Cubic unit cell is defined as unit cell for which a = b = c and a = b = g = 90°,
Simple cubic or primitive unit cell
In primitive unit cell one atom is present at each corner of the unit cell. Atom present at corner in primitive contributes 1/8th of its total volume to the unit cell, hence effective atom(s) in unit call is/are given by
where ‘r’ is the radius of atom and ‘a’ is edge length of the unit cell, because in primitive unit cell a = 2r,
= 0.52 or 52%
This indicates that the void fraction in primitive unit cell is 48% and rest is occupied by atoms.
Body centered cubic unit cell (BCC)
In BCC one atom is present at each corner and contributes 1/8th of its volume to the unit cell and one is present at centre and contributed 100% to the unit cell, total atoms can be calculated as
Packing fraction =0.68
If means in above unit cell 68% volume is occupied by atoms and rest 38% is void.
Face centred cubic unit cell- or cubic close packing (fcc or ccp). In this type of unit cell atoms are present at all corners and faces, corner atom contributes 1/8th and face centred atom contributes ½ of its total volume to unit cell, effective atoms can be calculated as.
The face diagonal is given by =aÖ2
or aÖ2 = 4r
= 0.74 or 74%
Hence 26% volume of unit cell is unoccupied.
Hexagonal close packing unit cell
In hcp atoms are present at corners, top and bottom faces and inside the unit cell, corner and face atoms contribute 1/6 and ½ of its total volume respectively and three atoms situated inside the unit cell effectively contribute 100%, atoms can be calculated as
Volume of unit cell
Area of base is formed by six equilateral triangles and each equilateral triangle has side length ‘a’ (a = 2r). The area of one triangle is , hence area of the base is
Height of the unit cell is distance between centres of top and bottom face atoms, which can be calculated as
In triangle ABC, AB = 2r
OB = r
AD = 2r
Volume of unit cell = Area of base x height
Hence in hcp and fcc. (or ccp) percent occupancy and void fraction present are same.
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