Answer:
The r/R ratio for an interstitial compound can be calculated using the following formula:
r/R = [(3V_i)/(4πV_h)]^(1/3)
where r is the radius of the interstitial atom, R is the radius of the host atom, V_i is the volume of the interstitial atom and V_h is the volume of the host atom.
For BCC interstitial compound, the interstitial site is located at the center of the cube, and the coordination number is 8. Therefore, the volume of the interstitial atom is:
V_i = (4/3)πr^3
The volume of the host atom is:
V_h = [(4/3)πR^3]/2
since there are only two atoms per unit cell.
Substituting these values into the formula, we get:
r/R = [(3V_i)/(4πV_h)]^(1/3)
r/R = [(3(4/3)πr^3)/(4π[(4/3)πR^3]/2)]^(1/3)
r/R = [(2r^3)/(R^3)]^(1/3)
r/R = (2/1)^(1/3)
r/R = 1.2599
For FCC interstitial compound, the interstitial site is located at the face center of the cube, and the coordination number is 12. Therefore, the volume of the interstitial atom is:
V_i = (4/3)πr^3
The volume of the host atom is:
V_h = [(4/3)πR^3]/4
since there are four atoms per unit cell.
Substituting these values into the formula, we get:
r/R = [(3V_i)/(4πV_h)]^(1/3)
r/R = [(3(4/3)πr^3)/(4π[(4/3)πR^3]/4)]^(1/3)
r/R = [(4r^3)/(R^3)]^(1/3)
r/R = (4/1)^(1/3)
r/R = 1.5874
Therefore, the r/R ratio for BCC interstitial compound is 1.2599, and for FCC interstitial compound is 1.5874.