Compute the radius r of an impurity atom that will just fit into an FCC octahedral site in terms of the atomic radius R of the host atom (without introducing lattice strains)

The Diameter of an atom that fits into the FCC is 2r,

Which leads to the final steps of 0414R

Explanation:

The diameter of an atom that will fit into the FCC site is (2r) which differentiates length edge unit cell for the radii of two atoms that are seen on both side of the site (R); which is 2r = a - 2R

The FCC is also a part of R, therefore a=2R √2

Solving for R, the equation is, r = a-2R/2 = 2R√2-2R/2 = 0414R