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Unit Cells: Number Of Atoms In A Unit Cell

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Unit Cells: Number Of Atoms In A Unit Cell - Lesson Summary

Every crystal lattice is made of an infinite number of unit cells, and every lattice site is occupied by a constituent particle that may be an atom, an ion or a molecule.
 
Consider three types of cubic unit cells
 
Simple or primitive cubic lattice:
 

 
A simple cubic lattice consists of 8 atoms at the 8corners of a cube. Each atom at the corners contributes 1/8 of its original volume to each unit cell. There are 8 atoms at the 8 corners. Number of atoms in each unit cell = 8 corner atoms *1/8 atoms per unit cell
 
                           8*1/8 =1

A body-centred unit cell:
It contains eight corner atoms, & there is one more atom at its body-centre.
 

 
It is clear that the 8 atoms at the corners contribute 1/8 of their actual volume each. In addition, one atom lies completely inside the cell. The number of atoms at the corners per unit cell = 8 corner atoms *1/8 atoms per unit cell
                           8*1/8 =1

There is one atom at the centre of the cube. The total number of atoms in a body-centred cubic unit cell =1 +1 =2

The face-centred cubic lattice: 
It consists of 8 corner atoms just like in the simple and body-centred cubic lattice. It also contains 6 more atoms at the 6 faces of the cube cell. The atoms touch along the diagonals, and each touches 4 corner atoms.
 
     
 
The atoms at the corners contribute 1/8 of their original volume each. And there are six atoms at the 6 faces, which contribute half of their original volume each. The number of atoms at the corners per unit cell = 8 corner atoms *1/8 atoms per unit cell
 
                                  8*1/8 =1
 
The number of atoms present at the faces per unit cell =6 atoms at the faces * ½ atom per unit cell.
 
                                   6*1/2 =3

The total number of atoms per face-centred cubic unit cell =1 +3 =4

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