Point Group vs. Space Group

Attribute	Point Group	Space Group
Definition	Refers to the symmetry operations of a molecule or crystal that leave at least one point fixed.	Refers to the symmetry operations of a crystal that leave at least one point fixed and also involve translation.
Number of Operations	Finite number of operations.	Infinite number of operations.
Translation	No translation involved.	Includes translation.
Application	Used to describe the symmetry of molecules.	Used to describe the symmetry of crystals.
Examples	C1, C2, C3, C4, C5, C6, C∞, D2, D3, D4, D5, D6, D∞, T, O, I	P1, P2, P3, P4, P6, Pm, Cm, Pn, Cn, Pmm, Pnn, Pmn, Pbc, Pcc, Pma, Pca, Pnc, Pmn2_1, Pba2, Pna2_1, Pnn2, Cmc2_1, Pmc2_1, Pcc2_1, Pma2_1, Pca2_1, Pnc2_1, Pmn21, Pba21, Pna21, Pnn21, Cmc21, Pmc21, Pcc21, Pma21, Pca21, Pnc21, Pmn2_1, Pba2_1, Pna2_1, Pnn2_1, Cmc2_1, Pmc2_1, Pcc2_1, Pma2_1, Pca2_1, Pnc2_1, Pmn21, Pba21, Pna21, Pnn21, Cmc21, Pmc21, Pcc21, Pma21, Pca21, Pnc21

Introduction

In the field of crystallography, the study of crystals and their structures, two important concepts are point group and space group. These concepts help us understand the symmetry and arrangement of atoms within a crystal lattice. While both point group and space group describe the symmetry of crystals, they differ in their scope and level of detail. In this article, we will explore the attributes of point group and space group, highlighting their similarities and differences.

Point Group

The point group of a crystal refers to the set of symmetry operations that leave the crystal invariant. These operations include rotations, reflections, and inversions. The point group is defined by the arrangement of atoms around a point, known as the symmetry center. The symmetry center can be an atom, an empty space, or a combination of both. The point group is denoted by a Schoenflies symbol or a Hermann-Mauguin symbol, which represents the symmetry elements present in the crystal.

One of the key attributes of a point group is its symmetry elements. These elements include rotation axes, mirror planes, inversion centers, and improper rotation axes. The rotation axes represent the rotational symmetry of the crystal, while the mirror planes reflect the crystal across a plane. The inversion centers involve a 180-degree rotation followed by an inversion through the center. The improper rotation axes combine rotation and reflection to create symmetry.

Another attribute of the point group is its point group operations. These operations describe the transformations that preserve the crystal's symmetry. For example, a rotation operation involves rotating the crystal by a certain angle around an axis, while a reflection operation reflects the crystal across a plane. The combination of these operations forms the point group of the crystal.

The point group also determines the physical properties of the crystal. It affects the crystal's optical, electrical, and thermal properties. For instance, crystals with high symmetry, such as those belonging to the cubic point group, exhibit isotropic behavior, meaning their properties are the same in all directions. On the other hand, crystals with lower symmetry may exhibit anisotropic behavior, where their properties vary with direction.

In summary, the point group of a crystal describes its symmetry elements, point group operations, and their impact on the crystal's physical properties.

Space Group

While the point group focuses on the symmetry around a single point, the space group considers the symmetry of the entire crystal lattice. The space group describes the arrangement of multiple unit cells in a crystal, taking into account translations, rotations, and reflections. It provides a more comprehensive understanding of the crystal's symmetry and structure.

One of the key attributes of a space group is its translational symmetry. Translations involve shifting the crystal along the three crystallographic axes. These translations, combined with the point group operations, create a repeating pattern of atoms in the crystal lattice. The space group identifies the specific translation vectors and symmetry operations that generate the crystal structure.

Another attribute of the space group is its symmetry operations. These operations include rotations, reflections, and translations. The rotations and reflections are similar to those in the point group, but they are applied to the entire crystal lattice rather than a single point. The translations describe the periodicity of the crystal structure and determine the unit cell size and shape.

The space group also provides information about the crystal's symmetry elements. These elements include screw axes, glide planes, and inversion centers. Screw axes involve a combination of rotation and translation, while glide planes combine reflection and translation. The presence of these symmetry elements in the space group affects the overall symmetry and arrangement of atoms in the crystal lattice.

Furthermore, the space group determines the crystal's crystallographic axes and lattice parameters. The crystallographic axes define the coordinate system used to describe the crystal structure, while the lattice parameters specify the size and shape of the unit cell. These parameters are crucial for understanding the crystal's physical properties and for performing calculations related to crystallography.

In summary, the space group of a crystal describes its translational symmetry, symmetry operations, symmetry elements, crystallographic axes, and lattice parameters, providing a comprehensive view of the crystal's symmetry and structure.

Comparison

While point group and space group both describe the symmetry of crystals, they differ in their scope and level of detail. The point group focuses on the symmetry around a single point, while the space group considers the symmetry of the entire crystal lattice.

The point group describes the symmetry elements and operations that leave the crystal invariant. It determines the physical properties of the crystal and is denoted by a Schoenflies or Hermann-Mauguin symbol. On the other hand, the space group describes the arrangement of multiple unit cells in the crystal, taking into account translations, rotations, and reflections. It provides information about translational symmetry, symmetry operations, symmetry elements, crystallographic axes, and lattice parameters.

While the point group is defined by the arrangement of atoms around a point, the space group considers the arrangement of atoms throughout the crystal lattice. The point group operations are applied to a single point, while the space group operations are applied to the entire crystal lattice. The point group determines the symmetry elements and operations that preserve the crystal's symmetry, while the space group identifies the translation vectors and symmetry operations that generate the crystal structure.

Both point group and space group play crucial roles in crystallography. They help us understand the symmetry and structure of crystals, which in turn provides insights into their physical properties and behavior. By studying the point group and space group of a crystal, scientists can predict and explain various phenomena, such as the optical, electrical, and thermal properties of materials.

In conclusion, while point group and space group share the common goal of describing the symmetry of crystals, they differ in their scope and level of detail. The point group focuses on the symmetry around a single point, while the space group considers the symmetry of the entire crystal lattice. Both concepts are essential for understanding the structure and properties of crystals, and their study continues to contribute to advancements in various scientific fields.