Packing in Solids

In a crystalline solid, the constituent particles are closely packed such that there is minimum vacant space. The particles are arranged regularly in a repeating pattern. All the particles are assumed to be equal-sized, rigid, and spherical.

The highest percentage of space occupied by these spheres is 74%. This means that there is always some space left in a cell. The spheres are arranged such that the vacant space is minimised. Close packing in solids results in the particles being very close to each other.

Let us have a detailed study about one dimensional, two dimensional, and 3-D packing in a solid-state. 

Coordination number

Coordination number in solid packing refers to the number of particles that touch an atom in the lattice arrangement. It is also defined as the number of nearest neighbours of a particle/atom.

Close packing in solids: one dimension

In 1-D close packing, there is only one way to arrange the constituent particles. They are placed next to each other in a row. Each sphere touches two of its adjacent spheres.

Therefore the coordination number in one-dimensional closed packing is 2.

Close packing in solids: two dimension

Stacking two rows of closely packed spheres results in a two-dimensional closed packing. There are two ways to achieve this:

Square closed packing in two dimensions.

The second row of spheres is placed exactly above the first row. The spheres of both rows are aligned vertically and horizontally. 

If the first row is called row ‘A’, the second row would also be called ‘A’ since both are identical. This arrangement is termed AAA type of arrangement. 

Each constituent particle/sphere is in direct contact with four other spheres. Hence, the coordination number in 2-D packing is 4.

The resultant would be a sphere if four immediate neighbouring spheres are considered and their centres are joined. Therefore this type of packing is known as square closed packing in two dimensions. 

The packing efficiency of the 2-D square unit cell is 39.27%.

Hexagonal closed packing in two dimensions.

In such a packing, the second row of the spheres is stacked above the first one staggered. The spheres of the second row fit perfectly in the depressions of the first row.

Let us consider the first row to be ‘A’-type. Since the second row is slightly different, it is called the ‘B’ type. 

Now, if a third layer is to be placed above the second ‘B’ layer, the spheres would fit into the depressions such that the constituent particles are aligned with the first layer. 

This will continue, and the arrangement of the lattice would be ABABAB…

Each sphere is adjacently connected to six other spheres. Hence, the coordination number would be 6.

We would obtain a hexagon by joining the centres of these six constituent particles. Therefore this type of packing is known as hexagonal closed packing in 2-D.

Hexagonal closed packing is more efficient than square packing as it has lesser free space. The percentage of space occupied is more. 

Close packing in three dimension

Three-dimensional lattices are obtained by stacking two-dimensional layers over one other. Two lattices are seen in the 3-D packing.

3-D closed packing from 2-D square close packed-layers

The second square packed layer is placed over the first such that the constituent particles of the second are precisely above the spheres of the first layer. The spheres are aligned horizontally and vertically, resulting in a cube.

The first layer is called ‘A’. since the second layer is identical to the first, it is also referred to as ‘A’, and so are the subsequent layers. The arrangement here is AAA…

The resultant lattice resembles a cube and is called a simple cubic lattice. Its unit cell is called the primitive cubic unit cell. 

3-D closed packing from 2-D hexagonal close packed-layers

In hexagonal packing, lattices can be formed in 2 ways:

• Placing the second layer over the first

Let us consider a two-dimensional hexagonal close-packed layer and name it ‘A’. The second layer is placed above the first layer such that the spheres fill in some depressions. The spheres of both layers are aligned differently. Therefore it is called layer ‘B’. 

But all the triangular voids are filled by the constituent particles of the second layer. Two different lattice arrangements arose due to this.

A tetrahedral void is seen wherever a particle/sphere of the second layer is above the void of the first layer (or vice versa). The voids are so named because a tetrahedron is obtained by joining the centres of these four spheres. 

An octahedral void is formed when the triangular voids of the second layer are above those of the first layer. One has its apex pointing upwards and the other pointing downwards. Six spheres surround octahedral voids. 

The number of both voids present depends upon the number of close-packed spheres.

If the number of close-packed spheres = n, the,

The number of octahedral voids = n

The number of tetrahedral voids = 2n.

• Placing the third layer over the second layer

Two possibilities exist in this scenario.

1. Covering tetrahedral voids

The spheres of the third layer cover the tetrahedral voids. As seen, the particles of the third layer are perfectly aligned with the spheres of the second layer. This pattern is repeated alternatively, giving rise to the ABABAB… arrangement. 

The structure formed is called a hexagonal close-packed (hcp) structure.

The coordination number is 12.

Metals like magnesium (Mg) and zinc (Zn) exhibit the hcp structure.

1. Covering octahedral voids

The third layer is stacked so that its spheres fill the octahedral voids. In this arrangement, the third layer is not aligned with the first two layers. Therefore it is referred to as ‘C’. 

When a fourth layer is placed, its spheres align precisely with the particles of the first layer. The resultant arrangement is ABCABCABC….

Cubic close-packed (ccp) structures or face-centred cubic (fcc) structures are formed in this arrangement. The coordination number of this structure is 12.

Their packing efficiency is 74%. Metals that crystallise in this structure include copper (Cu) and silver (Ag).

Conclusion

Here we learned about packing in solids. The article explores multiple layers of packing in solids, along with their detailed definitions. You can also learn the arrangements of layers of solids. Make sure to go through each explanation carefully.