Close Packing In Solids In Three Dimensions

For three-dimensional close packing, emphasis is placed on the closest packing possible with atoms. The two dominant three-dimensional close packings are hexagonal close packing (HCP) and cubic close packing (CCP), which is the same as face-centered cubic (FCC). The two forms of structures are interrelated with each other and are made from definite layers of atoms by repeating a pattern.

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This Story also Contains

1. Close Packing in One Dimension
2. Close Packing in Two Dimensions
3. Close Packing in Three Dimensions:
4. Voids and Their Locations in the Unit Cell :

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 the 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 the 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 a 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 the 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 the first layer. In this case, two types of voids are formed.
   
   

• When a sphere of the second layer is placed above the void of the layer, a tetrahedral void (T.V.) is formed. Because on joining the centers of these four spheres a tetrahedron is formed. Its coordination number is 4.
• When a triangular void formed in the first layer is not covered even in the second layer the 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 number of octahedral voids formed = N
The number of tetrahedral voids formed = 2N

1. Placing the third layer over the second layer
   There are two types of voids that 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 the 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:

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

Recommended topic video on(close packed structure)

Some Solved Examples

Example 1
Question: In two-dimensional close-packing structures, the possible coordination numbers of a molecule is:
1) 3, 5
2) (correct) 4, 6
3) 5, 7
4) 6, 3

Solution: In two-dimensional close-packing structures, a molecule is surrounded by 4 and 6 of its neighbor atoms in square close packing and hexagonal close packing, respectively. Hence, the answer is option (2).

Example 2
Question: A hexagonal close-packed structure and a cubic close-packed structure for a given element would be expected to have the same density as:
1) Both have a packing efficiency of 74%
2) Both have a coordination number of 12
3) Both have different packing efficiency and different coordination number
4) (correct) Both (1) and (2) are correct

Solution: Both hexagonal close-packed and cubic close-packed structures have a packing efficiency of 74% and both have the same coordination number of 12. Hence, the answer is option (4).

Example 3
Question: The number of atoms present in the Unit cell of the CCP lattice is:
1) 2
2) (correct) 4
3) 6
4) 8

Solution: The number of atoms present in the unit cell of the cubic close packing (CCP) lattice, also visualized as the Face Centered Cubic (FCC) lattice, is 4. Hence, the answer is option (2).

Example 4
Question: What is the Rank of an HCP unit cell?
1) 2
2) 4
3) (correct) 6
4) 8

Solution: The effective number of atoms in an HCP unit cell, calculated from the contributions of the atoms shared between unit cells, results in a rank of 6. Hence, the correct answer is option (3).

Example 5
Question: A compound forms hexagonal close-packed structure. What is the total number of voids in 0.5 mol of it? How many of these are tetrahedral voids?

1) (correct)$9.03 \times 10^{23}, 6.022 \times 10^{23}$

2)$8.26 \times 10^{23}, 6.022 \times 10^{23}$

3)$9.03 \times 10^{23}, 6.022 \times 10^{22}$

4)$9.03 \times 10^{23}, 5.34 \times 10^{23}$

Solution: The total number of voids in 0.5 mol of a compound forming a hexagonal close-packed structure is $9.03 \times 10^{23}, 6.022 \times 10^{23}$ calculated to be being tetrahedral voids. Hence, the answer is option (1).

Summary

Close-packing, in HCP, it is structured according to the ABAB pattern, while in CCP, it is according to the ABCABC pattern. They each show the highest possible packing density in which all of the atoms have 12 nearest neighbors around it, and they occupy around 74% of the available space. This dense packing gives the materials enhanced stability and strength, thus properties like hardness, ductility, and melting points are influenced as well.