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10.7 Cubic Unit Cell

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 cubic unit cell

Packing fraction volume of atom
packing fraction

packing fraction

where ‘r’ is the radius of atom and ‘a’ is edge length of the unit cell, because in primitive unit cell a = 2r,

Packing fraction primitive unit cell

packing fraction

= 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

body centered cubic unit cell

Packing fraction volume of atom,

In given unit cell body diagonal is given by = aÖ3

or aÖ3 = 4r

or body centered cubic unit cell
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Packing fraction solid state equation=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.

cubic-close-packing

Packing fraction packing fraction

The face diagonal is given by =aÖ2

or aÖ2 = 4r

or face-diagonal

face diagonal

Packing fraction packing fraction

= 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

hexagonal close packing unit cell

Packing friction volume of atom
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value of unit cell

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 value of unit cell, hence area of the base is

value of unit cell1

equilateral triangles

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

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= rÖ3

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AD = 2r

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Volume of unit cell = Area of base x height college chemistry help

Packing fraction general chemistry help

Hence in hcp and fcc. (or ccp) percent occupancy and void fraction present are same.

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