Crystal Structures
Introduction
The crystal structure of a material or the arrangement of atoms in a crystal structure
can be described in terms of its unit cell. The unit cells stacked in three-dimensional space
describe the bulk arrangement of atoms of the crystal. The crystal structure has a three
dimensional shape.
Fundamental Parameters
Let us see some fundamental parameters which are used to describe the crystal
structure.
No of atoms per unit cell
It is no of atoms possessed by unit cell.
Atomic radius ‘r’:
Atomic radius is defined as half the distance between two nearest neighbouring
atoms in a crystal
Co-ordination number:
It is the number of equidistant nearest neighbours that an atom has in its unit cell.
Packing factor or density of packing:
It is the ratio of the volume occupied by atoms in an unit cell to the total volume of
the unit cell.
Face Centered Cubic (FCC)
Face centered cubic unit cell having eight atoms at the eight corners of the cube and six
atoms at the centers of six faces. The figure shows atomic site model of FCC cubic cell. We
shall now determine the characteristics of the FCC cell.
Number of atoms per unit cell (Z)
Fig 1.5. Face Centered Cubic
In a FCC cell, eight atoms are located at the eight corners and six atoms are at the centers of
the six faces. Each face of the cell is common to two adjacent cells. Therefore each face centered
atom contributes only half of its volume to one cell. Each corner atom contributes 1/8 of it
content.
1
1
The total number of atoms in one unit cell = x8 + x 6 
8
2
Z= 4 atoms /unit cell
Coordination number (CN)
In a FCC cell each corner atom is in contact with the face center atom. It would be
simultaneously touching 4 atoms in XY plane, 4 atoms in YZ plane and 4 atoms in ZX plane. So
that total of 12 atoms. There fore, the coordination number is 12
CN=12
Atomic radius
r
In a FCC cell atoms are in contact along the face diagonal
of the cube, as shown in figure.
R
From figure, face diagonal can be written as,
A
Fig 1.6. To determine Atomic radius
AC
2
2
(4r)
16r2
r2
2
=AB +BC
2
= AD2+DC2+CG2
=a2+a2
= 2a2
= 2a2/16
D
A
r
2r
r
A
B
A
2a 2
r2 =
16
r=
C
A
a
a 2
4
Atomic packing factor (or) Packing fraction (APF)
Number of atoms per unit cell = 4
Zv
V
4
= r 3
3
Atomic packing factor (APF) =
Volume of the atom (v)
Volume of the unit cell (V)
APF
=
= a3
4r 3
4x
3
 4r  3


 2
r =
APF =
a 2
4

2
= 0.74
6
There fore 74% of the volume is occupied by atoms and remaining 26% of volume is
vacant.
Eg : Aluminium.