Question

Calculate the packing fraction of siniple cubic arrangement

Answer 1

In a simple cubic arrangement

Consider the cube with an edge length ‘a’ Volume of the cube with edge length a is = a x a x a = a³…………….(l)

Let ‘r’ ¡s the radius of the sphere

From the figure a = 2 r ⇒ r = \(\frac{a}{2}\)

:. volume of the sphere with radius r = \(\frac{4}{3}\) πr³

In a simple cubic arrangement. number of spheres belongs to a unit cell equal to one.

∴Total volume occupied by the spheres in sc unit cell = 1 x\(\Big(\frac{\pi a^3}{6}\Big)\) ^......(3)

Dividing 3 by 1

Packing fraction =

Only 52.31% of the available volume is occupied by the spheres in simple cubic packing, making in efficient use of available space and hence minimizing the attractive forces.