Question

Packing Fraction in BCC lattice is:
A. ${\displaystyle \frac{1}{6}}\pi$
B. ${\displaystyle \frac{\sqrt{2}}{6}}\pi$
C. ${\displaystyle \frac{\sqrt{3}}{8}}\pi$
D. ${\displaystyle \frac{\sqrt{3}}{2}}$

Answer 1

Hint: Packing efficiency is defined as the ratio of the volume occupied by atoms in a unit cell by the total volume of the unit cell and the efficiency of the body-centered cubic lattice is 68% and the coordination number of this BCC lattice structure is eight. Packing efficiency is also known as the atomic packing factor.

Complete answer:
It is the fraction of volume in a crystal structure that is filled up or occupied by particles constituent. Packing efficiency has no physical dimensions hence it is a dimensionless quantity.
In a Body-centered cubic unit cell, 8 atoms are present at each corner along with one center atom in the center of the cube. Each corner atom has only one-eighth of its volume within the unit cell. So BCC has a total of 2 lattice points per unit cell.
In BCC total number of atoms per unit cell is $1+({\displaystyle \frac{1}{8}}\times 8)=2$ ….(I)
In BCC, the center atom is touched by every corner atom. From one corner of the cube through the center and to the other corner a line is drawn and that passes through$4r$, $r$ is the radius of the atom and from the geometry length of the diagonal is $a\sqrt{3}$
 

So, the length of each side of the BCC structure can be related to the radius of the atom by $a={\displaystyle \frac{4r}{\sqrt{3}}}$
We know that volume of sphere $={\displaystyle \frac{4}{3}}\pi r^3$ ….(II)
Packing efficiency =${\displaystyle \frac{N_{atom}{V}_{atom}}{V_{unit\text{ cell}}}}$ …..(III)
$N_{atom}$is the number of atoms in a unit cell and $V_{atom}$ is the volume of each atom and the volume occupied by the unit cell is given by $V_{unit\text{ cell}}$
Substituting the values in equation (III)
Packing fraction$={\displaystyle \frac{2\times {\displaystyle \frac{4}{3}}\pi r^3}{{({\displaystyle \frac{4r}{\sqrt{3}}})}^3}}$
Packing fraction in BCC lattice $={\displaystyle \frac{\sqrt{3}}{8}}\pi$

Therefore the correct answer is option (C).

Note:
The smallest part of a component in a crystal is called a unit cell. Some of the types of crystal structures are monoclinic crystal structure, triclinic crystal structure, tetragonal crystal structure, orthorhombic crystal structure, hexagonal crystal structure and rhombohedron.