CHAPTER 3
THE STRUCTURE
OF
CRYSTALLINE SOLIDS
3.2 FUNDAMENTAL CONCEPTS
SOLIDS
AMORPHOUS
CRYSTALLINE
Atoms in a crystalline solid
are arranged in a repetitive
three dimensional pattern
Long Range Order
All metals are crystalline solids
Atoms in an amorphous
solid are arranged
randomly- No Order
Many ceramics are crystalline solids
Some polymers are crystalline solids
LATTICE
Lattice -- points arranged in a pattern that
repeats itself in three dimensions.
The points in a crystal lattice coincides
with atom centers
3-D view of a lattice
3.3
► Unit
UNIT CELL
cell -- smallest grouping which
can be arranged in three dimensions to
create the lattice. Thus the Unit Cell is
basic structural unit or building block
of the crystal structure
Unit cell & Lattice
Lattice
Unit Cell
3.4 METALLIC CRYSTALS
• 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 not directional.
-Nearest neighbor distances tend to be small in
order to have lower bonding energy.
• Have the simplest crystal structures.
Let us look at three such structures...
4
FACE CENTERED CUBIC STRUCTURE (FCC)
FACE CENTERED CUBIC STRUCTURE (FCC)
Al, Cu, Ni, Ag, Au, Pb, Pt
BODY CENTERED CUBIC STRUCTURE (BCC)
BODY CENTERED CUBIC STRUCTURE (BCC)
Cr, Fe, W, Nb, Ba, V
HEXAGONAL CLOSE-PACKED STRUCTURE HCP
Mg, Zn, Cd, Zr, Ti, Be
SIMPLE CUBIC STRUCTURE (SC)
Only Pd has SC structure
Number of atoms per unit cell
BCC
1/8 corner atom x 8 corners + 1 body center atom
=2 atoms/uc
FCC 1/8 corner atom x 8 corners + ½ face atom x 6
faces
=4 atoms/uc
HCP 3 inside atoms + ½ basal atoms x 2 bases + 1/ 6
corner atoms x 12 corners
=6 atoms/uc
Relationship between atomic radius and
edge lengths
For FCC:
For BCC:
For HCP
a = 2R√2
a = 4R /√3
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
r
a0
a0
Body Centered Cubic (BCC)
3a0  4r
2a0
a0
3a0
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
• APF for a simple cubic structure = 0.52
6
ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
a = 4R /√3
R
a
Unit cell contains:
1 + 8 x 1/8
= 2 atoms/unit cell
FACE CENTERED CUBIC STRUCTURE (FCC)
• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
• Coordination # = 12
ATOMIC PACKING FACTOR: FCC
• APF for a face-centered cubic structure = 0.74
a = 2R√2
a
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 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

= 4 x 63.5 g/mol / 16 √2(1.28 x 10-8 cm)3 x 6.023 x
1023 atoms/mol
Ans = 8.98 g/cm3
Experimentally determined value of density of Cu = 8.94 g/cm3
3.6 Polymorphism and Allotropy
►
Polymorphism  The phenomenon in some metals, as
well as nonmetals, having more than one crystal
structures.
►
When found in elemental solids, the condition is often
called allotropy.
►
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
BCC (From room temperature to 912 oC)
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.
►
One such scheme is based on the unit cell geometry, i.e.
the shape of the appropriate unit cell parallelepiped
without regard to the atomic positions in the cell.
►
Within this framework, an x, y, and z coordinate system is
established with its origin at one of the unit cell corners;
each x, y, and z-axes coincides with one of the three
parallelepiped edges that extend from this corner, as
illustrated in Figure.
The Lattice Parameters
Lattice parameters
a, b, c, , ,  are called the lattice
Parameters.
►
Seven different possible
combinations of edge
lengths and angles give
seven crystal systems.
►
Shown in Table 3.2
►
Cubic system has the
greatest degree of
symmetry.
►
Triclinic system has the
least symmetry.
The Lattice Parameters
Lattice parameters are
a, b, c, , ,  are called the lattice
Parameters.
3.7 CRYSTAL SYSTEMS
3.8 Point Coordinates in an Orthogonal
Coordinate System Simple Cubic
3.9 Crystallographic Directions in Cubic System
Determination of the directional indices in cubic
system:
Four Step Procedure (Text Book Method)
►
1.
2.
3.
4.
Draw a vector representing the direction within the unit
cell such that it passes through the origin of the xyz
coordinate axes.
Determine the projections of the vector on xyz axes.
Multiply or divide by common factor to obtain the three
smallest integer values.
Enclose the three integers in square brackets [ ].
e.g.
[uvw]
u, v, and w are the integers
Crystallographic Directions
z
Algorithm
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
y 3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvw]
x
ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]
-1, 1, 1 => [ 111 ]
where overbar represents a
negative index
families of directions <uvw>
Crystallographic Directions in Cubic System
[111]
[120]
[110]
Crystallographic Directions in Cubic System
Head and Tail Procedure for determining Miller
Indices for Crystallographic Directions
1.
2.
3.
4.
Find the coordinate points of head and tail
points.
Subtract the coordinate points of the tail
from the coordinate points of the head.
Remove fractions.
Enclose in [ ]
Indecies of Crystallographic Directions in Cubic System
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
_ _
B = [403]
C = [???]
D = [???]
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

D = [I2I]
B= [???]
A = [???]
Crystallographic Directions in Cubic System
[210]
Crystallographic Directions in Cubic System
CRYSTALLOGRAPHIC DIRECTIONS IN
HEXAGONAL UNIT CELLS
Miller-Bravais indices -- same as Miller
indices for cubic crystals except that there
are 3 basal plane axes and 1 vertical axis.
Basal plane -- close packed plane similar
to the (1 1 1) FCC plane.
contains 3 axes 120o apart.
Direction Indices in HCP Unit Cells –
[uvtw] where t=-(u+v)
Conversion from 3-index system to 4-index system:
[u v w ]  [uvtw]
' '
'
n
u  ( 2u '  v ' )
3
n
'
'
v  ( 2v  u )
3
t  (u  v )
w  nw
'
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
[ u 'v 'w ' ]  [ uvtw ]
a2
-
a3
a1
Fig. 3.8(a), Callister 7e.
1
u  (2 u ' - v ')
3
1
v  (2 v ' - u ')
3
t  - (u +v )
w  w'
Crystallographic Directions in Cubic System
Indices of a Family or Form
< 100 >  [100], [010], [001], [010], [001], [100]
< 111 >  [111], [11 1 ], [1 1 1], [ 1 11],
[ 1 1 1 ], [ 1 1 1], [ 1 1 1 ], [1 1 1 ]
< 110 >  [110], [011], [101], [110], [011], [101]
[110], [011], [101], [110], [011], [101]
3.10 MILLER INDICES FOR
CRYSTALLOGRAPHIC PLANES
►
►
1.
2.
3.
4.
Miller Indices for crystallographic planes are the
reciprocals of the fractional intercepts (with
fractions cleared) which the plane makes with
the crystallographic x,y,z axes of the three
nonparallel edges of the cubic unit cell.
4-Step Procedure:
Find the intercepts that the plane makes with the three
axes x,y,z. If the plane passes through origin change
the origin or draw a parallel plane elsewhere (e.g. in
adjacent unit cell)
Take the reciprocal of the intercepts
Remove fractions
Enclose in ( )
Crystallographic Planes
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
a
1
1/1
1
1
4.
Miller Indices
(110)
example
1. Intercepts
2. Reciprocals
3.
Reduction
a
1/2
1/½
2
2
4.
Miller Indices
(100)
b
1
1/1
1
1
c

1/
0
0
c
y
b
a
x
b

1/
0
0
c

1/
0
0
z
c
y
a
x
b
Crystallographic Planes
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
4.
Miller Indices
a
1/2
1/½
2
6
b
1
1/1
1
3
(634)
c
c
3/4
1/¾
4/3

4 a
x


b
Family of Planes {hkl}
Ex: {100} = (100), (010), (001), (100), (010), (001)
y
Miller Indecies of Planes in Crystallogarphic
Planes in Cubic System
Drawing Plane of known Miller Indices in a cubic unit
cell
Draw (011) plane
Miller Indecies of Planes in Crystallogarphic Planes in
Cubic System
Origin for A
Origin for B
Origin
for A
A = (IĪ0)
B = (I22)
A = (2IĪ)
B = (02Ī)
CRYSTALLOGRAPHIC PLANES AND
DIRECTIONS IN HEXAGONAL UNIT CELLS
Miller-Bravais indices -- same as Miller
indices for cubic crystals except that there
are 3 basal plane axes and 1 vertical axis.
Basal plane -- close packed plane similar
to the (1 1 1) FCC plane.
contains 3 axes 120o apart.
Crystallographic Planes (HCP)
► In
hexagonal unit cells the same idea is used
z
example
1. Intercepts
2. Reciprocals
3.
Reduction
a1
1
1
1
1
a2

1/
0
0
a3
-1
-1
-1
-1
c
1
1
1
1
a2
a3
4.
Miller-Bravais Indices
(1011)
a1
Adapted from Fig. 3.8(a), Callister 7e.
Miller-Bravais Indices for crystallographic planes
in HCP
_
(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
► Linear
Density “LD”
is defined as the number of atoms per unit
length whose centers lie on the direction
vector of a given crystallographic direction.
No. of atoms centered on direction
LD 
Length of direction vector
Linear Density
LD for [110] in BCC.
# of atom centered on the direction
vector [110]
= 1/2 +1/2 = 1
Length of direction vector [110] = 2 a
a = 4R/  3
1
LD 

2a
1

2 (4 R / 3 )
3
2 4R
[110]
 2a
Linear Density
►
LD of [110] in FCC
# of atom centered on the direction
vector [110] = 2 atoms
Length of direction vector [110] = 4R
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.
No of atoms centered on the plane
PD =
—————————————
Area of the plane
Planar Density of (110) plane in FCC
# of atoms centered on the
plane (110)
= 4(1/4) + 2(1/2) = 2
atoms
Area of the plane
= (4R)(2R  2) = 8R22
(111) Plane in FCC
2 atoms
1
PD110 

2
8R 2 4 R 2 2
a = 2R  2
4R
Closed Packed Crystal Structures
► FCC
and HCP both have:
CN = 12
and
APF = 0.74
APF= 0.74 is the most efficient packing.
Both FCC and HCP have Closed Packed Planes
FCC ----(111) plane is the Closed Packed Plane
HCP ----(0001) plane is the Closed Packed Plane
The atomic staking sequence in the above two
structures is different from each other
Closed Packed Structures
Closed Packed Plane Stacking in HCP
Hexagonal Close-Packed
Structure (HCP)
• ABAB... Stacking Sequence
• 3D Projection
c
a
• 2D Projection
A sites
Top layer
B sites
Middle layer
A sites
Bottom layer
Adapted from Fig. 3.3(a),
Callister 7e.
• Coordination # = 12
• APF = 0.74
• c/a = 1.633
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
FCC Stacking Sequence
• ABCABC... Stacking Sequence
• 2D Projection
B
B
C
A
B
B
B
A sites
C
C
B sites
B
B
C sites
• FCC Unit Cell
A
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.
Various stages in the solidification :
a) Small crystallite nuclei Growth
of the crystallites.
b) Obstruction of some grains that
are adjacent to one another is
also shown.
c) Upon completion of
solidification, grains that are
adjacent to one another is also
shown.
d) Grain structure as it would
appear under the microscope.
3.15 Anisotropy
►
The physical properties of single crystals of some
substances depend on the crystallographic direction in
which the measurements are taken.
►
For example, modulus of elasticity, electrical conductivity,
and the index of refraction may have different values in
the [100] and [111] directions.
►
This directionality of properties is termed anisotropy.
►
Substances in which measured properties are
independent of the direction of measurement are
isotropic.