11a Close packing in three dimensions

All real structures are three dimensional structures. They can be obtained by stacking two dimensional layers one above the other.
First place one square close packed layer and name it as A

Place a second square close packed layer above the first such that the spheres of both the layers align horizontally and vertically.
The second layer can also be named as A
Similarly place few more layers of spheres one above another. The spheres in all the layers align vertically and horizontally. So, this lattice has AAA… type pattern.
The lattice thus generated is simple cubic lattice and its unit cell is primitive cubic unit cell.

This type of packing is called three dimensional close packing from two dimensional square close packed layers.

Three dimensional structures can be generated by placing two dimensional layers one above the other.
Place first layer of hexagonal close packed layer ‘A’
Then, place second layer over the first layer such that the second layer fits in the depression of the first layer
Since the spheres of the two layers are aligned differently, second layer can be named as ‘B’
All the triangular voids of first layer are not covered by spheres of second layer. This gives rise to different arrangements.

When the triangular voids of first layer are covered by second layer or vice versa, it gives rise to a tetrahedral void because a tetrahedron is formed when centers of the four nearing spheres are joined.
When triangular voids of second layer are placed on triangular voids of first layer, they do not overlap.
Apex of one triangle points upward and apex of the other points downward. Such voids are surrounded by six spheres.
So, an octahedral void is formed because, when the center of the nearing spheres is joined, it forms an octahedron.
The number of these two types of voids depends upon the number of close packed spheres.
If the number of close packed sphere = N
The number of octahedral voids generated =N
The number of tetrahedral voids generated = 2N

When the third layer is placed over the second layer, there are two possibilities.

First possibility is Covering Tetrahedral voids.
Place 2 layers of spheres such that the second layer fits in the depression of first layer. First layer is named as A and the second as B

The third layer of spheres can be placed over the second layer such that the tetrahedral voids are covered.
Spheres of third layer exactly align with the spheres of first layer and can be named as ‘A’
Thus the pattern of spheres is repeated in alternate layers and can be named as ABAB…pattern.
This structure is known as hexagonal close packing (hcp).
Example: metals like Magnesium and Zinc have this kind of packing.

This type of packing is very efficient because of very less space between the spheres, that is packing efficiency of this lattice is 74%.
Each sphere is in contact with 12 nearest atoms. So, the coordination number is 12.

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