Calculate the efficiency of packing in case of a metal crystal for body-centred cubic
It can be observed from the above figure that the atom at the centre is in contact with the other two atoms diagonally arranged.
From ΔFED, we have:
b2 = a2 + a2
⇒ b2 = 2a2
⇒ b = `sqrt2a`
Again, from ΔAFD, we have:
c2 = a2 + b2
⇒ c2 = a2 + 2a2 (Since b2 = 2a2)
⇒ c2 = 3a2
⇒ `c = sqrt3a`
Let the radius of the atom be r.
Length of the body diagonal, c = 4π
⇒`sqrt3a = 4r`
or `r = (sqrt3a)/4`
Volume of the cube `a^3 = ((4r)/sqrt3)^3`
A body-centred cubic lattice contains 2 atoms.
So, volume of the occupied cubic lattice `2pi4/3 r^3`
:.Packing efficiency = `("Voulume occupied by two spheres in the unit cell")/"Total volume of unit"xx100%`
=`(8/3pir^3)/(64/(3sqrt3)r^3) xx 100%`