ME 3101 Material Science

1. 1
Ch 3:
CRYSTAL STRUCTURES &
PROPERTIES
(adapted from Callister)
Mir M. Atiqullah, Ph.D.
Focus for this chapter:
• How do atoms assemble into solid structures?
(e.g. metals)- even molecules of compounds
• How does the density of a material depend on
its structure?
• When do material properties vary with the
test sample (i.e., part) orientation/direction?
1
Ch3: Crystal Structures
2
• Non dense, random packing
• Dense, regular packing
Dense, regular-packed structures tend to have
lower energy. = stronger bonding!
Energy
r
typical neighbor
bond length
typical neighbor
bond energy
Energy
r
typical neighbor
bond length
typical neighbor
bond energy
ENERGY AND PACKING- (motivation for)

2. 2
• atoms pack in periodic, 3D arrays
• typical of:
3
Crystalline materials...
-metals
-many ceramics
-some polymers (partial)
• atoms have no periodic packing
• occurs for:
Noncrystalline materials...
-complex structures
-by rapid solidification
Si Oxygen
crystalline SiO2
noncrystalline SiO2"Amorphous" = Noncrystalline
Adapted from Fig. 3.18(b),
Callister 6e.
Adapted from Fig. 3.18(a),
Callister 6e.
What is a crystalline Material ?
4
• 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 non-directional.
-Nearest neighbor distances tend to be
small in order to lower bond energy.
• have the simplest crystal structures.
We will look closely at
four (4) such structures...
METALLIC CRYSTALS
A Unit Crystal Cell
Crystal Lattice & Lattice Constants
• Corners represent atomic
sites/positions.
• a,b,c,α,β,γ are lattice constant
that define the lattice structure
Unit cell: Smallest packing pattern of the atoms that
repeats…
Crystal lattice- a 3 D arrangement of points representing
atomic positions

3. 3
Unit cell
geometries
Unit cell=
smallest
repetitive
Pattern/stacking
These are
7 Basic types
14 variations
possible
Here are the
14 varieties
What is the difference between
crystal structure and crystal system?
Answer: A crystal structure is described by both the
geometry of, and atomic arrangements within, the
unit cell, whereas a crystal system is described only in
terms of the unit cell geometry.
For example, face-centered cubic and body-centered
cubic are crystal structures that belong to the cubic
crystal system.

4. 4
5
• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# of nearest neighbors)
(Courtesy P.M. Anderson)
SIMPLE CUBIC STRUCTURE (SC)
6
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
• APF for a simple
cubic structure = 0.52
APF =
a3
4
3
π (0.5a)31
atoms
unit cell
atom
volume
unit cell
volume
close-packed directions
a
R=0.5a
contains 8 x 1/8 =
1 atom/unit cell
Adapted from Fig. 3.19,
Callister 6e.
ATOMIC PACKING FACTOR-Simple Cubic
• Coordination # = 8
7
Adapted from Fig. 3.2,
Callister 6e.
(Courtesy P.M. Anderson)
• Close packed directions are cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
BODY CENTERED CUBIC Crystal (BCC)
Ex.
Manganese

5. 5
a
R
8
• APF for a body-centered
cubic structure = 0.68
Close-packed directions:
length = 4R
= 3 a
Unit cell contains:
1 + 8 x 1/8
= 2 atoms/unit cell
Adapted from
Fig. 3.2,
Callister 6e.
ATOMIC PACKING FACTOR: BCC
APF =
a3
4
3
π ( 3a/4)32
atoms
unit cell atom
volume
unit cell
volume
9
• Coordination # = 12
Adapted from Fig. 3.1(a),
Callister 6e.
(Courtesy P.M. Anderson)
• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
FACE CENTERED CUBIC STRUCTURE (FCC)
APF =
a3
4
3
π ( 2a/4)34
atoms
unit cell atom
volume
unit cell
volume
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
a
10
• For a face-centered cubic structure APF = 0.74
Close-packed directions:
length = 4R
= 2 a
Adapted from
Fig. 3.1(a),
Callister 6e.
ATOMIC PACKING FACTOR: FCC

6. 6
11
• ABCABC... Stacking Sequence
• 2D Projection
A sites
B sites
C sites
B B
B
BB
B B
C C
C
A
A
• FCC Unit Cell
A
B
C
FCC STACKING SEQUENCE
12
• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection
A sites
B sites
A sites
Bottom layer
Middle layer
Top layer
Adapted from Fig. 3.3,
Callister 6e.
HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
Atoms per unit HCP cell
= 3 + 12* 1/6 + 2* ½ = 6
APF = 0.74 ( same as FCC)
Challenge- Show this.
13
• Compounds: Often have similar close-packed structures.
• Close-packed directions
--along cube edges.
• Structure of NaCl
Which color is Na ?
(Courtesy P.M. Anderson) (Courtesy P.M. Anderson)
STRUCTURE OF COMPOUNDS: NaCl
Size of compound mol
affects the cell size

7. 7
Some Crystal molecules
diamond
Graphite – Hexagonal
SiO4-group Structural unit
of Quartz
14
ρ = n A
VcNA
# atoms/unit cell Atomic weight (g/mol)
Volume/unit cell
(cm3/unit cell)
Avogadro's number
(6.023 x 1023 atoms/mol)
THEORETICAL DENSITY, ρ
Balancing units:
... vol
g
atoms
mole
vol
cell
mole
g
cell
atoms
mole
atoms
cell
vol
mole
g
cell
atoms
=⋅⋅⋅=
⋅
⋅
=ρ
Given Crystal structure, atomic wt.,density!VC!R( using ?)
Example: Density of Copper
• 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)
Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3
Result: theoretical ρCu = 8.89 g/cm3
Data from Table inside front cover of Callister (see next slide):
Compare to actual: ρCu = 8.94 g/cm3
Why the difference? Existence of Isotopes- some atoms
with more neutrons in nucleus.

8. 8
15
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
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
------
------
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
------
------
Crystal
Structure
FCC
------
BCC
HCP
Rhomb
------
HCP
FCC
Hex
BCC
------
BCC
HCP
FCC
------
Ortho.
Dia. cubic
FCC
------
------
Adapted from
Table, "Charac-
teristics of
Selected
Elements",
inside front
cover,
Callister 6e.
Characteristics of Selected Elements at 20C
ρmetals ρceramics ρpolymers
16
ρ(g/cm3)
Graphite/
Ceramics/
Semicond
Metals/
Alloys
Composites/
fibers
Polymers
1
2
20
30
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).10
3
4
5
0.3
0.4
0.5
Magnesium
Aluminum
Steels
Titanium
Cu,Ni
Tin, Zinc
Silver, Mo
Tantalum
Gold, W
Platinum
Graphite
Silicon
Glass-soda
Concrete
Si nitride
Diamond
Al oxide
Zirconia
HDPE, PS
PP, LDPE
PC
PTFE
PET
PVC
Silicone
Wood
AFRE*
CFRE*
GFRE*
Glass fibers
Carbon fibers
Aramid fibers
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.
DENSITIES OF MATERIAL CLASSES
Polymorphism or Allotropy
Some metals/nonmetals may exist in more than one
crystal structure depending on both temperature and
pressure.
Expl: Carbon ! graphite, diamond, Bucky Ball C60
nanotube O2 O3
BCC iron ! high temp. ! FCC iron
This condition of multiple crystal structures is also
known as ALLOTROPY.
See p-48 on Tin-Disease
" Tin (Sn) at room Temp BCC. At T<13.2 C, it transforms into
a structure similar to diamond ( complex grid structure) with
lower density and larger volume. May cause
disintegration/crumbling ... Read the story of Russian
disaster in 1850.
See the full story in Text

9. 9
Review :
Polymophism --
Difference between theoretical and actual density of
materials—
Interstitial –
Crystal structure vs Crystal system
How many crystal system, structure
APF
Coordination number
Close packed direction, plane
Slip plane
Crystallographic Directions: Cubic cell
A cubic cell with usual coordinates
Example directions = vectors with
lattice constants (side of the cube) =1
Direction of Red arrow:
Temporary new origin O’ (if needed, translate
the vector in parallel so that it starts at origin or at a
corner and contained within unit cell)
Direction vector –1, 1, 0.
Clear fraction : -1, 1, 0.
Format :
O’
0]11[
Example direction: Green Arrow.
Temporary origin:
Direction vector:
Clear Fraction:
Format:
O”
½, 1, -1
1, 2, -2
O”
]221[
½
HCP Directions
a1, a2, a3, z ! 4 directions/coordinates
Same Method/procedure as cubic:
1. Vector in units of coordinates a1, a2, a3, z
2. Clear fraction
3. Format
Example: Red arrow direction.
Vector: 1 -½ -1½ -1 ?
Clear Fraction: 2 -1 -3 -2 (multiplying by 2)
Formatted: [ ?? ]

10. 10
Linear Density ( of atomic Packing)
ex: linear density of Al in [110] direction
Al - Lattice Constant a = 0.405 nm
Linear Density of Atoms ≡ LD =
Unit length of direction vector
Number of atoms
# atoms
length
1
3.5 nm
a2
2
LD
−
==
a
[110]
Directions having same linear densities are considered
EQUIVALENT DIRECTIONS.
Al crystal- BCC, HCP, FCC, SC ?
How about along [100] direction?
Crystal Planes
Using 3 Miller Indices:
Procedure-
1.If plane goes thru origin,
choose another parallel
plane
2.length of the intercepts
in terms of a, b, c.
3.Calc. Reciprocal
4.Change the numbers to
smallest integer
5. Indices are (hkl)
Ex: Fig (b)-
2. 1 1 ∞
3. 1 1 0
4. (1 1 0)
Ex: A plane in a cubic cell
z
x
y
a b
c
•
•
•
4. Miller Indices (634)
example
1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾
2 1 4/3
3. Reduction 6 3 4
(001)(010),
Define: Family of Planes {hkl} ! having same planar density
(100), (010),(001),Ex: {100} = (100),

11. 11
Example: Planes in cubic cell
Example 3.9: Miller Indices?
Fig. (a) Plane thru origin.
1. Move the plane/origin along y-axis.
2.intercepts: inf., -1, ½
3. Reciprocal: 0, -1, 2
4. Smallest integers: 0, -1, 2
5. Indices are (hkl): )210(
Problem 3.41 Plane A.
1. Move the origin to O’, all intercepts within 1 unit..
2. Intercepts: ½, -½ , infinity.
3. Reciprocal: 2, -2, 0
4. Integers: 2, -2, 0
5. Indices are (hkl):
O’
0)22(
Atomic Arrangements
What is the atomic packing in a given crystallographic
plane?
Packing of atoms often determine the characteristic of
that materials in certain application, e.g. catalyst.
Iron foil can be used as a catalyst. The atomic packing
of the exposed planes is important.
" Consider (100) and (111) crystallographic planes for Fe.
" What are the planar densities for each of these planes?
Planar Density: Example =(100) Iron
Solution: At T < 912°C iron has the BCC
structure.
(100)
Radius of iron R = 0.1241 nm
R
3
34
a =
Adapted from Fig. 3.2(c), Callister 7e.
2D repeat unit
=Planar Density =
a2
1
atoms
2D repeat unit
=
nm2
atoms
12.1
m2
atoms
= 1.2 x 1019
1
2
R
3
34area
2D repeat unit

12. 12
Planar Density of (111) Iron
1 atom in plane/ unit surface cell
33
3 2
2
R
3
16
R
3
42
a3ah2area =







===
atoms in plane
atoms above plane
atoms below plane
ah
2
3
=
a2
1
= =
nm2
atoms
7.0
m2
atoms
0.70 x 1019
3 2
R
3
16
Planar Density =
atoms
2D repeat unit
area
2D repeat unit
Family of Planes
Family – (most) have common characteristics – same height,
same hair, eye color etc.
Planes in a family, similarly must have same planar densities.
In a given crystal unit cell, all planes having same density
belong to the same family.
E.g. in a BCC cell a family of planes is {100} = (100), (010),
(001), (-100), (101), (0-10), (110), (00-1), (111).  3 are wrong
For a different shaped crystal, this may not be true.
Rule of Thumb for Cubic Cell only:
" Planes in the family have same indices, +ve or –ve, irrespective of
the sequence, are all equivalent. E.g. {123} includes planes (123),
(312), (-231) etc.
REVIEW
Family of planes – all planes have same ____?
If a crystalline direction goes thru origin (of the cell
coordinates)- what should be done?
If a crystal plane goes through the origin, then? _____?
APF =?
COORD. # = ?
Review – Cryst. Direction! [ 1 2 1],
Miller indices (h k l) ! ( 2
1 2),
family of planes (cubic){ 1 1 0} includes
(1 0 1), (1 1 0), (1 0 0), (0 0 1), (0 1 1)
(0 1 0), (1 0 1), any plane not in the
family?
Direction indices=[ ]
1/3

13. 13
17
• Some engineering applications require single crystals:
• Crystal properties reveal features
of atomic structure.
(Courtesy P.M. Anderson)
--Ex: Certain crystal planes in quartz
fracture more easily than others.
(True for large salt crystals.)
--diamond single
crystals for abrasives
--turbine blades
--silicon wafers
Fig. 8.30(c), Callister 6e.
(Fig. 8.30(c) courtesy
of Pratt and Whitney).
(Courtesy Martin Deakins,
GE Superabrasives, Worthington,
OH. Used with permission.)
CRYSTALS AS BUILDING BLOCKS
Solidification
leading to
polycrystalline
structure
Neucleation –
forming of
first solid
Boundary –
formation
Grain boundary
Under microscope
18
• Most engineering materials are polycrystals.
• 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).
Adapted from Fig. K, color
inset pages of Callister 6e.
(Fig. K is courtesy of Paul E.
Danielson, Teledyne Wah
Chang Albany)
1 mm
POLYCRYSTALS

14. 14
19
• Single Crystals- bulk is just
one crystal
-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)
-If grains are textured,
anisotropic.
E (diagonal) = 273 GPa
E (edge) = 125 GPa
200 µm
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.)
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].)
SINGLE VS POLYCRYSTALS-
Their effects
What is –ANISOTROPY ?
Basically It is the dependence of materials properties
on the (crystallographic) direction.
-e.g. strength of wood depends on the fiber
orientation
ISOTOPIC MATERIALS- whose properties are
INDEPENDENT of direction of measurement..
Polycrystalline Material – (generally) ISOTROPIC
Which of these are ISOTROPIC?: wood, steel, fabric,
polymer/plastic, fiberglass fishing rod, rubber, ceramics
X Ray Diffraction (XRD)
Did you know x-ray is the main technology to
determine/estimate crystal structure?
Why? – x-ray wavelengths ~ lattice constants
a, b, c etc.
How long are wavelengths of Xray compared
to visible light? Microwave? FM ?
Let’s look at the comparative chart next !

15. 15
Electro magnetic Spectrum
Where is the range for cell phones?
Constructive and destructive Interference
of light waves
In phase
Out of phase
20
• Incoming X-rays diffract from crystal planes.
• Measurement of:
Critical angles, θc,
for X-rays provide
atomic spacing, d.
Adapted from Fig.
3.2W, Callister 6e.
X-RAYS TO CONFIRM CRYSTAL STRUCTURE
reflections must
be in phase to
detect signal
spacing
between
planes
d
incom
ing
X-rays
outgoing
X-rays
detector
θ
λ
θ
extra
distance
travelled
by wave “2”
“1”
“2”
“1”
“2”

16. 16
X-ray Diffraction Calculations-Bragg’s law
222
cells,cubicAlso
sin2
law)s(Bragg'sin2
distanceextrathe
lkh
a
d
n
d
dn
QTSQn
hklhkl
hkl
++
=
⋅
=
⋅⋅=
+=
θ
λ
θλ
λ
Knowing type of cubic,
we can select the PEAK
and its angle and the
reflecting plane.
(Goniometer in a) Xray Diffractometer
T – source of x ray
C - Detector (goniometer, with angle measurement)
S – powder crystalline sample – with possibly many grains
reflecting x-ray.
2Ө = angle of diffraction
d=nλ/2sinθc
x-ray
intensity
(from
detector)
θ
θc
Bragg’s law Demo
How it works?
Material is prepared as fine powder – that still keeps
many crystals.
X-ray will be reflected from various dense reflecting
planes with relatively higher intensity.
Angles (2θ) of all these intense reflections are
measured using the goniometer (of the
diffractometer).

17. 17
Typical Diffraction Pattern
Material : Pb (SC, BCC or FCC ?) powder
Type h2+k2+l2
Simple Cubic (All but 7) 1, 2, 3, 4, 5, 6, 8
BCC (Even) 2, 4, 6, 8, 10, 12, 14, 16
FCC 3, 4, 8, 11, 12, 16
Crystals of various materials but same structure,
will diffract at different angles but in the same
sequence of planes.
TRUE/FALSE ?
• 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.
23
SUMMARY