# Mention different types of Bravais lattices possible in crystalline materials. Show that the atomic packing factor (APF) of FCC crystal structure is 0.74.

The French mathematician Bravais from geometrical considerations that there can be only 14 different ways in which similar points can be arranged in three dimensional space. Thus, the total number of space lattices belonging to all the seven crystal system put together is only 14. These are known as Bravias Lattice.

Atomic packing factor of FCC and BCC crystal: In the FCC unit cell illustrated, the atoms touch one another across a face-diagonal the length of which is 4 R. Since the unit cell is a cube, its volume is a3, where a is the cell edge length.

From the right triangle on the face,

a2 + a2 = (4R)2

a = ∫8R = 2R∫2

The FCC unit cell volume Vc may be computed from

Vc = a3 = (2R∫2)3 = 16R3∫2 ANS.

The APF (atomic packing factor) is defined as the function of solid sphere volume in a unit cell.

APF= Total sphere volume = Vs
Total unit cell volume = Vc

Both the total sphere and unit cell volumes may be calculated in terms of the atomic radius R.

Face-centred Cubic (F.C.C.) structure

The Volume for a sphere = 4/3 πR³

Since there are 4 atoms per FCC unit cell

Number of atoms in all six faces = ½ x 6 =3

Number of atoms in all corners = 1/8 x 8 = 1

Total number of atoms = 3 + 1 = 4

The total FCC sphere volume is

Vs = 4 x 4/3 πR³ = 16/3 πR³

The total unit cell volume is Vc = 16 R³ ∫2

The atomic packing factor is

APF = Vs/Vc = 16/3 πR³/16 R³∫2 = 0.74

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