# Hexagonal Close Packing in Three Dimensions

Table of Contents

### Hexagonal Close Packing in Three Dimensions

Let us mark the spheres in first layer as ‘A’. In the first layer there are some empty spaces called voids or holes. These are triangular in shape. These triangular voids are of two types marked as ‘a’ and ‘b’. All voids are equivalent but the spheres of second layer may place either on voids marked as ‘a’ or on marked as ‘b’.

Let us place the spheres on voids marked ‘b’ to make second layer which may labelled as B layer. Obviously, the voids marked ‘a’ remain unoccupied while building the second layer.

While building the second layer new voids are generated, mark them as ‘c’. It is clear that the two types of voids are not similar. The ‘b’ and ‘c’ type of voids are triangular but ‘a’ type of voids is double triangular (one each of first layer and second layer).

A simple triangular voids like ‘b’ and ‘c’ in a crystal is surrounded by four spheres and is called tetrahedral void. A double triangular void with apices in opposite direction (like ‘a’) is surrounded by six spheres and is called octahedral void.

Now the third layer can be built up by placing spheres above tetrahedral voids marked ‘c’ or octahedral voids marked ‘a’.

###### By Covering Tetrahedral Voids

When third layer is placed over the second layer in such a way that the sphere covers the tetrahedral or ‘c’ voids. Third layer become exactly identical to the first layer. This type of arrangement is referred to as ABABA…….arrangement. This type of packing is called hexagonal close packing (hcp). Coordination number of each sphere is 12. Metal like magnesium, zinc, beryllium, molybdenum etc. adopt this type of arrangement.

###### By Covering Octahedral Voids

When third layer is placed over the second layer in such a way that the sphere covers the octahedral or ‘a’ void. Third layer become differ to the first layer ‘A’ and second layer ‘B’. Let us call it as layer ‘C’. This type of arrangement is referred to as ABCABC…….arrangement. This type of packing is called cubic close packing (ccp) or face-centred cubic packing (fcc). Coordination number of each sphere is 12.

###### Note:

Let the number of closed packed spheres be N, then:

The number of tetrahedral voids generated = 2N

The number of octahedral voids generated = N

Total number of tetrahedral and octahedral voids = 3N

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