Bravais Lattice 

The basic lattice arrangements are Bravais lattices. All other lattices can be reduced to Bravais lattices. By translation, Bravais lattices move on a certain basis such that they all line up on the same foundation.

There are 14 Bravais lattices in three dimensions:

1. Simple Cubic

2. Face Centered Cubic

3. Body-Centered Cubic

4. Hexagonal

5. Rhombohedral

6. Simple tetragonal

7. Body-centered Tetragonal

8. Simple Orthorhombic

9. Face-Centered Orthorhombic

10. Body-Centered Orthorhombic

11. Base-Centered Orthorhombic

12. Simple Monoclinic

13. Base-Centered Monoclinic

14. Triclinic

Importance of Bravais Lattices in Crystallography

A lattice and foundation define crystal structures. What is repeated in the crystal structure is the basis. The lattice is the method by which it is reproduced. (If you need a refresher on what crystals are, check out this basic explanation.)

Most materials are crystals, and crystals have a pattern of atoms that repeats. The basic crystal structure is described by Bravais lattices (there are additional point groups and space groups, but I’ll get to those later).

The underlying crystal structure of a material can have a significant impact on its properties. A full explanation of the distinctions between Face-Centered Cubic and Body-Centered Cubic crystal formations can be found here.

1D Bravais Lattice 

In one dimension, there is only one bravais lattice. Although discussing this is simple, I believe it is the most straightforward approach to illustrate how more complex arrangements collapse into the basic bravais lattices.

As we know there is only one direction, anything that repeats must do so at a specific distance in that direction. There is just one lattice because there is only one variable (the distance something will repeat). This is known as the “linear” Bravais lattice.

2D Bravais Lattices

We have five basic 2D Bravais lattices.

1. Square, a=b, θ=90°

2.Hexagonal, a=b, θ=120°

3. Rectangular, a≠b, θ=90°

4. Centered Rectangular

A key point  the primitive cell is hexagonal or square but with less symmetry

 only centered, not face body or base

Rhomboidal, a≠b, θ≠90°.

 Square

The 2D square Bravais lattice is a square-tiled lattice that entirely covers a space. The vectors a and b are equal and at right angles to each other.

Hexagonal

The hexagonal 2D Bravais lattice is also known as the rhombic lattice. You might think of them as triangular, but maintaining translational symmetry necessitates the use of two triangles (one up, one down).

This lattice employs hexagons or rhombuses to tile a space (I like “hexagonal” because it implies more symmetry). The vectors a and b are at a 120-degree angle to each other.

(The hexagonal Bravais lattice is a subset of the centered-rectangular lattice in principle, but it is given its name because the perfect 120° angle has superior symmetry and produces hexagons).

Rectangular

The rectangular 2D Bravais lattice tiles a rectangle-filled region. The vectors a and b are at right angles, but their magnitudes are different.

Another way to think about this lattice is if a and b were the same length but not at right angles, with a third lattice point in the middle.

Rhombus-shaped (Oblique)

The rhomboidal 2D Bravais lattice uses rhomboids to tile a space. This lattice has non-90° angles between vectors a and b of varying lengths.

In most places I checked, this lattice was referred to as “oblique,” but I thought it would make more sense if I named it by its real name. Rhomboids are parallelograms having two different side lengths and non-90° angles. There is no such thing as an “oblique” 2D shape

3 D Bravais Lattice 

Each lattice has six faces, twelve edges, and eight vertices. We may characterise these polyhedrons using three vectors that correspond to three of the 12 edges (there will be three sets of four matching edges due to the polygon’s four-sided structure). That’s why, as long as they come from three different edge sets, we only need to specify three different vectors). The cube is the lattice shape with the most symmetry. Each of the 12 edges is the same length and has the same angle to each other (90°), which can be expressed as three vectors of equal length and 90° to each other.

Conclusion 

We conclude that it provides information about the periodic array in which the crystal’s repeated components are placed. The units themselves could be an atom, a group of atoms, a molecule, an ion, or something else. However, regardless of the exact units, the Bravais lattice summarises just the geometry of the underlying periodic structure.