Close Packing in two Dimensions:
To understand the packing of constituent particles of crystals, let us consider the packing of hard spheres of equal size. The spheres can be arranged side by side touching each other in a row (horizontal alignment). The rows can be combined in the following two ways with respect to the first row to build a crystal plane-
(i) The spheres are packed in such a way that the rows have a horizontal as well as vertical alignment. In this arrangement, the spheres are found to form squares. This type of packing is also called square close packing.
(ii) The spheres are packed in such a way that the spheres in the second row are placed in the depressions between the spheres of the first row. Similarly, the spheres in the third row are placed in the depressions between the spheres of the second row and so on. This gives rise to hexagonal close packing of spheres.
A comparison of two ways of packing of spheres shows that in arrangement I, the spheres are less closely packed than in arrangement II. It has been calculated that in arrangement I, only 52.4% of the available space is occupied by the spheres. In the second arrangement, 60.4% of the space is occupied. Therefore, the arrangement II is more efficient and leaves less space unoccupied by spheres. Thus, arrangement II is more economical and it represents a close packing of spheres.
It can be seen that in arrangement (I), each sphere is in contact with four other spheres as shown in Fig. (I). On the other hand, each sphere is in contact with six other similar spheres in arrangement II as shown in Fig.
Fig: (I) Fig: (II)
The number of spheres which are touching a given sphere is called the co-ordination number. Thus, the co-ordination number of each sphere in arrangement I is four and in arrangement II is six. Thus, the arrangement II represents closest packing of spheres in a layer.
Close Packing in Three Dimensions:
We can now build other layers over the first layer to extend the packing in three dimensions. Let us mark the spheres in the first layer as ‘A’. It is clear from Fig. 2(a) that there are two types of voids or hollows in the first layer. These are marked as ‘a’ and ‘b’. All the hollows are equivalent but the spheres of second layer may be placed either on hollows which are marked ‘a’ or on other set of hollows marked ‘b’. It may be noted that it is not possible to place spheres on both types of hollows. Let us place the spheres on hollows marked ‘b’ to make the second layer which may be labelled as ‘B’ layer. Obviously the holes marked ‘a’ remain unoccupied while building the second layer. The second layer is indicated as dotted circles in Fig. 2 (b).
Fig. 2.
When a third layer is to be added, again there are two types of hollows available. One type of hollows marked ‘a’ are unoccupied hollows of the first layer. The other types of hollows are hollows in the second layer marked as ‘c’. Thus, there are two alternatives to build the third layer. These are:
(i) The third layer of spheres may be placed on the hollows of second layer marked as ‘c’. In this arrangement, the spheres of the third layer lie directly above those in the first layer. In other words, third layer becomes exactly identical to the first layer labelled as ‘A’. This is shown in Fig. 2.5. This type of packing is referred to as ABABA…. arrangement. This type of packing is also known as hexagonal close packing. It is abbreviated as hcp.
(ii) The second way to pack spheres in the third layer is to place them over hollows marked ‘a’ (unoccupied hollows of first layer). This gives rise to a new layer labelled as C. However, it can be shown that the spheres in the fourth layer will correspond to those in the first layer. This is shown in Figs. 2.6 (a) and 2.6 (b). This gives the ABCABCA…. type of arrangement. It is also known as cubic close packing and is abbreviated as ccp. For simplicity ccp arrangement can be drawn as shown in Fig. 2.6 (c). It is clear from Fig. 2.6 (c) that there is a sphere at the centre of each face of the cube. Therefore, this arrangement is also known as face centred cubic arrangement and is abbreviated as fcc. It may be noted that both types of packing are equally economical though these have different forms. In both cases, 74% of the available volume is occupied by the spheres. 
Fig: 2.5 (a) ABABA… Fig: 2.6 (a) ABCABCA…