A bcc lattice is made up of hollow spheres of X. Spheres of solid 'Y' are present in hollow spheres of X. The radius of 'Y' is half of the radius of 'X' . Calculate the ratio of the total volume of spheres of 'X' unoccupied by Y in a unit cell and volume of the unit cell ?

A bcc lattice is made up of hollow spheres of X. Spheres of solid 'Y'

1.	A bcc lattice is made up of hollow spheres of X. Spheres of solid 'Y' are present in hollow spheres of X. The radius of 'Y' is half of the radius of 'X' . Calculate the ratio of the total volume of spheres of 'X' unoccupied by Y in a unit cell and volume of the unit cell ?
Answer» Solution :Let the RADIUS of HOLLOW sphere X be r. As spheres X have BCC structure, edge length (a)=`"4r"/sqrt3 "" (therefore "for bcc", r=sqrt3/4 a)` `therefore` VOLUME of the unit cell =`a^3=("4r"/sqrt3)^3` Radius of sphere Y=`r/2` (Given ) `therefore` Volume of sphere `X=4/3pir^3` Volume of sphere `Y=4/3 pi(r/2)^3` `therefore` Volume of X unoccupied by Y in unit cell =`2xx[4/3pir^3-4/3pi(r/2)^3]` ( `because` bcc lattice has 2 spheres per unit cell ) `=2xx4/3pir^3(1-1/8)=2xx4/3pir^3xx7/8` `therefore "Volume of X unoccupied by Y in unit cell"/"Volume of unit cell"=(2xx4/3pir^3xx7/8)/((4r)/sqrt3)^3=(7pisqrt3)/64`