Identical spheres of radius R form an fcc lattice and smaller spheres of radius exactly (sqrt3/2-1)R are placed in every tetrahedral void. Take pi=3.14. The fraction of the unit cell volume occupied by all the spheres is closest to

JEE Advanced levelSingle correct optionDifficulty 3 of 3ss124

Identical spheres of radius $R$ form an fcc lattice and smaller spheres of radius exactly $(3/2 - 1) R$ are placed in every tetrahedral void. Take $π = 3.14$ . The fraction of the unit cell volume occupied by all the spheres is closest to

a = 22 R

, so

a^{3} = 162 R^{3}

. Small sphere radius

= 0.2247 R

; 8 tetrahedral voids per cell. Total sphere volume

= \frac{4}{3} π R^{3} (4 + 8 \times 0.224 7^{3}) = \frac{4}{3} π R^{3} \times 4.091

. Fraction

= 0.757

(without the small spheres it would be 0.740).