Closed Packed Structure Part -2

Packing efficiency

Packing Efficiency in the (FCC) Face – Centred Cubic Unit Cell

A face-centred cubic unit cell seems to be the most densely occupied unit cell. A single atom can also be discovered at the centre of each of the cube’s faces. Because each atom only makes up half of the cell, these face-centred atoms are shared between two adjacent unit cells.

Packing Efficiency in the (BCC) Body – Centred Cubic Unit Cell

Having eight atoms in each corner and one in the middle, the BCC is essentially equal to a basic cubic unit cell. BCC has an open structure. The atom in the centre belongs to the Unit cell where it is found exclusively.

Packing Efficiency of Bcc Unit Cell Formula

Important points regarding calculation of atoms in BCC are as follows;

  1. 8 (eight) corners generate 1/8 atoms at each corner that is equals to 8 × 1/8 = 1 atom.
  2. At the centre of body, one atom is equal to one atom.
  3. Two atoms form body-centred cubic unit cell.

Let, 

a = edge length

c = body diagonal length

b = length of diagonal

Packing Efficiency = 68% 

Packing Efficiency of BCC unit cell is 68%.

Packing Efficiency of Simple Cubic Unit Cell

r=a/2

a = 2r

Therefore, 

Packing Efficiency =volume occupied by one atom total volume of unit cell ×100   

Packing Efficiency =43r3 (2r)3 ×100 

Packing Efficiency = 52.4%

Packing efficiency of simple cubic unit cell is 52.4%.

Conclusion

Unit cell is the smallest collection of atoms or molecules which forms a crystal when repeated at regular intervals in 3D.

The proportion of the total space in a unit cell which is occupied by constituent particles like atoms, ions, or molecules packed within the lattice is described as packing efficiency.

Packing efficiency is determined in three ways which are as follows;

  1. Close packing (Hexagonal close packing and cubic close packing)
  2. Cubic Structures which Focus on Body
  3. Cubic lattice structures (body centred)

Types of Unit Cell

Number of Atoms in Unit Cell

Coordination Number

Relation Between "a" and "r"

Packing Efficiency 

Free Space

Simple Cubic (SCC)

1

6

a = 2r

52.4%

47.6%

Body-Centred Cubic (BCC)

2

8

a = (4/ 3) r

68%

32%

Face-Centred Cubic (FCC)

4

12

a = 22 r

74%

26%

 

Calculations Involving Unit Cells Dimensions

 Density of Unit Cell

A unit cell is a three-dimensional structure occupying one, two or more atoms. With the help of dimensions of unit cells, we can evaluate the density of the unit cell. To do so let’s consider a unit cell of edge length ‘a’, therefore the volume of the cell will be ‘a3’. Also, density is defined as the ratio of the mass of unit cell and volume of the unit cell.

So we can write 

Mass of unit cell varies with number of atoms “n” and mass of a single atom “m”. Mathematically mass of unit cell is the product of number of atoms “n” and mass of one atom “m” i.e.

 Mass of Unit Cell = n × m

Also from quantitative aspect of atoms, mass of one atom can be written in terms of Avogadro Number ( NA ) and molar mass of atom ( M), that is,

Volume of Unit Cell = a3

Placing the required values in equation 1 we get

Therefore if we know molar mass of atom “M”, number of atoms “n”, the edge length of unit cell “a” we can evaluate the density of unit cell.