Solid State
4.0 Close Packing in Crystals
4.0 Close Packing in Crystals
In order to understand the packing of the constituent particles in a crystal, it is assumed that these particles are hard spheres of identical size. The packing of the crystals is done such that they occupy the maximum available space and hence the crystal has maximum density.
There are two common ways in which spheres of each size can be packed as shown below:
In arrangement $(I)$, the spheres are packed in such a way that their centres are at the centres of an equilateral triangle. Each sphere is surrounded by six other similar spheres. This arrangement is called hexagonal close packing.
This arrangement can be extended in three dimensions by adjusting spheres on the top of hollows or voids of the two dimensional layers
In the base layer shown in fig. (a), the spheres are marked as $A$ and the two types of voids between the spheres are marked as $'a'$ and $'b'$. The efficient way of placing the spheres in second layer is to place them in the $'a'$ voids of the first layer, the $'b'$ voids remaining unoccupied.
There are two types of voids in the second layer i.e. $'c'$ and $'d'$. The $'a'$ and $'b'$ voids in the first layer are triangular while only $'c’$ voids of the second layer are triangular. The $'d'$ voids are combination of two triangular voids (one each of first and second layer) with the vertex of one triangle upwards and the vertex of other triangle downwards.
The void surrounded by four spheres and placed at an angle of $109^° 28'$ is known as tetrahedral voids.
Now, there are two ways to build up the third layer:
$1.$ When the third layer is placed over the second layer so as to cover the tetrahedral or $'c'$ voids, a three-dimensional closest packing is obtained where the spheres in every third layer are vertically aligned to the first layer. This arrangement is called $ABAB...$ pattern or hexagonal (HCP) close packing (calling first layer as $A$ and second layer $B$).
$2.$ When the third layer is placed over the second layer such that the spheres cover the octahedral or $‘d’$ voids, a layer $C$ different from $A$ and $B$ is formed. This pattern is called $ABCABC…$ pattern or cubic close packing (CCP).
In both HCP and CCP methods of packing, a sphere is in contact with six other spheres in its own layer. It touches three spheres in the layer above and three spheres in the layer below. Thus a sphere is in direct contact with $12$ other spheres. In other words, the coordination number of that sphere is $12$. Coordination number is the number of closest neighbours of any constituent particle.
The HCP and CCP arrangements can be also be shown as below
