If the radius of the octahedral void is r and the radius of the atoms in the packing is R, derive relationship between r and R.

Explanation

An octahedral void is formed between six solid spheres touching each other. These six spheres can be considered in three layers. Four in one layer, one below this layer and one above this layer. The four spheres in the central layer are at the corners of a corners of a square as shown. The sphere fitting into the octahedral void is shown by shaded circle. A sphere above and a sphere below this small sphere have not be shown.

`DeltaABC` is a right angled triangle with `angleBAC=90^(@)`
`BC^(2)=AB^(2)+AC^(2)` (Pythagoras Theorem)
`(2R)^(2)=(R+r)^(2)+(R+r)^(2)`
`2R^(2)=(R+r)^(2)`
`sqrt(2)R=R+r`
`sqrt(2)=1+r/(R)`
`r/(R)=sqrt(2)-1=1.414-1=0.414`
r = 0.414 R