Definition and Explanation of Hexagonal Close Packing
The three main states of matter are solids, liquids and gas. The matter is made up of all the elements present in the universe which includes atoms, molecules and ions. In the case of solid, these atoms are closely packed with each other which prevents them from moving within a substance due to which solids are rigid and have a fixed shape.
Further Solids are Divided into Two Categories-
Crystalline solids (also known as true solids)- In crystalline solids, atoms, molecules and ions are arranged in a symmetrical pattern.
Amorphous solids (also known as pseudo solids)- These have no characteristic symmetry.
Each unit cell combines and forms a network called a crystal lattice. There are fourteen types of crystal lattices which are further classified into seven crystal systems on the basis of arrangement of atoms-
Cubic
Hexagonal
Tetragonal
Rhombohedral
Orthorhombic
Monoclinic
Triclinic
The Two Types of Packing Present in Solids are-
Cubic close packing (ccp)- AAAA type arrangement
Hexagonal close packing (hcp)- ABAB type arrangement
Let us study the hexagonal close packing in detail.
Solid materials show specific crystal structures. They can be classified as crystalline and amorphous on the basis of arrangement of their constituent particles. A crystalline solid consists of a large number of small crystals, each of them having a definite characteristic geometrical shape. In these crystals, arrangement of constituent particles is in order and repetitive in three dimensions.
We can accurately predict the arrangement of particles of another region, if we know the arrangement of particles in one region of the crystal. Thus, a crystal has a long range order which means that there is a regular pattern of arrangement of particles which repeats itself periodically over the entire crystal.
A smallest group of constituent particles which has overall symmetry of the crystal and by repetition of it in three dimensions, the entire lattice can be built is called unit cell of that crystal.
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In the Three-Dimensional Crystal Structure, Unit Cell is Characterised By –
Its dimensions along the three edges a, b and c. These edges may or may not be mutually perpendicular.
Angles between the edges, α (between b and c), β (between a and c) and γ (between a and b). Thus, a unit cell is characterised by six parameters a, b, c, α , β and γ.
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Hexagonal close packing, or hcp in short, is one of the two lattice structures which are able to achieve the highest packing density of ~74%, the other being face centred cubic (fcc) structure. This packing structure is found in metals such as zinc, cadmium, cobalt and titanium.
When the layers of spheres are packed in such a way that the spheres in alternating layers overlie one another, then the packing is known as hexagonal close packing or hcp.
Hexagonal Close Packing is Very Common in the following Metals:
Beryllium
Cadmium
Magnesium
Titanium
Zinc
Zirconium
Structure
Hexagonal close packing structure consists of alternating layers of spheres or atoms arranged in a hexagon, with one additional atom at the centre. Another layer of atoms is sandwiched between these two hexagonal layers which is in a triangular form and the atoms of this layer fill the tetrahedral holes created by the top and bottom layers. Such a structure can be pictorially represented as below:
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As you can see in the image above, the lattice structure is essentially an alternative pattern of layers ‘a’ and ‘b’. The series for this lattice structure can also be written as …..-a-b-a-b-a-b-…..and so on.
There are three layers in the hexagonal packing structure. There are six atoms each arranged in a shape of a hexagon in the top and the bottom layer and a seventh atom in the middle of the hexagon is present. There are three layers in the middle layer between the atoms of the top and bottom layers.
In hexagonal close packing, there are two types of voids, octahedral and tetrahedral voids.
Hcp Structure Unit Cell
We have shown the hcp structure diagram in the previous section. In this section, we will discuss the hcp unit cell in a little more detail. Hcp unit cell can be represented as follows:
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In this unit cell, the atoms in the middle layer are not shared with any other unit cell, while the atoms in upper and bottom layers are shared with adjacent unit cells. The central sphere or atom in top and bottom layers is shared with 1 other unit cell, hence they only contribute ½ to one unit cell. The atoms at the hexagonal vertices in top and bottom layers are shared with 5 other unit cells, hence they contribute just 1/6 to one unit cell. Using this understanding, we can compute the number of atoms per unit cell in a hcp structure as below:
Number of atoms fully contributing to the unit cell = 3 (all in the middle layer)
Number of atoms contributing ½ to the unit cell = 2 (1 in top and 1 in bottom layer, respectively)
Number of atoms contributing 1/6 to the unit cell = 12 (6 in top and 6 in bottom layer, respectively)
Total number of atoms in the unit cell =\[ 3 + (\frac{1}{2} \times 2 )+ (\frac{1}{6} \times 12 ) = 3 + 1 + 2 = 6 \]
Hence, one unit cell of hcp structure comprises 6 atoms. The coordination number, i.e. number of atoms surrounding the central atom, for hcp structure is 12. The coordination number can also be visualised as below:
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The central atom in layer B is surrounded by 6 atoms from layer B itself, while it is also surrounded by 3 atoms each from top and bottom layers. Hence the total number of surrounding atoms is 6 + 3 + 3 = 12, which gives us a coordination number of 12 for hcp structure.
As we showed in Figure 4, a hexagonal close packed structure has 2 lattice parameters, namely ‘a’ and ‘c’. ‘a’ is also called a basal parameter and ‘c’ is also called the height parameter. We can compute the volume of a unit cell using these 2 parameters as follows:
Volume of Hcp Unit Cell = \[ \frac{3\sqrt{3}a^{2}c}{2}\]
The ratio between the space occupied by spheres and empty space in a hcp structure is approximately 74%:26%. The amount of space occupied is given by the formula
\[\frac{\pi}{3\sqrt{2}}\]
which approximately translates to 0.74048.
The shape of the unit cell of a hexagonal close packed lattice structure is a hexagonal prism. The angle between two equal axes or sides with lengths ‘a’ is 120o while the angle between height ‘c’ and side ‘a’ is 90o.
This ends our coverage on the topic “Hexagonal Close Packing”. We hope you enjoyed learning and were able to grasp the concepts. We hope after reading this article you will be able to solve problems based on the topic.
The unit cell describes the complete structure of a solid that can be regarded as an endless repetition of the unit cell.
The volume of a unit cell is calculated from its shape and dimensions.
The volume of a hexagonal unit cell is the product of the height of the cell and the area of the base. Since the atoms for a closed-packed structure at the corners of the base of the unit-cell are in contact.
a = b = 2r
In hexagonal close packing, there are two lattice parameters-
‘a’ which represents the basal parameter
‘c’ which represents the height parameter
Volume = \[ \frac{3\sqrt{3}a^{2}c}{2}\]
To know more about Hexagonal Close Packing, study the concept in detail. Learn more about this topic and answer questions related to it easily in an exam. Practice with the images given about the different crystal structures to understand the differences.
FAQs on Hexagonal Close Packing
1. What are the properties of hexagonal close packing crystal structure?
The properties of the hexagonal close packing crystal structure are-
The packing factor of an hcp crystal structure is 0.74.
The packing efficiency is 74% in the case of hcp, while 26% is empty space.
The coordination number of an hcp crystal structure is 12.
The hcp structure contains 6 atoms per unit cell.
The number of octahedral interstitial sites in the hexagonal close-packed structure is 6.
The number of tetrahedral interstitial sites in the hexagonal close-packed structure is 12.
The stacking sequence of a hexagonal packed structure is ABABAB.
2. What is the difference between hcp and ccp?
HCP | CCP | |
Full-Form | The term ‘hcp’ means hexagonal close-packed crystal structure. | The term ‘ccp’ means cubic close-packed crystal structure. |
Unit Cell | The unit cell of a hexagonal close-packed structure has 6 members. | The unit cell of a cubic close-packed structure also has 4 members. |
Structure | The spheres in the third layer are arranged as the spheres in the first layer. Therefore, the tetrahedral holes of the first and third layers are covered by the spheres of the middle layer. | The spheres of the first second layer are placed onto half of the depressions of the first layer. The third layer is completely different from the first and the second layer. The third layer is placed onto the depressions of the second layer |
Repeating Structure | 2 layers of spheres repeat in the hcp structure. | 3 layers of spheres repeat in the ccp structure. |
3. Mention some similarities between hexagonal close packing and cubic close packing.
Some similarities between hexagonal close packing and cubic close packing are-
There are 12 neighbouring spheres in both hexagonal close packing and cubic close packing.
In both hexagonal close packing and cubic close packing, 74% of the volume is occupied by the spheres of the atoms while 26% is occupied by empty spaces.
The coordination number of both the crystal structures- hcp and ccp is 12.
4. Calculate the packing efficiency of the hcp crystal structure.
Packing efficiency is given by-
\[\text{Packing efficiency} = \frac{\text{volume occupied by one atom}}{\text{total volume of unit cell}} \times 100\]
For hcp,
In a hexagonal close-packed structure
a = b = 2r
\[c = \frac{4\sqrt{2}}{3}\]
where r is the atomic radius
No of atoms per unit cell-
Contribution of corner spheres to unit cell= \[12 \times \frac{1}{6} = 2\]
Contribution of spheres inside the unit cell= 3 \[\times\] 1= 3
Contribution of face corner spheres to unit cell= \[2 \times \frac{1}{2} = 1\]
Contribution to all types of spheres to unit cell= 6
Volume of an hcp unit cell-
Volume= Base area \[\times\] height
Base Area = \[ 6 \times \frac{\sqrt{3}}{4} \times (2r)^{2} \]
Height of Unit cell = \[4r \times \sqrt{\frac{2}{3}} \]
Total Volume = \[6\sqrt{3}r^{2} \times 4r \sqrt{\frac{2}{3}} \]
= \[24 \times \sqrt{2} \times r^{3} \]
Therefore, Packing efficiency in hcp
\[\text{Packing efficiency} = \frac{\text{volume occupied by one atom}}{\text{total volume of unit cell}} \times 100\]
\[V = 6 \times \frac{4}{3} \pi \times r^{3} \]
\[= 8 \pi r^{3}\]
\[\text{Packing efficiency} = \frac{8 \pi r^{3}}{24 \times \sqrt{2} \times r^{3}} \times 100\]
\[= \frac{\pi}{3\sqrt{2}} \times 100\]
= 0.74 \[\times \] 100 = 74%
5. What are amorphous solids?
Solids having no definite form are known as amorphous solids. In amorphous solids, the constituent particles do not possess a regular 3-D arrangement. Amorphous solids are also known as supercool liquids since their molecules are arranged randomly. For example- gels, rubber, ceramics, etc. Amorphous solids are neither geometric nor crystalline If amorphous solids are heated or cooked, they become crystalline at some point.