Crystal Structures - Class 1
Crystal Structures - Class 1
Crystal Structures - Class 1
OF
CRYSTALLINE SOLIDS
3.2 FUNDAMENTAL CONCEPTS
SOLIDS
AMORPHOUS CRYSTALLINE
Atoms in an amorphous Atoms in a crystalline solid
solid are arranged are arranged in a repetitive
randomly- No Order three dimensional pattern
Long Range Order
All metals are crystalline solids
Many ceramics are crystalline solids
Some polymers are crystalline solids
LATTICE
4
FACE CENTERED CUBIC STRUCTURE (FCC)
FACE CENTERED CUBIC STRUCTURE (FCC)
For FCC:
a = 2R√2
For BCC:
a = 4R /√3
For HCP
a = 2R
c/a = 1.633 (for ideal case)
Note: c/a ratio could be less or more than the ideal value of
1.633
Face Centered Cubic (FCC)
2a0 4r
r
2r a0
r
a0
Body Centered Cubic (BCC)
3a 0 4r
2a0
3a0
a0
Coordination Number
► The number of touching or nearest
neighbor atoms
► SC is 6
► BCC is 8
► FCC is 12
► HCP is 12
ATOMIC PACKING FACTOR
6
ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
a = 4R /√3
Unit cell contains:
1 + 8 x 1/8
R = 2 atoms/unit cell
a
FACE CENTERED CUBIC STRUCTURE (FCC)
• Coordination # = 12
ATOMIC PACKING FACTOR: FCC
• APF for a face-centered cubic structure = 0.74
a = 2R√2
Unit cell contains:
6 x 1/2 + 8 x 1/8
a
= 4 atoms/unit cell
3.5 Density Computations
►
► Density of a material can be determined theoretically
from the knowledge of its crystal structure (from its
Unit cell information)
► Density mass/Volume
► Mass is the mass of the unit cell and volume is the
unit cell volume.
► mass = ( number of atoms/unit cell) “n” x mass/atom
► mass/atom = atomic weight “A”/Avogadro’s Number
“NA”
► Volume = Volume of the unit cell “Vc”
THEORETICAL DENSITY
Example problem on Density Computation
Problem: Compute the density of Copper
Given: Atomic radius of Cu = 0.128 nm (1.28 x 10-8 cm)
Atomic Weight of Cu = 63.5 g/mol
Crystal structure of Cu is FCC
Solution: = n A / Vc NA
n= 4
Vc= a3 = (2R√2)3 = 16 R3 √2
NA = 6.023 x 1023 atoms/mol
► Examples:
Graphite is the stable polymorph at ambient conditions,
whereas diamond is formed at extremely high
pressures.
Pure iron is BCC crystal structure at room temperature,
which changes to FCC iron at 912oC.
POLYMORPHISM AND ALLOTROPY
Fe
FCC (at Temperature above 912 oC)
912 oC
Fe (BCC) Fe (FCC)
3.7 Crystal Systems
► Since there are many different possible crystal structures,
it is sometimes convenient to divide them into groups
according to unit cell configurations and/or atomic
arrangements.
Lattice parameters
a, b, c, , , are called the lattice
Parameters.
► Seven different possible
combinations of edge
lengths and angles give
seven crystal systems.
e.g. [uvw]
x [uvw]
[111]
[120]
[110]
Crystallographic Directions in Cubic System
Head and Tail Procedure for determining
Miller Indices for Crystallographic Directions
Direction A
Head point – tail point
(1, 1, 1/3) – (0,0,2/3)
1, 1, -1/3
Multiply by 3 to get smallest
integers
3, 3, -1
A = [33Ī]
Direction B
Head point – tail point
(0, 1, 1/2) – (2/3,1,1)
-2/3, 0, -1/2
Multiply by 6 to get smallest
integers
_ _ C = [???] D = [???]
B = [403]
Indices of Crystallographic Directions in Cubic System
Direction C
Head Point – Tail Point
(1, 0, 0) – (1, ½, 1)
0, -1/2, -1
Multiply by 2 to get the smallest integers
C = [0I2]
Direction D
Head Point – Tail Point
(1, 0, 1/2) – (1/2, 1, 0)
1/2, -1, 1/2
Multiply by 2 to get the smallest
integers
B= [???] A = [???]
D = [I2I]
Crystallographic Directions in Cubic System
[210]
Crystallographic Directions in Cubic System
CRYSTALLOGRAPHIC DIRECTIONS IN
HEXAGONAL UNIT CELLS
3
t (u v)
w n w'
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
Origin for A
Origin for B
Origin
for A
example a1 a2 a3 c
1. Intercepts 1 -1 1
2. Reciprocals 1 1/ -1 1
1 0 -1 1 a2
3. Reduction 1 0 -1 1
a3
_
(1211)
Miller-Bravais Indices for crystallographic
directions and planes in HCP
Atomic Arrangement on (110) plane in FCC
Atomic Arrangement on (110) plane in BCC
Atomic arrangement on [110] direction
in FCC
3.11 Linear and Planar Atomic Densities
a = 4R/ 3
1 1 3
LD
2a 2 (4 R / 3 ) 2 4R
[110]
2a
Linear Density
► LD of [110] in FCC
# of atom centered on the direction
vector [110] = 2 atoms
LD = 2 /4R
LD = 1/2R
Linear density can be defined as
reciprocal of the repeat distance
‘r’
LD = 1/r
Planar Density
► Planar Density “PD”
is defined as the number of atoms per unit
area that are centered on a given
crystallographic plane.
2 atoms 1
PD110
2
8R 2 4 R 2 2 a = 2R 2
4R
Closed Packed Crystal Structures
A
• FCC Unit Cell B
C
Closed Packed Plane Stacking in FCC
Crystalline and Noncrystalline Materials
3.13 Single Crystals
► For a crystalline solid, when the periodic and repeated
arrangement of atoms is perfect or extends throughout
the entirety of the specimen without interruption, the
result is a single crystal.
► All unit cells interlock in the same way and have the
same orientation.
► Single crystals exist in nature, but may also be produced
artificially.
► They are ordinarily difficult to grow, because the
environment must be carefully controlled.
► Example: Electronic microcircuits, which employ single
crystals of silicon and other semiconductors.
Polycrystalline Materials
3.13 Polycrytalline Materials
Polycrystalline crystalline solids
composed of many small
crystals or grains.
c) Upon completion of
solidification, grains that are
adjacent to one another is also
shown.