Unit – VI B

## Packing in Solids (Three Dimensional)

A three-dimensional close-packed structure can be generated by stacking (placing) layers of close-packed spheres in two dimensions one over the other. This can be done in following ways.

### Three-dimensional close packing from a two-dimensional square layer:

• In such arrangement, the second layer is placed over the first layer such that the spheres of the upper layer are exactly above those of the first layer. In this arrangement spheres of both the layers are perfectly aligned horizontally as well as vertically as shown. Similarly, we may place more layers one above the other.
• If the arrangement of spheres in the first layer is called ‘A’ type, all the layers have the same arrangement. Thus this lattice has AAA…. type pattern. The lattice thus generated is the simple cubic lattice, and its unit cell is the primitive cubic unit cell.
• The coordination number for such arrangement is 6.

### Three-dimensional close packing from two-dimensional hexagonal close-packed layers:

• This close-packed structure can be generated by placing layers one over the other.
• Placing second layer over the first layer Let us take a two-dimensional hexagonal close packed layer ‘A’ and place a similar layer above it such that the spheres of the second layer are placed in the depressions of the first layer. Since the spheres of the two layers are aligned differently, let us call the second layer as B.
• Placing the third layer over the second layer can be done by two ways.

#### a) Tetrahedral Voids or Tetrahedral Hole:

• It can be observed from Fig.  that not all the triangular voids of the first layer are covered by the spheres of the second layer.
• This gives rise to two different types of holes or voids. Wherever a sphere of the second layer is above the void of the first layer (or vice versa) a tetrahedral void is formed. These voids are called tetrahedral voids because a tetrahedron is formed when the centres of these four spheres are joined.
• They have been marked as ‘T’ in Fig.  One such void has been shown separately in Fig.

#### b) Octahedral Voids:

• Wherever a sphere of the second layer is above the void of the first layer (or vice versa) a tetrahedral void is formed. These voids are called tetrahedral voids because a tetrahedron is formed when the centres of these four spheres are joined.
• It can be observed from Fig. that not all the triangular voids of the first layer are covered by the spheres of the second layer. This gives rise to two different arrangements.
• At other places, the triangular voids in the second layer are above the triangular voids in the first layer, and the triangular shapes of these do not overlap. One of them has the apex of the triangle pointing upwards and the other downwards. These voids have been marked as ‘O’ in Fig.
• Such voids are surrounded by six spheres and are called octahedral voids. One such void has been shown separately in Fig.

#### Number of Voids:

• The number of these two types of voids depends upon the number of close packed spheres.
• Let the number of close packed spheres be N, then, The number of octahedral voids generated = N and the number of tetrahedral voids generated = 2N

### A) Covering Tetrahedral Voids:

• Tetrahedral voids of the second layer may be covered by the spheres of the third layer. In this case, the spheres of the third layer are exactly aligned with those of the first layer. Thus, the pattern of spheres is repeated in alternate layers. This pattern is often written as ABAB ……. pattern. This structure is called hexagonal close packed (hcp) structure.
• This sort of arrangement of atoms is found in many metals like magnesium and zinc.

### B) Covering Octahedral Voids:

• The third layer may be placed above the second layer in a manner such that its spheres cover the octahedral voids. When placed in this manner, the spheres of the third layer are not aligned with those of either the first or the second layer. This arrangement is called ‘C’ type.
• Only when fourth layer is placed, its spheres are aligned with those of the first layer as shown. This pattern of layers is often written as ABCABC ……….. This structure is called cubic close packed (ccp) or face-centred cubic (fcc) structure. Metals such as copper and silver crystallise in this structure.

Notes:

• Both these types of close packing are highly efficient and 74% space in the crystal is filled.
• In either of them, each sphere is in contact with twelve spheres. Thus, the coordination number is 12 in either of these two structures.