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

Step 1. In a bcc cell, spheres touch along the body diagonal. From the cube's geometry: face diagonal $AC = \sqrt{2}\,a$ (Pythagoras on a face), and body diagonal $AG = \sqrt{AC^2+a^2} = \sqrt{2a^2+a^2} = \sqrt{3}\,a$.

Step 2. The spheres touch along this body diagonal, which spans 4 radii: $\sqrt{3}a = 4r \Rightarrow r = \dfrac{\sqrt{3}}{4}a$.

Step 3. Volume of one sphere: $\dfrac{4}{3}\pi r^3 = \dfrac{4}{3}\pi\left(\dfrac{\sqrt{3}}{4}a\right)^3 = \dfrac{\sqrt{3}\pi a^3}{16}$.

Step 4. A bcc cell contains 2 spheres, so total sphere volume $= 2\times\dfrac{\sqrt{3}\pi a^3}{16} = \dfrac{\sqrt{3}\pi a^3}{8}$. …

Derive r in terms of a from the body-diagonal touching condition, compute the sphere volume, multiply by …

• Using the face-diagonal relation (as in fcc) instead of the body-diagonal relation for bcc -- bcc spheres touch along the BODY diagonal, not the face …