Coordination Number and Atomic Packing Factor - Civil Engineering (CE) PDF Download

Coordination number and Atomic packing factor Definition

(1) Coordination number:
It is the number of equidistant nearest neighbors that an atom has in a crystal structure.

(2) Nearest neighbour distance:
It is the distance between two nearest neighbors in a crystal structure.

(3) Atomic packing factor (APF) or Packing fraction:
It is the fraction of space occupied by atoms in a unit cell. It is defined as the ratio of volume occupied by atoms in unit cell to the volume of the unit cell. If the number of atoms per unit cell are n and if V_a is the volume of atoms in the unit cell and V is the volume of the unit cell then,

Calculation of Atomic packing factor for different crystal systems
 Simple Cubic Structure
In simple cubic structure each atom consists of 6 equidistant nearest neighbours. Hence its co-ordination number is 6.
Eight unit cells share each atom at the corner. Hence only 1/8th of the volume of the atom lies in each cell. Since the atoms are present only at corners, the number of atoms per unit cell is given by

In this structure the atoms touch each other along the sides of the cube. Therefore a=2R,where R is the radius of each atom

Hence atoms occupy 52.35% off the volume of the unit cell.
Body Centered Cubic (BCC) Structure:
Each atom has 8 equidistant nearest neighbors. Hence the co-ordination number is 8. Since

there are eight atoms at corners and 1 atom at the body center, the no of atoms per unit cell is given by 

Also in this structure the atoms touch each other along the body diagonal. Therefore   Where R is the radius of the atom

Hence atoms occupy 68.02% of the volume of the unit
Face Centered Cubic (FCC) Structure: 
In FCC structure in addition to atoms at corners, atoms are present at face centers. Each 

atom consists of 12 equidistant nearest neighbours. Hence the coordination number is 12. The number of atoms per unit cell is

In this structure atoms touch each other along the face diagonal. Therefore

Where R is the atomic radius

Hence atoms occupy 74.05% of the volume of the unit cell.