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Keywords: Unit cell, Types of Unit Cell, FCC unit cell, BCC unit cell, Primitive cubic unit cell, body-centered cubic unit cell, face-centered cubic unit cell
Types of Unit Cell
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Two or more unit cells together constitute a solid crystal lattice. Based on the position of atoms in a three-dimensional structure, unit cells are categorized into three types:
- Primitive Cubic Unit Cell
When lattice points are only present at the corners, it is called a primitive cubic unit cell. This is more often used as a simple cubic unit cell.
- Face-centered Cubic Unit Cell
In the case of the face-centered cubic unit cell or FCC, lattice points are present at two locations- the center of each face, and the corner of a cube.
- Body-centered Cubic Unit Cell
The presence of lattice points in the case of a body-centered cubic unit cell or BCC is also at two locations- the center of each body and the corner of a cube.
Calculation of Number of Particles Per Unit Cell
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The number of particles per unit cell is calculated based on each atom's contribution at a lattice point.
Calculation for Primitive Cubic Unit Cell:
For primitive cubic unit cells, atoms are only positioned at the corners and each atom is joined by eight unit cells.
- Unit cells are arranged in a way that four unit cells are located in the same layer and the four are positioned below the upper layer.
- Therefore, the contribution of each unit cell, in this case, is 1/8th of an atom.
Calculation for Face-centered Cubic Unit Cell:
Due to the presence of lattice points at two locations (face and corners), the contribution of each unit cell, in this case, is ½ of an atom.
Calculation for Body-centered Cubic Unit Cell:
In the body-centered cubic unit cell, the contribution of each unit cell in a body-centered cubic unit cell is 1 due to the presence of an atom at the center of the body.
Lattice Systems
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Different types of crystal structures form lattice systems. Few parameters are responsible for the formation of crystal structure.
Parameters of Lattice Systems
In the lattice systems, a unit cell's size and shape are determined by lattice system parameters.
- Length of edges (a,b,c)
- Angles of edges (∝,β,γ)
There are mainly seven main types of crystal structures which are as follows:-
- Cubic
- Tetragonal
- Orthorhombic
- Monoclinic
- Triclinic
- Rhombohedral
- Hexagonal
The table below shows the different parameters in the seven crystal structures along with their examples:
| Crystal Class | Angles | Distance | Examples |
| Cubic | ∝=β=γ=90o | a=b=c | Diamond, Copper |
| Tetragonal | ∝=β=γ=90o | a=b≠c | SnO2, TiO2 |
| Orthorhombic | ∝=β=γ=90o | a≠b≠c | Rhombic sulfur, KNO3 |
| Hexagonal | ∝=β=90o, γ=120o | a=b≠c | Graphite, Magnesium |
| Rhombohedral / Trigonal | ∝=β=γ≠90o | a=b=c | Calcite, Cinnabar |
| Monoclinic | ∝=β=90o, γ≠90o | a≠b≠c | Monoclinic sulfur, Na2SO4.10H2O |
| Triclinic | ∝≠β≠γ≠90o | a≠b≠c | Potassium dichromate, CuSO4.5H2O |
Sample Questions
Ques. What do you understand about packing efficiency? (5 marks)
Ans. The percentage volume of unit cells preoccupied by atoms, ions, or molecules is called its packaging efficiency. The value of packaging efficiency varies for primitive, FCC, and BCC unit cells.
- The formula of packing efficiency in a unit cell is given by:
Packaging Efficiency = Volume Occupied by Constituent Particles / Total Volume
From the above formula, it can be concluded that,
- For a simple cubic unit cell, packaging efficiency is 52.36%.
- In the case of a body-centered cubic unit cell, packaging efficiency is 68%.
- The packaging efficiency of the lattice in a face-centered cubic unit cell is 74%.
Ques. What are the defining characteristics of a unit cell? (3 marks)
Ans. Three types of unit cells exist which have common characteristics such as:
- Seven crystal structures are used for the study of lattice systems.
- Angle and length of edges are the two important parameters defining a unit cell's size and shape.
- For cubic crystal structure, all the angles are mutually perpendicular and equal to 90o.
Ques. What are the major differences between crystal lattice and unit cells? (3 marks)
Ans. Following are some major differences between crystal lattice and unit cells-
| Crystal Lattice | Unit Cells |
|---|---|
| A crystal lattice is defined by its unit cells. | A crystalline solid is defined by the properties of unit cells |
| Numerous cells together form a crystal lattice. | Unit cells are the simplest unit and can not be further split. |
| For analysis and studies, crystal lattice can be easily isolated. | Isolation of unit cells is not possible. |
Ques. How packaging efficiency of a FCC unit cell is calculated? (5 marks)
Ans. A face-centered cubic unit cell has a crystal lattice at each face of the center and corners of a cube.
- AbC is a right-angled triangle, so here Pythagoras theorem can be applied.
- According to the question, the unit cell is face-centered. Therefore,
The volume occupied by particles = πa3 / 3√2
- The formula for calculating packaging efficiency is:
Packaging Efficiency = Volume Occupied by Constituent Particles / Total Volume
Packaging efficiency = πa3 / 3√2 X 100
= 74%
Ques: What is the reason behind the formation of crystal lattice? (2 marks)
Ans: Ions are drawn together by electrostatic attraction to produce a minimum gap between them. The crystal lattice is created as a result. In a crystal lattice, the arrangement of ions is orderly and geometrical.
Ques. If lithium metal crystalizes in the BCC unit cell and the length of the side of the unit cell is 250pico-meter, then find the atomic radius of lithium. (3 marks)
Ans. In the question, length is given. I..e., 250pm
In the BCC unit cell, the derived formula for calculating atomic radius is given as:
r = √3/4 a
Where r is the radius of an atom,
a is the length of the side of an atom.
Now, according to the question, the atomic radius of the lithium atom = √3/4 a
=> 1.73 X 250 (pm) / 4 = 108.125
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