Question

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

Answer 1

Derivation of relation between r and R

A sphere fitted into the octahedral void is shown by shaded circle. The spheres present in other layers are not shown in the figure.

$\therefore \Delta ABC$ is a right angled triangle.

$\therefore$ We apply pythagoras theorem.

$A{C}^2=A{B}^2+B{C}^2(2R{)}^2=(R+r{)}^{+}(R+r{)}^2=2(R+r{)}^{2}4R^2=2(R+r{)}^{2}2R^2=(R+r{)}^2(\sqrt{2}R{)}^2=(R+r{)}^2\sqrt{2}R=R+rr=\sqrt{2}R-Rr=(\sqrt{2}-1)Rr=(1.414-1)Rr=0.414R$