# Space Group and Symmetry Operations

Here we learn symmetry operations allowed in conventional crystals and the presentation of space group.

## Bravais Lattice

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.

• Body-centered Lattice: Add a point to the center of each parallelepiped.
• Face-centered Lattice: Add a point to the center of each face surrounding parallelepipeds.
• Bace-centered Lattice: Add a point to the center of parallel rectangles.

### Classification of Bravais Lattices

Fourteen Types of Bravais Lattices

#### Seven Types of Simple 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.

#### Three Types of Body-Centered Lattices

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.

#### Two Types of Face-centered Lattices

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.

#### Two Types of Base-centered Lattices

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.

### Summary: Seven Lattice Systems and Fourteen Types of Bravais Lattice

• cubic crystal system
• simple cubic lattice
• body-centered cubic lattice
• face-centered cubic lattice
• tetragonal crystal system
• simple tetragonal lattice
• body-centered tetragonal lattice
• orthorhombic crystal system
• simple orthorhombic lattice
• base-centered orthorhombic lattice
• body-centered orthorhombic lattice
• face-centered orthorhombic lattice
• monoclinic crystal system
• simple monoclinic lattice
• base-centered monoclinic lattice
• hexagonal crystal system
• hexagonal lattice
• trigonal crystal system
• hexagonal lattice
• rhombohedral lattice
• triclinic crystal system
• triclinic lattice

Note that a part of the triclinic crystal system has the same lattice as the hexagonal crystal system.

## Symmetry Operations in Crystals

Space group is used to classify the crystals from the viewpoint of atomic arrangement. The point group is based on the following symmetry operations.

• n (1,2,3,4,6): 360/n-degree rotation
• m: mirror reflection
• n (n: 1,2,3,4,6): 360/n-degree rotoinversion

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.

#### Screw

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 nq by using two integers n and q. nq denotes the combination of 360/n-degree rotation and translation of q/n of a lattice vector.

#### Glide

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.

• a-glide: mirror reflection and translation by a/2.
• b-glide: mirror reflection and translation by b/2.
• c-glide: mirror reflection and translation by c/2.
• e-glide: mirror reflection and translation by a half of either of the two lattice vectors parallel to the mirror plane.
• n-glide: mirror reflection and translation by a half of the sum of the two lattice vectors parallel to the mirror plane.
• d-glide: mirror reflection and translation by a quarter of the sum of the two lattice vectors parallel to the mirror plane.

## Notation of Space Group

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.

• P: Primitive lattice
• I: Body-centered lattice
• F: Face-centered lattice
• A: Base-centered lattice with lattice points at the center of a plane
• B: Base-centered lattice with lattice points at the center of b plane
• C: Base-centered lattice with lattice points at the center of c plane
• R: Rhombohedral 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, 21/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

Miller indices give a notation system in crystallography. There are four types of notations.

• (h k l): the family of planes orthogonal to h a* + k b* + lc*
• {h k l}: the set of all planes that are equivalent to (h k l) by the symmetry of the lattice.
• [h k l]: the direction parallel to h a + k b + lc
• <h k l>: the set of all the axes that are equivalent to [h k l] by the symmetry of the lattice.

Here a negative integer is often represented by a overline, as (1 1 0). This is identical to (1 − 1 0).

### Notation for Cubic Space Groups

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>.

### Notation for Tetragonal Space Groups

Symmetry operations must be described in the order of  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.

### Notation for Orthorhombic Space Groups

Symmetry operations must be described in the order of  and/or (100), [010> and/or (010), and  and/or (001). The space inversion symmetry is represented as the presence of the reflection or glide planes normal to the , , and .

### Notation for Hexagonal Space Group

Symmetry operations must be described in the order of  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.

### Notation for Trigonal Space Group

Symmetry operations must be described in the order of , <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 .

### Notation for Monoclinic Space Group

Set the axis with symmetry operation as b. The space inversion symmetry is represented as the combination of rotation/screw and reflection/glide symmetries.

### Notation for Triclinic Space Groups

P1 and P1 for centrosymmetric and noncentrosymmetric crystals, respectively.

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Taka-hisa Arima