Bravais lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850),[1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by where the ni are any integers, and ai are primitive translation vectors, or primitive vectors, which lie in different directions (not necessarily mutually perpendicular) and span the lattice.

The choice of primitive vectors for a given Bravais lattice is not unique.

The Bravais lattice concept is used to formally define a crystalline arrangement and its (finite) frontiers.

A crystal is made up of one or more atoms, called the basis or motif, at each lattice point.

Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups.

A unit cell is defined as a space that, when translated through a subset of all vectors described by

There are clearly many choices of cell that can reproduce the whole lattice when stacked (two lattice halves, for instance), and the minimum size requirement distinguishes the primitive cell from all these other valid repeating units.

Despite this rigid minimum-size requirement, there is not one unique choice of primitive unit cell.

To have the smallest cell volume, a primitive unit cell must contain (1) only one lattice point and (2) the minimum amount of basis constituents (e.g., the minimum number of atoms in a basis).

All primitive unit cells with different shapes for a given crystal have the same volume by definition; For a given crystal, if n is the density of lattice points in a lattice ensuring the minimum amount of basis constituents and v is the volume of a chosen primitive cell, then nv = 1 resulting in v = 1/n, so every primitive cell has the same volume of 1/n.

(Again, these vectors must make a lattice with the minimum amount of basis constituents.

In this case, a conventional unit cell easily displaying the crystal symmetry is often used.

In two dimensions, any lattice can be specified by the length of its two primitive translation vectors and the angle between them.

One can impose conditions on the length of the primitive translation vectors and on the angle between them to produce various symmetric lattices.

Thus, lattices can be categorized based on what point group or translational symmetry applies to them.

From there, there are 4 further combinations of point groups with translational elements (or equivalently, 4 types of restriction on the lengths/angles of the primitive translation vectors) that correspond to the 4 remaining lattice categories: square, hexagonal, rectangular, and centered rectangular.

Note: In the unit cell diagrams in the following table the lattice points are depicted using black circles and the unit cells are depicted using parallelograms (which may be squares or rectangles) outlined in black.

The area of the unit cell can be calculated by evaluating the norm ‖a × b‖, where a and b are the lattice vectors.

Similarly, all A- or B-centred lattices can be described either by a C- or P-centering.

This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.

This can be seen by imagining moving the unit cell slightly in the negative direction of each axis while keeping the lattice points fixed.

This shows that only one of the eight corner lattice points (specifically the front, left, bottom one) belongs to the given unit cell (the other seven lattice points belong to adjacent unit cells).

In addition, only one of the two lattice points shown on the top and bottom face in the Base-centered column belongs to the given unit cell.

Finally, only three of the six lattice points on the faces in the Face-centered column belong to the given unit cell.

The volume of the unit cell can be calculated by evaluating the triple product a · (b × c), where a, b, and c are the lattice vectors.

The seven sided polygon (heptagon) and the number 7 at the centre indicate the seven lattice systems.