Question

Which of the following is correct relation between radius of octahedral void (r) and radius of atom (R) in closed packed structure?

• A. $R=0.732r$
• B. $r=0.732R$
• C. $R=0.414r$
• D. $r=0.414R$

Answer 1

The correct option is D $r=0.414R$

A sphere fitting into octahedral void is shown in the figure (green colour)
A sphere above and sphere below this green coloured sphere is not shown :

$△ABC$ is a right angled triangle.
$AB=AC=(R+r)$ as shown :


Applying pythagoras theorem :
$B{C}^2=A{B}^2+A{C}^2$
$(2R{)}^2=(R+r{)}^2+(R+r{)}^{2}2R^2=(R+r{)}^2(\sqrt{2}R{)}^2=(R+r{)}^2\sqrt{2}R=R+rr=\sqrt{2}R-R$

$r=0.414R$