Question

Define Packing efficiency? What is packing efficiency of hcp.

Answer 1

The packing efficiency is the fraction of the crystal (or unit cell) actually occupied by the atoms. It must always be less than 100% because it is impossible to pack spheres (atoms are usually spherical) without having some empty space between them.

Mathematically Packing Efficiency is: Packing efficiency is defined as the percentage of space occupied by constituent particles packed inside the lattice. It can be calculated with the help of geometry in three structures namely: HCP and CCP structures.

Answer 2

Answer:

Packing efficiency :

The percentage of the total space filled by the particles in the three dimensional close packing is known as the packing efficiency.

Packingefficiency of HCP =

74.06 %

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Steps to find packing efficiency of HCP

Let us consider HCP unit cell with edge length "a" and face diagonal "b"

From the figure triangle ACD

$ac {}^{2} = ad {}^{2} + cd {}^{2}$

$b {}^{2} = \: a {}^{2} + a {}^{2} = 2 a {}^{2} \\ b \: = \sqrt{2} a$

But face diagonal

b = 4r

$4r = \sqrt{2a}$

$a = \frac{4}{ \sqrt{2} } r = 2 \sqrt{2} r$

We have 4 atoms in FCC unit cell

Volume occupied will be

$4 \times \frac{4}{3} \pi \: r {}^{3}$

Packing efficiency will be

$\frac{volume \: of \:4 \: spherical \: atoms \: in \: unit \: cell}{volume \: of \: cube} \times 100$

$\frac{4 \times \frac{4}{3}\pi \: r {}^{3} }{a {}^{3} } \times 100$

$\frac{16\pi \: r {}^{3} }{3 \times( 2 \sqrt{2}r) {}^{3} } \times 100$

$\frac{16\pi}{3 \times 16 \sqrt{2} } \times 100$

$here \: r{}^{3} \: get \: cancelled$

$packing \: effiency \: = 74.06\%$

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Additional points :

Packing efficiency of BCC = 68 %

Packing efficiency of SCC = 52.4 %

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