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 .

Explanation

Derivation of relation between r and R .
Octahedral void is shown in the following diagram as a shaded circle .

`therefore Delta ABC` is a right angled triangle.
`therefore ` We apply pythagoras theorem .
`AC^(2) = AB^(2) + BC^(2)`
`(2R)^(2) = (R + r)^(2) + (R + r)^(2) = 2 (R + r)^(2)`
`4R^(2) = 2 (R + r)^(2)`
`2 R^(2) = (R + r)^(2)`
`(sqrt2 R)^(2) = (R + r)^(2)`
`sqrt2 R = R + r `
` r = sqrt2 R - R`
` r = ( sqrt2 -1) R`
`r = (1.414 - 1) R`
` r = 0.414` R .