination # = 12
• APF = 0.74
• c / a = 1.633
6 atoms/unit cell ex: Cd, Mg, Ti, Zn
Chapter 3- 12

• Compounds: Often have similar close-packed structures.
• Structure of Na Cl • Close-packed directions
--along cube edges.
(Courtesy P.M. Anderson) (Courtesy P.M. Anderson)
Chapter 3- 13

Arthur C. Clarke's Three Laws:
1)When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.
2)The only way to discover the limits of the possible is to go beyond them into the impossible.
3)Any sufficiently advanced technology is indistinguishable from magic. (from Profiles of the Future, 1961)
Chapter 3-


Example: Copper
Data from Table inside front cover of Callister (see next slide):
• 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 cm)
Result: theoretical 
Cu = 8.89 g/cm3
Compare to actual: 
Cu = 8.94 g/cm3
Chapter 3- 14


R a atoms unit cell

= volume unit cell
2 52.00
a
3
6.023 x 10 23
• Ex: Cr (BCC)
A = 52.00 g/mol
R = 0.125 nm n = 2 a = 4 R / 3 = 0.2887 nm g mol
 theoretical
 actual atoms mol
= 7.18 g/cm 3
= 7.19 g/cm 3
Chapter 3 -

Characteristics of Selected Elements at 20C
Element
Aluminum
Argon
Barium
Beryllium
Boron
Bromine
Cadmium
Calcium
Carbon
Cesium
Chlorine
Chromium
Cobalt
Copper
Flourine
Gallium
Germanium
Gold
Helium
Hydrogen
Symbol
Al
Ar
Ba
Be
B
Br
Cd
Ca
C
Cs
Cl
Cr
Co
Cu
F
Ga
Ge
Au
He
H
At. Weight
(amu)
26.98
39.95
137.33
9.012
10.81
79.90
112.41
40.08
12.011
132.91
35.45
52.00
58.93
63.55
19.00
69.72
72.59
196.97
4.003
1.008
Density
(g/cm3)
2.71
------
3.5
1.85
2.34
------
8.65
1.55
2.25
1.87
------
7.19
8.9
8.94
------
5.90
5.32
19.32
------
------
Atomic radius
(nm)
0.143
------
0.217
0.114
------
------
0.149
0.197
0.071
0.265
------
0.125
0.125
0.128
------
0.122
0.122
0.144
------
------
Adapted from
Table, "Characteristics of
Selected
Elements", inside front cover,
Callister 6e.
Chapter 3- 15

 metals
•  ceramics
•  polymers
Why?
Metals have...
• close-packing
(metallic bonding)
• large atomic mass
Ceramics have...
• less dense packing
(covalent bonding)
• often lighter elements
Polymers have...
• poor packing
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
Data from Table B1, Callister 6e.
Chapter 316

• Some materials may exist in more than one crystal structure, this is called polymorphism .
• If the material is an elemental solid, it is called allotropy .
An example of allotropy is carbon, which can exist as diamond, graphite, and amorphous carbon.
Chapter 3-

Hexagonal (h-BN): Soft sp 2
Properties of BN change with crystal structure
Cubic (c-BN) : Exteremely Hard sp 3
Chapter 3-

B
36
N
36
B
12
N
12
4.3 Å
C
60
B
N
C
6.8 Å
7.2 Å
Chapter 3-

• It is necessary to specify a particular point/location/atom/direction/plane in a unit cell
• We need some labeling convention. Simplest way is to use a 3-D system, where every location can be expressed using three numbers or indices .
– a , b , c and α, β, γ z
β
γ
α y x
Chapter 3-

• Crystallographic direction is a vector [uvw]
– Always passes thru origin 000
– Measured in terms of unit cell dimensions a, b, and c
– Smallest integer values
• Planes with Miller Indices (hkl)
– If plane passes thru origin, translate
– Length of each planar intercept in terms of the lattice parameters a, b, and c .
– Reciprocals are taken
– If needed multiply by a common factor for integer representation
Chapter 3-

x a
z c
111
Point coordinates for unit cell center are a /2, b /2, c /2 ½ ½ ½ y
000 z b


2c
Point coordinates for unit cell corner are 111
 b b
 y
Translation: integer multiple of lattice constants  identical position in another unit cell
Chapter 3-

z
y
Algorithm
1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unit cell dimensions a , b , and c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[ uvw ] x ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]
-1, 1, 1 => [ 111 ] where overbar represents a negative index families of directions < uvw >
Chapter 3-

• Linear Density of Atoms  LD =
Number of atoms
Unit length of direction vector
[110] ex: linear density of Al in [110] direction a = 0.405 nm a
# atoms
LD
= length
2
2 a
=
3.5 nm
-
1
Chapter 3-

z
Algorithm a
2
1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unit cell dimensions a
1
, a
2
, a
3
, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
a
3 [ uvtw ] a
1
Adapted from Fig. 3.8(a), Callister 7e.
a
2
2 ex:
½, ½, -1, 0 => [ 1120 ] a
3 dashed red lines indicate projections onto a
1 and a
2 axes a
1
2 a a
1
2
a
3
Chapter 3-

• Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u ' v ' w ') as follows.
z a
3 a
2 a
1
-
[ u ' v ' w ' ]

[ uvtw ] u
=
1
3
( 2 u ' v ' ) v
=
1
3
( 2 v ' u ' ) t
= w
=
( u + v ) w '
Fig. 3.8(a), Callister 7e.
Chapter 3-

Adapted from Fig. 3.9, Callister 7e.
Chapter 3-

• Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same
Miller indices.
• Algorithm
1. Read off intercepts of plane with axes in terms of a , b , c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no commas i.e., ( hkl )
Chapter 3-

z example
1. Intercepts
2. Reciprocals
3. Reduction a b c
1 1

1/1 1/1 1/

1 1 0
1 1 0 a c
4. Miller Indices (110) x z example
1. Intercepts
2. Reciprocals
3. Reduction a b c
1/2
 
1/½ 1/ 
1/

2 0 0
2 0 0 c
4. Miller Indices (100) a x b b
Chapter 3y y

z example a b c
1. Intercepts 1/2 1 3/4
2. Reciprocals
1/½ 1/1 1/¾
2 1 4/3
3. Reduction 6 3 4 a
4. Miller Indices (634) x c


 b y
Family of Planes { hkl }
Ex: {100} = (100), (010), (001), (100), (010), (001)
Chapter 3-

• In hexagonal unit cells the same idea is used z example
1. Intercepts
2. Reciprocals
3. Reduction a
1 a
2 a
3
1

-1 1
1 1/

-1 c
1
1 0 -1 1
1 0 -1 1
4. Miller-Bravais Indices (1011) a
2 a
3 a
1
Adapted from Fig. 3.8(a), Callister 7e.
Chapter 3-

• We want to examine the atomic packing of crystallographic planes
• Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important. a) Draw (100) and (111) crystallographic planes for Fe.
b) Calculate the planar density for each of these planes.
Chapter 3-

Solution: At T < 912  C iron has the BCC structure.
2D repeat unit
(100) a
=
4
3
3
R
Adapted from Fig. 3.2(c), Callister 7e.
atoms
2D repeat unit 1
= Planar Density = area
2D repeat unit a
2
Radius of iron
1
4
3
3
R
2
R
= 12.1
atoms nm 2
= 0.1241 nm
= 1.2 x 10 19 atoms m 2
Chapter 3-

Solution (cont): (111) plane
1 atom in plane/ unit surface cell
2 a atoms in plane atoms above plane atoms below plane atoms
2D repeat unit
Planar Density = area
2D repeat unit area
=
2 ah
=
3 a
2 = h
=
3
2 a
3


4
3
3
R


2
=
16
3
3
R
2
1
16
3
3
R 2
= 7.0
atoms nm 2
= 0.70 x 10 19 atoms m 2
Chapter 3-

• In a single crystal material the periodic and repeated arrangement of atoms is
PERFECT This extends throughout the entirety of the specimen without interruption.
• Polycrystalline material , on the other hand, is comprised of many small crystals or grains . The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries .
Chapter 3-

Chapter 3-

• Some engineering applications require single crystals:
--diamond single --turbine blades crystals for abrasives
(Courtesy Martin Deakins,
GE Superabrasives,
Worthington, OH. Used with permission.)
Fig. 8.30(c), Callister 6e.
(Fig. 8.30(c) courtesy of Pratt and Whitney).
• Crystal properties reveal features of atomic structure.
--Ex: Certain crystal planes in quartz fracture more easily than others.
(Courtesy P.M. Anderson)
Chapter 3- 17

• Most engineering materials are polycrystals.
Adapted from Fig. K, color inset pages of
Callister 6e.
(Fig. K is courtesy of
Paul E. Danielson,
Teledyne Wah Chang
Albany)
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).
Chapter 3- 18

• Single Crystals
-Properties vary with direction: anisotropic .
-Example: the modulus of elasticity (E) in BCC iron:
Data from Table 3.3,
Callister 6e.
(Source of data is
R.W. Hertzberg,
Deformation and
Fracture Mechanics of
Engineering Materials,
3rd ed., John Wiley and Sons, 1989.)
• Polycrystals
-Properties may/may not vary with direction.
-If grains are randomly oriented: isotropic .
(E poly iron
= 210 GPa)
-If grains are textured , anisotropic.
200 m m
Adapted from Fig.
4.12(b), Callister 6e.
(Fig. 4.12(b) is courtesy of L.C. Smith and C. Brady, the
National Bureau of
Standards,
Washington, DC [now the National Institute of Standards and
Technology,
Gaithersburg, MD].)
Chapter 3- 19

• Different directions in a crystal have a different APF .
• For instance, atoms along the edge of FCC unit cell are more separated than along the face diagonal. This causes anisotropy in the properties of crystals.
• For example, the deformation amount depends on the direction in which a stress is applied, other properties are thermal conductivity, optical properties, magnetic properties, hardness, etc.
• In some polycrystalline materials, grain orientations are random, hence bulk material properties are isotropic, i.e. equivalent in each direction
• Some polycrystalline materials have grains with preferred orientations
(texture) , so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties.
Chapter 3-

• The Young’s modulus defines the amount of elastic strain induced on a material when stressed, slope of a stress-strain curve.
Ultimate Strength
Rupture
Question: Plot tensile Young’s Modulus E (θ) as a function of θ for graphite single crystal. θ is the angle between c- and a- axis of the hexagonal crystal system graphite assumes.
1/E(θ) = S
11 cos 4 θ + S
33 sin 4 θ + (S
44
+2S
13
).(cos 2 θ sin 2 θ)
Basal Planes of graphite a
θ
Modulus of Elasticity
Young’s Modulus
S (Compliance, GPa -1) C (Stiffness, GPa)
S11= 0.00098
S33= 0.0275
S44= 0.25
S12= -0.00016
S13= -0.00033
C11= 1060
C22= 36.5
C44= 4
C12= 180
C13= 15 c
Chapter 3-

X-RAYS TO CONFIRM CRYSTAL STRUCTURE
• Incoming X-rays diffract from crystal planes.
Adapted from Fig.
3.2W, Callister 6e.
• Measurement of:
Critical angles, q c
, for X-rays provide atomic spacing, d.
Chapter 3- 20

• Atoms can be arranged and imaged
Photos produced from the work of C.P. Lutz,
Zeppenfeld, and D.M.
Eigler. Reprinted with permission from
International Business
Machines Corporation, copyright 1995.
Carbon monoxide molecules arranged on a platinum (111) surface.
Iron atoms arranged on a copper (111) surface. These
Kanji characters represent the word
“atom”.
Chapter 3- 21

• Demonstrates " polymorphism "
The same atoms can have more than one crystal structure.
Chapter 3- 22

• 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.
Chapter 323

• Allotropy
• Amorphous
• Anisotropy
• Atomic packing factor (APF)
• Body-centered cubic (BCC)
• Coordination number
• Crystal structure
• Crystalline
• Face-centered cubic (FCC)
• Grain
• Grain boundary
• Hexagonal close-packed (HCP)
• Isotropic
• Lattice parameter
• Non-crystalline
• Polycrystalline
• Polymorphism
• Single crystal
• Unit cell
Chapter 3-

Chapter 3-

Reading: Read and work on the examples of
Chapter 3 multiple times. Visit the Virtual
Materials Science Engineering Website link listed in your book and go thru the different
Crystal Structures !
Core Problems: 3.14, 3.15, 3.20, 3.31, 3.39,
3.40, 3.54
Bonus Problems: 3.26 (?), 3.32,
3.41,3.42,
3.51,3.53
Due Date: 6 March 2008
Chapter 30