Determine the packing efficiency of the unit cell in hexagonal close-packed (hcp) structure of particles.

Explanation

The unit cell in an hcp structure of particles is a hexagonal unit cell as shown in the given figure. The number of particles in the hexagonal unit cell = 6. If radius of each the

particles making up the unit cell is = r, then total volume for 6 particles = `6xx4/3pir^(3)`.
The area of the base of a hexagonal unit cell = `=6xxsqrt3/4xx4r^(2)=6sqrt3r^(2)`.
The height of this unit cell = `4rsqrt(2/3)`
Therefore, the volume of unit cell
= area of the base `xx` height = `6sqrt3r^(2)xx4rsqrt(2/3)=24sqrt2r^(3)`
`therefore" "` Packing efficiency of the hexagonal unit cell
`=("Volume of 6 particles")/("Volume of unit cell")xx100=(6xx4/3pir^(3))/(24r^(3)sqrt2)xx100`
= 74.01%