Crystal Lattice

The constituent particles in crystalline solids are arranged in a regular and definite geometry in all three dimensions. This three dimensional arrangement of constituent particles is called crystal lattice. in other words ” In the diagram of the crystal lattice, the constituent particles (atomic molecules or ions) are represented by dots. By joining these points by straight lines, a diagram of the crystal lattice is formed. These points are called lattice points

• Each constituent particle is represented by a point, which is called a lattice point or lattice site.
• Depending on the composition of crystalline solids, the constituent particles may be atoms, ions or molecules.
• The lattice points are joined by straight lines to express the geometry of the lattice.
• The plane formed in the crystal lattice is called the crystal plane.

Unit Cell

The smallest part of the crystal lattice which is the image of the whole crystal is called a unit cell. i.e. the smallest unit in the entire crystal structure that is equally distributed in all directions.

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Parameters Of A Unit Cell

Some parameters are required to represent the unit cell. These parameters are the characteristic properties of that unit cell. These parameters are the length (axial distance) of the three edges of the unit cell and

1. The axial lengths of the three sides of a unit cell are represented by a, b and c respectively, which may or may not be perpendicular to each other.
2. The angles between the edges are represented by α, β and γ. α is the angle between the sides b and c, β is the angle between the sides a and c and γ is the angle between the sides a and b. These are called axial angles.

Hence, the unit cell is represented by 6 parameters, a, b, c and α, β, γ.

Primitive and Centred Unit Cells

Broadly speaking, a cell is divided into two parts.

(i) Primitive Unit Cell (ii) Centred Unit Cell

Primitive Unit Cell

In this type of unit cell, the constituent particles (lattice points) are present only at the corners of the cell. The primitive unit cell is also called simple unit cell.

Centred Unit Cell

If the constituent particles in a unit cell are present at positions other than the corners, then this unit cell is called Centred unit cell. Types of Centred Unit Cells Centred Unit Cells are of three types.

Body Centred Unit Cell – If the constituent particle (lattice point) is present in the center of the unit cell in addition to the corners of the unit cell, then it is called in-Centred unit cell.

Face Centred Unit Cell – In this type of unit cell, the constituent particles are located at the center of each face apart from the corners.

End Centred Unit Cell – In this type of unit cell, the constituent particles are located at the center of any two opposite faces apart from the corners.

Seven Crystal Systems

Seven types of primitive unit cells are found in crystals. These unit cells are differentiated by their parameters a, b, c, α, β and γ.

1. Cubic Primitive Unit Cell

The three axial distances are equal, and are perpendicular to each other, that is, each axial angle is 90°.

(a = b = c & α = β = γ = 90°)

2. Tetragonal Primitive Unit Cell

All three axes are perpendicular to each other, but two axes are equal and one is unequal.

(a = b ≠ c and α = β = γ = 90°)

3. Orthorhombic Primitive Unit Cell

The three axial distances are unequal but perpendicular to each other.

(a ≠ b ≠ c & α = β = γ = 90°)

4. Monoclinic Primitive Unit Cell

The three axial distances are unequal. Two axial angles differ by 90° and one axial angle by 90°.

(a ≠ b ≠ c & α = β = 90°, γ ≠ 90°)

5. Hexagonal Primitive Unit Cell

Two axes are equal but third axes are unequal. Similarly, two axial angles are of 90° and one axial angle is of 120°.

(a = b ≠ c & α = β = 90°, γ = 120°)

6. Rhombohedral Primitive Unit Cell

All three axes are not equal to 90°. are of length, the axial angles between them are also equal, but are not 90°.

(a = b = c & α = β = γ ≠ 90°)

7. Triclinic Primitive Unit Cell

In this structure, the three axes are of unequal length. All three angles are unequal and none of them is of 90°.

(a ≠ b ≠ c & α ≠ β ≠ γ ≠ 90°)

Table of Seven Crystal Systems

Crystal Group	Axial Distances	Axial Angle	Example
Cubic Primitive Unit Cell	a = b = c	α = β = γ = 90°	NaCl, KCl, Diamond, Cu, Ag, zinc blend (ZnS)
Tetragonal Primitive Unit Cell	a = b ≠ c	α = β = γ = 90°	white tin, SnO2, TiO2, CaSO4
Orthorhombic Primitive Unit Cell	a ≠ b ≠ c	α = β = γ = 90°	rhombus gadfly, KNO3, K2SO, BaSO4, PbCO3
Monoclinic Primitive Unit Cell	a ≠ b ≠ c	α = β = 90°, γ ≠ 90°	monotonous sulfur, Na2SO4, 10H2O, PbCrO4
Hexagonal Primitive Unit Cell	a = b ≠ c	α = β = 90°, γ = 120°	graphite, ZnO, Cds, PbI2
Rhombohedral Primitive Unit Cell	a = b = c	α = β = γ ≠ 90°	calcite (CaCO3), Quartz, NaNO3, cinnabar (HgS), Sb
Triclinic Primitive Unit Cell	a ≠ b ≠ c	α ≠ β ≠ γ ≠ 90°	CuSO4, 5H2O, K2Cr2O7, H3BO3

Bravais Lattice

The French mathematician Breve reported in 1848 that the seven crystal communities could be divided into 14 different types of stereoscopic lattices based on the arrangement of the constituent particles in their unit cells. These space lattice are called Brave lattices. Brave traps are arranged in the following table. is shown in the table. which unit cells in a crystal community.

Crystal Group	Types of Stereoscopic Lattices (Unit Cells)	Number Of Stereoscopic Lattices
Cubic Primitive Unit Cell	Primitive, Body Centred, Face Centered	3
Tetragonal Primitive Unit Cell	Primitive, Body Centred,	2
Orthorhombic Primitive Unit Cell	Primitive, body centered, End centered, Face centered	4
Monoclinic Primitive Unit Cell	Primitive, End Centred	2
Hexagonal Primitive Unit Cell	Primitive	1
Rhombohedral Primitive Unit Cell	Primitive	1
Triclinic Primitive Unit Cell	Primitive	1
		TOTAL 14

Number Of Constituent Particles In A Unit Cell

In a long crystalline structure, the constituent particles of each unit cell are also related to other unit cells present around it. That is, a constituent particle is divided into more than one unit cell. Now we will see which part of each constituent particle (atomic ion or molecule) belongs to a particular unit cell. For the sake of simplicity, we will take cubic unit cells as an example and consider the constituent particle as an atom. The constituent particles are shown in the figure as spheres of equal size. The number of constituent particles in the unit cell is calculated as follows.

Each constituent particle (sphere) present at the corners of the unit cell is shared by eight unit cells. Therefore, in each unit cell, 1/8th part of it is related. The constituent particle present at the center of the face of the unit cell is divided by two unit cells. So in each unit cell only 1/2 th part of it is related.

The constituent particle present at the center of the unit cell does not share with any other unit cell. Therefore, only its whole part contributes to the unit cell. The constituent particle present at the edge of the unit cell is divided by four unit cells. Therefore, each unit cell contributes only 1/4th of this particle. Hence, the contribution of the constituent particles of the unit cell is as follows.

corner particle = 1/8
panel component = 1/2
center constituent particle = 1
edge particle = 1/4

Each unit cell is represented in three ways.

Open structures

The small sphere present at a position represents the center of the atom and not the actual size of the atom, these structures are called open or open structures. It is easy to understand the arrangement of particles from such structures.

Space Filling Representation

In this representation the actual size of the atoms is shown. It shows the actual structure of the unit cell. This structure gives an idea of ​​how the particles are packed in the unit cell.

Showing Actual Portion of atoms

The part which each particle contributes in a unit cell is shown. The count of particles in all three types of unit cells is as follows.

Simple Cubic Unit Cell

In the primitive cubic unit cell, the atoms are present at the corners of the positive.

Hence the total number of atoms in the unit cell = 1/8 × 8 = 1

Body Centred Cubic Unit Cell (bbc)

In the unit cell, 8 atoms are present at the corners of the cube and one atom is present at the center of the cube.

Hence, the total number of atoms in the internally Centred cubic unit cell is –

8 corner atoms × 1/8 contribution of each atom = 1
Contribution of center atom = 1
Total number of atoms in the unit cell = 1 + 1 = 2

Face Centred Cubic (fcc) Unit Cell)

In fcc unit cell 8 atoms are present at the corners of the cube and 6 atoms are present at the center of each face.

Hence the total number of atoms in the fcc unit cell –

8 corner atoms × 1/8 contribution of each atom = 1
6 Atoms at the Center of the Face x 1/2 contribution of each atom = 3

Total number of atoms in the unit cell = 4