💎Mathematical Crystallography Unit 3 – Point Groups & Crystal Classes

Key Concepts

• Point groups describe the symmetry of molecules and crystals by identifying all symmetry operations that leave the object unchanged
• Crystal classes are determined by the combination of symmetry elements present in a crystal structure
• Symmetry operations include rotation, reflection, inversion, and improper rotation which can be combined to form point groups
• The 32 crystallographic point groups are divided into 7 crystal systems based on their symmetry elements and lattice parameters
• Hermann-Mauguin notation and Schoenflies symbols are used to represent point groups and symmetry operations
  • Hermann-Mauguin notation uses a combination of numbers and letters to describe symmetry elements (2, m, 4̄)
  • Schoenflies symbols use letters and subscripts to denote symmetry operations (C2, Ci, D4)
• Understanding point groups is essential for analyzing crystal structures, predicting properties, and solving crystallographic problems
• The relationship between point groups and space groups is crucial for understanding the symmetry of crystal structures

Symmetry Operations

• Rotation (Cn) involves rotating an object by 360°/n around an axis, where n is the order of rotation (C2, C3, C4)
  • Rotation axes can be parallel to crystallographic axes (a, b, c) or inclined at specific angles
• Reflection (σ) involves reflecting an object across a mirror plane, resulting in a mirror image
  • Mirror planes can be perpendicular to rotation axes (vertical) or contain rotation axes (horizontal, dihedral)
• Inversion (i) involves inverting an object through a point, resulting in a centrosymmetrically related object
• Improper rotation (Sn) combines rotation and reflection, rotating an object by 360°/n followed by reflection through a plane perpendicular to the rotation axis
• Identity (E) is a trivial symmetry operation that leaves the object unchanged
• Symmetry operations can be combined to form point groups, with each point group having a unique set of symmetry elements
• The order of a point group is the number of symmetry operations it contains, ranging from 1 to 48

Point Group Classification

• Point groups are classified based on the presence and combination of symmetry elements
• The 32 crystallographic point groups are divided into 7 crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic
  • Triclinic system has no symmetry elements other than identity (1, 1̄)
  • Monoclinic system has one 2-fold rotation axis or one mirror plane (2, m, 2/m)
  • Orthorhombic system has three perpendicular 2-fold rotation axes or mirror planes (222, mm2, mmm)
• Point groups within each crystal system are further classified based on the specific combination of symmetry elements
• Laue classes are subgroups of point groups that share the same set of symmetry operations without considering translational symmetry
• Centrosymmetric point groups contain an inversion center, while non-centrosymmetric point groups lack an inversion center
• Enantiomorphic point groups occur in pairs, related by a mirror reflection or inversion, and are important for studying chiral molecules and crystals

Crystal Systems

• The 7 crystal systems are defined by the minimum symmetry requirements and the relationships between lattice parameters (a, b, c, α, β, γ)
  • Triclinic: a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°
  • Monoclinic: a ≠ b ≠ c, α = γ = 90° ≠ β
  • Orthorhombic: a ≠ b ≠ c, α = β = γ = 90°
  • Tetragonal: a = b ≠ c, α = β = γ = 90°
  • Trigonal: a = b = c, α = β = γ ≠ 90°
  • Hexagonal: a = b ≠ c, α = β = 90°, γ = 120°
  • Cubic: a = b = c, α = β = γ = 90°
• Each crystal system contains a unique set of point groups, with higher symmetry systems having more point groups
• The relationship between crystal systems and Bravais lattices is important for understanding the translational symmetry of crystal structures
• Crystal systems can be primitive (P) or centered (A, B, C, I, F), depending on the presence of additional lattice points
• The symmetry of a crystal system determines the possible crystal habits and physical properties of the material

Notation and Symbols

• Hermann-Mauguin notation and Schoenflies symbols are the two main notation systems used to represent point groups and symmetry operations
• Hermann-Mauguin notation:
  • Uses a combination of numbers, letters, and symbols to describe symmetry elements
  • The principal axis is listed first, followed by symmetry elements perpendicular to it and then symmetry elements parallel to it
  • Examples: 2/m (monoclinic), 4mm (tetragonal), 6̄m2 (hexagonal)
• Schoenflies symbols:
  • Use letters and subscripts to denote symmetry operations
  • The principal axis is represented by a letter (C, D, T, O, I), with subscripts indicating the order of rotation
  • Additional symbols are used for mirror planes (s, h, v) and improper rotation axes (S)
  • Examples: C2h (monoclinic), D4h (tetragonal), D3d (trigonal)
• International Tables for Crystallography use Hermann-Mauguin notation as the standard for describing point groups and space groups
• Knowing both notation systems is important for communicating and understanding crystallographic information across different sources

Applications in Crystallography

• Point groups are used to predict and analyze the physical properties of crystals, such as optical activity, piezoelectricity, and ferroelectricity
  • Non-centrosymmetric point groups can exhibit optical activity and piezoelectricity
  • Certain point groups (10 polar groups) allow for ferroelectricity
• Symmetry information is crucial for structure determination using X-ray, neutron, or electron diffraction techniques
  • Systematic absences in diffraction patterns are related to the presence of certain symmetry elements
  • The symmetry of the crystal determines the number and intensity of diffraction peaks
• Point groups are used to describe the symmetry of molecules, which is important for understanding chemical bonding, reactivity, and spectroscopic properties
• Symmetry-adapted perturbation theory (SAPT) uses point group symmetry to simplify the calculation of intermolecular interactions
• Symmetry considerations are important for designing materials with specific properties, such as nonlinear optical materials or ferroelectric memory devices

Problem-Solving Techniques

• Identifying the point group of a molecule or crystal:
  • Analyze the symmetry elements present in the object
  • Determine the highest-order rotation axis and the presence of mirror planes or inversion center
  • Use flow charts or decision trees to systematically identify the point group
• Determining the crystal system from the point group:
  • Identify the minimum symmetry requirements for each crystal system
  • Match the point group to the corresponding crystal system based on its symmetry elements
• Applying symmetry operations to molecules or crystal structures:
  • Use matrices or geometric transformations to apply rotation, reflection, or inversion operations
  • Determine the coordinates of symmetry-equivalent atoms or molecules
• Analyzing diffraction patterns using symmetry information:
  • Identify systematic absences based on the presence of certain symmetry elements (glide planes, screw axes)
  • Determine the possible space groups consistent with the observed diffraction pattern and symmetry constraints
• Utilizing symmetry to simplify calculations or measurements:
  • Apply symmetry-adapted basis functions to reduce the complexity of quantum mechanical calculations
  • Use symmetry to predict the number and orientation of independent components for physical properties (elastic constants, optical indicatrix)

Advanced Topics

• Relationship between point groups and space groups:
  • Space groups combine point group symmetry with translational symmetry (lattice centering, glide planes, screw axes)
  • There are 230 space groups, which are classified into 73 symmorphic and 157 non-symmorphic groups
  • Symmorphic space groups can be obtained by combining point groups with Bravais lattices
  • Non-symmorphic space groups contain glide planes or screw axes, which involve both point group symmetry and translation
• Irreducible representations and character tables:
  • Irreducible representations are the simplest matrix representations of a point group that cannot be further reduced
  • Character tables summarize the trace (sum of diagonal elements) of each irreducible representation for each symmetry operation
  • Character tables are used to determine the symmetry-adapted linear combinations (SALCs) of atomic orbitals or vibrational modes
• Symmetry-breaking phenomena:
  • Certain physical phenomena, such as ferroelectricity or magnetic ordering, can break the symmetry of a crystal structure
  • Symmetry breaking can lead to phase transitions and the emergence of new physical properties
  • Landau theory is used to describe symmetry-breaking phase transitions based on changes in the order parameter
• Quasicrystals and incommensurate structures:
  • Quasicrystals possess long-range order but lack translational periodicity, exhibiting forbidden rotational symmetries (5-fold, 8-fold, 12-fold)
  • Incommensurate structures have a mismatch between the periodicities of different sublattices, leading to modulated structures
  • The symmetry of quasicrystals and incommensurate structures is described using higher-dimensional crystallography and superspace groups