1.6 Point groups and space groups

Point groups and space groups are crucial concepts in solid state physics, describing symmetry in molecules and crystals. Point groups focus on symmetry operations that leave a point fixed, while space groups include translational symmetry. Together, they form the foundation for understanding crystal structures.

These symmetry concepts have wide-ranging applications in materials science. They help explain physical properties, determine selection rules for transitions, and analyze phonon dispersion and electronic band structures. Understanding symmetry is key to predicting and manipulating material behavior in various fields.

Definition of point groups

• Point groups are mathematical groups that describe the symmetry of a molecule or crystal in three dimensions
• They consist of a set of symmetry operations that leave at least one point fixed in space
• Point groups are essential for understanding the physical properties and behavior of molecules and crystals

Symmetry operations in point groups

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• Symmetry operations in point groups include rotation, reflection, inversion, and improper rotation
• Rotation symmetry ([object Object],[object Object]) involves rotating the object by $360^\circ/n$ about an axis
• Reflection symmetry ($\sigma$) involves reflecting the object across a plane
• Inversion symmetry ([object Object],[object Object]) involves inverting the object through a point
• Improper rotation symmetry ([object Object],[object Object]) is a combination of rotation and reflection

Classification of point groups

• Point groups are classified based on their symmetry elements and the relationships between them
• There are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic
• Each crystal system has a specific set of point groups associated with it
• The total number of three-dimensional point groups is 32

Schoenflies notation for point groups

• Schoenflies notation is a system used to describe point groups using symbols
• The symbols include $C_n$ for n-fold rotation, $\sigma$ for reflection, $i$ for inversion, and $S_n$ for improper rotation
• Examples of Schoenflies notation include [object Object],[object Object] (two-fold rotation with vertical reflection planes) and [object Object],[object Object] (four-fold rotation with horizontal reflection plane)

Hermann-Mauguin notation for point groups

• Hermann-Mauguin notation is another system used to describe point groups using symbols
• It is based on the symmetry elements present in the point group and their orientation relative to the crystal axes
• Examples of Hermann-Mauguin notation include [object Object],[object Object] (two-fold rotation with perpendicular mirror plane) and [object Object],[object Object] (four-fold rotation with mirror planes parallel to the rotation axis)

Definition of space groups

• Space groups are mathematical groups that describe the symmetry of a crystal in three dimensions, including both point group symmetry and translational symmetry
• They consist of a combination of symmetry operations that leave the crystal structure invariant
• Space groups are crucial for understanding the arrangement of atoms in a crystal and its resulting properties

Symmetry operations in space groups

• Symmetry operations in space groups include all the point group symmetry operations, as well as translation, screw axis, and glide plane operations
• Translation symmetry involves shifting the crystal by a lattice vector
• Screw axis symmetry is a combination of rotation and translation along the rotation axis
• Glide plane symmetry is a combination of reflection and translation parallel to the reflection plane

Classification of space groups

• Space groups are classified based on their symmetry elements and the relationships between them
• There are 230 unique three-dimensional space groups, which are divided into 7 crystal systems and 14 Bravais lattices
• Each space group is characterized by a unique combination of symmetry operations and a specific arrangement of atoms in the unit cell

Hermann-Mauguin notation for space groups

• Hermann-Mauguin notation is used to describe space groups using symbols
• The notation includes the symbols for the point group symmetry elements, followed by the symbols for the translational symmetry elements
• Examples of Hermann-Mauguin notation for space groups include $P2_1/c$ (primitive monoclinic with a two-fold screw axis and a glide plane) and $Fm\bar{3}m$ (face-centered cubic with full octahedral symmetry)

Number of space groups in 2D and 3D

• In two dimensions, there are 17 unique space groups, also known as wallpaper groups
• These 2D space groups describe the possible symmetries of patterns on a plane
• In three dimensions, there are 230 unique space groups, which describe the symmetries of crystal structures
• The number of space groups in 3D is determined by the combination of the 32 crystallographic point groups and the 14 Bravais lattices

Relationship between point groups and space groups

• Point groups and space groups are closely related, as space groups are built upon the foundation of point group symmetry
• Point groups describe the local symmetry of a molecule or crystal, while space groups describe the global symmetry of the entire crystal structure
• Understanding the relationship between point groups and space groups is essential for a comprehensive analysis of crystal symmetry

Point groups as subgroups of space groups

• Every space group contains one or more point groups as subgroups
• The point group of a space group is obtained by removing all translational symmetry elements from the space group
• For example, the space group $[P4mm](https://www.fiveableKeyTerm:p4mm)$ has the point group $C_{4v}$ as a subgroup

Space groups as extensions of point groups

• Space groups can be viewed as extensions of point groups, incorporating translational symmetry elements
• The translational symmetry elements in a space group are combined with the point group symmetry elements to create the full symmetry of the crystal structure
• For example, the point group $D_{2h}$ can be extended to the space group [object Object],[object Object] by adding translational symmetry elements

Wyckoff positions and site symmetry

• Wyckoff positions are sets of equivalent points in a unit cell that are related by the symmetry operations of the space group
• Each Wyckoff position has a specific site symmetry, which is the point group symmetry of the position
• The site symmetry of a Wyckoff position is a subgroup of the point group of the space group
• Wyckoff positions and site symmetry are important for understanding the arrangement of atoms in a crystal and the resulting physical properties

Applications of point groups and space groups

• Point groups and space groups have numerous applications in solid state physics, chemistry, and materials science
• They provide a framework for understanding the symmetry-dependent properties of molecules and crystals
• Some key applications include the analysis of crystal structures, selection rules for transitions, phonon dispersion relations, and electronic band structures

Symmetry in crystal structures

• Space groups are used to describe the symmetry of crystal structures
• The symmetry of a crystal determines the arrangement of atoms in the unit cell and the overall lattice structure
• Examples of crystal structures with different space groups include diamond ($Fd\bar{3}m$), graphite ([object Object],[object Object]), and perovskite ($Pm\bar{3}m$)

Symmetry-based selection rules for transitions

• Point group symmetry plays a crucial role in determining the selection rules for electronic, vibrational, and rotational transitions in molecules and crystals
• Selection rules dictate which transitions are allowed or forbidden based on the symmetry of the initial and final states
• Examples of symmetry-based selection rules include the Laporte rule (transitions between states of opposite parity are forbidden in centrosymmetric molecules) and the spin selection rule (transitions between states of different spin multiplicity are forbidden)

Symmetry in phonon dispersion relations

• Space group symmetry influences the phonon dispersion relations in crystals
• Phonon modes at high-symmetry points in the Brillouin zone are classified according to the irreducible representations of the point group of the wavevector
• The symmetry of the phonon modes determines their degeneracy and the shape of the dispersion curves
• Examples of high-symmetry points in the Brillouin zone include the $\Gamma$ point (center), X point (edge center), and L point (hexagonal face center)

Symmetry in electronic band structures

• Space group symmetry also affects the electronic band structure of crystals
• The symmetry of the crystal determines the degeneracy and connectivity of the electronic bands
• High-symmetry points in the Brillouin zone are used to label the electronic states and to analyze the band structure
• Examples of electronic band structures with different symmetries include the direct bandgap in GaAs (zinc blende structure, $F\bar{4}3m$) and the indirect bandgap in Si (diamond structure, $Fd\bar{3}m$)

Determination of point groups and space groups

• Determining the point group and space group of a molecule or crystal is essential for understanding its symmetry and physical properties
• Several experimental and computational methods are used to determine the symmetry of a system
• These methods include X-ray diffraction, spectroscopy, and computational symmetry analysis

Experimental methods for symmetry determination

• Experimental techniques such as X-ray diffraction, neutron diffraction, and electron diffraction are used to determine the crystal structure and symmetry
• Spectroscopic methods, such as infrared and Raman spectroscopy, can provide information about the point group symmetry of molecules and crystals
• Polarized light microscopy can be used to identify the crystal system and point group based on the optical properties of the sample

X-ray diffraction and space group determination

• X-ray diffraction is the most common method for determining the space group of a crystal
• The diffraction pattern provides information about the lattice parameters, crystal system, and the presence of certain symmetry elements
• Systematic absences in the diffraction pattern can be used to identify the space group
• Examples of systematic absences include the absence of (0k0) reflections for a 2_1 screw axis along the b-axis and the absence of (h00) reflections for an a-glide plane perpendicular to the a-axis

Computational methods for symmetry analysis

• Computational methods, such as group theory and first-principles calculations, are used to analyze the symmetry of molecules and crystals
• Group theory can be used to determine the point group and space group of a system based on its symmetry elements
• First-principles calculations, such as density functional theory (DFT), can provide information about the electronic structure and symmetry-dependent properties of a system
• Examples of computational symmetry analysis include the use of the

  FINDSYM

  program to identify the space group of a crystal structure and the use of the

  Bilbao Crystallographic Server

  to generate the irreducible representations of a space group

Consequences of symmetry breaking

• Symmetry breaking occurs when a system undergoes a phase transition that lowers its symmetry
• The breaking of symmetry can lead to the emergence of new physical properties and phenomena
• Examples of symmetry breaking include ferroelectricity, piezoelectricity, and magnetism

Ferroelectricity and noncentrosymmetric point groups

• Ferroelectricity is a property of certain materials that exhibit a spontaneous electric polarization that can be reversed by an external electric field
• Ferroelectric materials belong to noncentrosymmetric point groups, which lack an inversion center
• Examples of ferroelectric materials include barium titanate (BaTiO3, point group 4mm) and lead zirconate titanate (PZT, point group 4mm)

Piezoelectricity and polar point groups

• Piezoelectricity is the ability of certain materials to generate an electric charge in response to applied mechanical stress
• Piezoelectric materials belong to polar point groups, which have a unique polar axis
• Examples of piezoelectric materials include quartz (SiO2, point group 32) and zinc oxide (ZnO, point group 6mm)

Magnetism and time-reversal symmetry breaking

• Magnetism arises from the breaking of time-reversal symmetry in a material
• Magnetic materials have a spontaneous magnetic moment that can be aligned by an external magnetic field
• The magnetic symmetry of a material is described by its magnetic point group and magnetic space group
• Examples of magnetic materials include iron (Fe, point group m3m, space group $Im\bar{3}m$) and nickel (Ni, point group m3m, space group $Fm\bar{3}m$)