Close Packing In Solids In Three Dimensions - Practice Questions & MCQ

Constituent particles are closest packed in space. Closer the packing, more will be the stability of the crystal. Let we consider all the constituents are identical as solid spheres. In two-dimensional arrangement, spheres are arranged in rows to form a layer and packing in different layers form three-dimensional arrangement which is known as crystal or space lattice. We shall discuss the mode of packing in detail.

Close Packing in One Dimension
Arrangement of different atoms in a row touching each other forms one dimension or edge.

Close Packing in Two Dimensions
The rows of particles can be stacked in two ways 

• Square Close Packing: Spheres are packed in such a way that they align together vertically as well as horizontally and center of all spheres are in a straight line. Here each sphere is in contact with four other spheres in the same plane. This is termed as square close packing.
  
  

• Hexagonal Close Packing: When the second row is arranged in the depression of the first row and all atoms align diagonally to each other. Each atom is in contact with 6 other spheres in the same plane. It is known as hexagonal close packing. It is more efficient mode of packing than the square close packing in a layer in two dimensions. Here coordinate number is 6. In this layer triangular voids are formed.
  
  

Close Packing in Three Dimensions: When layers are arranged over each other they form three- dimensional packing.
(a) Three-dimensional close packing from two-dimensional square close-packed layers: It is layer packing in which the second layer is placed over the first layer in such a way that all the spheres are exactly above each other and all the spheres align horizontally as well as vertically. This arrangement forms AAA…. type of lattice. It forms a simple cubic lattice and its unit cell is primitive cubic unit cell.


(b) Three-dimensional close packings from two-dimensional hexagonal close-packed layers: When layers containing hexagonal close packing are arranged over each other, two types of arrangements are feasible.

1. Placing second layer over the first layer Start with a two-dimensional hexagonal close-packed layer 'A' and arrange another similar layer B on it in such a way that spheres of 2nd layer are placed in the depressions of first layer. In this case, two types of voids are formed.
   
   

• When a sphere of second layer is placed above the void of layer, a tetrahedral void (T.V.) is formed. Because on joining the centres of these four spheres a tetrahedron is formed. Its coordination number is 4. 
• When a triangular void formed in first layer is not covered even in the second layer then triangular shapes of these voids do not overlap, these are called octahedral voids (O.V.) which are surrounded by 6 spheres. Its coordination number (C.N) is 6. 

          Number of voids formed depends on the number of close-packed spheres.
          Let the number of close-packed spheres = N
          The numbers of octahedral voids formed = N
          The number of tetrahedral voids formed = 2N

1. Placing third layer over the second layer
   There are two types of voids which are to be covered in the third layer. These are the octahedral voids (a) which remain unoccupied for two consecutive layers and tetrahedral voids (c) formed in the second layer. 
   If third layer is formed in such a way that tetrahedral voids (c) are covered. In this way, the spheres of the third layer lie directly above those in first layer. It means the third layer becomes exactly identical to the first layer. This type of packing is ABABAB….. arrangement and it is known as hexagonal close packing (hcp).
   e.g., Mg, Zn, Cd, Be etc.
   
   
   
   If third layer is formed in such a way that spheres of third layer must cover octahedral voids (a). It forms a new third layer C. It forms ABCABC ... type arrangement called cubic closed packing (ccp). In ccp each unit cell is face-centred type. e.g., Ag, Cu, Fe, Ni, Pt etc.
   
   
   In both the arrangements i.e., hcp and ccp, each lattice point is in contact with 12 more nearest spheres which is called their coordination number (C.N.)
   
   

Voids and Their Locations in the Unit Cell :

Voids
It is the space left after different types of packings like hcp, ccp due to the spherical nature of atoms that is, the three-dimensional interstitial gaps are called voids. These are of the following types: 

1. Trigonal Voids: It is the vacant space touching three spheres that is, it is a two-dimensional void formed when three spheres are in the same plane whose centres are at corners of the triangle. 

2. Tetrahedral Voids: It is the vacant space touching four spheres that is the void whose surrounding spheres are located at the corners of a regular tetrahedron. In general, the number of these voids In a unit cell is double the number of effective atoms in that unit cell.

3. Octahedral Voids: It is the vacant space touching six spheres that is void resulting from overlapping of two trigonal voids of adjacent layer whose surrounding spheres are located at the corners of the octahedron. In general, the effective number of these voids in a unit cell is equal to the number of effective atoms present in the unit cell.

4. Cubic voids: It is the vacant space touching eight spheres located at the corners of a cube.
   
   

Location Of Voids in Unit Cell : 

1. Tetrahedral Voids: These voids are located at the body diagonals, two in each body diagonal at one-fourth of the distance from each end. Total number of these voids per unit cell = 8 

2. Octahedral Voids: These voids are located at the middle of the cell edges and at the centre of a cubic unit cell.

Total number of octahedral voids = 1/4 x 12 + 1 = 4
So in ccp, the total number of voids per unit cell = 8 + 4 = 12

Size of voids
V_oct = 0.414 X r
V_tetra = 0.214 X r
V_tn = 0.115 X r

V_oct > V_tetra > V_tn
Here r is the radius of the biggest sphere