A metallic crystal cystallizes into a lattice containing a sequence of layers `ABABAB…`. Any packing of spheres leaves out voids in the lattice. What percentage by volume of this lattice is empty spece?

Explanation

ABABAB……sequence gives hexagonal close packing (hcp) and the number of atoms for unit cell is 6.
`"Packing fraction"=("Volume of six spheres")/("Area of the base"xx"height(C)")`
`"Volume of six spheres"=6xx(4)/(3)pir^(3)=8pir^(3)`
`"Area of base"=6xx(sqrt3)/(4)(2r)^(2)=6r^(2)sqrt3`
`"Height(C)"=4rsqrt((2)/(3))`
`"Packing fraction"=(8pi^(3))/(6r^(2)xxsqrt3xx4rxxsqrt((2)/(3)))`
`=(pi)/(3sqrt2)=(3.143)/(3xx1.4142)=0.74.`
Volume of empty space `=1-0.74=0.26`
Percentage void `=0.26xx100=26%`