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
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%
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/Na (M is molar mass)
therefore, density of the unit cell =mass/volume
=z×m/a3 = z×M/a3×Na
d = zM/a3Na
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:- 8 corners atoms × 1/8 atoms per unit cell=1 atom
- 6 face centered atoms × 1/2 atoms per cell=3 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 = a3Mass 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/Na (M is molar mass)
therefore, density of the unit cell =mass/volume
=z×m/a3 = z×M/a3×Na
d = zM/a3Na