3.

CRYSTAL STRUCTURES
(Chapter 3)

3.1. Unit Cell

Crystalline material – a solid composed of atoms, ions, or molecules arranged in a

pattern which is repetitive in three dimensions.
All metals, many ceramics and certain polymers form crystalline structure.

Crystal structures: Regular, repeated arrangements of atoms

 Unit cell - repeated unit
 Crystal lattice - repeated pattern

The unit cell is the smallest group of atoms possessing the symmetry of the crystal
which, when repeated in all directions, will develop the crystal lattice.

Lattice parameters (a, b, c), and angles () define the size and shape of the unit cell.
z

b y
a
x, y, z – coordinate axes
x

Material structures are defined by:

 Shape of the cell
 Arrangement of atoms within cell
There are 7 crystal systems, and 14 possible types of crystallographic lattices.

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 3.2 Metallic Crystal Structures

Simple Cubic
Atomic radius:

2r  a

One atom pr unit cell!

• Number of atoms per unit cell (N)

• Atomic packing factor (APF)

APF = (volume of atoms in a unit cell) / (Total unit cell volume)

• Coordination Number (CN) - is the number of atoms touching a particular atom,

or the number of nearest neighbors

Body-Centered Cubic (BCC)

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  Atoms at each corner of the cube and one in the center
 The center atom touches each corner atom, but these do not touch each other.
 Fe, Cr, W, Mo

Face-Centered Cubic (FCC)

4r
a
2

 Atoms at each corner of the cube and one in the center of each face
 Each face atom touches its nearest corner atom
 Cu, Al, Ag, Au, Fe

Hexagonal Close-Packed (HCP)

 Atoms at each corner of the hexagon, one in the middle of the basal plane, and three in the
center of the hexagon
 The greatest possible density
 Zr, Mg, Ti

Number of atoms per unit cell

Atoms may be shared by more than one unit cell.
 Each corner atom is shared by 8 unit cells
 Each face center atom is shared between two unit cells.

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 BCC
N = 8 x (1/8) +1 = 2
8 atoms at the corners x 1/8 = 1 atom
1 center atom = 1 atom

Total = 2 atoms

FCC
N = 8 x (1/8) + 6 x (1/2) = 4
8 atoms at the corners x 1/8 = 1 atom
6 face-centered x ½ = 3 atoms
________________
Total = 4 atoms
HCP
N = 12 x (1/6) + 2 x (1/2) + 3 = 6
12 atoms at the corners x 1/6 = 2 atoms
2 face centered x ½ = 1 atom
3 center atoms = 3 atoms
Total = 6 atoms

Coordination Number (CN)

The coordination number is the number of atoms touching a particular atom, or the
number of nearest neighbors.

BCC: CN = 8
FCC: CN = 12 (close packed - maximum)
HCP: CN = 12 (close packed - maximum)

3.2 Crystallographic Indices (points, directions, and planes)

Need labeling conventions to describe crystal structures in detail. In particular:
 Directions
 Planes within crystals
The method is independent of cell type.

Point coordinates
The position of any point within unit cell may be
specified in terms of its coordinates as fractional
multiplies of the unit cell edge lengths.

• The number refers to the distance from the

origin in terms of lattice parameters.
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 Crystallographic Directions

Labeling conventions have been established in which three integers or indices are used to
designate planes and directions.

A crystallographic direction is defined as a line between two points, or a vector.

1. Draw a vector from the origin of the coordinate system.

- Any vector may be translated to the origin

2. Define the components of the vector on each of the three coordinate axes

3. Reduction to integers (multiply or divide all the coordinates by a common

factor).
To reduce all the coordinates to the smallest possible integer values.

The three indices, not separated by commas, are enclosed in square brackets.
Negative numbers should be written with a bar over the number.

x y z
[u v w]

z
[011]
[111]

-y y

[100]
[110]
x

Family <100>

[100] [100]
[010] [010]
[001] [001]

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 z
u v w

a/2 b 0
c
1/2 1 0

b y 1 2 0
a

x [1 2 0]

Translation

[110] x

Any vector may be translated to the origin .

Crystallographic Planes

Atoms are arranged in planes. Planes are labeled by Miller indices: (hkl)

1. Identify the points at which the plane intercepts the x, y, and z-axes in terms of
the number of lattice parameters. (If the plane passes through the origin, another
plane must be constructed by translation.)

2. Take reciprocals of these intercepts

3. Clear fractions but do not reduce to lowest integers

4. Enclose the resulting numbers in parentheses ( ). Negative numbers should be

written with a bar over the number.

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 x y z x y z x y z

Intersection  1   -1 1 1 1 3
2 2 4

Reciprocal 1 1 1 1 1 2 2 1 (4) x3
 1   -1 1 3
Reduction 6 3 4

Miller indices (010) (0 1 2) (634)

(010) z
(100

(010)
- y
y

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 (111)

What are the Miller indices of this plane? z

Translation

012
y

Equivalent planes

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 C

(210)

HCP (hexagonal) structure

HCP: directions and planes

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 Close-Packed Crystal Structures

 Atomic packing factor (APF): FCC 0.74; HCP – 0.74

 FCC and HCP are close packed structures.
- BCC is not.

 Consider the FCC <111> plane (closed packed planes)

 FCC uses ABCABC… stacking.

 Consider HCP <0001> plane (closed packed planes)

 Whereas, HCP used ABABAB…

A
B (111)
(0001) C
C
o
m
m
o
n

M
i
l
l
e
r

i
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d
i
 HCP FCC
• A plane • A plane
• B plane placed in a voids of plane A • B plane placed in a voids of plane A
• Next plane - A • Next plane placed in voids positions C
Stacking ABAB… • Stacking ABCABC…

HCP FCC

 Effects materials behaviour during plastic deformation

 FCC alloys (Al, Cu) are very ductile
 HCP alloys (Mg) – more brittle

3.4 Polymorphism (Allotropy)

 Polymorphism - material may have more than one crystal structure

 If the change in structure is reversible, then the polymorphic change is known as allotropy
 Different properties

The best known example for allotropy is iron. When iron crystallizes at 1536oC it is BCC. ( -
iron), at 1394oC the structure changes to FCC ( -iron or austenite), and at 911oC it again
becomes BCC. ( -iron or ferrite).

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 T oC

1536

1394 Cooling curve for

pure iron

911

3.5 Amorphous structure

An amorphous material is material in

.
which there is no long-range order of
the positions of the atoms

Amorphous structure of glass (SiO2)

Window glass
Ceramic
Many polymers

Crystalline Amorphous Mixed

usually
Metals rarely never
(e.g. steel, brass)
often often often
Ceramics
(e.g. alumina) (e.g. soda glass) (e.g. silicon nitride)

never usually sometimes

Polymers
(e.g. polyethylene) (e.g. nylon)

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 3.6 Polycrystals

 Single crystal – all unit cells are packed in the same way (large peace of material with
the same orientation)

 Polycrystal – the crystallographic orientation varies from grain to grain. Most materials
contain many crystals called GRAINS.

The GRAIN BOUNDARY (GB) is the interface where two crystals with different
crystallographic orientations are joined together.

Example of regular polyhedra Microstructure of a polycrystalline representing

grain shapes in zirconium, Optical Microscopy, 350 x
polycrystalline materials

3.7 Anisotropy

 Often, the properties of a material are different in different directions.

o Anisotropy
- e.g. conductivity
- mechanical properties

 Highly anisotropic crystals include:

- graphite (hexagonal with a large c/a value).
- mica (sheet silicate).

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