## Packing in Solids (One and Two Dimensional)

### The Principle of Packing in Solids:

• In a crystal particle (atoms, molecules or ions) occupy the lattice points in the crystal lattice.
• The constituent particles try to get closely packed together so that the maximum closely packed structure is attained.
• The maximum closely packed structure is such that the minimum empty space is left.
• Due to this arrangement, a maximum possible density of the crystal is achieved.
• Close packing imparts stability to the crystal.

### Close Packing in One Dimension:

• There is only one way of arranging spheres in a one-dimensional close-packed structure, that is to arrange them in a row and touching each other.
• In this arrangement, each sphere is in contact with two of its neighbours. The number of nearest neighbours of a particle is called its coordination number. Thus, in a one-dimensional close-packed arrangement, the coordination number is 2.

### Close Packing in Two Dimensions:

• A two-dimensional close-packed structure can be generated by stacking (placing) the rows of close-packed spheres. This can be done in two different ways.

#### a) Square Close Packing:

• The second row may be placed in contact with the first one such that the spheres of the second row are exactly above those of the first row. The spheres of the two rows are aligned horizontally as well as vertically.
• If we call the first row as ‘A’ type row, the second row being exactly the same as the first one, is also of ‘A’ type. Similarly, we may place more rows to obtain AAA type of arrangement.
• In this arrangement, each sphere is in contact with four of its neighbours. Thus, the two-dimensional coordination number is 4.
• Also, if the centres of these 4 immediate neighbouring spheres are joined, a square is formed. Hence this packing is called square close packing in two dimensions.

#### b) Hexagonal Close Packing:

• In this arrangement, the second row may be placed above the first one in a staggered manner such that its spheres fit in the depressions of the first row.
• If the arrangement of spheres in the first row is called ‘A’ type, the one in the second row is different and may be called ‘B’ type. When the third row is placed adjacent to the second in a staggered manner, its spheres are aligned with those of the first row. Hence this row is also of ‘A’ type. The spheres of the similarly placed fourth row will be aligned with those of the second row (‘B’ type). Hence this arrangement is of ABAB type.
• In this arrangement, there is less free space and this packing is more efficient than the square close packing. Each sphere is in contact with six of its neighbours and the two-dimensional coordination number is 6.
• The centres of these six spheres are at the corners of a regular hexagon hence this packing is called two-dimensional hexagonal close packing.
• It can be seen in Figure that in these rows there are some voids (empty spaces). These are triangular in shape. The triangular voids are of two different types. In one row, the apexes of the triangles are pointing upwards and in the next layer downwards.