Calculate the efficiency of packing in case of a metal crystal for body centred cubic lattice.

Explanation

In `DeltaEFD`
`b^2=a^2+a^2=2a^2`
`b=sqrt2a`
In `DeltaAFD`
`c^2=a^2+b^2=3a^2`
`c=sqrt3a`
In bee arrangement, body diagonal e is equal to 4r, where r is the radius of sphere (atom).
So, `sqrt3a=4r`
`a=(4r)/(sqrt3)`
`r=(sqrt3)/4a`
In bee arrangement no. of spheres (atoms) per unit cell is 2. So, the total volume of 2 spheres `=2xx3/4pir^3`
Where, volume of cube, `(a^3)=(4/(sqrt3)r)^3`
`underset("(In percentage)")("So, Packing efficiency")= ("Volume of two spheres in unit cell")/("Total volume of unit cell")`
`=(2xx[4/3]pir^3)/([(4/(sqrt3)r)]^3)=(8/3pir^3xx100)/((64//3sqrt3)r^3)=68%`