Question

calculate the packing efficiency of a cubic lattice when an atom located at the diagonal remains in touch with the other two atoms​

Answer 1

Face Centred Cubic Crystal

a. Number of atoms in unit cell = 14

b. Number of atoms per unit cell = 4

c. Relation between edge length ‘a’ and radius of atom ‘r’ = $\sqrt{2}a = 4r$ ; a = $\dfrac{4r}{\sqrt{2}}$

d. Number of nearest neighbors [ coordination number ] = 12

e. Distance between nearest neighbours = $\dfrac{a}{\sqrt{2}}$

f. Packing fraction : The fraction of total space os lattice which is occupied by the atom's volume. It is the volume of atom constituting unit cell per the volume of unit cell.

Answer 2

Answer:

The packing efficiency of a cubic lattice when an atom located at the diagonal remains in touch with the other two atoms is 74%.

Explanation:

A cubic lattice whose atom at the diagonal is in contact with the other two atoms is a face-centered cubic unit cell.

A face-centered cubic cell has four atoms per unit cell.

Also, a=2√2r where

a= edge length, r= radius of the atom.

The total volume of the unit cell= a³

Packing Efficiency =

(Volume of four spheres / Total volume of the unit cell)×100

Packing efficiency = $(\frac{\frac{16}{3}\pi r^{3}}{16\sqrt{2}r^{3}}) 100$ = 74%

Hence the packing efficiency of the given cubic lattice is 74%.