Close Packing in One, Two and Three Dimensions

Solids exhibit a regular as well as repeating pattern of constituent particles. These constituent particles are rigid incompressible solids of equal shape and size. The goal is to be packed as closely as possible in order to achieve maximum stability. Another reason for close packing is to reduce the gaps (voids) among particles. This close packing is achieved in three different types:

• Close Packing in One-Dimension 
• Close Packing in Two-Dimensions
• Close Packing in Three-Dimensions

Key Terms: Close Packing, Crystal, Voids, Lattice, Sphere, Hexagonal Close Packing, Square Close Packing, Polonium

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Close Packing in One-Dimension

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Close Packing in One-Dimension is done by aligning the constituent particles in a single row. The particles will be connected with each other.

Coordination Number refers to the number of nearest neighbors of a particle. Since each sphere is surrounded by two neighboring spheres, the coordination number of one-dimensional close packing is 2.

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Close Packing in Two-Dimensions

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Close Packing in Two-Dimensions can be achieved in the following ways:

• Square Close Packing: In Square Close Packing, multiple rows of sphere particles are stacked one above the other. All the rows are in contact with each other. By assuming the first row to be ‘A’, the second row will also be ‘A’ as all the rows are identical. Similarly, by adding more rows, we get an AAA-type arrangement. Since each sphere is surrounded by four neighboring spheres, the coordination number of square close packing is 4.
• Hexagonal Close Packing: In Hexagonal Close Packing, one row is stacked above the other in such a way that the second row aligns itself in the depressions of the first row. By assuming the first row to be ‘A’, the second row will be ‘B’ as it is aligned differently. Similarly, the third row will align itself in the depression of the second row and hence, directly in line with the spheres in the first row. Therefore, it will also be of ‘A’ type. By continuing in this manner, we will get an ABAB-type arrangement.

Hexagonal Close Packing is more stable than Square Close Packing as it has less free space. Since each sphere is surrounded by six neighboring spheres, the coordination number of hexagonal close packing is 6.

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Close Packing in Three-Dimensions

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Close Packing in Three-Dimensions can be achieved in the following ways:

• From Two-Dimensional Square Close Packing: In this type of close packing, one layer of square close packing is aligned exactly above the other. More and more layers can be aligned on top of one another. All the spheres are connected with each other.
  By assuming the arrangement in the first layer to be ‘A’, the second layer will also be ‘A’ as all the layers are identical. Similarly, by adding more layers, we get an AAA-type arrangement. This results in a simple cubic lattice.
  Since each sphere is surrounded by 6 spheres in neighboring, the coordination number will be 6. The percentage of occupied and empty space in this packing is 52% and 48% respectively. Only Polonium shows this type of close packing among all the elements in the periodic table.
• From Two-Dimensional Hexagonal Close Packing: In this type of close packing, one layer of hexagonal close packing is aligned over the other.
  In hexagonal close packing, two types of triangular voids are formed: One with the apex pointing upwards and one with the apex pointing downwards. Let these voids be ‘a’ and ‘b’. When the second layer of hexagonal close packing is placed over the first, both voids cannot be filled.
  Now, assume that the second layer is placed in the depressions of the first layer. Let the first layer be ‘A’ and the second is ‘B’. This results in the formation of two different voids: Tetrahedral and Octahedral Voids. The third layer can be aligned according to one of these two voids.
• Covering Tetrahedral Voids - When the sphere of the second layer is aligned above the void of the first layer, a tetrahedral Void is formed. To cover these voids, the spheres in the third layer are in line with those of the first layer. Here, an ABAB-type arrangement is formed. This arrangement is known as the HCP structure (Hexagonal Close Packed). Metals like Zinc and Magnesium exhibit this kind of close packing.
• Covering Octahedral Voids: Octahedral Void is formed when the triangular void in the first layer is above the triangular void in the second layer but their apexes point in different directions. The spheres align neither with the first layer nor with the second layer to cover these voids. Here, an ABCABC-type arrangement is formed. This arrangement is called Cubic Close Packed (CCP) or Face-Centred Close (FCP) structure. Metals like Copper and Silver exhibit this kind of close packing.

Three-Dimensional close packing is highly efficient, covering 74% of the space. The coordination number of this type of close packing is 12.

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Things to Remember

• Close packing in solids is done to attain maximum stability. The packing in which the percentage of filled space is high and the percentage of the empty space is low is favorable. 
• Close Packing in One-Dimension is done by aligning the constituent particles in a single row. The particles will be in contact with each other.
• Since each sphere is surrounded by two neighboring spheres, the coordination number of one-dimensional close packing is 2.
• In Square Close Packing, multiple rows of sphere particles are stacked one above the other. Here, an AAA-type arrangement is formed. 
• Since each sphere is surrounded by four neighboring spheres, the coordination number of square close packing is 4. 
• In Hexagonal Close Packing, one row is stacked above the other in such a way that the second row aligns itself in the depressions of the first row. Here, an ABAB-type arrangement is formed. 
• Since each sphere is surrounded by six neighboring spheres, the coordination number of hexagonal close packing is 6.
• In Three-Dimensional Close Packing from Two-Dimensional Square Close Packing, one layer of square close packing is aligned exactly above the other. Here, an AAA-type arrangement is formed. This results in a simple cubic lattice.
• The coordination number in the above type is 6. The percentage of occupied space in this packing is 52%. Only Polonium shows this type of close packing among all the elements in the periodic table.
• Tetrahedral Void is formed when the sphere of the second layer is aligned above the void of the first layer.
• To cover the Tetrahedral voids, the spheres in the third layer are in line with those of the first layer. Here, an ABAB-type arrangement is formed. This arrangement is called Hexagonal Close Packed (HCP) structure. Metals like Zinc and Magnesium exhibit this kind of close packing.
• Octahedral Void is formed when the triangular void in the first layer is above the triangular void in the second layer but their apexes point in different directions.
• To cover Octahedral voids, the spheres align neither with the first layer nor with the second layer. Here, an ABCABC-type arrangement is formed. 
• This arrangement is called Cubic Close Packed (CCP) or Face-Centred Close (FCP) structure. Metals like Copper and Silver exhibit this kind of close packing.

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