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.

In the given figure, let the sphere have the centre, O and is fitted in the octahedral void.

As given, radius of the sphere fitted in the octahedral void = r 

And the radius of the atoms in close packing = R

Here, angle AOD = 900 

In triangle AOD,

DA2 = OA2 + OD2 

⇒  (R+R)2 = ( R + r )2+ (R+r)2

⇒  4R2 = 2(R+r)2

⇒  2R2 = (R+r)2 

⇒  √2 R = (R+r)

⇒  R + r = √2R

⇒  r = √2R – R 

⇒  r = R (1.414 - 1) 

⇒  r = 0.414 R

This is the required relation between r and R.