Question

The vacant space in the bcc unit cell is -

Answer 1

Body-Centered Cubic (BCC) Unit Cell

• The Body-Centered Cubic (BCC) crystal structure is one of the ways for atoms to arrange themselves in metals.
• The BCC crystal structure has one atom per lattice point at each corner of the cube and the center of the cube.
• BCC is nearly close-packed and has great stability because of its high number of nearest neighbors and next-nearest neighbors.

Step 1: Packing efficiency of BCC unit cell

• Packing efficiency of BCC unit cell is given by,

$\mathrm{Packing}\mathrm{efficiency}$$=$ $\frac{\mathrm{Volume}\mathrm{occupied}\mathrm{by}\mathrm{two}\mathrm{spheres}\times 100}{\mathrm{Volume}\mathrm{of}\mathrm{unit}\mathrm{cell}}$

$=$ $\frac{2\times {\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3\times 100}{a^3}$

$=$ $68\%$

• The $\%$ volume occupied by atoms in a unit cell is $68\%$

Step 2: $\%$ Vacant space in BCC unit cell

• $\%$ vacant space is given by,

$\%$ $\mathrm{Vacant}\mathrm{space}$ $=$ $100$ $-$ $\%$ $\mathrm{of}\mathrm{the}\mathrm{volume}\mathrm{occupied}\mathrm{by}\mathrm{atoms}$

$=$ $100–68$

$=$$32\%$

• $\%$Vacant space is $32\%$

Therefore, the $\%$ vacant space in the BCC unit cell is$32\%$.