# explain the packing efficiency of fcc,bcc,hcp?

Dear Student,

**HCP and CCP Structures**

In both types of packing hexagonal closed packing or cubic closed packing the packing efficiency is same. In order to calculate packing efficiency in cubic closed structure, we consider a unit cell with edge length “a” and face diagonal AC to be “b”.

**Image 2: CCP Structure**

Looking at the ABCD face of the cube, we see a triangle ABC. Let the radius of the each sphere ball be ‘r’. Correlating radius and edge of the cube, we proceed as follows:

In △ ABC, by Pythagoras theorem we can write:

AC^{2} = BC^{2} + AB^{2}

Since AC = b and BC = AB = a, substituting the values in the above relation we get

b^{2 }= a^{2} + a^{2}

b^{2} = 2a^{2}

b = √2 a….. (1)

also edge b in terms of radius ‘r’ equals

b = 4r … (2)

From (1) and (2) we can write

4r = √2 a

or

a = 2 √2 r

As there are 4 spheres in a ccp structure unit cell, the total volume occupied by them will be

4 × 4 / 3 π r^{3}

Also, total volume of a cube is (edge length)^{3} that is, ( a^{3}) or in terms of r it is (2 √2 r)^{3}, therefore packing efficiency will be:

Therefore packing efficiency in fcc and hcp structures is 74%

Similarly in HCP lattice the relation between radius ‘r’ and edge length of unit cell “a” is r = 2a and number of atoms is 6.

**Body-Centred Cubic Structures**

In body-centred cubic structures, the three atoms are diagonally arranged. To find the packing efficiency we consider a cube with edge length a, face diagonal length b and cube diagonal as c.

**Image 3: BCC structure**

In △ EFD according to Pythagoras theorem

b^{2 }= a^{2} + a^{2}

b^{2} = 2a^{2}

b = √2 a

Now in △ AFD according to Pythagoras theorem

c^{2 }= a^{2} + b^{2} = a^{2} + 2a^{2}

c^{2} = 3a^{2}

c = √3 a

If the radius of each sphere is ‘r’ then we can write

c = 4r

√3 a = 4r

r = √3/ 4 a

As there are two atoms in the bcc structure the volume of constituent spheres will be

2 × (4/3) π r^{3}

Therefore packing efficiency of the body-centred unit cell is 68%.

Regards.

**
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