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# Calculate the efficiency of packing in case of a metal crystals for : face-centered cubic (with the assumptions that atoms are touching each other)

$\begin{array}{1 1}74\%\\64\%\\84\%\\34\%\end{array}$

Packing efficiency in face-centered cubic (with the assumptions that atoms are touching each other)
Let us calculate the efficiency in ccp or fcc structure
Let the unit cell edge length be 'a' and face diagonal AC=b
In $\Delta ABC$
$AC^2=b^2=BC^2+AB^2$
$b^2=a^2+a^2=2a^2$
$b=\sqrt 2a$
If r is the radius of the sphere
We find $b=4r$
$4r=\sqrt 2a$
$a=\large\frac{4r}{\sqrt 2}$$=2\sqrt 2r We know that in fcc structure has effectively 4 spheres Total volume of the four spheres is equal to 4\times \large\frac{4}{3}$$\pi r^3$ and the volume of the cube $a^3=(2\sqrt 2 r)^3$
Therefore packing efficiency =$\large\frac{\text{Volume occupied by four spheres in unit cell}}{\text{Total volume of the unit cell}}$$\times 100\% \Rightarrow \large\frac{4\times \Large\frac{4}{3}\pi r^3\times 100}{(2\sqrt 2 r)^3}$$\%$
$\Rightarrow \large\frac{\Large\frac{16}{3}\normalsize \pi r^3\times 100}{16 \sqrt 2r^3}$$\%=74\%$