Chapter 3: The Structure of Crystalline Solids
ISSUES TO ADDRESS...
• How do atoms assemble (arrange) into solid structures?
(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?
Structure Levels: Subatomic  now: Atomic
Chapter 3 - 1
Energy and Packing
• Non dense, random packing
Energy
typical neighbor
bond length
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 - 2
Materials and Packing
Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline
crystalline SiO2
Si
Oxygen
noncrystalline SiO2
Chapter 3 - 3
Crystal Systems
Unit cell: smallest repetitive volume which
contains the complete lattice pattern of a crystal.
There are 6 lattice parameters
Their different combinations lead to:
7 crystal systems (shapes)
14 crystal lattices
In metals: Cubic and Hexagonal
a, b, and c are the lattice constants
Fig. 3.4, Callister 7e.
Chapter 3 - 4
Table 3.2
Lattice Parameter Relationships and Figures Showing
Unit Cell Geometries for the Seven Crystal Systems
See other shapes
Chapter 3 - 5
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 - 6
Metallic Crystal Structures
• Tend to be densely packed.
• 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.
- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures...
Chapter 3 - 7
Parameters in Studying Crystal Structures
•
•
•
•
Number and Position of Atoms in unit cell
Coordination No.
No. of touching-neighbor atoms
Cube edge (length)
Atomic Packing Factor (APF):
Fraction of volume occupied by atoms
Fraction of empty space = 1 – APF
APF = Total volume of atoms in the unit cell
Volume of the unit cell
Chapter 3 - 8
Simple Cubic Structure (SC)
• Rare due to low packing density (only Po* has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
* Po: Polonium
Chapter 3 - 9
Atomic Packing Factor (APF)
Volume of atoms in unit cell*
APF =
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
atoms
unit cell
a
R=0.5a
APF =
volume
atom
4
p (0.5a) 3
1
3
a3
close-packed directions
contains 8 x 1/8 = 1 atom/unit cell
volume
unit cell
Chapter 3 - 10
Body Centered Cubic Structure (BCC)
Atoms touch each other along cube diagonals.
examples: Cr, Fe(), W, Ta, Mo*
Coordination # = 8
No of Atoms per unit cell = 1 center + 8 corners x 1/8 =2 atoms
--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
* Cr: Chromium, W: Tungsten, Ta: Tantalum, Mo: Molybdenum
Chapter 3 - 11
Atomic Packing Factor: BCC
• APF for a body-centered cubic structure = 0.68
3a
a
2a
R
No. atoms
Close-packed directions:
length = 4R = 3 a
a
4
volume
atom
p ( 3a/4) 3
2
unit cell
3
APF =
volume
3
a
unit cell
Chapter 3 - 12
Face Centered Cubic Structure (FCC)
Atoms touch each other along face diagonals.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
Coordination # = 12
No of atoms per unit cell = 6 face x 1/2 + 8 corners x 1/8 = 4 atoms
--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
Chapter 3 - 13
Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74
maximum achievable APF
2a
Close-packed directions:
length = 4R = 2 a
a
atoms
volume
4
3
p ( 2a/4)
4
unit cell
atom
3
APF =
volume
3
a
unit cell
Chapter 3 - 14
FCC Stacking Sequence
• ABCABC... Stacking Sequence
• 2D Projection
B
B
C
A
B
B
B
A sites
C
C
B sites
B
B
C sites
• FCC Unit Cell
A
B
C
Chapter 3 - 15
Hexagonal Close-Packed Structure
(HCP)
• ABAB... Stacking Sequence
• 3D Projection
c
• 2D Projection
A sites
Top layer
B sites
Middle layer
A sites
Bottom layer
a
• Coordination # = 12
• APF = 0.74
• c/a = 1.633
No. of atoms/unit cell = 6
ex: Cd, Mg, Ti, Zn
Cd: Cadmium, Mg: Magnesium, Ti: Titanium, Zn: Zinc
Chapter 3 - 16
Polymorphism
Polymorphous Material:
Same Material with two or more distinct crystal structures
iron system
Crystal Structure is a Property:
It may vary with temperature
liquid
 Allotropy /polymorphism
Examples
Titanium: , -Ti
BCC
1538ºC
-Fe
Carbon: diamond, graphite
FCC
1394ºC
-Fe
912ºC
BCC
-Fe
Heating
Chapter 3 - 17
Theoretical Density, r
Density = r =
r =
where
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
= 6.023 x 1023 atoms/mol
Chapter 3 - 18
Theoretical Density, r
Example: For Cr
Known: A = 52.00 g/mol and R = 0.125 nm
Crystal Structure: BCC
For BCC: n = 2 and a = 4R/ 3 = 0.2887 nm
Calculate:
atoms
unit cell
r=
volume
unit cell
Compare
2 52.00
g
mol
a3 6.023 x 1023
rtheoretical = 7.18 g/cm3
ractual
= 7.19 g/cm3
R
a
atoms
mol
Chapter 3 - 19
Densities of Material Classes
In general
rmetals > rceramics > rpolymers
30
Why?
20
Metals have...
• close-packing (metallic bonding)
10
Ceramics have...
• less dense packing
• often lighter elements
Polymers have...
r (g/cm3 )
• often large atomic masses
• low packing density
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
5
4
3
2
1
0.5
0.4
0.3
Metals/
Alloys
Platinum
Gold, W
Tantalum
Silver, Mo
Cu,Ni
Steels
Tin, Zinc
Titanium
Aluminum
Magnesium
Water
Graphite/
Ceramics/
Semicond
Composites/
fibers
Polymers
Based 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).
Zirconia
Al oxide
Diamond
Si nitride
Glass -soda
Concrete
Silicon
Graphite
PTFE
Silicone
PVC
PET
PC
HDPE, PS
PP, LDPE
Glass fibers
GFRE*
Carbon fibers
CFRE*
Aramid fibers
AFRE*
Wood
Chapter 3 - 20
Structure-Properties
Materials-Structures
• Amorphous: Random atomic orientation
• Crystalline:
– Single Crystal Structure:
One Crystal
Perfectly repeated unit cells.
– Polycrystalline
Many crystals
Properties
Anisotropy: depend on direction
Isotropy: does not depend on direction
Chapter 3 - 21
Single Crystals
• Some engineering applications require single crystals:
--diamond single crystals for abrasives
--turbine blades
• Properties of crystalline materials often related to crystal structure.
Ex: Quartz fractures more easily
along some crystal planes than others.
Chapter 3 - 22
Properties for Single and Polycrystals
E (diagonal) = 273 GPa
Single Crystals
-Properties vary with direction: anisotropic.
-Example: the modulus of elasticity (E) in BCC iron:
Polycrystals
- Properties may/may not vary with direction.
-If grains are randomly oriented: isotropic.
(Epoly iron = 210 GPa)
E (edge) = 125 GPa
200 mm
-If grains are textured, anisotropic.
Chapter 3 - 23
Polycrystals
• Most engineering materials are polycrystals.
Anisotropic
1 mm
Isotropic
• Each "grain" is a single crystal (composed of many unit cells).
• If grains are randomly oriented,
overall component properties are not directional.
• Grain sizes typ. range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).
Chapter 3 - 24
Point Coordinates
z
Point coordinates for unit cell center are
111
c
a/2, b/2, c/2
y
000
a
x
b
½½½
Point coordinates for unit cell corner are
111

z
2c
Translation:
integer multiple of lattice constants 
identical position in another unit cell



b
y
b
Chapter 3 - 25
Crystallographic Directions
Vectors or lines between two points
Described by indices
Algorithm (for obtaining indices)
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 (multiply by a factor)
4. Enclose in square brackets, no commas
[uvw]
(note: If the value is –ve put the – sign over the number) x
Instead of 1 and 2: you can
1. Determine coordinates of head (H) and tail (T)
2. Subtract:
head – tail
Then as before (steps 3 and 4)
Ex 2: -1, 1, 1 => [ 111 ]
z
y
Ex 1: 1, 0, ½

2, 0, 1

[ 201 ]
(overbar represents a negative index)
families of directions <uvw>
Chapter 3 - 26
Linear Density
• Linear Density of Atoms  LD =
Number of atoms
Unit length of direction vector
ex: linear density of Al in [110] direction
[110]
a = 0.405 nm
# atoms
LD =
length
2
= 3.5 nm -1
2a
a
Chapter 3 - 27
HCP Crystallographic Directions
Algorithm
z
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
a2
[uvtw]
-
a3
a2
a1
ex:
½, ½, -1, 0
-a3
a2
2
=>
[ 1120 ]
a3
dashed red lines indicate
projections onto a1 and a2 axes
a1
2
a1
Chapter 3 - 28
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 - 29
Crystallographic Planes
Chapter 3 - 30
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 (for obtaining Miller Indices)
If the plane passes through the origin, choose another origin
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 - 31
Crystallographic Planes
z
Example 1
1. Intercepts
2. Reciprocals
3.
Integers
a
1
1/1
1
1
4.
Miller Indices
(110)
Example 2
1. Intercepts
2. Reciprocals
3.
Integers
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 - 32
Crystallographic Planes
z
Example 3
1. Intercepts
2. Reciprocals
3.
4.
Integers
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
Family of Planes {hkl}
Ex: {100} = (100), (010), (001), (100), (010), (001)
Chapter 3 - 33
Crystallographic Planes (HCP)
• In hexagonal unit cells the same idea is used
z
example
1. Intercepts
2. Reciprocals
3.
Integers
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
Chapter 3 - 34
Planar Density
Concept
The atomic packing of crystallographic planes
No. of atoms per area of a plane
Importance
Iron foil can be used as a catalyst.
The atomic packing of the exposed planes is important.
Example
a) Draw (100) and (111) crystallographic planes
for Fe (BCC).
b) Calculate the planar density for each of these planes.
Chapter 3 - 35
Planar Density of (100) Iron
Solution: At T < 912C iron has the BCC structure.
2D repeat unit
(100)
a=
4 3
R
3
Radius of iron R = 0.1241 nm
atoms
2D repeat unit
Planar Density =
area
2D repeat unit
1
a2
=
1
4 3
R
3
atoms
atoms
19
= 1.2 x 10
2 = 12.1
2
nm
m2
Chapter 3 - 36
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 - 37
Determination of Crystal Structure
Wave: has an amplitude and wave length (l)
Diffraction: phase relationship between two (or more) waves that
have been scattered.
Scattering:
–
–
In-phase
Out-of-phase
Constructive
Destructive
Detecting scattered waves after encountering an object:
Identify positions of atoms
‫مثل الرؤية البصرية‬
Diffraction occurs when a wave encounters a series of regularly
spaced atoms – IF:
1)
The obstacles are capable of scattering the wave
2)
Spacing between them nearly = wave length
Wave length depend on the source (i.e. material of the anode).
Chapter 3 - 38
X-Ray Diffraction
• Diffraction gratings must have spacings comparable to
the wavelength of diffracted radiation.
• Can’t resolve spacings  l
Chapter 3 - 39
Constructive versus Destructive Diffraction
Chapter 3 - 40
Constructive X-Ray Diffraction
Chapter 3 - 41
X-Ray Experiment
Chapter 3 - 42
X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
reflections must be in phase for
a detectable signal
extra distance traveled by wave 2
q
d Interplaner spacing
spacing between planes
Constructive diffraction occurs when
the second wave travels a distance = n l
From Geometry
extra traveled distance = 2 d sin q
Then
nl
d=
2 sin qc
In-phase
X-ray intensity
q
l
Constructive
q
qc
Chapter 3 - 43
Calculations
Measurement of critical angle, qc, allows computation of planar spacing, d.
nl
d=
2 sin qc
d=
a
(h2+k2+l2)0.5
Bragg’s Law, n: order of reflection
for plane (h k l)
Atoms in other planes cause extra scattering
For validity of Bragg’s law
For BCC
h+k+l = even
For FCC
h, k, l all even or odd
Sample calculations
Given
the plane indices and
Crystal structure,
get
a (from C.S. and R) and d (from indices)
Then with wavelength get angle or wavelength
Chapter 3 - 44
Solved Problem
3.58 Determine the expected diffraction angle for the first-order reflection
from the (310) set of planes for BCC chromium
when monochromatic radiation of wavelength 0.0711 nm is used.
BCC
d=
a
for plane (h k l)
(h2+k2+l2)0.5
For (310)
Brag’s Law
Chapter 3 - 45
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
(200)
Diffraction angle 2q
Diffraction pattern for polycrystalline -iron (BCC)
Chapter 3 - 46
y
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 - 47
SUMMARY
• 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).
• X-ray diffraction is used for crystal structure and
interplanar spacing determinations.
Chapter 3 - 48