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

ABAB….. Type of packing means hexagonal close packing. A unit cell ( with lifted layers) is shown in the fig 1.61.
suppose the radius of each sphere =r
Volume of the unit cell = Base area ` xx` height (h)
Base area of regular hexagon = Area of six equilateral triangle s.
each with side a ( i.e, 2r) =` 6 xx sqrt3/4 a^(2)`
Height h =2 ` xx` Distance between closest packed layers.
` 2xx sqrt(2/3) a`
voluvme of unit cell =` ( 6 xx sqrt3/4 a^(2)) ( 2xx sqrt(2/3) a) = 3sqrt2 a^(3)`
putting a = 2r, volume of the unit cell ` = 3 sqrt 2 ( 2r)^(3) = 24 sqrt(2) r^(3)`
No. of atoms in hcp per unit cell =` 12xx 1/6` (corners) ` + 2 xx 1/2` ( face centres) + 3( in the body) = 6
Volume of six spheres ` = 6 xx 4/3 pi r^(3) = 8 pi r ^(3)`
packing fraction = `( 8 pi r^(3))/(24 sqrt2 r^(3)) = pi/( 3sqrt2) = 3.143/(3xx 1.414) = 0.74`
% volume occupied = 74 %
% volume empty space = 26 %