Download Chapter 3: The Structure of Crystalline Solids

What is the difference between atomic
structure and crystal structure?
Chapter 6
METAL STRUCTURE
ENT 145 Materials Engineering
Atomic structure relates to the number of protons and
neutrons in the nucleus of an atom, as well as the
number and probability distributions of the constituent
electrons.
Crystal structure pertains to the arrangement of atoms
in the crystalline solid material.
Chapter 3 - 1
Materials and Packing
Metallic Crystal Structures
Crystalline materials...
• atoms are situated in a repeating/
periodic array.
• typical of: -metals
-many ceramics
-some polymers
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline
Chapter 3 - 2
• Tend to be densely packed.
• Reasons for dense packing:
crystalline SiO2
Adapted from Fig. 3.23(a),
Callister & Rethwisch 8e.
Si
Oxygen
- 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...
noncrystalline SiO2
Adapted from Fig. 3.23(b),
Callister & Rethwisch 8e.
Chapter 3 - 3
Unit Cells
Simple Cubic Structure (SC)
• subdivide the structure into small repeat entities
• represent the symmetry of crystal structures- all atom position
In the crystal may be generated by translation of the unit cells
along each of its edges
• basic structural unit or building block
Adapted from Fig. 3.2,
Callister & Rethwisch 8e.
Hard sphere model
Aggregate of atoms
Chapter 3 - 4
Chapter 3 -
• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
Click once on image to start animation
(Courtesy P.M. Anderson)
Chapter 3 - 6
1
Atomic Packing Factor (APF)
APF =
Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals.
Volume of atoms in unit cell*
Volume of unit cell
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
*assume hard spheres
ex: Cr, W, Fe (), Tantalum, Molybdenum
• Coordination # = 8
• APF for a simple cubic structure = 0.52
atoms
unit cell
a
R=0.5a
1
APF =
4
3
volume
atom
p (0.5a) 3
a3
close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
volume
unit cell
(Courtesy P.M. Anderson)
Adapted from Fig. 3.24,
Callister & Rethwisch 8e.
• APF for a body-centered cubic structure = 0.68
Chapter 3 - 8
Exercise 1. Show for the body-centered cubic (BCC) crystal
structure that the unit cell edge length a and the atomic radius R
are related through a =4R / √3 .
3a
Using the triangle NOP
(NP)2 = a2 + a2 = 2a2
And then for triangle NPQ,
a
(NQ) 2 = (QP) 2 + (NP) 2
But NQ = 4R, R being the atomic radius.
Also, QP = a. Therefore,
(4R)2 = a2 + 2a2
or
a = 4R / √3
2a
R
Close-packed directions:
length = 4R = 3 a
a
atoms
volume
4
p ( 3a/4 ) 3
2
unit cell
atom
3
APF =
volume
3
a
unit cell
Chapter 3 - 9
Chapter 3 - 10
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: Al, Cu, Au, Pb, Ni, Pt, Ag
• Coordination # = 12
Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74
maximum achievable APF
Close-packed directions:
length = 4R = 2 a
2a
a
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 8e.
Adapted from Fig. 3.1, Callister & Rethwisch 8e.
Click once on image to start animation
(Courtesy P.M. Anderson)
2 atoms/unit cell: 1 center + 8 corners x 1/8
Chapter 3 - 7
Atomic Packing Factor: BCC
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
Adapted from Fig. 3.2,
Callister & Rethwisch 8e.
Click once on image to start animation
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
Chapter 3 - 11
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
atoms
volume
4
p ( 2a/4 ) 3
4
unit cell
atom
3
APF =
volume
a3
unit cell
Chapter 3 - 12
2
Theoretical Density, r
Density = r =
r =
where
Theoretical Density, r
• Ex: Cr (BCC)
A = 52.00 g/mol
R = 0.125 nm
n = 2 atoms/unit cell
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
= 6.022 x 1023 atoms/mol
Chapter 3 - 15
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
atoms
unit cell
r=
volume
unit cell
R
a
2 52.00
a3 6.022 x 1023
a = 4R/ 3 = 0.2887 nm
g
mol
rtheoretical = 7.18 g/cm3
ractual
atoms
mol
= 7.19 g/cm3
Chapter 3 - 16
Exercise 2.
Iron has a BCC crystal structure, an atomic radius of 0.124 nm,
and an atomic weight of 55.85 g/mol. Compute and compare its
theoretical density with the experimental value found in the
front section of the book.
Chapter 3 - 17
Chapter 3 - 18
Exercise 3.
Rhodium has an atomic radius of 0.1345 nm and a density of
12.41 g/cm3. Determine whether it has an FCC or BCC crystal
structure.
Chapter 3 - 19
Chapter 3 - 20
3
Polycrystalline
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
Graphite/
Ceramics/
Semicond
Composites/
fibers
Polymers
B ased 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).
Adapted from Fig. K,
color inset pages of
Callister 5e.
(Fig. K is courtesy of
Paul E. Danielson,
Teledyne Wah Chang
Albany)
Zirconia
Titanium
Aluminum
Magnesium
Al oxide
Diamond
Si nitride
Glass -soda
Concrete
Silicon
G raphite
1
1 mm
PTFE
Silicone
PVC
PET
PC
HDPE, PS
PP, LDPE
0.5
0.4
0.3
Glass fibers
GFRE*
Carbon fibers
CFRE*
A ramid fibers
AFRE *
Wood
Data from Table B.1, Callister & Rethwisch, 8e.
Chapter 3 - 21
Various stages in the solidification of polycrystalline
overall component properties are not directional.
• Grain sizes typically range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).
E (diagonal) = 273 GPa
Data from Table 3.3,
Callister & Rethwisch
8e. (Source of data is
R.W. Hertzberg,
Deformation and
Fracture Mechanics of
Engineering Materials,
3rd ed., John Wiley and
Sons, 1989.)
E (edge) = 125 GPa
• Polycrystals
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
200 mm
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
Upon completion of solidification, grains having
irregular shapes have formed
Chapter 3 - 22
Single vs Polycrystals
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
Growth of the crystallite
Isotropic
• Nb-Hf-W plate with an electron beam weld.
• Each "grain" is a single crystal.
• If grains are randomly oriented,
• Single Crystals
Small crystallite nuclei
Anisotropic
• Most engineering materials are polycrystalline.
Adapted from Fig.
4.14(b), Callister &
Rethwisch 8e.
(Fig. 4.14(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].)
Grain structure as it would appear
under microscope
Polymorphism
Chapter 3 - 24
Crystal Systems
• Two or more distinct crystal structures for the same
material (allotropy/polymorphism)
iron system
titanium
liquid
, -Ti
1538ºC
-Fe
BCC
carbon
1394ºC
diamond, graphite
-Fe
FCC
Unit cell: smallest repetitive volume which
contains the complete lattice pattern of a crystal.
7 crystal systems
14 crystal lattices
912ºC
BCC
a, b, and c are the lattice constants
-Fe
Fig. 3.4, Callister & Rethwisch 8e.
Chapter 3 - 25
Chapter 3 - 26
4
Point Coordinates
z
Point coordinates for unit cell
center are
111
c
a/2, b/2, c/2
000
a
x
½½½
y
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 -
Point Coordinates
Chapter 3 - 28
Point Coordinates
tableun_03_p57
Crystallographic Directions
Algorithm
z
y
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 - 31
figun_03_p58
5
Crystallographic Planes
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. If the plane passes through selected origin, either
another parallel plane must be constructed within
the unit cell or new origin must be construct.
2. Read off intercepts of plane with axes in
terms of a, b, c
3. Take reciprocals of intercepts
4. Reduce to smallest integer values
5. Enclose in parentheses, no commas i.e., (hkl)
Adapted from Fig. 3.10,
Callister & Rethwisch 8e.
Chapter 3 - 33
Chapter 3 - 34
Crystallographic Planes
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
y
c

1/
0
0
example
1. Intercepts
2. Reciprocals
b
a
x
b

1/
0
0
z
c
z
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


y
b
c
y
a
Family of Planes {hkl}
b
Ex: {100} = (100), (010), (001), (100), (010), (001)
x
Chapter 3 - 35
X-Ray Diffraction
Chapter 3 - 36
X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
extra
distance
travelled
by wave “2”
q
q
Chapter 3 - 37
Adapted from Fig. 3.20,
Callister & Rethwisch 8e.
spacing
dhkl between
planes
Braggs Law
• Diffraction gratings must have spacings comparable to
the wavelength of diffracted radiation.
• Can’t resolve spacings  
• Spacing is the distance between parallel planes of
atoms.

reflections must
be in phase for
a detectable signal
dhkl =
dhkl =
n
2 sin q
a
√h2 + k2 + l2
 = wavelength
dhkl = Interplanar
spacing
a=
lattice parameter
n = order of reflection
Chapter 3 - 38
6
Exercise 3.
Using the data for molybdenum in Table 3.1, compute
the interplanar spacing for the (111) set of planes
X-Ray Diffraction Pattern
z
z
Intensity (relative)
c
a
x
z
c
b
y (110)
a
x
c
b
y
a
x (211)
b
y
(200)
Diffraction angle 2q
Diffraction pattern for polycrystalline -iron (BCC)
Adapted from Fig. 3.22, Callister 8e.
Chapter 3 - 39
Chapter 3 - 40
Exercise 4.
Determine the expected diffraction angle for the firstorder reflection from the (113) set of planes for FCC
platinum when monochromatic radiation of wavelength
0.154 nm is used.
Chapter 3 - 41
Chapter 3 - 42
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).
• Crystallographic points, directions and planes are
specified in terms of indexing schemes.
Crystallographic directions and planes are related
to atomic linear densities and planar densities.
Chapter 3 - 43
Chapter 3 - 44
7
SUMMARY
ANNOUNCEMENTS
• Materials can be single crystals or polycrystalline.
Material properties generally vary with single crystal
orientation (i.e., they are anisotropic), but are generally
non-directional (i.e., they are isotropic) in polycrystals
with randomly oriented grains.
• Some materials can have more than one crystal
structure. This is referred to as polymorphism (or
allotropy).
Reading:
Core Problems:
Self-help Problems:
• X-ray diffraction is used for crystal structure and
interplanar spacing determinations.
Chapter 3 - 45
Chapter 3 - 46
8