Fcc unit cell

Introduction to unit cell
A regular three dimensional arrangement of points in space is called a crystal lattice. A unit cell is the smallest of a crystal lattice which, when repeated in different directions generates the entire lattice

Face centered cubic unit cell

A face centered cubic unit cell contains atoms at all the corners and at the centre of all the faces of the cube. Each atom located at the face centre is shared between two adjacent unit cells and only half of each atom belongs to a unit cell. Thus, in a face centered cubic unit cell:

1. 8 corners atoms × 1/8 atoms per unit cell=1 atom
2. 6 face centered atoms × 1/2 atoms per cell=3 atoms

Therefore, total no. of atoms per unit cell = 4 atoms

Packing efficiency of fcc unit cell

Packing efficiency is the percentage of total space filled by the particles. Let us calculate the packing efficiency of fcc unit cell. Let the unit cell edge be ‘a’  and face diagonal be ‘b’.
We know that b=√2a
If r is the radius of the sphere, we find
    b = 4r =√2a or a = 2√2r
we know that each unit cell in fcc structure, has effectively 4 spheres. Total volume of four spheres is equal to 4×(4/3)πr³ and the volume of the cube is a³ or (2√2r)³.
Therefore,
Packing efficiency of fcc unit cell
=volume of 4 spheres ×100/volume of unit cell  %
=4×(4/3)πr³×100/(2√2r)³
=74%

Density of unit cell

Volume of unit cell = a³
Mass of the unit cell =number of atoms in unit cell×mass of each atom
=z × m
Where z is the number of atoms in unit cell and m is the mass of single atom.
Mass of an atom present in a unit cell:
    m = M/N_a   (M is molar mass)
therefore, density of the unit cell =mass/volume
         =z×m/a³ = z×M/a³×N_a
     d   =    zM/a³N_a

Summay

The number of atoms in a fcc unit cell is four and these are present at all corners as well as at the centre of all faces of the cube