14 Bravais Lattices and Their Derivation

14 Bravais Lattices and Their Derivation

Introduction

• The internal structure of crystals is based on the regular arrangement of atoms, ions, or molecules.
• To represent this arrangement, scientists use a three-dimensional pattern called a space lattice.
• In 1848, the French scientist Auguste Bravais showed that only fourteen distinct types of lattices are possible in three-dimensional space.
• These arrangements are known as the 14 Bravais Lattices.
• They provide the basic framework for all known crystal structures and are widely used in crystallography and mineralogy.

What are Bravais Lattices?

• A Bravais lattice is a regular and repeating arrangement of lattice points in three-dimensional space.
• Every lattice point has an identical environment around it.
• The repetition of these lattice points forms the complete crystal structure.
• Bravais lattices help explain how atoms are arranged inside crystals.

Derivation of Bravais Lattices

• Crystals are grouped into seven crystal systems based on the lengths of their axes and the angles between them.
• Different arrangements of lattice points within these crystal systems produce different lattice types.
• By combining crystal geometry with lattice-point arrangements, only 14 unique lattices are possible.
• These fourteen lattices represent all possible periodic arrangements of points in crystalline materials.

The Seven Crystal Systems

The fourteen Bravais lattices are derived from the following seven crystal systems:

1. Cubic System

• All three axes are equal in length.
• All angles are 90°.
• This system possesses high symmetry.

The cubic system contains:

• Simple Cubic (P)
• Body-Centred Cubic (I)
• Face-Centred Cubic (F)

Total lattices = 3

2. Tetragonal System

• Two horizontal axes are equal.
• The vertical axis is different.
• All angles are 90°.

The tetragonal system contains:

• Simple Tetragonal (P)
• Body-Centred Tetragonal (I)

Total lattices = 2

3. Orthorhombic System

• All three axes are unequal.
• All angles are 90°.

The orthorhombic system contains:

• Simple Orthorhombic (P)
• Base-Centred Orthorhombic (C)
• Body-Centred Orthorhombic (I)
• Face-Centred Orthorhombic (F)

Total lattices = 4

4. Monoclinic System

• All axes are unequal.
• Two angles are 90° and one angle differs from 90°.

The monoclinic system contains:

• Simple Monoclinic (P)
• Base-Centred Monoclinic (C)

Total lattices = 2

5. Triclinic System

• All three axes are unequal.
• All angles are unequal and none are 90°.

The triclinic system contains:

• Simple Triclinic (P)

Total lattices = 1

6. Hexagonal System

• Three horizontal axes are equal.
• One vertical axis is different.
• Specific angular relationships exist between the axes.

The hexagonal system contains:

• Simple Hexagonal (P)

Total lattices = 1

7. Rhombohedral (Trigonal) System

• All axes are equal.
• All angles are equal but not 90°.

The rhombohedral system contains:

• Rhombohedral (R)

Total lattices = 1

List of the 14 Bravais Lattices

Cubic System

• Simple Cubic (P)
• Body-Centred Cubic (I)
• Face-Centred Cubic (F)

Tetragonal System

• Simple Tetragonal (P)
• Body-Centred Tetragonal (I)

Orthorhombic System

• Simple Orthorhombic (P)
• Base-Centred Orthorhombic (C)
• Body-Centred Orthorhombic (I)
• Face-Centred Orthorhombic (F)

Monoclinic System

• Simple Monoclinic (P)
• Base-Centred Monoclinic (C)

Triclinic System

• Simple Triclinic (P)

Hexagonal System

• Simple Hexagonal (P)

Rhombohedral System

• Rhombohedral (R)

Types of Lattice Centering

The fourteen lattices are formed using different centering arrangements.

Primitive Lattice (P)

• Lattice points are present only at the corners of the unit cell.
• It is the simplest lattice arrangement.

Body-Centred Lattice (I)

• Lattice points occur at the corners and one point at the center of the unit cell.

Face-Centred Lattice (F)

• Lattice points occur at the corners and at the center of each face.

Base-Centred Lattice (C)

• Lattice points occur at the corners and at the centers of two opposite faces.

Rhombohedral Lattice (R)

• A special lattice arrangement found only in the trigonal crystal system.

Importance of Bravais Lattices

• Provide the foundation for crystal structure studies.
• Help classify all crystalline materials.
• Explain the arrangement of atoms inside minerals.
• Assist in X-ray crystallographic analysis.
• Support mineral identification and crystal classification.
• Play a major role in geology, mineralogy, chemistry, and materials science.
• Help scientists understand crystal symmetry and crystal properties.

Applications of Bravais Lattices

• Crystal structure determination.
• Mineralogical investigations.
• X-ray diffraction studies.
• Material science research.
• Semiconductor technology.
• Metallurgical studies.
• Nanotechnology and advanced material development.
• The 14 Bravais lattices represent all possible three-dimensional arrangements of lattice points in crystalline materials. Derived from the seven crystal systems, they provide a systematic method for understanding crystal structures and symmetry. Their study forms a fundamental part of crystallography and serves as the basis for modern mineralogical and materials science research.