Question

How much space is filled in cubic close packing?

Answer 1

Packing of motifs in crystals:

• The structure of any crystalline solid depends upon the packing of its motifs within the space lattice.
• The motifs in a crystal are very closely packed in order to have maximum possible attractive interactions between them and to reach a state of the lowest possible energy.
• The number of equidistant nearest neighbors of a given motif in a crystal is known as the coordination number.

Cubic close packing(CCP):

• Close-packing of spheres is possible when the spheres of the third layer called C residing in the depressions of the second layer B are not aligned vertically above the spheres of either B or the first layer A.
• A repetition of this stacking sequence builds up a structure with a layer sequence ABCABABCA…..
• This is known as cubic close packing or CCP.
• The structure thus built up corresponds to a cubic arrangement of spheres in which a sphere is present also at the center of each face of the unit cube.
• Hence, the CCP arrangement is identical to the face-centered cubic (fcc) packing.
• Therefore, the three-dimensional close-packing arrangement of identical spheres in which the spheres of the fourth layer are positioned directly above those of the first, those of the fifth layer lie directly above those of the second, and so on is called cubic close packing (CCP) or face-centered cubic packing or the ABCABABCA..... packing

Packing efficiency:

Step 1: The volume of the unit cell

In face-centered cubic packing, it can be shown by simple geometrical means that, since three spheres along with the face diagonal touch each other,

So the face diagonal is $4\mathrm{r}$

As per geometry, the face diagonal of a cubic cell having an edge $a$ is$\sqrt{2}\mathrm{a}$

Therefore, the relationship between the edge length $a$ and the radius $r$ of the unit cell is $a=\sqrt{8}r$
The volume of the unit cell is $a^3=\left(\sqrt{8}{r}^3\right)=8\sqrt{8}{r}^3$

Step 2: The volume occupied by 4 atoms

That is the volume occupied by 1 atom is $\frac{4}{3}{\mathrm{\pi r}}^3$

A CCP unit cell has 4 atoms per unit cell.
The volume occupied by 4 atoms is $4\times \frac{4}{3}\pi r^3=\frac{16}{3}\pi r^3$

Step 2: The packing efficiency

$\text{The packing efficiency}=\frac{\text{Volume occupied by atoms in a unit cell}}{\text{total volume of unit cell}}\times 100$
The packing efficiency =$=\frac{\frac{16}{3}\pi r^3}{8\sqrt{8}{r}^3}\times 100=74.0\mathrm{\%}$

Therefore, $74\%$ of space is filled in cubic close packing.