In hexagonal system of crystals,a frequently encountered arrangement of atoms is described as a hexagonal prism.Here, the top and bottom of the cell are angular hexagons and three atoms are sandwitched in between them .A space filling model of this structure , called hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwitched in between them. A space filling model of this structure, called hexagonal closed packed (HCP) , is consituted of a sphere on a flat surface surrounding in the same plane by six identical spheres as closed as possible . Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these sphere touches three spheres of the bottom layer. Finally , the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be 'r'.
The empty space in this HCP unit cell is

Options

A. 0.74
B. 0.476
C. 0.32
D. 0.26

Answer

D

Explanation

Packing fraction
`=("volume of the atomos in onunitcell")/("volume of one unit cell")`
`=(6xx4//3pi^(3))/(24sqrt(2)r^(3))=(pi)/(3sqrt(2))=0.74=74%`
Pakcing fraction =74%
Empty space=26%
Hence, (4) is correct option.