Bravais lattice is a lattice formed by replacing the smallest repeating unit of an atomic array with a point.
When considering the crystal system, the parallelepiped with a better shape is selected.
The parallelepiped is not necessarily the smallest repeating unit.
The lattice formed by the corners of the parallelepiped is not always a Bravais lattice.
There are following three ways to add points, while satisfying the periodicity.
Fourteen Types of Bravais Lattices
Every crystal system has a simple lattice. Hence seven types of simple lattices are different classes of Bravais lattices.
The triclinic system does not have any symmetry operations other than inversion.
It is meaningless to define a larger 'unit cell'.
Therefore, all the triclinic system is defined as the simple triclinic lattice.
The unit cell of the hexagonal system is composed of two regular triangular pillars.
The speciality of the trianglar pillar is broken if another lattice point is added at the center of the parallelepiped or at the center of each plain.
Therefore, all the hexagonal system is defined as the simple trigonal lattice.
Body-centered lattice of cubic, tetragonal, orthorhombic, or monoclinic system is distinct from the simple lattice of the same symmetry.
Here, the body-centered lattice of monoclinic system can be expressed as another base-centered (see below) monoclinic lattice. One should use the base-centered lattice.
The bodh-centered lattice of a rhombohedral system can be expressed as another simple rhombohedral lattice.
Face-centered cubic or orthorhombic system is distinct from the simple or body-centered lattice of the same symmetry.
The face-centered tetragonal lattice can be expresed as a body-centered tetragonal lattice of a half volume.
One should hence use the body-centered tetragonal lattice.
The face-centered monoclinic lattice can be expressed as a base- or bode-centered monoclinic lattice of a half volume. One should hence use the base-centered monoclinic lattice.
The face-centered rhombohedral lattice can be expressed as another simple rhombohedral lattice.
One cannot make a base-centered cubic or rhombohedral lattice.
The base-centered tetragonal lattice can be expressed as another simple tetragonal lattice.
The base-centered orthorhombic or monoclinic lattice system is distincet from other lattice systems.
Here, the base face should be a rectangle.
Note that a part of the triclinic crystal system has the same lattice as the hexagonal crystal system.
Space group is used to classify the crystals from the viewpoint of atomic arrangement. The point group is based on the following symmetry operations.
The translation of all the atoms after the rotation or reflection is neglected when considering the point group.
In contrast, the translation is taken into account for classifying a crystal into the space group.
Two kinds of symmetry operations are added.
The combination of a rotation about an axis and a translation parallel to the axis is called a screw operation.
Screw operation is expressed as n_{q} by using two integers n and q.
n_{q} denotes the combination of 360/n-degree rotation and translation of q/n of a lattice vector.
The combination of a mirror reflection and a translation in a direction parallel to the mirror plane is called glide.
Glide operation is expressed as follows.
There are 230 three-dimensional space groups.
All the space groups are summarized in the book 'International Tables for Crystallography A'.
Space groups are represented by a capital letter denoting the Bravais lattice followed by several sympols denoting symmetry operations.
The capital letter denotes the type of Bravais lattice.
Following symbols denote rotation, rotoinversion, screw, mirror reflection, and glide.
The symbols are arranged in the prescribed order from the viewpoint of the directions of the rotation/rotoinversion axes and normal axes to the reflection/glide planes.
If there is a reflection or glide plane perpendicular to a rotation or screw axis, write the symbol for the rotation or screw operation, a slash (/), and the symbol for the reflection or glide operation.
For example, 2_{1}/c indicates a 180-degree screw axis and a c-glide plane perpendicular the axis.
The notation of the spatial inversion depends on the crystal system.
Miller indices give a notation system in crystallography. There are four types of notations.
Here a negative integer is often represented by a overline, as (1 1 0). This is identical to (1 − 1 0).
Symmetry operations must be described in the order of <100> and/or {100}, <111>, and <110> and/or {110}. The space inversion symmetry is described by three-fold rotoinversion 3 along <111>.
Symmetry operations must be described in the order of [001] and/or (001), <100> and/or {100}, and <110> and/or {110}. The space inversion symmetry is represented as the presence of the (001) reflection or glide plane.
Symmetry operations must be described in the order of [100] and/or (100), [010> and/or (010), and [001] and/or (001). The space inversion symmetry is represented as the presence of the reflection or glide planes normal to the [100], [010], and [001].
Symmetry operations must be described in the order of [001] and/or (001), <110> and/or {110}, and <110> and/or {110}. The space inversion symmetry is represented as the presence of the (001) reflection or glide plane.
Symmetry operations must be described in the order of [001], <110> and/or {110}, and <110> and/or {110} in the hexagonal setting. If the lattice is rhombohedral, the third part must be omitted. The space inversion symmetry is represented by three-fold rotoinversion 3 along [001].
Set the axis with symmetry operation as b. The space inversion symmetry is represented as the combination of rotation/screw and reflection/glide symmetries.
P1 and P1 for centrosymmetric and noncentrosymmetric crystals, respectively.
Taka-hisa Arima