Answer to Question #157413 in Molecular Physics | Thermodynamics for Vincent

Question

Define packing fraction and find its value for a body-centred cubic structure

Answer 1

Solution:

Packing faction or Packing efficiency is the percentage of total space filled by the particles:

In body centred cubic unit cell

In 

Let DF= b

and we know that

ED=EF= a (edge length)

Now,

b² = a² + a² = 2a²

In 

Let, AF = c

We know that

FD = b

& AD = a (edge length)

Now,

c2 = a² + b² = a² + 2a² = 2a²

or c = $\sqrt{\smash[b]{3}}a$

we know that c is body diagonal. As the sphere at the centre touches the sphere at the corner. Therefore body diagonal c = 4r

i.e. $\sqrt{\smash[b]{3}}a$ = 4r

or r = $(\dfrac{\sqrt{\smash[b]{3}}}{4})$ a

or a = $\tfrac{4r}{\sqrt{\smash[b]{3}}}$

∴ Volume of the unit cell = a3 = ($\tfrac{4r}{\sqrt{\smash[b]{3}}}$)³ = $\tfrac{64r^3}{3\sqrt{\smash[b]{3}}}$

No. of spheres in bcc = 2 

∴ volume of 2 spheres = 2 × $\tfrac{4}{3\pi r^3}$

$Packing \;efficiency=\dfrac{Volume \;occupied \;by\;two\;spheres\;in\;the\;unit\;cell}{Total\;volume\;of\;the\;unit\;cell }*100$

The packing efficiency of BCC is 68%.