Post on 08-Aug-2018
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The Structure of
CHAPTER 3
Chapter 3-
Crystalline Solids
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ISSUES TO ADDRESS...ISSUES TO ADDRESS...ISSUES TO ADDRESS...ISSUES TO ADDRESS...
How do atoms assemble into solid structures?
(focus on metals)
How does the density of a material depend on
Chapter 3-
When do material properties vary with the
sample (i.e., part) orientation?
1
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Level of Structure in Metals
Macrostructure 10 XMicrostructure 1500 X
Chapter 3-
Substructure 105 X (SEM)
Crystal Structure X ray
Electronic Structure Spectrometer
Nuclear Structure Nuclear
Spectrometer
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Material Structure Level &
Chapter 3-Crystallization
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How do atoms assemble into solid
structures? (focus on metals)
SOLID MATERIALS
Chapter 3-
Crystalline Materials Non-Crystalline Materials
Single Crystal Poly-Crystal
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atoms have no periodic packing
occurs for:
Noncrystalline materials...
-complex structures
-rapid cooling
MATERIALS AND PACKING
Chapter 3-
Si Oxygen
noncrystalline SiO2
"Amorphous" = Noncrystalline
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Single Crystal (Quartz)
Chapter 3-
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atoms pack in periodic, 3D arrays
typical of:
Crystalline materials...
-metals-many ceramics
-some polymers
MATERIALS AND PACKING
Chapter 3-crystalline SiO2
Si Oxygen
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tend to be densely packed.
have several reasons for dense packing:
-Typically,Typically,Typically,Typically, onlyonlyonlyonly oneoneoneone elementelementelementelement isisisis present,present,present,present, sosososo allallallall atomicatomicatomicatomic
METALLIC CRYSTALS
Chapter 3- 4
rararara areareareare eeee samesamesamesame....----MetallicMetallicMetallicMetallic bondingbondingbondingbonding isisisis notnotnotnot directionaldirectionaldirectionaldirectional....
----NearestNearestNearestNearest neighborneighborneighborneighbor distancesdistancesdistancesdistances tendtendtendtend totototo bebebebe smallsmallsmallsmall inininin
orderorderorderorder totototo lowerlowerlowerlower bondbondbondbond energyenergyenergyenergy....
have the simplest crystal structures.
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Metallic Solids (Imp. Props)
High Strength
Ductile
High Density
Chapter 3-
Good electrical and thermalconductivity
Good workability
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Crystal Structure Terminology
Crystal Lattice (Regular periodic
arrangement of atoms)
Chapter 3-
Unit cell
Lattice parameters
cba ,, & ,,
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7 Crystal Systems
CUBIC
TETRAGONAL ORTHORHOMBIC
Chapter 3-
MONOCLINIC
TRICLINIC
HEXAGONAL
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CUBIC
Crystal Structure
cba ==
Chapter 3-
90===
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TETRAGONAL
CRYSTAL STRUCTURE
cba =
Chapter 3-
90===
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ORTHORHOMBIC
Crystal Structure
cba
Chapter 3-
90===
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RHOMBOHEDRAL
Crystal Structure
cba ==
Chapter 3-
90==
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MONOCLINIC
Crystal Structure
cba
Chapter 3-
== 90
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TRICLINIC
Crystal Structure
cba
Chapter 3-
90
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HEXAGONAL
Crystal Structure
Chapter 3-
120,90 ===
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SIMPLE CUBIC STRUCTURE
Chapter 3-
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Rare due to poor packing (only Po has this structure)
SIMPLE CUBIC STRUCTURE (SC)
Chapter 3- 5Ex.: Po structure
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Effective No. of Atoms in a unit cell
a
Chapter 3-
close-packed directions
R=0.5a
contains 8 x 1/8 =1 atom/unit cell
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APF =Volume of atoms in unit cell*
ATOMIC PACKING FACTOR
Chapter 3- 6
o ume o un ce
*assume hard spheres
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Data for Simple Cubic Structure
Coordination No. (CN) 6
a-r Relation a = 2r
Chapter 3-
Effective No. of atoms per unit cell
(EN) 1/8 X 8 = 1
Atomic Packing Factor (APF) 0.5
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Body Centered Cubic (BCC)
Chapter 3-
BODY CENTERED CUBIC
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--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
BODY CENTERED CUBIC
STRUCTURE (BCC)
Chapter 3- 7Ex Structure of Cr, Iron
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Data for BCC Structure
Coordination No. (CN) 8
a-r Relation ar 34 =
Chapter 3-
Effective No. of atoms per unit cell
(EN) 1/8 X 8 + 1 = 2
Atomic Packing Factor (APF) 0.68
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Face Centered Cubic FCC
Chapter 3-
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--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
FACE CENTERED CUBIC
STRUCTURE (FCC)
Chapter 3- 9Ex.: Structure of Al, Cu, Au
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Data for FCC Structure
Coordination No. (CN) 12
a-r Relation ar 24 =
Chapter 3-
Effective No. of atoms per unit cell
(EN) 1/8(8)+1/2(6) = 4
Atomic Packing Factor (APF) 0.74
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Hexagonally Packed Structure
HCP
Chapter 3-
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CLOSELY PACKED STRUCTURE (HCP)
B
Chapter 3-A
Stacking
A-B-A-B-A-B
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ABAB... Stacking Sequence
3D Projection
HEXAGONAL CLOSE-PACKED
STRUCTURE (HCP)
Chapter 3- 12
B sites
A sites
Ex.: Structure of Mg & Al
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Data for HCP Structure
Coordination No. (CN) 12
a-r Relation a = 2r
Chapter 3-
Effective No. of atoms per unit cell
(EN) = 1/6(12) + (2) + 1(3) = 6
Atomic Packing Factor (APF) = 0.74
CLOSELY PACKED STRUCTURE (FCC)
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CLOSELY PACKED STRUCTURE (FCC)
B
Stacking
A-B-C-A-B-C-A-B-C
Chapter 3-AC
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AB
ABCABC... Stacking Sequence
2D Projection
FCC STACKING SEQUENCE
Chapter 3-
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Crystallographic Points
Study of
Chapter 3-
Crystallographic Directions
Crystallographic Planes
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Crystallographic Points
Chapter 3-
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z
b
a
Pointq r s
Chapter 3-
x
yc
qa
rb
sc
q r s
L t 1 P i t i it ll
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z
a = 0.48 nm
b = 0.46 nm
Locate 1 Point in a unit cell
Chapter 3-
x
y
c =
0.40 nm 0.12
0.46
0.20
1
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Crystallographic Points in a SC Unit Cell
z
b
a 58
Chapter 3-
x
yc
2 3
4
1
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Concept of
Chapter 3-
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Equivalent Crystallographic Pointsare the points to arrive at same
Chapter 3-
corresponding point by unittranslations.
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0 0 0 All Corner Points
Set of Equivalent Points in a SC Unit Cell
Chapter 3-
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Crystallographic Points in a BCC Unit Cell
z
b
a 58
Chapter 3-
x
yc
2 3
4
1
9
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Set of Equivalent Points in a BCC Unit Cell
0 0 0 All Corner Points
All Body Centered
Chapter 3-
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Lattice Points in a FCC Unit Cellz
b
a 5
7
8
Chapter 3-
x
yc
2 3
41
9
10
11
Set of Equivalent Points in a FCC Unit Cell
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Set of Equivalent Points in a FCC Unit Cell
0 0 0 All Corner Points
0 All z-face centered
Chapter 3-
0 All y-face centered
0 All x-face centered
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Crystallographic Direction
Chapter 3-
ec or
Procedure to Draw a Crystal Direction
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Procedure to Draw a Crystal Direction
in a Unit Cell Note intercepts on x-, y- & z-axis in terms of edge lengths.
Divide the intercepts by corresponding edge lengths
Reduce them to smallest integers
Enclose these inte ers in S r bracket e. . [u v w].
Chapter 3-
Note[u v w] for one dirn, for family of dirns.
Negative index is noted by a bar on the top.
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Crystallographic Direction in a cubic Unit Cell
z
a
a
Chapter 3-
x
yO
A
OA [1 0 0]
C t ll hi Di ti i bi U it C ll
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z
a
a
Crystallographic Direction in a cubic Unit Cell
Chapter 3-
x
yO
A
OA [1 1 0]
C t ll hi Di ti i bi U it C ll
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z
a
a
Crystallographic Direction in a cubic Unit Cell
Chapter 3-
x
yO
OA [1 1 1]
Di ti i bi U it C ll
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z
a
a
Direction in a cubic Unit Cell
Chapter 3-
x
ya/2
OA
OA [2 0 1]
Direction in a cubic Unit Cell
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z
a
a
Direction in a cubic Unit Cell
Chapter 3-
x
y
a/2
O
A
a/2
OA [1 2 1]
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Equivalent Directions in a cubic Unit CellOR Family of Directions
are the directions which are crystallographically
Chapter 3-
equ va en ; .e. av ng same a om c pac ng
(spacing)
Equivalent Directions in a cubic Unit Cell
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q
along edges of cubeza
a
Chapter 3-
x
yO
A
OA [1 0 0]
Equivalent Directions in a cubic Unit Cell
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along edges of cubeza
a
Chapter 3-
x
yO
A
AO [1 0 0]
Equivalent Directions in a cubic Unit Cell
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along edges of cubeza
a
Chapter 3-
x
yO
A
OA [0 1 0]
Equivalent Directions in a cubic Unit Cell
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along edges of cubeza
a
Chapter 3-
x
yO
A
AO [0 1 0]
Equivalent Directions in a cubic Unit Cell
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along edges of cubeza
aA
Chapter 3-
x
yO
OA [0 0 1]
Equivalent Directions in a cubic Unit Cell
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along edges of cubeza
aA
Chapter 3-
x
yO
AO [0 0 1]
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Equivalent Directionsalong edges
Famil Re resentation
Chapter 3-
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
Chapter 3-
x
yO
A
OA [1 1 0]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
Chapter 3-
x
yO
A
AO [1 1 0]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
Chapter 3-
x
y
O
A
OA [1 1 0]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
Chapter 3-
x
y
O
A
AO [1 1 0]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
A
Chapter 3-
x
yO
OA [1 0 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
A
Chapter 3-
x
yO
AO [1 0 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
A
Chapter 3-
x
y
O
OA [1 0 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
A
Chapter 3-
x
y
O
AO [1 0 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cubeA
Chapter 3-
x
y
O
OA [0 1 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cubeA
Chapter 3-
x
y
O
AO [0 1 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
A
Chapter 3-
x
y
O
OA [0 1 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along face-diagonal of cube
A
Chapter 3-
x
y
O
AO [0 1 1]
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Equivalent Directionsalong Face Diagonals
Famil Re resentation
Chapter 3-
Equivalent Directions in a cubic Unit Cell
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z
a
a
along Body-Diagonal of cube
A
Chapter 3-
x
yO
OA [1 1 1]
Equivalent Directions in a cubic Unit Cell
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z
a
a
along Body-Diagonal of cube
A
Chapter 3-
x
yO
AO [1 1 1]
Equivalent Directions in a cubic Unit Cell
along Body-Diagonal of cube
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z
a
a
along Body Diagonal of cube
A
Chapter 3-
x
y
O
OA [1 1 1]
Equivalent Directions in a cubic Unit Cell
along Body-Diagonal of cube
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z
a
a
along ody iagonal of cube
A
Chapter 3-
x
y
O
AO [1 1 1]
Equivalent Directions in a cubic Unit Cell
along Body-Diagonal of cube
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z
a
a
g y g
A
Chapter 3-
x
yO
OA [1 1 1]
Equivalent Directions in a cubic Unit Cell
along Body-Diagonal of cube
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z
a
a
g y g
A
Chapter 3-
x
yO
AO [1 1 1]
Equivalent Directions in a cubic Unit Cell
along Body-Diagonal of cube
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z
a
a
g y g
A
Chapter 3-
x
y
O
OA [1 1 1]
Equivalent Directions in a cubic Unit Cell
along Body-Diagonal of cube
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z
a
a
g y g
A
Chapter 3-
x
y
O
AO [1 1 1]
Equivalent Directions
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q
along Body Diagonals
Famil Re resentation
Chapter 3-
Important Observation in Cubic Unit Cell
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In case of cubic unit cell,directions having same indices
irrespective of ORDER & SIGNare equivalent.
Chapter 3-
E.g. [1 1 0], [0 1 1], [1 0 1] are
equivalent
Draw a [1 1 0] direction in a Cubic Unit Cell
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z
a
a
Chapter 3-
x
yO
A
-a
[1 1 0] O
A
[1 1 0]
Shift of Origin to accommodate ve direction
in unit cell itselfO
) [ ]
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z
a
a
5
3
) [u v w] [- + +]
[+ - +]
+ + -
Chapter 3-
x
y
a
12
4
[- - +]
[+ - -]
[- + -]
[- - -]
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Cr stallo ra hic Planes
Chapter 3-
Procedure to Draw a Crystallographic
Plane in a Unit Cell
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Note intercepts on x-, y- & z-axis in terms of edgelengths.
Divide the intercepts by corresponding edge lengths Write the reciprocals
Chapter 3-
Enclose these integers in parenthesis e.g. (h k l). Note
(h k l) for one plane {h k l} for family of planes
Negative indices are noted by a bar on the top
Crystal Planes in a cubic Unit Cell along
faces of unit cellz
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z
a
a
B
Chapter 3-
x
y
A
D
ABCD (1 0 0)
z
Crystal Planes in a cubic Unit Cell along
faces of unit cell
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z
a
a
C
D
Chapter 3-
x
yA
B
ABCD (0 1 0)
z
Crystal Planes in a cubic Unit Cell along
faces of unit cell
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z
a
a
A
D
Chapter 3-
x
y
ABCD (0 0 1)
z
Crystal Planes in a cubic Unit Cell
along face-Diagonal of cube
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z
a
a
B
C
Chapter 3-
x
y
A
D
ABCD (1 1 0)
z
Crystal Planes in a cubic Unit Cell
along face-Diagonal of cube
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z
a
a
C
D
Chapter 3-
x
y
B
A
ABCD (1 1 0)
ABCD (1 1 0)OR
z
Crystal Planes in a cubic Unit Cell
along face-Diagonal of cube
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z
a
a C
B
Chapter 3-
x
y
A
D
ABCD (0 1 1)
z
Crystal Planes in a cubic Unit Cell
along face-Diagonal of cube
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z
a
a
C
B
Chapter 3-
x
y
A
D
ABCD (0 1 1)
ABCD (0 1 1)OR
z
Crystal Planes in a cubic Unit Cell
along face-Diagonal of cube
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z
a
a
BC
Chapter 3-
x
yAD
ABCD (1 0 1)
ABCD (1 0 1)OR
z
Crystal Planes in a cubic Unit Cell
along face-Diagonal of cube
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z
a
a
AB
Chapter 3-
x
y
O
ABCD (1 0 1)
C
D
z
Crystal Planes in a cubic Unit Cell
along 3 face diagonals of a cube
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za
aC
Chapter 3-
x
y
A
B
ABC (1 1 1)
z
Determine the Miller Indices for the plane
ABCD
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za
a
Chapter 3-
x
y
A
a/2
BDa
ABCD (0 1 2)
z
Construct a (0 1 1) plane within a
Cubic Unit Cell
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a
a
Chapter 3-
x
yO
z
For (0 1 1) plane adjacent Unit Cell in ve y-
direction is required
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a
a
Chapter 3-
x
y
A
-yO
z
(0 1 1) plane w.r.t Cubic Unit Cell
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a
a
(0 1 1)
Chapter 3-
x
y-y
0
O
a
Transfer the coordinate System to
accommodate y axis
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a
a
Chapter 3-
x
y
z
O
-y
a
(0 1 1) plane within a Cubic Unit Cell
C
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a
a
C
B
Chapter 3-
x
y
z
O
-y
A
D
z
Construct a (1 1 1) plane within a
Cubic Unit Cell
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a
a
C
O
Chapter 3-
x
y
z
Construct a (1 1 1) plane within a
Cubic Unit Cell
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a
a
O
A
B
Chapter 3-
x
y
C
(1 1 1)
Equivalent Planes OR Family of Planes
are the planes which are Crystallographically
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p y g p y
equivalent; i.e. having the same atomic packing
(spacing)
Chapter 3-
{1 0 0}
{1 1 0} {1 1 1}
In case of cubic unit cell, planes
Important Observation in Cubic Unit Cell
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, phaving same indices irrespective
of ORDER & SIGN are equivalent.
Chapter 3-
E.g. (0 0 1), (0 1 0), (1 0 0) areequivalent
Important Observation in Cubic Unit Cellz
a
a
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B aC
Chapter 3-
x
y
A
DABCD (1 0 0
OA [1 0 0
O
Miller indices are same for a plane and
direction, which are perpendicular to each
other
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Density of a crystal Structure
Chapter 3-
Theoretical Density of a crystal Structure
nA
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Ac
Th
NV
nA=
Chapter 3-
n Effective No. of atoms per unit cell
A Atomic weight, gram per mole
Vc Volume of unit cell
Na Avogadro's No. = 6.023 XXXX 1023
Data from Table inside front cover of Callister
Theoretical Density of Cu
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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 = 10-7 cm)
Chapter 3-
R = 0.128 x 10- cm
Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10
-23cm3
Compare to actual: Cu = 8.94 g/cm3
Result: theoretical Cu = 8.89 g/cm3
metals ceramics polymersGraphite/
Ceramics/Semicond
Metals/Alloys Composites/fibersPolymers
30Why?
DENSITIES OF MATERIAL CLASSES
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3)
20Based on data in Table B1, Callister
*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on
60% volume fraction of aligned fibersin an epoxy matrix).10
SteelsCu,Ni
Tin, Zinc
Silver, Mo
TantalumGold, WPlatinum
Zirconia
Why?Metals have... close-packing
(metallic bonding) large atomic mass
Chapter 3- 16
(g
/c
1
2
3
4
0.3
0.40.5
Magnesium
Aluminum
Titanium
Graphite
Silicon
Glass-sodaConcrete
Si nitrideDiamondAl oxide
HDPE, PSPP, LDPE
PC
PTFE
PETPVCSilicone
Wood
AFRE*
CFRE*
GFRE*
Glass fibers
Carbon fibers
Aramid fibers
... less dense packing
(covalent bonding) often lighter elements
Polymers have... poor packing
(often amorphous) lighter elements (C,H,O)
Composites have... intermediate values
Atomic arrangements
(110) plane of FCC
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A B C
D E F
A
BC
Chapter 3-
DE
F
Atomic packing in(110) planeReduced sphereFCC unit cell with(110) plane
Atomic packing in (100) plane of FCC
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Chapter 3-
Atomic Packing in (111) plane of BCC
Atomic arrangements
(110) plane of BCC
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A/
B/
C/E/
A/ B/
C/
D/
Chapter 3-
D/
Reduced sphereBCC unit cell with(110) plane
Atomic packing in(110) plane of aBCC structure
E/
Atomic Packing Factor: BCC APF for a body-centered cubic structure = 0.68
a3
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a
a2
a3
Chapter 3-119
APF =
4
3 ( 3a/4 )3
2
atoms
unit cell atom
volume
a3
unit cell
volume
length = 4R =
Close-packed directions:
3aaR
From
Fig. 4.2(a)
Callisters MaterialsScience and Engineering,
Adapted Version.
APF for a face-centered cubic structure = 0.74
Atomic Packing Factor: FCC
maximum achievable APF
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Close-packed directions:
length = 4R = 2a
Unit cell contains:
2a
Chapter 3-120
APF =
4
3
( 2a/4 )34
atoms
unit cell
atom
volume
a3
unit cell
volume
6 x1/2 + 8 x1/8
= 4 atoms/unit cellaFrom
Fig. 4.1(a),
Callisters Materials Science and
Engineering,
Adapted Version.
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Chapter 3-
Linear Density formula
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VectorDirectionoLen thvectordirectiononcenteredatomsofNumberLD
.........=
Chapter 3-
z
a
a
Linear density in a SC unit cell along
Edge
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a
a
Chapter 3-
x
y
LD = 1/a = 1/2R
z
Linear density in a SC unit cell along
Face Diagonal
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a
a
Chapter 3-x
y
a
LD = 1/ 2a = 1/2 2R
za
Linear density in a SC unit cell along Body
Diagonal LD
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a
Chapter 3-
x
y
a
LD
= 1/ 3a = 1/2 3R
z
a
Linear density in a BCC unit cell along Body
Diagonal LD
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a
Chapter 3-
x
y
a
LD
= 2/ 4R = 1/ 2R
Linear Density
Linear Density of Atoms LD =Unit length of direction vector
Number of atoms
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ex: linear density of Al in [110]direction
=
[110]
Chapter 3-127
a# atoms
length
13.5 nm
a2
2LD
========
z
a
Linear density in a FCC unit cell along
Face Diagonal LD
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a
a
Chapter 3-x
y
a
LD = 2/ 2a = 1/2R
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PLANAR DENSITY
Chapter 3-
Planar Density Formula
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PlaneaoAreaPlaneaoncenteredAtomsofNumberPD
.........=
Chapter 3-
Planar Density in a SC unit cell
PD {1 0 0}
za
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a
Chapter 3-
x
z
PD{1 00}= 1 /a2 = 1/4R2
Planar Density in a SC unit cell
PD {1 1 0}
za
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a
Chapter 3-
x
z
PD{1 1 0}= 1 /a 2 a =
1/4R2 2
Planar Density in a SC unit cell
PD {1 1 1}
za
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a
Chapter 3-
x
z
PD{1 1 1}= 1/2 /( 3 a2/2) =
1/4R2 3
Planar Density in a BCC unit cell
PD {1 0 0}
za
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a
Chapter 3-
x
z
PD{1 00}= 1 /a2 = 3/16R2
Planar Density in a BCC unit cell
PD {1 1 0}
za
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a
Chapter 3-
x
z
PD{1 10}=2/ 2a2 = 3 2/16R2
Planar Density in a BCC unit cell
PD {1 1 1}
za
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a
Chapter 3-
x
z
PD{1 1 1}= 1/2 /( 3 a2/2) =
3 /16R2
Planar Density in a FCC unit cell
PD {1 0 0}
za
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a
Chapter 3-
x
z
PD{1 00}= 2 /a2 = 1/4R2
Planar Density in a FCC unit cell
PD {1 1 0}
za
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a
Chapter 3-
x
z
PD{1 10}=2/ 2a2 = 2/8R2
Planar Density in a FCC unit cell
PD {1 1 1}
za
a
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a
Chapter 3-
x
z
PD{1 1 1}= 2 /( 3 a2/2) = 1/2 3 R2
Polymorphism & Allotropy
Metals/Nonmetals may have more than one crystalstructure. This phenomenon is called as polymorphism
d th diti i ll d ll t
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and the condition is called as allotropy
Chapter 3-
e.g. 1. Carbon: Graphite at ambient condition
Diamond at extreme pressure
2. Iron: BCC Iron at room temperature
FCC Iron at 912 oC.
Demonstrates "polymorphism"The same atoms can
have more than one
crystal structure
DEMO: HEATING AND
COOLING OF AN IRON WIRE
T t C
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crystal structure.Temperature, C
1536BCC Stable
Liquid
Chapter 3- 22
BCC Stable
FCC Stable
914
1391
shorter
longer!shorter!
longer
Tc 768 magnet falls off
heat up
cool down
ANISOTROPY
Directionality of a property is called as an
ANISOTROPY
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ANISOTROPY.
If properties are independent of direction, then
Chapter 3-
.
Degree of anisotropy is dependent on structuresymmetry.
Higher the symmetry, lower the degree of anisotropy
and lower the symmetry, higher the degree of anisotropy.
Atoms may assemble into crystalline or
amorphous structures.
SUMMARY
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We can predict the density of a material,provided we know the atomic weight, atomic
Chapter 3-
BCC, HCP).
Material properties generally vary with singlecrystal orientation (i.e., they are anisotropic),
but properties are generally non-directional
(i.e., they are isotropic) in polycrystals withrandomly oriented grains.
23