## Crystallographic Symmetry

*by* Ritika

The Crystallographic Symmetry and the applications related to it are vital for people related to various fields namely mathematicians, physicists, and chemists and so on and many researches and investigation related to these have been carried by them. The operations related to crystallographic symmetry could fall in any of the different kinds of available isometries. So first it is essential to have a fair knowledge about isometries and its kinds which will be dealt in the below section. Generally an object is referred as symmetric with respect to a transformation only if the object appears to be in a state that is identical to its initial state after the transformation. In terms of crystallography the types of symmetry is in terms of an apparent movement of the object such as some type of rotation or translation.

### About Isometries

Isometries define in simple term is one which maps each point present in the point space onto exactly one image point. In addition it also refers to the mapping of the point space onto itself thereby leaving all distances and angles also invariant.

### Kinds of Isometries

The two broad categories in which the kinds of isometries fall are namely:

- Proper Isometries
- ImProper Isometries

The proper isometries are also called as isometries of the first kind and the improper isometries are also termed as isometries of the second kind.

### The kinds of isometries are as below

- Identity referred as I
- Translation referred as T
- Rotation
- Screw Rotation
- Inversion
- Rotoinversion
- Reflection
- Glide Reflection

In the above, the first four isometries fall in the category of proper isometries, also called isometries of the first kind. This is because in all the first four kinds of isometries while mapping they preserve the handedness of object. In contrast, the kinds of isometries from 5 to 8 in above list change the handedness and so fall in the improper isometries category.

### About Space Group and Site Symmetry

The space group S of crystal pattern is used to refer all the symmetry operations associated with the crystal pattern. The set of all elements of S in other words the space group which leaves a fixed point F is termed as site symmetry group of F with regard to the space group S.

### Operations Related to Crystallographic Site-Symmetry

Symmetry around a position in a molecule or crystal is very vital and this is determined by various attributes like chemical bonds. This in turn has an influence on physical and chemical attributes of the crystal or substance. The elements of crystallographic site symmetry from types of isometries are

#### Identity

This is a member of crystallographic site-symmetry group as this operation has order one and as it leaves any fixed point.

#### Inversion

The order of this operation is two and this is could also be a member of crystallographic site-symmetry group.

#### Rotation

There are two distinct methods of rotational symmetry operations. They are termed as Hermann-Mauguin nomenclature and chönflies nomenclature which fall under either proper rotations or improper rotations. The proper rotations move an object without changing the handedness of the object but in contrast the improper rotation has in it a component for proper rotation and in addition also inverts the handedness of the object. The extension to the concept of rotation symmetry is to include in each rotation operator a translation component which result in operations termed as screw rotations.

#### Reflections

The above operations namely identity, inversion, rotation, rotoinversions and reflections are symmetry operations of space group.

It is also possible to combine symmetry operations which would result in the generation of other symmetry operations. This is technically referred to as group of symmetry operations. It is called point groups because the symmetry elements of these operations all pass through a single point of the object. For instance, operations related to rotation symmetry could be combined with translations of part of the unit cell which would result in relating entirely new symmetry operations. Thus it is possible to combine proper and improper rotation operations with rules of groups being followed which would result in generation of 32 crystallographic point groups.

### Symmetry Element

The crystallographic symmetry operation could be viewed geometrically using the geometric element also termed as symmetry element. Here symmetry element refers to a point or line or plane related to the symmetry. This varies based on the operation involved in symmetry namely if the operation of symmetry is rotoinversion the symmetry point would be the inversion point. But the two operations namely the identity operation and the translation operation do not have a symmetry element. It is also possible that a single symmetry element may have associated with it more than a single symmetry operation but in contrast, the symmetry element associated with a symmetry operation is unique.

### Crystal Systems with Symmetry Order

The symmetry in crystals structure well seen in crystal faces is because of the ordered internal arrangement of atoms in a crystal structure. The study of symmetry in crystallography helps in variety of applications and some to name are it helps to characterize crystals, identify repeating parts of molecules and also simplify many calculations in research related to crystals.

In general, crystals are solids that have an atomic structure with long-range, 3-dimensional order. If one carefully notices the several crystals from the same material it would be visible that though they may have different sizes it results that all crystals have the same shape or habit. Crystalline materials are separated into 7 crystal different systems. They are:

- Triclinic
- Monoclinic
- Orthorhombic
- Tetragonal
- Trigonal
- Hexagonal
- Rhombohedral
- Hexagonal
- Cubic

In the above seven crystal systems the lowest order of symmetry is triclinic. The tetragonal, Trigonal, and hexagonal crystal systems have one axis of higher symmetry.

Thus to generate a lattice with higher symmetry then it could be achieved by selecting the lattice vectors in such a way that one or more lattice points are also on the center of a face of the lattice or inside of the unit cell. Always determination of space group and diffraction pattern for a crystal is straight forward. If suppose it is not possible to uniquely determine the space group then structure solution and refinement steps are taken for all space groups by lowering the symmetry until one could complete the structure determination perfectly.