In this lecture, we are going to learn about crystal systems and Bravais Lattices. What are the types of Crystal systems? So Let’s start by knowing what crystal systems are.

**Crystal Systems**

Crystals are classified into 7 Crystal systems on the basis of:

- axial lengths a, b, c
- Interfacial (axial) angles α, β, γ, and
- directions of the axis of symmetry

The 7 Crystal systems are:

1. Cubic | 2. Tetragonal | 3. Orthorhombic | 4. Monoclinic |

5. Triclinic | 6. Trigonal or Rhombohedral | 7. Hexagonal |

**Bravais Lattice**

- Bravais in 1880 showed that there are 14 possible types of space lattices in the seven crystal systems of crystal.

- According to Bravias, there are only 14 possible ways of arranging points in space lattice such that all the lattice points have exactly the same surroundings. That is these 14 lattices are called the Bravias Lattice.

- The possible types of Bravais lattice of the seven crystal systems are explained in brief in this section.

*1. Cubic Crystal System*

*1. Cubic Crystal System*

In a cubic crystal system, the three crystal axes are perpendicular to each other and the axial lengths ( repetitive unit) is the same along all the there axis as shown in the figure below.

a \;= b\; =\; c

\alpha = \beta = \gamma = 90^{\circ}

**Bravais Lattices:**

In cubic crystal systems, the cubic lattices are of three types. They are:

- Simple Cubic (SC)
- Body-Centered Cubic (BCC)
- Face Centered Cubic (FCC).

*2. Tetragonal Crystal System*

*2. Tetragonal Crystal System*

In the tetragonal system, the three crystal axes are perpendicular to each other. Two of the three axial lengths are the same, but the third axial length is different as shown in the figure below.

a \;= b\;\neq\; c

\alpha = \beta = \gamma = 90^{\circ}

**Bravais Lattice:**

In a tetragonal system, two types of Bravias lattices are possible. They are:

- Simple and
- Body-Centered

*3. Orthorhombic Crystal System*

*3. Orthorhombic Crystal System*

In the orthorhombic crystal system, the three crystal axes are perpendicular to each other. In this system, all three axial lengths are of unequal lengths as shown in the figure below.

a \;\neq b\;\neq\; c

\alpha = \beta = \gamma = 90^{\circ}

**Bravais Lattice:**

In an orthorhombic crystal system, four different types of Bravais lattice are possible.

- Simple
- Body-Centered
- Face Centered
- Base Centered

*4. Monoclinic Crystal System*

*4. Monoclinic Crystal System*

In the monoclinic crystal system, two of the crystal axes are perpendicular to each other, but the third is obliquely inclined. The axial lengths are different along the three axes as shown in the figure below.

a \;\neq b\;\neq\; c

\alpha = \beta = 90^{\circ} \neq\gamma

**Bravais Lattices:**

In a monoclinic crystal system, only two types of Bravias lattices are possible.

- Simple and
- Base Centered

*5. Triclinic Crystal System*

*5. Triclinic Crystal System*

In the triclinic crystal system, none of the crystal axes is perpendicular to any of the others. The axial lengths are different along the three axes shown in the figure below.

a \;\neq b\;\neq\; c

\alpha \neq \beta \neq \gamma\neq 90^{\circ}

**Bravais Lattices:**

- Only one type of Bravias lattice is possible in this triclinic crystal system which is the simple lattice.

*6. Trigonal or Rhombohedral Crystal System*

*6. Trigonal or Rhombohedral Crystal System*

In the trigonal crystal system, the three axes are inclined to each other at an angle other than 90o. The three axial lengths are equal along three axes as shown in the figure below.

a \;= b\; =\; c

\alpha = \beta = \gamma \neq 90^{\circ}

**Bravais Lattice:**

- Only a simple lattice type is said to exist for this trigonal crystal system.

*7. Hexagonal Crystal System*

*7. Hexagonal Crystal System*

In a hexagonal crystal system, two of the crystal axes are 90^{o} apart while the third is perpendicular to both of them.

The axial lengths are the same along the axes that are 90^{o} apart, but the axial lengths along the third axis are different as shown in the figure below.

a \;= b\; \neq\; c

\alpha = \beta =90^{\circ} and \; \gamma = 120^{\circ}

**Bravais Lattice:**

- For the hexagonal system also, only a simple lattice type is said to exist.

**7 Crystal Systems and 14 Bravais Lattices**

System | Bravais Lattice | Unit cell Characteristics | Characteristic symmetry elements | Example |
---|---|---|---|---|

Cubic | Simple (P) Body-Centered (I) Face-Centered (F) | a \;= b\; =\; c \alpha = \beta = \gamma = 90^{\circ} | Four 3-fold rotation axes (along cube diagonal) | NaCl, CaF_{2}, NaClO_{3} |

Tetragonal | Simple (P) Body-Centered (I) | a \;= b\;\neq\; c \alpha = \beta = \gamma = 90^{\circ} | One 4-fold roataion axis | NiSO_{4}, SnO_{2} |

Orthorhombic | Simple (P) Base-Centered (C) Body-Centered (I) Face-Centered (F) | a \;\neq b\;\neq\; c \alpha = \beta = \gamma = 90^{\circ} | Three mutually orthogonal 2-fold rotation axes | KNO_{3}, BaSO_{4}, MgSO_{4} |

Monoclinic | Simple (P) Base-Centered (C) | a \;\neq b\;\neq\; c \alpha = \beta = 90^{\circ} \neq\gamma | One 2-fold rotation axis | Na_{2}SO_{4}, FeSO_{4} |

Triclinic | Simple (P) | a \;\neq b\;\neq\; c \alpha \neq \beta \neq \gamma\neq 90^{\circ} | None | CuSO4, K_{2}Cr_{2}O_{7} |

Trigonal (Rhombohedral) | Simple (P) | a \;= b\; =\; c \alpha = \beta = \gamma \neq 90^{\circ} | One 3-fold rotation axis | CaSO_{4}, Calcite |

Hexagonal | Simple (P) | a \;= b\; \neq\; c \alpha = \beta =90^{\circ} and \; \gamma = 120^{\circ} | One 3-fold rotation axis | SiO_{2}, Agl, quartz |

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