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7/25/2019 Ch03 Crystalline Solids
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AL- FATIHAH
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2MME 2503Materials Science & Engineering
CHAPTER 3:
THE STRUCTURE OF CRYSTALLINE SOLIDS
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?
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MME 2503Materials Science & Engineering 3
MATERIALS AND PACKING
atoms pack in periodic, 3D arraysCrystallinematerials...
-metals
-many ceramics
-some polymers
atoms have no periodic packing
Noncrystallinematerials...
-complex structures-rapid cooling
crystalline SiO2
noncrystalline SiO2"Amorphous" = Noncrystalline
Si Oxygen
typical of:
occurs for:
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Chapter 3 -MME 2503Materials Science & Engineering
4
CRYSTAL SYSTEMS
7 crystal systems
14 crystal lattices
Unit cell: smallest repetitive volume whichcontains the complete lattice pattern of a crystal.
a, b, and care the lattice constants
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Chapter 3 -MME 2503Materials Science & Engineering 5
CRYST L SYSTEMS
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MME 2503Materials Science & Engineering 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.
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7
SIMPLE CUBIC STRUCTURE (SC)
Rare due to low packing density(only Po has this structure)
Close-packed directionsare cube edges.
Coordination #= 6
(# nearest neighbors)
MME 2503Materials Science & Engineering
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8
ATOMIC PACKING FACTOR (APF)
APF for a simple cubic structure = 0.52
APF =a3
4
3p (0.5a) 31
atoms
unit cellatom
volume
unit cell
volume
APF =Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
close-packed directions
a
R=0.5a
contains 8 x 1/8 =1 atom/unit cell
MME 2503Materials Science & Engineering
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9
BODY CENTERED CUBIC STRUCTURE (BCC)
Coordination # = 8
Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded
ex: Cr, W, Fe (), Tantalum, Molybdenum
2 atoms/unit cell: 1 center + 8 corners x 1/8
MME 2503Materials Science & Engineering
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10
ATOMIC PACKING FACTOR: BCC
a
APF =
4
3p( 3a/4 )32
atoms
unit cell atom
volume
a3
unit cell
volume
length = 4R=
Close-packed directions:
3 a
APF for a body-centered cubic structure = 0.68
aR
a2
a3
MME 2503Materials Science & Engineering
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11
FACE CENTERED CUBIC STRUCTURE (FCC)
Coordination # = 12
Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
MME 2501 - Engineering Materials
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MME 2501 - Engineering Materials 12
ATOMIC PACKING FACTOR: FCC APF for a face-centered cubic structure = 0.74
maximum achievable APF
APF =
43p ( 2a/4 )34
atoms
unit cell atomvolume
a3
unit cell
volume
Close-packed directions:
length = 4R= 2 a
Unit cell contains:6 x1/2 + 8 x1/8
= 4 atoms/unit cella
2 a
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Chapter 3 -MME 2501 - Engineering Materials
13
A sites
B B
B
BB
B B
Csites
C C
CA
B
Bsites
ABCABC... Stacking Sequence
2D Projection
FCC Unit Cell
FCC Stacking Sequence
B B
B
BB
B B
Bsites
C C
CA
C C
CA
A
BC
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MME 2501 - Engineering Materials 14
HEXAGONAL CLOSE-PACKED STRUCTURE
(HCP)
Coordination # = 12
ABAB... Stacking Sequence
APF = 0.74
3D Projection 2D Projection
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
c/a= 1.633
c
a
A sites
Bsites
A sites Bottom layer
Middle layer
Top layer
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Chapter 3 -MME 2501 - Engineering Materials
15
THEORETICAL DENSITY, R
where n= number of atoms/unit cell
A=atomic weight
VC= Volume of unit cell = a3for cubic
NA= Avogadros number
= 6.023 x 1023atoms/mol
Density = =
VCNA
nA =
CellUnitofVolumeTotal
CellUnitinAtomsofMass
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Chapter 3 -
Ex: Cr (BCC)
A=52.00 g/mol
R= 0.125 nm
n= 2
MME 2501 - Engineering Materials
16
THEORETICAL DENSITY, R
theoretical
a= 4R/ 3 = 0.2887 nm
actual
aR
=a3
52.002
atoms
unit cell mol
g
unit cell
volume atomsmol
6.023x1023
= 7.18 g/cm3
= 7.19 g/cm3
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Chapter 3 -MME 2501 - Engineering Materials
17
DENSITIES OF MATERIAL CLASSES
metals>
ceramics>
polymers
Why?
(
g/cm
)3
Graphite/Ceramics/
Semiconductor
Metals/Alloys
Composites/fibers
Polymers
1
2
2 0
30
*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% 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
TantalumGold, WPlatinum
Graphite
Silicon
Glass-sodaConcrete
Si nitrideDiamond
Al oxide
Zirconia
HDPE, PSPP, LDPE
PC
PTFE
PET
PVCSilicone
Wood
AFRE*
CFRE*
GFRE*
Glass fibers
Carbon fibers
Aramid fibers
Metalshave... close-packing
(metallic bonding)
often large atomic massesCeramicshave...
less dense packing
often lighter elements
Polymershave...
low packing density(often amorphous)
lighter elements (C,H,O)
Compositeshave... intermediate values
In general
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MME 2501 - Engineering Materials 18
CRYSTALS AS BUILDING BLOCKS
Someengineering applications require single crystals:
Properties of crystalline materials
often related to crystal structure.
--Ex: Quartz fractures more easily
along some crystal planes thanothers.
--diamond single
crystals for abrasives
--turbine blades
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MME 2501 - Engineering Materials 19
POLYCRYSTALS
Mostengineering materials are polycrystals.
Nb-Hf-W plate with an electron beam weld.
Each "grain" is a single crystal. If grains are randomly oriented,
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).
1 mm
Isotropic
Anisotropic
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MME 2501 - Engineering Materials 20
SINGLE VS POLYCRYSTALS
Single Crystals
-Properties vary withdirection: 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.
200 mm
E (diagonal) = 273 GPa
E (edge) = 125 GPa
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Chapter 3 -
Two or more distinct crystal structures for thesame material (allotropy/polymorphism)
titanium
, -Ti
carbon
diamond, graphite
MME 2501 - Engineering Materials
21
POLYMORPHISM
BCC
FCC
BCC
1538C
1394C
912C
-Fe
-Fe
-Fe
liquid
iron system
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Chapter 3 -
Point coordinates for unit
cell center are
a/2, b/2, c/2
Point coordinates for unitcell corner are 111
Translation: integer multipleof lattice constants identical position inanother unit cell
MME 2501 - Engineering Materials
22
POINT COORDINATESz
x
y
a b
c
000
111
y
z
2c
b
b
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Chapter 3 -MME 2501 - Engineering Materials
23
CRYSTALLOGRAPHIC DIRECTIONS
1. Vector repositioned (if necessary) to passthrough 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]
ex:1, 0, => 2, 0, 1 => [201]
-1, 1, 1
z
x
Algorithm
where overbar represents a
negative index
[111]=>
y
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Chapter 3 -MME 2501 - Engineering Materials
26
CRYSTALLOGRAPHIC PLANES
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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.
Algorithm1. Read off intercepts of plane with axes in
terms of a, b, c
2. Take reciprocals of intercepts3. Reduce to smallest integer values
4. Enclose in parentheses, no
commas i.e., (hkl)
MME 2501 - Engineering Materials
27
CRYSTALLOGRAPHIC PLANES
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Chapter 3 -MME 2501 - Engineering Materials
28
CRYSTALLOGRAPHIC PLANESz
x
ya b
c
4. Miller Indices (110)
example a b cz
x
ya b
c
4. Miller Indices (200)
1. Intercepts 1 1
2. Reciprocals 1/1 1/1 1/1 1 0
3. Reduction 1 1 0
1. Intercepts 1/2 2. Reciprocals 1/ 1/ 1/
2 0 03. Reduction 2 0 0
example a b c
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Chapter 3 -MME 2501 - Engineering Materials
29
CRYSTALLOGRAPHIC PLANES
z
x
ya b
c
4. Miller Indices (634)
example1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/ 1/1 1/
2 1 4/3
3. Reduction 6 3 4
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Chapter 3 -
Linear Density of Atoms LD =
MME 2501 - Engineering Materials
31
LINEAR DENSITY
ex: linear density of Al in [110]
direction
a= 0.405 nm
a
[110]
Unit length of direction vector
Number of atoms
# atoms
length
13.5 nma2
2LD -==
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Chapter 3 -
We want to examine the atomic packingof 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.
MME 2501 - Engineering Materials
32
CRYSTALLOGRAPHIC PLANES
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Chapter 3 -
Solution: At T < 912C iron has the BCCstructure.
MME 2501 - Engineering Materials
33
PLANAR DENSITY OF (100) IRON
(100)
Radius of iron R= 0.1241 nm
R3
34a =
Adapted from Fig. 3.2(c), Callister 7e.
2D repeat unit
=Planar Density =a2
1
atoms
2D repeat unit
=nm2
atoms12.1
m2
atoms= 1.2 x 1019
1
2
R3
34area
2D repeat unit
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Chapter 3 -
Solution (cont): (111) plane
MME 2501 - Engineering Materials
34
PLANAR DENSITY OF (111) IRON
1 atom in plane/ unit surface cell
333
2
2
R3
16R
3
42a3ah2area =
===
atoms in plane
atoms above plane
atoms below plane
ah23=
a2
1
= =nm2
atoms7.0
m2
atoms0.70 x 1019
3 2R3
16Planar Density =
atoms
2D repeat unit
area
2D repeat unit
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Chapter 3 -
Diffraction gratings must have spacingscomparable to the wavelength of diffractedradiation.
Cant resolve spacings Spacing is the distance between parallel planes of
atoms. MME 2501 - Engineering Materials
35
X-RAY DIFFRACTION
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Chapter 3 -MME 2501 - Engineering Materials
36
X-RAYS TO DETERMINE CRYSTAL
STRUCTURE
X-rayintensity(fromdetector)
q
qc
d=n
2sinqc
Measurement ofcritical angle, qc,allows computation of
planar spacing, d.
Incoming X-rays diffractfrom crystal planes.
Adapted from Fig. 3.19,
Callister 7e.
reflections mustbe in phase fora detectable signal
spacingbetweenplanes
d
q
qextra
distancetravelledby wave 2
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Chapter 3 -MME 2501 - Engineering Materials
37
X-RAY DIFFRACTION PATTERN
(110)
(200)
(211)
z
x
ya b
c
Diffraction angle 2q
Diffraction pattern for polycrystalline -iron (BCC)
Intensity(relative)
z
x
ya b
c
z
x
ya b
c
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MME 2501 - Engineering Materials 38
SUMMARY
Atoms may assemble into crystallineoramorphousstructures.
We can predict the densityof a material, provided we
know the atomic weight, atomic radius, and crystal
geometry(e.g., FCC, BCC, HCP).
Common metallic crystal structures are FCC, BCC, and
HCP. Coordination numberand atomic packing factor
are the same for both FCC and HCP crystal structures.
Crystallographic points, directionsand planesare
specified in terms of indexing schemes.
Crystallographic directions and planes are related
to atomic linear densitiesand planar densities.
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MME 2501 Engineering Materials 39
SUMMARY
Some materials can have more than one crystal
structure. This is referred to as polymorphism(or
allotropy).
Materials can be single crystalsor 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.
X-ray diffractionis used for crystal structure andinterplanar spacingdeterminations.
Recommended