$\begin{array}{1 1}68\%\\78\%\\38\%\\58\%\end{array} $

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Packing efficiency in body-centered cubic structures :

It is clear that atom at the centre will be in touch with the order two atoms diagonally arranged.

In $\Delta EFD$

$b^2=a^2+a^2=2a^2$

$b=\sqrt 2a$

Now in $\Delta AFD$

$c^2=a^2+b^2=a^2+2a^2=3a^2$

$c=\sqrt 3a$

The length of the body diagonal c is equal to 4r,where r is the radius of the sphere(atom),as all the three spheres along the diagonal touch each other.

Therefore,$\sqrt 3a=4r$

$a=\large\frac{4r}{\sqrt 3}$

Also we can write,

$r=\large\frac{\sqrt 3}{4}$$a$

In this type of structure,total number of atoms is 2 and their volume is $2\times \large\frac{4}{3}$$\pi r^3$

Volume of the cube $a^3=\big(\large\frac{4r}{\sqrt 3}\big)^3$

Packing efficiency =$\large\frac{\text{ Volume occupied by two spheres in the unit cell}}{\text{Total volume of the unit cell}}$$\times 100\%$

$\Rightarrow \large\frac{2\times \Large\frac{4}{3}\normalsize \pi r^3\times 100}{\big(\Large\frac{4r}{\sqrt 3}\big)^3}$$\%$

$\Rightarrow \large\frac{(8/3)\pi r^3\times 100}{\Large\frac{64}{3\sqrt 3}\normalsize r^3}$$\%$

$\Rightarrow 68\%$

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