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

Explanation

ABAB… type of packing means hexagonal close packing . A unit cell (with lifted layers) .
Suppose the radius of the each sphere = r
Volume of the unit cell = Base area x height (h)
Base of area of regular hexagon = Area of six equilateral triangles, each with side a (i.e., 2r) =`6xxsqrt3/4a^2`
Height h = 2 x Distance between closest packed layers
`=2xx sqrt(2/3)a`
`therefore` Volume of the unit cell =`(6xxsqrt3/4a^2)(2xxsqrt(2/3)a)=3sqrt2a^3`
Putting a=2r, Volume of the unit cell =`3sqrt2 (2r)^3 =24sqrt2r^3`
No. of atoms in hcp per unit cell =`12xx1/6`(corners ) +`2xx1/2` (faces centres ) + 3 (in the body )=6
Volume of six spheres =`6xx4/3pir^3=8pir^3`
`therefore` Packing fraction=`(8pir^3)/(24sqrt2r^3)=pi/(3sqrt2)=3.143/(3xx1.414)=0.74`
% volume occupied =74%
% volume empty space =26 %