Q.If the radius of the octahedral void is r and radius of the atoms in close-packing is R, derive relation between r and R.

Consider four close-packed spheres (radius $R$) forming a square around the void (radius $r$). Along the face: edge $=2R$, diagonal $=2(R+r)$. Since diagonal $=\sqrt2\times$ edge: …

Step 1 – Geometry of the octahedral void.

Take the four spheres (each radius $R$) lying in one plane around the void; their centres form a square. The void sphere of radius $r$ sits at the centre of this square, touching all four.

Step 2 – Side and diagonal of the square.

Adjacent spheres touch, so the side of the square is

The diagonal passes through the void sphere and two opposite spheres:

Step 3 – Relate side and diagonal. …

Model the octahedral void by the square of four touching spheres in one plane; equate the square's diagonal ($2(R+r)$) …

• Using the tetrahedral relation ($r=0.225R$) instead of deriving the octahedral one. …