Packing factor of Diamond Cubic Crystal Structure

Diamond cubic crystal structure is a type of crystal structure that is commonly found in many elements and compounds such as diamond, silicon, germanium, etc. In this crystal structure, each atom is covalently bonded to four nearest neighbors, forming a tetrahedral structure.

Definition of packing factor

Packing factor is a measure of how efficiently the atoms are arranged in a crystal structure. It is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell.

Calculation of packing factor for diamond cubic crystal structure

The diamond cubic crystal structure can be thought of as two interpenetrating face-centered cubic (FCC) lattices. Each FCC lattice contains 4 atoms at the corners of the unit cell and 1 atom at the center of the unit cell. Therefore, the total number of atoms in the unit cell is 8.

The volume of the unit cell can be calculated as follows:

a^3 = 4r^3

where a is the edge length of the unit cell and r is the radius of the atom. For the diamond crystal structure, the radius of the atom is half the distance between two nearest neighbor atoms, which is equal to a/2√2.

Substituting the value of r in the above equation, we get:

a = 2r√3

Volume of the unit cell = a^3 = 16r^3√3

The volume occupied by each atom is equal to (4/3)πr^3. Therefore, the total volume occupied by the 8 atoms in the unit cell is:

8 × (4/3)πr^3 = (32/3)πr^3

The packing factor can be calculated as follows:

Packing factor = (Total volume occupied by atoms) / (Volume of the unit cell)

= [(32/3)πr^3] / [16r^3√3]

= 0.34

Therefore, the packing factor of diamond cubic crystal structure is 0.34 or 34%.