Chapter3 - Chapter 3 Crystal Structures Introduction...

•
Introduction
•
Bravais Lattice
•
Crystals with One Atom per Lattice Point
•
Miller Indices
•
Densities and Packing Factors of Crystalline
Structures
•
Interstitial Positions and Sizes
•
Crystals with Multiple Atoms per Lattice Site
•
Liquid Crystals
•
Single Crystals and Polycrystalline Materials
•
Polymorphism
•
X-Ray Diffraction
Chapter 3
Crystal Structures

•
Many materials crystallize in a regular array
with basic building blocks being at repeated
regular intervals.
•
The structure is determined by the bond
character and energy minimization.
•
Metals, ceramics, polymers can all form
crystal structures.
•
Properties are dependent upon the type of
bond and the structure.
Chapter 3
Crystal Structures

Lattice points
Unit cell
Square Rectangular Hexagonal
a = b, φ = 90° a ≠ b, φ = 90° a = b, φ = 120°
Bravais Lattice and Unit Cells

Tiling a General Parallelogram
The tiles are space filling satisfying the need
for translational periodicity of a lattice

Lack of Translational Periodicity Using a
Pentagon

Basis-
Single character
Basis-
Double character
Basis-
Two characters with
different orientations
Lattice + Basis = Crystal structure
Bravais Lattice and Unit Cells

a = b = c, α = β = γ = 90°
a = b ≠ c, α = β = γ = 90°
a ≠ b ≠ c, α = β = γ = 90°
≠
a = b ≠ c,
α = β = 90°, γ = 120°
Cubic
Tetragonal
Orthorhombic
Rhombohedral
a ≠ b ≠ c,
α = γ = 90°≠ β
a = b = c,
α = β = γ ≠ 90°
a ≠ b ≠ c,
α ≠ β ≠ γ ≠ 90°
Monoclinic
Triclinic
Hexagonal
Bravais Lattice and Unit Cells

x
y
z
Bravais Lattice and Unit Cells- Face Centered
Cubic (FCC)
a
1
a
3
a
2
Each lattice point
is surrounded by
12 equivalent lattice
points.
Three alternative
vectors define a
unit cell that is
primitive, one
lattice point per
unit cell.

Equivalence of lattice points
Position A is in the center
of the black cube , and a
corner of the blue cube
Position B is in the center
of the blue cube , and a
corner of the black cube
Bravais Lattice and Unit Cells

Body Center Cubic (BCC) Lattice
Each lattice point
is surrounded by
8 equivalent lattice
points.
Three alternative
vectors define a
unit cell that is
primitive, one
lattice point per
unit cell.

Basis
Lattice
Alternate Descriptions

What is meant by the terms cross product and triple scalar product ?
ˆ
ˆ ˆ
o
a b c V
a a ui vj wk
⋅ × =
 
= + +
 
r
r r
r
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ]
3
3 3
100 010 001
010 001 0 1 0 100
0 0 1
100 100
o
o o
a b c a
i j k
a a
⋅ × = ⋅ ×
 
 
× = =
 
 
 
⋅ =
r
r r
[ ]
[ ]
[ ]
ˆ
ˆ ˆ
1 0 0 100
ˆ
ˆ ˆ
0 1 0 010
ˆ
ˆ ˆ
0 0 1 001
o o
o o
o o
a a i j k a
b a i j k a
c a i j k a
 
= + + =
 
 
= + + =
 
 
= + + =
 
r
r
r
o
a
[ ]
100
[ ]
010
[ ]
001
Non-primitive unit cell

[ ] [ ] [ ]
[ ] [ ]
[ ]
3
3 3
101 110 011
8
110 011 1 1 0 111
0 1 1
101 111
8 4
o
p p p
o o
p p p
a
a b c
i j k
a a
a b c
⋅ × = ⋅ ×
 
 
 
× = =
 
 
 
 
 
⋅ × = ⋅ =
 
r
r r
r
r r
o
a
[ ]
101
2
o
a
[ ]
110
2
o
a
[ ]
011
2
o
a
Primitive unit cell

Face Centered Cubic (FCC) Phase
Lattice
Hard sphere packing
Fractions of
atoms in the
unit cell
Distance between
the centers of
adjacent atoms
Copper is an
FCC material