Difference Between Point Group and Space Group

Key Difference – Point Group vs Space Group
 

The terms point group and space group are used in crystallography. Crystallography is the study of the arrangement of atoms in a crystalline solid. The crystallographic point group is a set of symmetry operations that leave at least one point unmoved. A symmetry operation is an act of obtaining the original image of an object even after moving it. The symmetry operations used in point groups are rotations and reflections. A space group is the 3D symmetry group of a configuration in space. A symmetry group is the group of all transformations obtained without varying the composition during the group operation. The key difference between point group and space group is that there are 32 crystallographic point groups whereas there are 230 space groups that are created by the combination of 32 point groups and 14 Bravais lattices.

CONTENTS

1. Overview and Key Difference
2. What is Point Group
3. What is Space Group
4. Side by Side Comparison – Point Group vs Space Group in Tabular Form
5. Summary

What is Point Group?

The crystallographic point group is a set of symmetry operations that leave at least one point unmoved. The symmetry operations described in point groups are rotations and reflections. In point group symmetry operations, one central point in the object is kept unmoved (fixed) while moving other faces of the object to the positions of features of the same kind. There, the macroscopic features of the object should remain same before and after the symmetry operation.

For any given object, there is a certain number of symmetry operations possible (with defined geometrical relations among symmetry operations). The object is said to have the symmetry described by the point group. Therefore, different objects having different point symmetries are described by different point groups.

In the notation of point groups, there are two systems in use;

1. Schoenflies Notation

In Schoenflies notation system, point groups are named as C_nv, C_nh, D_nh, T_d, O_h, etc. The different symbols used in this notation system are given below.

• n is the highest number of rotation axes
• v is the vertical mirror plane (mentioned only when there are no horizontal mirror planes)
• h is the horizontal mirror planes
• T is a tetrahedral point group
• is an octahedral point group

For example, C_n is used indicate that the point group has n-fold rotation axis. When it is given as C_nh, it means there is a C_n along with a mirror plane (reflection plane) perpendicular to the axis of rotation. In contrast, C_nv is C_n with a mirror plane parallel to the axis of rotation. If the point group is given as S_2n, it indicates that the point group has only a 2n-fold rotation-reflection axis.

2. Hermann-Mauguin Notation

The Hermann-mauguin notation system is commonly used for space groups. But, it is also are used for crystallographic point groups. It gives the highest rotation axis. For example, the point group having only 2-fold rotation axis is denoted as 2. The point group given as C_2h by Schoenflies notation is given as 2/m in Hermann-mauguin notation system in which the symbol ‘m’ indicates a mirror plane and the slash symbol indicates that the mirror plane is perpendicular to the two-fold axis. Following table shows different notations of point groups for different lattice systems.

There are 32 point groups. The simplest point groups are 1, 2, 3, 4, 5 and 6. All these point groups comprise only one rotation axis. For rotary-inversions, there are axes named -1, m, -3, -4 and -6. Other 22 point groups are combinations of these point groups.

What is Space Group?

A space group is the 3D symmetry group of a configuration in space. There are 230 space groups. These 230 groups are a combination of 32 crystallographic point groups (mentioned above) and 14 Bravais lattices. The Bravais lattices are given in the below table.

A space group gives a description of symmetry of a crystal. Space groups are combinations of translational symmetry of unit cell and symmetry operations such as rotation, rotary-inversion, reflection, screw axis and glide plane symmetry operations.

What is the Difference Between Point Group and Space Group?

Summary – Point Group vs Space Group

Point groups and space groups are terms described under crystallography. The crystallographic point group is a set of symmetry operations all of which leave at least one point unmoved. A space group is the 3D symmetry group of a configuration in space. The difference between point group and space group is that there are 32 crystallographic point groups whereas there are 230 space groups (created by the combination of 32 point groups and 14 Bravais lattices).

Reference:

1.“2: Symmetry Operations and Symmetry Elements.” Chemistry LibreTexts, Libretexts, 6 May 2017. Available here
2.“Crystallographic point group.” Wikipedia, Wikimedia Foundation, 28 Feb. 2018. Available here   

Image Courtesy:

1.’Ice Ih Space Group’By Dbuckingham42 – Own work, (CC BY-SA 4.0) via Commons Wikimedia