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.
HCP Structure
Hexagonal closed packed structure is one of the most common metallic structures. About
25 metals exhibit this structure. The unit cell of a closed packed hexagonal structure as shown in
Fig 2.7. There are three layers of atoms in it. The unit cell has one atom at each of the 12 corners of the hexagonal prism, one atom at the centre of the two hexagonal faces and three atoms are symmetrically arranged in the body of the cell. The center atom has six nearest neighboring atoms in the same plane. Height of the unit cell is maintained as ‘c’.

Number of atoms per unit cell (Z)
All corner atom is shared by 6 other unit cell , ie. Each corner atom gives 1/6 th of its share.
1
Number atoms in Hexagonal upper plane=
6 x = 1
1
Number atoms in Hexagonal lower plane=
6 x = 1
Each central atom is shared by two unit cells,
1
The total number of atoms in both upper and lower plane =
2 x
 
=1
Three central atoms in the unit cell, which are not shared by any other adjacent unit cells.
The Total number of atoms in one unit cell = 1 + 1+1+3
Z = 6 atoms/unit cell
Coordination number (CN)
Consider the top layer of the HCP structure; the central atom has 6 nearest neighboring atoms in the same plane. Below and the top of the center atom have two planes with three atoms each. In total, there are 12 nearest neighbors
CN = 12
Atomic radius (r)
The atoms are in contact along the edges of the hexagon .There fore,
2r = a
r = a / 2
Atomic packing factor (or) Packing fraction (APF)
Calculation of C/a ratio
Consider the triangle AOB in the bottom layer of the hexagonal. Exactly above the next layer atom lies at C.

A a
X
C
B
Y
O
A a
30 0
C
X
O
Y
B
Fig 1.8 To Calculate C/a ratio
C= Height of the unit cell, a= Distance between two neighboring atoms.
Consider

ABY,
Cos 30
0
=
AY
AB
AB Cos 30
0
= AY a Cos 30
0
= AY a
2
3
= AY
AY = a
2
3
But AX=
2
3
Y
AX =
2
3
x a
2
3
AX
AC
2
= a
3
In triangle AXC
=AX
2
+CX
2

a
2
= [ a
3
]
2
+[ c
2
]
2 a
2  a 2
 c
3 4
2 c
2
 a
2  a
2
4 c
2

2 a
4 3
2
3 c
2 a
2

8
3 c a

8
3
Volume of the unit cell (v)
Area of the base = 6x Area of the triangle AOB
Area of the triangle AOB =
1
2
BO x AY
=
1
2 x a x AY
=
1
2 x a x a
2
3
= a
2

3
2 2
Area of the base = 6 a
2

2 2
3 a
2
3 3
2
Volume of the unit cell = Area x Height
V =
3 a 2 3
2

C
` =
3 a
2
3

8
2 3

=
3 a 2 3

2 2
2 3 x a
V = 3 a
2
3 a
Packing fraction (APF) =
Zv
V

6

4
3
 r
3
3 a
3
2

24
 r
3


3
APF
2

3

1
24 r
3
0 .
74
2
[
 a = 2r]
There fore 74% of the volume is occupied by atoms and remaining 26% of volume is vacant.
Eg : Magnesium