Bravais Lattice

Introduction

The atoms/molecules in a solid crystalline substance are grouped uniformly and periodically over all three dimensions. Crystal structure refers to the atomic arrangement of a crystal.

To make crystal symmetries easier to understand, a unit of atom (a single atom or a group of atom) that repeat in three dimensions in the crystal can be represented as a unit. The resultant points in space are called space lattice if each such atom or unit of atoms in a crystal is substituted by a point in space. A lattice point is a point in a space lattice, and a basis or pattern is an atom or a unit of atoms.

A space lattice is a geometrical arrangement of crystal whereby each lattice point has the same surroundings. A lattice is termed Bravais lattice if the surroundings of each lattice point are identical, or whether the atom or all the atoms at lattice points are identical. A non-Bravais lattice, on the other hand, is defined as one in which the atom or atoms at the lattice points are not the same.

Crystal Systems in Bravais Lattice

The symmetry of space lattices is used to classify them. Bravais demonstrated in 1948 that 14 lattices are enough to explain all crystals. Bravais lattices are a group of 14 lattices that are divided into seven crystal systems based on cell parameters. Primitive lattice (P), basecentered lattice (B), body-centered lattice (I) and face-centered lattice (F) are the four types of Bravais lattices (C).

Below are descriptions of the 7 crystal systems and Bravais lattices:

Cubic crystal system: All unit cell edge lengths are equal and at right angles to each other in this crystal system, i.e. $\mathrm{\alpha = \beta = \gamma = 90^{\circ}}$ and a = b = c. There are three Bravais lattices in the cubic system: simple (basic), body centred, and face centred (Figure 1). Cu, Au, Ag, diamond, and others are examples of cubic systems. Lattice atoms or points are present at the cube's corners in a basic cubic lattice. Atoms are present in the corners of a bodycentered cube, and one atom is totally present in the centre.

Tetragonal Crystal System: Two of the unit cell edges' lengths are equivalent in this crystal arrangement, while the third length is different. The three edges are perpendicular to each other, i.e., $\mathrm{\alpha = \beta = \gamma = 90^{\circ}}$ and a = b ≠ c. There are two Bravais lattices in the tetragonal system: simple and body-centered. Figure 1 depicts representative of $\mathrm{SnO_2, \:TiO_2}$, and other tetragonal crystal systems are examples.

Orthorhombic Crystal System: The unit cell edge lengths in this crystal structure are distinct and perpendicular to one another, i.e., $\mathrm{\alpha = \beta = \gamma = 90^{\circ}}$ and a ≠ b ≠ c. This system has four Bravais lattices. They're straightforward, with a focus on the face, body, and base. Figure 1 depicts them $\mathrm{SnSO_4.\:K_2SO_4,\:BaSO_4}$, and other orthorhombic crystal systems are examples.

Monoclinic Crystal System: The unit cell edge lengths differ in this crystal structure. Two unit cell edges are perpendicular to the third edge, but not perpendicular to each other, a ≠ b ≠ c and $\mathrm{\alpha = \gamma = 90^{\circ}≠ \beta}$. There are two Bravais lattices in this crystal system; both are base centred and simple. Figure 1 depicts them. $\mathrm{Na_3AlF_6 \:(cryolite),\: CaSO_4.2H_2O \:(gypsum)}$, and other monoclinic crystal systems are examples.

Triclinic Crystal System: The unit cell edge lengths in this crystal structure are variable and not perpendicular, i.e., $\mathrm{\alpha \:
eq\: \beta \:
eq\: \gamma \:
eq\: 90^{\circ}}$ and $\mathrm{a \:
eq\: b \:
eq\: c}$, and all angles are different. This crystal can only be found in a primordial cell. Figure 1 depicts representative of $\mathrm{CuSO_4.5H_2O,\:K_2Cr_2O_7}$, and other triclinic crystal systems are examples.

Rhombohedral or Trigonal Crystal System: The lengths of the unit cell edges are all equal in this crystal structure. The angles between the axes are equal but not exactly 90 degrees, i.e. a = b = c and $\mathrm{\alpha = \beta = \gamma\:
eq\:90^{\circ}}$. As seen in Fig. 1, the Bravais lattice is quite simple. Sb, Bi, As and others are examples of Rhombohedral crystal systems.

Hexagonal Crystal System: Two sides of the unit cell edge lengths are equal in this crystal arrangement, with a 120° angle between them. These two edges do not have the same length and are perpendicular to the third edge, i.e., $\mathrm{\alpha \: = \: \beta \:= \: 90^{\circ} ; \: \gamma \:= \: 120^{\circ}}$ and a = b ≠ c. Only the primitive Bravais lattice exists. Figure. 1 illustrates this. These crystal systems’ atoms are organised in a hexagonal tight pack.

Lattice	Examples	Edge Length	Angles between faces	Types
Triclinic	$\mathrm{CuSO_4.5H_2O.}$ $\mathrm{H_3PO_3}$	a ≠ b ≠ c	$\mathrm{\alpha\: eq\:\beta\: eq\:\gamma\: eq\: 90^{\circ}}$	Primitive
Monoclinic	Sulphur	a ≠ b ≠ c	$\mathrm{\alpha\:=\:\gamma\:=\:90^{\circ} eq \:\beta}$	End-centred, Primitive
Rhombohedral	$\mathrm{HgS,\: CaCO_3}$	a = b = c	$\mathrm{\alpha\:=\:\beta\:=\:\gamma\: eq\:90^{\circ}}$	Primitive
Hexagonal	CdS, ZnO and Graphite	a = b ≠ c	$\mathrm{\alpha\:=\:\beta\:=\:90^{\circ};\:\gamma\:=\:120^{\circ}}$	Primitive
Orthorhombic	$\mathrm{KNO_3,\:BaSO_4,}$ Rhombic Sulphur	a ≠ b ≠ c	$\mathrm{\alpha\:=\:\beta\:=\:\gamma\:=\:90^{\circ}}$	End-Centred, Face-Centred, Body-Centred, Primitive
Tetragonal	$\mathrm{CaSO_4, SnO_2,TiO_2}$ White Tin	a = b ≠ c	$\mathrm{\alpha\:=\:\beta\:=\:\gamma\:=\:90^{\circ}}$	Body-Centred, Primitive
Cubic	ZnS, NaCl, Copper	a = b = c	$\mathrm{\alpha\:=\:\beta\:=\:\gamma\:=\:90^{\circ}}$	Face-Centred, Body-Centred, Primitive

Conclusion

A "Bravais lattice" is a key notion in the characterization of crystalline solids. An endless distribution of points (or atoms) in space is called a Bravais lattice. When seen from any lattice point, the lattice seems to be identical. In three dimensions, there are 14 distinct Bravais lattices that are divided into seven different crystal systems.

FAQs

Q1. Define Bravais and non-Bravais Lattice.

Ans: When the environment of each lattice point are identical, or if the atom or all the atoms at lattice points are identical, the lattice is called a Bravais lattice. In contrast, a non-Bravais lattice is one in which the atom or atoms at the lattice points are not the same.

Q2. What are the seven different types of Crystal Systems?

Ans: The seven different types of Crystal Systems are as follows:

• Cubic
• Tetragonal
• Orthorhombic
• Monoclinic
• Triclinic
• Trigonal or Rhombohedral
• Hexagonal

Q3. How many types of Bravais Lattice are found in Monoclinic Crystal System?

Ans: Two types of Bravais Lattice are found in Monoclinic Crystal systems. They are End-Centred and Primitive.

Q4. Give examples of Triclinic Crystal Systems.

Ans: The examples of Triclinic Crystal Systems include: $\mathrm{H_3PO_3}$ and $\mathrm{CuSO_4.5H_2O}$

Q5. What are criteria of edge length and angles between the faces for a crystal to be Hexagonal?

The criteria of edge length and angles between the faces for a Hexagonal Crystal System:

(i) $\mathrm{a\:=\:b \:
eq\: c}$

(ii) $\mathrm{\alpha \:=\:\beta\: =\: 90^{\circ}; \:\gamma\: =\: 120^{\circ}}$