Question

The correct statements for cubic close packed (ccp) three dimensional structure is;

• A. The number of octahedral voids per atom are 4
• B. The packing efficiency of the lattice is nearly 74%
• C. The number of tetrahedral voids per atom are 4
• D. The unit cell edge length is $2\sqrt{3}$ times the radius of the atom

Answer 1

The correct option is B The packing efficiency of the lattice is nearly 74%

Packing efficiency of ccp lattice is 74%
Thus, it is a correct statement.
In a fcc lattice, the number of atom per unit cell is 4.
Thus, it has 4 octahedral voids and 8 tetrahedral voids in an unit cell
$\text{Number of octahedral void per atom in unit cell}=\frac{4}{4}=1$
Option (a) is incorrect
$\text{Number of tetrahedral void per atom in unit cell}=\frac{8}{4}=2$.
Thus, option (c) is incorrect

(d) Edge length, $a=\frac{4}{\sqrt{2}}r=2\sqrt{2}r$
Thus, option (d) is incorrect.

Explanation:


Edge length = a
Radius = r
$A{C}^2=A{B}^2+B{C}^2$
$(4r{)}^2=a^2+a^2=2a^2$
$4r=\sqrt{2}a$
$\Rightarrow r=\frac{\sqrt{2}}{4}a=\frac{a}{2\sqrt{2}}\Rightarrow a=2\sqrt{2}r$
In ccp structure, number of spheres is 4.
Hence,
$\text{Volume of 4 spheres}=4\left(\frac{4}{3}\pi r^3\right)$
Total volume of unit cell $=a^3=(2\sqrt{2}r{)}^3$
$\text{Packing efficiency (P.F)}=\frac{4\times \frac{4}{3}\times \pi r^3}{(2\sqrt{2}R{)}^3}P.F=\frac{\pi }{3\sqrt{2}}\approx 0.74$