Packing in Solids, Voids

Close packing means the arrangement of spheres of a solid crystal to occupy the most amount of available space while leaving the least amount of free space. The close packing relates to a maximum density state. The tighter the close packing in the solid crystal more will be its stability. In AA crystal, the coordination number also referred to as ligancy, is the number of atoms, ions, or molecules that a central atom or ion has as its nearest neighbours.

One Dimensional Close Packing In Solids

The only way to arrange the spheres in a one-dimensional close packing is by placing them in a single row or line where the spheres are in contact. Each sphere will connect with two other spheres, one on the right and one on the left, and have the coordination number two.

Two Dimensional Close Packing In Solids

In the 2D close packing in solids, one row is stacked above the other. There are two ways of making this arrangement.

1. Square Close Packing

The second row is placed in between the first and the third row. The spheres in the second row come just above the first row. The spheres are aligned horizontally and vertically in all the rows.

Only similar types of spheres are present in the rows. If the rows are called A-type rows, and similarly, the second row is also A-type as they are similar. Here the A-type rows are stacked one above the other. This arrangement can be called a AAA type arrangement.

Each sphere is in close contact with four other spheres, one above, one below, one to the right and one to the left. Hence its coordination number is four. And also, when the centres of the adjacent neighbouring spheres are connected, it forms a square shape, and hence the name square closes packing in solids in two dimensions.

2. Hexagonal Close Packing

In the Hexagonal Close Packing arrangement, the second row is placed above the first row in a staggered fashion to fit in the depression of the spheres of the first row.

In the hexagonal close packing arrangement, there are two types of rows. If one type of row is called an A-type row and the one stacked in the depressions of the first row will be called B type row. More rows are stacked one above the other in a staggered manner, thus making this arrangement the ABAB type.

There is less vacant space between the spheres, and the close packing is tight and more efficient than the square close packing. Here, a single sphere comes in close connection with six other spheres, two each from the above and below row and one from the right and left sides and hence makes the coordination number six.

When the centres of these six closely connected neighbouring spheres are combined, it gives rise to a regular hexagonal shape and hence it is named hexagonal close packing. There are some empty spaces or voids between the rows, triangular in shape. There are two types of these triangular voids. One whose apex will be pointing upwards, whereas, in the other row, its apex will be pointing downwards.

Three Dimensional Close Packing In Solids

By placing the close-packed two-dimensional spheres one above the other, the three-dimensional close packing is achieved. There are also two ways of arranging them.

• From a two dimensional square layer

In this three dimensional arrangement, the second layer is placed exactly above the first layer. This arrangement will vertically and horizontally align these layers. More layers are similarly arranged, one above the other.

If the first layer is A-type, this arrangement will be called the AAA type arrangement. Thus the lattice generated will be a simple cubic. It has a primitive cubic unit cell.

The coordination number for cubic close packing will be 4. The total volume occupied by the spheres will be 52%. Or in other terms, the packing efficiency is 52%.

The only metal that crystallises in this form is polonium.

•  From a two dimensional hexagonal layer

This arrangement is obtained by stacking one layer of 2-D hexagonal layers over another so that the spheres of the above layer are in the depression of the below one.

As the first and second layers are in different alignments, we can call the first layer A and B. When the layers are stacked, we notice a tetrahedral void is formed when the second layer is placed above the first layer.

Whereas in other places, we can notice that the triangular voids in the first layer are below the triangular voids of the second layer so that they do not extend over one another, thus forming the octahedral voids. The octahedral voids are covered by six spheres and have a coordination number of six.

We can place the third layer above the second layer in two different ways.

• Covering the Tetrahedral Voids

The spheres of the third layer can envelop the tetrahedral voids of the second layer. Here the covering of the tetrahedral void takes place. If this arrangement occurs, the third layer will be placed the same way as the first layer. Thus this alternating pattern repeats throughout the entire packing. This pattern is also called the ABAB pattern, also called hexagonal close packing (hcp). Magnesium and Zinc show this arrangement.

• Covering the Octahedral Voids

In this arrangement, the third layer is placed above the second layer, covering the second layer’s octahedral voids. When placed like this, the third layer is not in alignment with the first layer, and hence it is called a C type.

When the fourth layer is placed above the third layer, it aligns with the first layer.

This arrangement is often called the ABCABC type, and the structure formed is face centred cubic(fcc) or cubic close-packed (ccp). Copper and Silver are the elements that crystallise in this lattice.

Packing Efficiency

The percentage of total space in a unit cell occupied by constituent particles such as atoms, ions, or molecules packed within the lattice is called packing efficiency. In three-dimensional space, these particles settle the total amount of space. The fraction of a solid’s total volume occupied by a spherical atom that can be calculated is called the packing efficiency of a solid.

Factors determining the packing efficiency are:

• The volume of a unit cell
• The number of atoms in a lattice structure
• The volume of the atoms

Packing efficiency can be calculated using the formula:

Packing Efficiency = Number of atoms Volume occupied by one share 100%Total volume of the unit cell

Conclusion

Close packing in solids refers to the tight packing of ions or atoms of a solid to minimise the empty spaces. Close packing is directly proportional to stability. There are different types of close packing. One dimensional is the simple arrangement of atoms in a row. In two dimensions, there are again two ways.

Placing the spheres of one row directly above the other and the second one is by putting the sphere in the depression of the previous row. In three dimensions, hexagonal close packing (hcp) is formed by covering the tetrahedral voids and the formation of cubic close packing (ccp) by wrapping the octahedral voids. The packing efficiency gives the percentage of total space in a unit cell occupied by the constituent particles.