Q.Calculate the percentage efficiency of packing in case of body centered cubic crystal.

By relating the atomic radius to the unit cell edge along the body diagonal (where atoms actually touch in a bcc lattice) and dividing the total volume of the 2 atoms per cell by the cell's volume, the packing efficiency works out to about 68%.

Step 1 — relate $r$ and $a$: in a body-centred cubic (bcc) unit cell, atoms touch each other along the body diagonal (the corner atom, the body-centre atom, and the opposite corner atom are all mutually touching). The body diagonal has length $\sqrt3\,a$ (where $a$ is the edge length), and it spans $4r$ (radius of the corner atom + diameter of the body-centre atom + radius of the opposite corner atom $= r + 2r + r$): $\sqrt3\,a = 4r \quad\Rightarrow\quad r = \dfrac{\sqrt3}{4}a$

Step 2 — number of atoms per unit cell: a bcc unit cell has 8 corner atoms (each shared among 8 cells, contributing $8\times\frac{1}{8}=1$) plus 1 body-centre atom (contributing fully, $=1$), giving a total of $n=2$ atoms per unit cell.

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