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.
Body centered cubic (BCC)
This type of crystal structure consists of the unit cell has one atom at each corner of the
cube and one at the body center of the cube.
Fig 1.4. Body centered cubic unit cell
Number of atoms per unit cell (Z)
There are 8 atoms at 8 corners, each shared by 8 unit cells and one atom at the body center.
The atom at the centre of the body of the cell wholly belongs to the unit cell.
The total number of atoms in one unit cell =
1
x8 + 1
8
Z = 2 atoms /unit cell
Coordination number (CN)
For any corner atom of the
E
unit cell, the nearest atoms are the
F
body centered atom is surrounded by
A
D
8 unit
A
cells having 8 body centered
atoms. Hence coordination number is
H
D
G
4r
8.
CN=8
Atomic radius(r)
C
a
G
Each corner atom touches the central atom. The unit cell for BCC is shown in figure. Consider
atoms at A, G and at the center of the cell. These atoms lie in one straight line along the body
diagonal AG of the cube. Consider side of the unit cell is ‘a’ Fig .4
To determine Atomic radius
From Fig .4,
AG2
=AD2+DG2
= AD2+DC2+CG2
(4r) 2
=a2+a2+a2
16r2
= 3a2
r2
= 3a2/16
r2
=
r=
3a 2
16
a 3
4
Atomic packing factor (or) Packing fraction (APF)
Number of atoms per unit cell = 1
Atomic packing factor APF
=
4 3
r
3
Volume of the atom (v)
=
Volume of the unit cell (V)
= a3
APF
APF
=
=
APF =
Zv
V
1x
4
r 3
3
4r 3
2x
3
 4r  3
 
 3
a3
r =
a 3

4
3
= 0.68
8
There fore 68% of the volume is occupied by atoms and remaining 32% of volume is
vacant.
Eg: Chromium