Question

The packing efficiency of the face centered cubic (fcc), body centered cubic (bcc) and simple primitive cubic (sc) lattices follows the order :

• A. $fcc>bcc>sc$
• B. $bcc>fcc>sc$
• C. $sc>bcc>fcc$
• D. $bcc>sc>fcc$

Answer 1

The correct option is A $fcc>bcc>sc$

For simple cubic (sc):


$Z_{eff}\text{for sc =1}$
So,

$\text{The total volume occupied by sphere}V=1\times \frac{4}{3}\times \pi r^3$
$\text{Volume of the cube}=a^3=(2r{)}^3$
$\text{Packing efficiency (P.F)}=\frac{1\times \frac{4}{3}\times \pi r^3}{(2r{)}^3}\times 100\%P.F=\frac{\pi \times 100}{6}\%=52.33\%$
For Face centered cubic (fcc) :

Here,
$\sqrt{2}\times a=4ra=2\sqrt{2}\times r$

$Z_{eff}\text{for FCC}=4$
So,

$\text{The total volume occupied by sphere}V=4\times \frac{4}{3}\times \pi r^3$
$\text{Volume of the cube}=a^3=(2\sqrt{2}r{)}^3$
$\text{Packing efficiency (P.F)}=\frac{4\times \frac{4}{3}\times \pi r^3}{(2\sqrt{2}r{)}^3}\times 100\%P.F=\frac{\pi \times 100}{3\sqrt{2}}\%\approx 74\%$


For Body centered cubic (bcc):

Here,
$\sqrt{3}\times a=4ra=\frac{4r}{\sqrt{3}}$

$Z_{eff}\text{for BCC}=2$
So,

$\text{The total volume occupied by sphere}V=2\times \frac{4}{3}\times \pi r^3$
$\text{Volume of the cube}=a^3={\left(\frac{4r}{\sqrt{3}}\right)}^3$
$\text{Packing efficiency (P.F)}=\frac{2\times \frac{4}{3}\times \pi r^3}{{\left(\frac{4r}{\sqrt{3}}\right)}^3}\times 100\%P.F=\frac{\sqrt{3}\pi \times 100}{8}\%\approx 68\%$

Hence, the correct order is:
$fcc>bcc>sc$