2.2 Point groups and symmetry operations

Point groups are the backbone of crystal symmetry, describing how atoms arrange themselves in 3D space. They're like the secret code that unlocks a crystal's structure, properties, and behavior. There are 32 unique point groups, each with its own set of symmetry operations.

Understanding point groups is crucial for predicting how crystals will react to light, electricity, and other forces. By identifying a crystal's point group, we can forecast its shape, optical properties, and even its potential for applications like piezoelectricity.

Point Groups and Crystal Symmetry

Definition and Significance of Point Groups

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• Point groups mathematically describe symmetry operations leaving at least one point unmoved in crystal structures
• 32 crystallographic point groups represent all possible combinations of symmetry elements in three-dimensional crystals
• Point groups fundamentally describe symmetry of crystal structures and their physical properties
• Symmetry of a crystal's point group determines its morphology, optical properties, and behavior under applied fields (piezoelectricity, ferroelectricity)
• Denote point groups using Hermann-Mauguin symbols or Schoenflies notation providing concise representation of symmetry elements
  • Hermann-Mauguin symbols use letters and numbers (e.g. 4mm, 6/m)
  • Schoenflies notation uses letters and subscripts (e.g. C4v, D6h)

Crystal Systems and Point Group Classification

• Seven crystal systems form basis for classifying crystals into point groups
  • Triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, cubic
• Each crystal system divides into point groups based on specific combination of symmetry elements
  • Triclinic system: two point groups (1, -1)
  • Monoclinic system: three point groups (2, m, 2/m)
  • Orthorhombic system: three point groups (222, mm2, mmm)
  • Tetragonal system: seven point groups (4, -4, 4/m, 422, 4mm, -42m, 4/mmm)
  • Trigonal system: five point groups (3, -3, 32, 3m, -3m)
  • Hexagonal system: seven point groups (6, -6, 6/m, 622, 6mm, -6m2, 6/mmm)
  • Cubic system: five point groups (23, m-3, 432, -43m, m-3m)
• Organize point groups into families based on highest-order rotation axis
  • Cyclic, dihedral, tetrahedral, octahedral, icosahedral families
• Divide point groups into centrosymmetric and non-centrosymmetric categories
  • Centrosymmetric crystals possess inversion center
  • Non-centrosymmetric crystals lack inversion center
  • Categorization impacts crystal properties (piezoelectricity, pyroelectricity)

Symmetry Operations in Point Groups

Rotational Symmetry

• Rotation symmetry operation rotates object about axis by specific angle
• Common rotations include 2-fold, 3-fold, 4-fold, and 6-fold
  • 2-fold rotation: 180° rotation (order 2)
  • 3-fold rotation: 120° rotation (order 3)
  • 4-fold rotation: 90° rotation (order 4)
  • 6-fold rotation: 60° rotation (order 6)
• Rotation axes represented by symbols in point group notation
  • 2-fold: 2
  • 3-fold: 3
  • 4-fold: 4
  • 6-fold: 6

Reflection and Inversion Symmetry

• Reflection symmetry operation mirrors object across plane creating mirror image
  • Mirror planes denoted by 'm' in point group notation
  • Types of mirror planes: vertical, horizontal, diagonal
• Inversion symmetry operation inverts all points through center of symmetry
  • Effectively rotates object 180° and reflects through point
  • Inversion center denoted by '-1' or 'i' in point group notation
• Rotoinversion combines rotation and inversion
  • Rotate object then invert through point on rotation axis
  • Denoted by '-n' where n is the rotation order (e.g. -4, -3)

Complex Symmetry Operations

• Improper rotation (rotoreflection) combines rotation with reflection
  • Rotate object then reflect in plane perpendicular to rotation axis
  • Denoted by overbar symbol in some notations (e.g. $\bar{4}$)
• Identity operation leaves object unchanged
  • Included in all point groups as fundamental symmetry operation
  • Represented by '1' in point group notation
• Screw axes combine rotation with translation
  • Important in space groups but not in point groups
  • Example: 21 axis rotates 180° and translates 1/2 unit cell length

Classifying Crystals by Point Groups

Systematic Approach to Classification

• Identify all symmetry elements present in crystal structure
• Use highest-order rotation axis as primary axis for describing point group
• Determine presence of mirror planes, inversion centers, other symmetry elements in relation to primary axis
• Systematically eliminate point groups not matching observed symmetry elements
• Consider crystal's physical properties for additional confirmation
  • Pyroelectricity indicates polar point group (10 polar point groups)
  • Optical activity suggests non-centrosymmetric point group (21 non-centrosymmetric point groups)

Tools for Point Group Determination

• Visual aids assist in process of point group determination
  • Stereographic projections represent three-dimensional symmetry in two dimensions
  • Symmetry diagrams illustrate arrangement of symmetry elements
• Use point group flow charts for systematic identification
  • Start with crystal system, then narrow down based on symmetry elements
• Software tools automate point group determination from crystal structure data
  • Examples: PLATON, CrystalMaker, Mercury

Identifying Point Groups of Crystals

Practical Steps for Point Group Identification

• Observe crystal morphology and measure interfacial angles
  • Use goniometer for precise angle measurements
  • Compare measured angles to theoretical values for different point groups
• Analyze optical properties using polarized light microscopy
  • Observe extinction patterns, interference figures
  • Determine optical indicatrix shape (uniaxial, biaxial)
• Perform X-ray diffraction experiments
  • Analyze systematic absences in diffraction pattern
  • Determine Laue class from diffraction symmetry
• Consider physical properties related to symmetry
  • Piezoelectricity indicates non-centrosymmetric point group
  • Ferroelectricity suggests polar point group

Advanced Techniques and Considerations

• Use electron diffraction for small crystals or nanostructures
  • Convergent beam electron diffraction (CBED) reveals point group symmetry
• Apply group theory concepts to analyze vibrational spectra
  • Infrared and Raman spectroscopy provide information on molecular symmetry
• Consider symmetry breaking due to external factors
  • Temperature changes can induce phase transitions altering point group
  • Applied electric or magnetic fields may lower symmetry temporarily
• Utilize computational methods for complex structures
  • Density functional theory (DFT) calculations predict stable crystal structures
  • Symmetry analysis software automates point group determination from atomic coordinates