Q.Sodium metal crystallizes in bcc structure with the edge length of the unit cell $4.3 \times 10^{-8}$cm. Calculate the radius of sodium atom.

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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