Transcript Chapter 3

Why do we care about crystal
structures, directions, planes ?
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 3- 1
Energy and Packing
• Non dense, random packing
Energy
typical neighbor
bond energy
• Dense, ordered packing
r
Energy
typical neighbor
bond length
typical neighbor
bond energy
r
Dense, ordered packed structures tend to have
lower energies.
Chapter 3-
COOLING
typical neighbor
bond length
MATERIALS AND PACKING
Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers
LONG RANGE ORDER
crystalline SiO2
Adapted from Fig. 3.18(a),
Callister 6e.
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2
Adapted from Fig. 3.18(b),
Callister 6e.
SHORT RANGE ORDER
Chapter 3- 3
Unit Cell Concept
•
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 .
a
b
Lattice Parameter : Repeat
distance in the unit cell, one
for in each dimension
Chapter 3-
Crystal Systems
• 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
a
c
Lattice Parameter : Repeat
distance in the unit cell, one
for in each dimension
Chapter 3-
Section 3.3 – Crystal Systems
Unit cell: smallest repetitive volume which
contains the complete lattice pattern of a crystal.
7 crystal systems
14 crystal lattices
a, b, and c are the lattice constants
Fig. 3.4, Callister 7e.
Chapter 3 -
Section 3.4 – Metallic Crystal Structures
• 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 -
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 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
WEB SITE
http://professor.wiley.com/CGI-BIN/LANSAWEB?PROCFUN+PROF1+PRFFN22
Chapter 3-
SIMPLE CUBIC STRUCTURE (SC)
• 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
(Courtesy P.M. Anderson)
• Coordination # = 6
(# nearest neighbors)
Chapter 3- 5
ATOMIC PACKING FACTOR
• APF for a simple cubic structure = 0.52
Adapted from Fig. 3.19,
Callister 6e.
Chapter 3- 6
BODY CENTERED CUBIC
STRUCTURE (BCC)
• 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 3- 7
ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
3a
a
R
a
2a
Chapter 3- 8
FACE CENTERED CUBIC
STRUCTURE (FCC)
• 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
ATOMIC PACKING FACTOR: FCC
• APF for a body-centered cubic structure = 0.74
a
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
Chapter 3- 10
FCC STACKING SEQUENCE
• ABCABC... Stacking Sequence
• 2D Projection
A
B
B
C
A
B
B
B
A sites
C
C
B sites
B
B
C sites
• FCC Unit Cell
Chapter 3- 11
HEXAGONAL CLOSE-PACKED
STRUCTURE (HCP)
• 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
STRUCTURE OF COMPOUNDS: NaCl
• Compounds: Often have similar close-packed structures.
• Structure of NaCl
• Close-packed directions
--along cube edges.
(Courtesy P.M. Anderson)
(Courtesy P.M. Anderson)
Chapter 3- 13
Quotes from People in the know
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-
THEORETICAL DENSITY, 
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)
-7 cm)
• atomic radius R = 0.128 nm (1 nm = 10
Result: theoreticalCu = 8.89 g/cm3
Compare to actual: Cu = 8.94 g/cm3
Chapter 3- 14
Theoretical Density, 
• Ex: Cr (BCC)
A = 52.00 g/mol
R = 0.125 nm
n=2
R
atoms
unit cell
=
volume
unit cell
a
2 52.00
a3 6.023 x 1023
a = 4R/ 3 = 0.2887 nm
g
mol
theoretical = 7.18 g/cm3
actual
atoms
mol
= 7.19 g/cm3
Chapter 3 -
Characteristics of Selected Elements at 20C
At. Weight
Element
Symbol (amu)
Aluminum
Al
26.98
Argon
Ar
39.95
Barium
Ba
137.33
Beryllium
Be
9.012
Boron
B
10.81
Bromine
Br
79.90
Cadmium
Cd
112.41
Calcium
Ca
40.08
Carbon
C
12.011
Cesium
Cs
132.91
Chlorine
Cl
35.45
Chromium Cr
52.00
Cobalt
Co
58.93
Copper
Cu
63.55
Flourine
F
19.00
Gallium
Ga
69.72
Germanium Ge
72.59
Gold
Au
196.97
Helium
He
4.003
Hydrogen
H
1.008
Density
(g/cm 3 )
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
Adapted from
-----Table, "Charac-----teristics of
0.149 Selected
Elements",
0.197 inside front
0.071 cover,
0.265 Callister 6e.
-----0.125
0.125
0.128
-----0.122
0.122
0.144
----------- Chapter 3- 15
DENSITIES OF MATERIAL CLASSES
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 3- 16
POLYMORPHISM & ALLOTROPY
• 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-
Boron Nitride Allotropy
Hexagonal (h-BN): Soft
sp2
Properties of BN
change with crystal
structure
Cubic (c-BN) : Exteremely Hard
sp3
Chapter 3-
Fullerenes
B36N36
B12N12
4.3 Å
B
N
C
C60
7.2 Å
6.8 Å
Chapter 3-
Crystallographic Points, Directions, and
Planes
• 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.
z
– a, b, c and α, β, γ
α
β
y
γ
x
Chapter 3-
Crystallographic Points, Directions, and
Planes
• 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-
Section 3.8 Point Coordinates
z
Point coordinates for unit cell
center are
111
c
a/2, b/2, c/2
y
000
a
x
½½½
b
Point coordinates for unit cell
corner are 111

z
2c



b
y
Translation: integer multiple of
lattice constants  identical
position in another unit cell
b
Chapter 3-
Crystallographic Directions
z
Algorithm
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
y 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
• 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
# atoms
a
LD =
length
2
= 3.5 nm -1
2a
Chapter 3-
HCP Crystallographic Directions
z
Algorithm
a2
-
a3
a1
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvtw]
a
2
ex:
½, ½, -1, 0
-a3
a2
2
Adapted from Fig. 3.8(a), Callister 7e.
=>
[ 1120 ]
a3
dashed red lines indicate
projections onto a1 and a2 axes
a1
2
a1
Chapter 3-
HCP Crystallographic Directions
• Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are
related to the direction indices (i.e., u'v'w') as follows.
z
[ u 'v 'w ' ]  [ uvtw ]
a2
-
a3
a1
1
u = (2 u ' - v ')
3
1
v = (2 v ' - u ')
3
t = - (u +v )
w = w'
Fig. 3.8(a), Callister 7e.
Chapter 3-
Crystallographic Planes
Adapted from Fig. 3.9, Callister 7e.
Chapter 3-
Crystallographic Planes
• 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-
Crystallographic Planes
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
a
1
1/1
1
1
4.
Miller Indices
(110)
example
1. Intercepts
2. Reciprocals
3.
Reduction
a
1/2
1/½
2
2
4.
Miller Indices
(100)
b
1
1/1
1
1
c

1/
0
0
c
y
b
a
x
b

1/
0
0
c

1/
0
0
z
c
y
a
b
x
Chapter 3-
Crystallographic Planes
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
4.
Miller Indices
a
1/2
1/½
2
6
b
1
1/1
1
3
(634)
c
c
3/4
1/¾
4/3

4 a
x


b
Family of Planes {hkl}
Ex: {100} = (100), (010), (001), (100), (010), (001)
Chapter 3-
y
Crystallographic Planes (HCP)
• In hexagonal unit cells the same idea is used
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
a1
1
1
1
1
a2

1/
0
0
a3
-1
-1
-1
-1
c
1
1
1
1
a2
a3
4.
Miller-Bravais Indices
(1011)
a1
Adapted from Fig. 3.8(a), Callister 7e.
Chapter 3-
Crystallographic Planes
•
•
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-
Planar Density of (100) Iron
Solution: At T < 912C iron has the BCC structure.
2D repeat unit
(100)
Planar Density =
area
2D repeat unit
1
a2
=
4 3
R
3
Radius of iron R = 0.1241 nm
Adapted from Fig. 3.2(c), Callister 7e.
atoms
2D repeat unit
a=
1
4 3
R
3
atoms
atoms
19
= 1.2 x 10
2 = 12.1
2
nm
m2
Chapter 3-
Planar Density of (111) Iron
Solution (cont): (111) plane
1 atom in plane/ unit surface cell
2a
atoms in plane
atoms above plane
atoms below plane
h=
3
a
2
2
atoms
2D repeat unit
 4 3  16 3 2
2
area = 2 ah = 3 a = 3 
R  =
R
3
 3

1
atoms =
= 7.0
2
Planar Density =
area
2D repeat unit
16 3
3
R
2
nm
0.70 x 1019
atoms
m2
Chapter 3-
Single Crystals and Polycrystalline
Materials
•
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-
Example of Polycrystalline Growth
Chapter 3-
CRYSTALS AS BUILDING BLOCKS
• Some engineering applications require single crystals:
--diamond single
crystals for abrasives
(Courtesy Martin Deakins,
GE Superabrasives,
Worthington, OH. Used
with permission.)
--turbine blades
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
POLYCRYSTALS
• Most engineering materials are polycrystals.
1 mm
Adapted from Fig. K,
color inset pages of
Callister 6e.
(Fig. K is courtesy of
Paul E. Danielson,
Teledyne Wah Chang
Albany)
• 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 VS POLYCRYSTALS
• Single Crystals
Data from Table 3.3,
Callister 6e.
(Source of data is
R.W. Hertzberg,
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
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.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
200 mm
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
Anisotropy and Texture
• 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-
Sample Question
•
The Young’s modulus defines the amount of elastic strain induced on a material when
stressed, slope of a stress-strain curve.
Question: Plot tensile Young’s Modulus E (θ) as a
Rupture
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) = S11 cos4θ + S33sin4θ + (S44+2S13).(cos2θ sin2θ)
Basal Planes of
graphite
Necking
Strain Hardening
Yield Strength
Ultimate Strength
a
θ
c
Modulus of Elasticity
Young’s Modulus
S (Compliance, GPa -1)
S11= 0.00098
S33= 0.0275
S44= 0.25
S12= -0.00016
S13= -0.00033
C (Stiffness, GPa)
C11= 1060
C22= 36.5
C44= 4
C12= 180
C13= 15
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, qc,
for X-rays provide
atomic spacing, d.
Chapter 3- 20
SCANNING TUNNELING
MICROSCOPY
• 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
DEMO: HEATING AND
COOLING OF AN IRON WIRE
• Demonstrates "polymorphism"
The same atoms can
have more than one
crystal structure.
Chapter 3- 22
SUMMARY (I)
• 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 3- 23
Summary (II)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
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-
Midterm Dates And Places
• Midterm 1: Place and Time
• Midterm 2: Place and Time.
• Final: To be decided by the Faculty of
Engineering.
Chapter 3-
ANNOUNCEMENTS
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 3- 0