CRYSTAL SYSTEMS AND BRAVAIS LATTICES

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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 is crystal systems are?

Crystal Systems

Crystals are classified into seven systems on the basis of:

  1. axial lenghts a, b, c
  2. Interfacial (axial) angles α, β, γ, and
  3. directions of axis of symmetry

The seven basic crystal systems are:

1. Cubic2. Tetragonal3. Orthorhombic4. Monoclinic
5. Triclinic6. Trigonal or Rhombohedral7. 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

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}

cubic-crystal-system

Bravais Lattices:

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

  1. Simple Cubic (SC)
  2. Body Centered Cubic (BCC)
  3. Face Centered Cubic (FCC).

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}

tetragonal-crystal-system

Bravais Lattice:

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

  1. Simple and
  2. Body Centered

3. Orthorhombic Crystal System

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

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

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

orthorhombic-crystal-system

Bravais Lattice:

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

  1. Simple
  2. Body Centered
  3. Face Centered
  4. Base Centered

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

monoclinic-crystal-system

Bravais Lattices:

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

  1. Simple and
  2. Base Centered

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}

triclinic-crystal-system

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

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}

trigonal-crystal-system

Bravais Lattice:

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

7. Hexagonal Crystal System

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

The axial lengths are the same along the axes that are 90o 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}

hexagonal-crystal-system

Bravais Lattice:

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

Crystal Systems and Bravais Lattices

SystemBravais LatticeUnit cell CharacteristicsCharacteristic symmetry elementsExample
CubicSimple (P)
Body-Centered (I)
Face-Centered (F)
a \;= b\; =\; c
\alpha = \beta = \gamma = 90^{\circ}
Four 3-fold rotation axes (along cube diagonal)NaCl, CaF2, NaClO3
TetragonalSimple (P)
Body-Centered (I)
a \;= b\;\neq\; c
\alpha = \beta = \gamma = 90^{\circ}
One 4-fold roataion axisNiSO4, SnO2
OrthorhombicSimple (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 axesKNO3, BaSO4, MgSO4
MonoclinicSimple (P)
Base-Centered (C)
a \;\neq b\;\neq\; c
\alpha = \beta = 90^{\circ} \neq\gamma
One 2-fold rotation axisNa2SO4, FeSO4
TriclinicSimple (P)a \;\neq b\;\neq\; c
\alpha \neq \beta \neq \gamma\neq 90^{\circ}
NoneCuSO4, K2Cr2O7
Trigonal (Rhombohedral)Simple (P)a \;= b\; =\; c
\alpha = \beta = \gamma \neq 90^{\circ}
One 3-fold rotation axisCaSO4, Calcite
HexagonalSimple (P)a \;= b\; \neq\; c
\alpha = \beta =90^{\circ} and \; \gamma = 120^{\circ}
One 3-fold rotation axisSiO2, Agl, quartz

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