# Close packing of solid spheres

Close packing of identical solid spheres

The solids which have non-directional bonding, their structures are determined on the basis of geometrical consideration. For such solids, it is found that the lowest energy structure is that in which each particle is surrounded by the greatest possible number of neighbours. In order to understand the structures of such solids, let us consider the particles as hard sphere of equal size in three directions. Although there are many ways to arrange the hard spheres but the one in which maximum available space is occupied will be economical which is known as closest packing.

Now we describe the different arrangements of spherical particles of equal size. When the spheres are packed in a plane, they may arrange themselves in following two ways:

(i) The centers of the spheres lie one below another. This type of arrangement is called square close packing. In such packing one sphere touches four other spheres. In this case 52.4% of the volume is occupied. The remaining 47.6% of the volume is empty and is called void volume.

(ii) The spheres of the second row are placed in such a way that they fit themselves in the depressions between the spheres of the first row. The spheres of third row are placed in such a way that they are exactly below the spheres of first row. Evidently the spheres of fourth row will be below the spheres of second row and so on. This type of packing is called hexagonal close packing. In this case 60.4% volume is occupied and 39.6% volume is void volume. Therefore this type of packing is more stable than the square close packing.

In hexagonal close packing, there are two types of voids (open space or space left) which are divided into two sets and for convenience. The spaces marked are curved triangular spaces with tips pointing upwards whereas spaces marked are curved triangular spaces with tips pointing downwards.

Now we extend the arrangement of spheres in three dimensions by placing second close packed layer (hexagonal close packing) (B) on the first layer (A). The spheres of second layer may be placed either on space denoted by or ‘c’. It may be noted that it is not possible to place spheres on both types of voids (i.e., b and c). Thus half of the voids remain unoccupied by the second layer. The second layer also has voids of the types and in order to build up the third layer, there are following two ways:

(1)  In one way, the spheres of the third layer lie on the spaces of second layer (B) in such a way that they lie directly above those in the first layer (A). In other words we can say that the third layer becomes identical to the first layer. If the arrangement is continued indefinitely in the same order this is represented as AB AB AB ……………….. This type of arrangement represents hexagonal close packing (hep) symmetry (or structure), which means that the whole structure has only one 6-fold axis of symmetry (i.e., the crystal has same appearance on rotation through an angle of $60^0$).

Another view of hexagonal packing (cp) of spheres

(2)   In second way, the spheres of the third layer (C) lie on the second layer (B) in such a way that they lie over the unoccupied spaces ‘C’ of the first layer (A). If this arrangement is continuous indefinitely in the same order this is represented as ABC ABC ABC ABC………..This type of arrangement represents cubic close packed (ccp) structure. This structure has 3-fold axes of symmetry which pass through the diagonal of the cube. Since in this system, there is a sphere at the center of each face of the unit cell hence this structure is also known as face-centred cubic (fcc) structure.

It may be noted that in ccp (or fcc) structures each sphere is surrounded by 12 spheres hence the coordination number of each sphere is 12. The spheres occupy 74% of the total volume and 26% is the empty space in both (hcp and ccp) structures.

There is another possible arrangement of packing of spheres known as body centred cubic (bcc) arrangement. This arrangement is observed in square close packing

(which is slightly more open than hexagonal close packing). In bcc arrangement the spheres of the second layer lie at the space (hollows or voids) in the first layer. Thus each sphere of the second layer touches four spheres of the first layer. Now spheres of the third layer are placed exactly above the spheres of first layer. In this way each sphere of the second layer touches eight spheres (four of Ist layer and four of III rd layer). Therefore coordination number of each sphere is 8 in bcc structure. The spheres occupy 68% of the total volume and 32% of the volume is empty space.

FCC structure

Summary of hcp, ccp (fcc) AND bcc Structures

 S .No Properties Hcp Cpp Bcc 1. Arrangement of particles Closely packed Closely packed Not closely packed 2. Volume occupied 74 % 74% 68% 3. Coordination number 12 12` 8 4. Type of packing AB AB AB ABC ABC ——- 5. Malleability Less malleable Malleable Malleable 6. Ductility Ductile Ductile Ductile 7. Examples Mg, Xn, Cd, Ti, V, Cr, Mo Cu, Ag, Au, Ca, Sr, Pt Li, Na, K, Rb, Cs , Fe

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