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A unit cell can be defined as the smallest group of atoms, the repetition of which forms the entire crystal lattice. It is hence the building block of the crystal. It is a small portion of the crystal with its full overall symmetry. The crystal can be thought of as the repetition of unit cells in three-dimensional space. Each corner of a unit cell is a lattice point.
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Key takeaways: Solids, crystalline, density, mass, volume, atoms, symmetry, crystal, lattice point, crystal lattice, Unit cell
Understanding Lattice
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Lattice can be defined as the three-dimensional arrangement of points in space. This means that when the constituent particles of a crystal are arranged in a three-dimensional fashion with each particle being represented as a point, the arrangement formed is known as the crystal lattice.
Arrangement of particles
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|---|---|---|
| Atomic and Molecular masses | Avogadro’s number | Solid State |
| Covalent Bonds | Impurity Defects | Atomic Spectra |
Characteristics of a Unit Cell
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The smallest portion of a crystal lattice is known as the unit cell. These cells represent the pattern of the crystal which has three dimensions. Given below are the characteristics of unit cells
- The dimensions along three edges are termed as a, b, and c.
- The mentioned edges are not always necessarily perpendicular to each other.
- Angles between these edges are termed as:
- α: Angle between b and c
- β: Angle between a and c
- γ: Angle between b and c
Types of Unit Cells
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There are two categories under which unit cells can be classified which includes the following:
- Primitive Unit Cell: When the constituent particles are present only in the corners of the unit cell, it is known as the Primitive Unit Cell.
- Centered Unit Cell: In addition to corners, when constituent particles are present elsewhere in the unit cell, it is known as the Centered Unit Cell. It is further classified into three categories. They are:
- Body-Centered: There exists one constituent particle in the body center apart from the ones in the corner.
- Face-Centered: There exists one constituent particle at the center of each face apart from the ones in the corner.
- End-Centered: There exists one constituent particle at the center of any two opposite faces apart from the ones in the corner.
Simple Cubic Unit Cell
Density of a Unit Cell
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Consider a unit cell with an edge ‘a’. The number of atoms in the unit cell is denoted by ‘z’. Mass of an atom is denoted by ‘m’. Its molar mass is denoted by M. Avogadro’s number is denoted by NA.
Its volume will be a3.
The density of a unit cell can be calculated with the ratio of mass and volume.
Density of unit cell = Mass of unit cell / Volume of unit cell
Mass of unit cell = (Z*M) / Na
Volume of unit cell = a³
Density of unit cell = (Z*M) / (Na*a³)
Density of unit cell = [(Z*M)/Na]/a³
The mass of a unit cell will be equal to the number of atoms in that unit cell times the mass of each atom.
i.e., Mass of the unit cell = number of atoms in that unit cell x mass of each atom = z x m
Mass of an atom can be calculated by the formula
m = M / NA
Density = Mass / Volume = m / V =\(\frac{z \times m}{a^3}\)= \(\frac{z \times M}{a^3 \times NA}\)
- Density of a primitive cell:
z = 1
Density = \(\frac{1 \times M}{a^3 \times NA}\)
- Density of Body-centered cubic unit cell:
z = 2
Density =\(\frac{2 \times M}{a^3 \times NA}\)
- Density of Face-centered cubic unit cell:
z = 4
Density = \(\frac{4 \times M}{a^3 \times NA}\)
- Density of End-centered cubic unit cell:
z = 8
Density = \(\frac{8 \times M}{a^3 \times NA}\)
Density of unit cell = Mass of unit cell / Volume of unit cell
= (Z*M) / (Na*a³)
Mass of unit cell = (Z*M) / Na
Volume of unit cell = a³
Density of unit cell = [(Z*M)/Na]/a³
Things to Remember
- The density of the unit cell is equal to the substance density.
- Irregularities in the normal arrangement of atoms are called Crystal Defects. They can be point defects or line defects.
- Crystalline solids exhibit properties in accordance with the nature of interactions between their constituent particles.
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Sample Questions
Ques. Define the terms lattice and lattice point. (2 marks)
Ans. Lattice can be defined as the three-dimensional arrangement of unit cells of a crystal. In other words, lattice is basically a series of unit cells brought together in a distinct pattern.
Every point that exists in the lattice of a crystal is called a lattice point. These points when joined together with the use of straight lines bring out the geometry of the lattice.
Ques. Differentiate between crystalline and amorphous solids. (4 marks)
Ans. The differences are:
| Crystalline Solids | Amorphous Solids |
|---|---|
| Possess definite geometrical shape | Possess irregular shape |
| The melting point is sharp and at a given temperature | Melts gradually over a range of temperature |
| They are anisotropic in nature | They are isotropic in nature |
| True solids | Pseudo solids |
Ques. How can you justify that a primitive unit cell has only one atom? (2 marks)
Ans. In a primitive unit cell, atoms are present only at the corners. Each of these atoms is shared by eight adjacent unit cells. Hence, only 1 / 8th of an atom belongs to a particular unit cell.
We know that each cubic unit cell possesses eight corners.
Thus, the total number of atoms in one unit cell will be
8 x 1 / 8
= 1 atom.
Ques. What are crystal systems? (4 marks)
Ans. Crystal system is a method of classifying crystals on the basis of their edge lengths and axial angles. They are of seven types, namely:
- Cubic:
a = b = c
α = β = γ = 90°
- Tetragonal:
a = b ≠ c
α = β = γ = 90°
- Orthorhombic:
a ≠ b ≠ c
α = β = γ = 90°
- Hexagonal:
a = b ≠ c
α = β = 90°, γ = 120°
- Rhombohedral:
a = b = c
α = β = γ ≠ 90°
- Monoclinic:
a ≠ b ≠ c
α = γ = 90°, β ≠ 90°
- Triclinic:
a ≠ b ≠ c
α ≠ β ≠ γ ≠ 90°
Ques. Silver crystal exists in face-centered cubic structure. Its edge length is 408.7 pm. Given that the atomic mass of silver is 108 g/mol, find its density. (3 marks)
Ans. Given,
a = 408.7 pm
M = 108 g/mol
NA = 6.023 x 1023 mol-1
Type of structure = face centered cubic
Density = (z x M) / (NA x a3)
= (4 x 108) / (6.023 x 1023 x 408.7 x 408.7 x 408.7 x 10-30 cm3)
Density of silver = 10.51 g/cm3.
Ques. Copper crystal exists in face-centered cubic unit cell. Given that its edge length is 360.8 pm and density is 8.92 c/cm3, find out its atomic mass. (3 marks)
Ans. Given,
a = 360.8 pm
density = 8.92 g/cm3
NA = 6.023 x 1023 mol-1
Type of crystal structure = face centered cubic
In fcc, z = 4
Density = (z x M) / (NA x a3)
M = (Density x NA x a3) / z
= (8.92 x 6.023 x 1023 x (3.608 x 10-8)3) / 4
M = 63.07 g/mol
Ques. A crystal of silver exists in the fcc structure. Given that its density is 10.51 g/cm3 and molar mass is 106 g/mol, calculate the volume of its unit cell. (4 marks)
Ans. Given,
M = 108 g/mol
Density= 10.51 g
NA = 6.023 x 1023 mol-1
Type of structure = face centered cubic
In fcc, z = 4
Density = (z x M) / (NA x a3)
a3 = (z x M) / (N X Density)
= (4 x 108) / (10.51 x 6.023 x 1023)
Volume = 6.825 x 10-23 cm3
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