2.3 Hermann-Mauguin notation and stereographic projection
5 min read•august 16, 2024
is a key system for describing crystal symmetry. It uses numbers and letters to represent axes, mirror planes, and other symmetry elements in crystals, making it easier to understand complex structures.
Stereographic projections visualize 3D crystal symmetry on a 2D plane. By plotting symmetry elements as specific symbols, these projections help identify point groups and analyze crystal structures, bridging theory and practical crystal analysis.
Hermann-Mauguin Notation
Fundamentals of Hermann-Mauguin Notation
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- Hermann-Mauguin notation standardized system for describing crystallographic point groups and space groups
- Combines numbers and letters to represent symmetry elements in crystal structures
- Writes symbols in specific order rotational symmetries listed first, followed by mirror planes and centers
- Distinguishes between proper rotation axes (n) and improper rotation axes (n̄)
- Describes both two-dimensional and three-dimensional point groups
- Concisely represents complex symmetry arrangements in crystals
- Crucial for interpreting crystallographic data and communicating crystal structures in scientific literature
- Developed by Carl Hermann and Charles-Victor Mauguin in the early 20th century
- Widely adopted in crystallography due to its clarity and efficiency in describing symmetry
Applications and Importance
- Enables precise communication of crystal structures among researchers
- Facilitates comparison of different crystal structures by providing a standardized description
- Used in crystallographic databases to catalog and organize structural information
- Essential for understanding and predicting physical properties of crystals (optical, electrical, magnetic)
- Aids in the analysis of diffraction patterns obtained from X-ray crystallography
- Crucial in materials science for designing and engineering new materials with specific symmetry properties
- Helps in identifying potential applications of crystals in various fields (optics, electronics, pharmaceuticals)
- Allows for systematic classification of crystals based on their symmetry characteristics
Symbols in Hermann-Mauguin Notation
Rotational Symmetry and Basic Symbols
- Numbers 1, 2, 3, 4, and 6 indicate order of rotation axes
- Letter 'm' represents
- Letter 'n' denotes in space groups
- Number '1̄' (1 with an overbar) represents inversion center
- Symbol '/' separates generators of , indicating relative orientations
- Proper rotation axes denoted by simple numbers (2, 3, 4)
- Improper rotation axes indicated by numbers with overbars (3̄, 4̄)
- Position and order of symbols provide information about relative orientations of symmetry elements
- Special symbols '.' or '_' indicate absence of certain symmetry elements in specific directions
Advanced Symbol Combinations and Interpretations
- Combination of and mirror plane perpendicular to it written as nm (2m, 3m, 4m, 6m)
- Rotation axis with mirror plane parallel to it denoted as n/m (2/m, 4/m, 6/m)
- Symbol 2mm represents two perpendicular mirror planes intersecting along a 2-fold axis
- 4mm indicates a 4-fold axis with four mirror planes intersecting at 45° angles
- 3m represents a 3-fold axis with three mirror planes at 60° angles
- Symbol 6̄ denotes a 3-fold rotoinversion axis (combination of 3-fold rotation and inversion)
- 4̄2m represents a 4-fold rotoinversion axis with two perpendicular mirror planes
- Interpretation of complex symbols requires understanding of how individual symmetry elements combine
Stereographic Projections of Point Groups
Construction and Basic Principles
- Stereographic projection represents three-dimensional crystal symmetry on two-dimensional plane
- Created by intersecting symmetry elements with reference sphere and projecting onto plane
- Symmetry elements represented by specific symbols dots for rotation axes, solid lines for mirror planes, open circles for inversion centers
- Construction process involves plotting symmetry elements according to orientations and multiplicities in crystal structure
- Choice of projection direction (typically along principal symmetry axis) affects appearance and interpretation
- Can be constructed for all 32 crystallographic point groups, each with unique arrangement of symmetry elements
- Requires understanding of both point group symmetry and projection geometry
- Uses stereographic net (Wulff net) as a tool for precise plotting of symmetry elements
Interpretation and Analysis of Projections
- Provides visual representation of all symmetry elements present in crystal structure
- Arrangement and types of symmetry elements used to deduce crystal's point group
- Rotation axes identified by examining multiplicity and arrangement of equivalent points
- Mirror planes appear as straight lines, often bisecting sets of equivalent points
- Inversion centers typically located at center of projection, produce characteristic patterns of symmetry-related points
- Presence or absence of certain symmetry elements in specific orientations helps narrow down possible point group
- Comparison of unknown crystal's projection with standard projections of 32 point groups allows accurate identification
- Analysis requires consideration of both visible symmetry elements and those implied by combination of visible elements
- Symmetry-equivalent points on projection connected by great circles or small circles
Identifying Symmetry Elements vs Point Groups
Recognizing Individual Symmetry Elements
- Rotation axes identified by sets of equivalent points arranged in circular patterns
- 2-fold axes create pairs of symmetry-related points
- 3-fold axes produce triangular arrangements of equivalent points
- 4-fold axes result in square patterns of symmetry-related points
- 6-fold axes generate arrangements of equivalent points
- Mirror planes appear as straight lines dividing projection into symmetrical halves
- Inversion centers produce characteristic centrally symmetric patterns
- Improper rotation axes (rotoinversion) combine features of rotation and inversion
- Careful examination of point distributions and their relationships crucial for accurate identification
Deducing Point Groups from Symmetry Combinations
- Combination of observed symmetry elements narrows down possible point groups
- Presence of single mirror plane indicates point group m or 2/m
- Multiple intersecting mirror planes suggest higher symmetry groups (mm2, 4mm, 6mm)
- Observation of 3-fold axis with mirror plane indicates point group 3m
- Combination of 4-fold axis and two sets of mirror planes at 45° suggests 4/mmm
- Presence of inversion center with other elements points to centrosymmetric groups
- Absence of mirror planes or inversion center indicates possibility of enantiomorphic groups
- Consideration of crystal system (, , orthorhombic) helps in narrowing down options
- Systematic approach comparing observed symmetry with standard point group descriptions leads to accurate identification
Key Terms to Review (23)
A-axis: The a-axis is one of the three principal axes in a crystal lattice that defines the unit cell's dimensions and orientation. In crystallography, this axis typically corresponds to the longest edge of the unit cell in a monoclinic or orthorhombic crystal system, or can vary based on symmetry in different crystal systems. Understanding the a-axis is crucial for accurately describing crystal structures and their relationships to various symmetry operations and projection techniques.
B-axis: The b-axis is one of the three principal axes used to describe the geometry of a crystal lattice in crystallography, specifically representing the direction along the second axis. It plays a vital role in defining the unit cell dimensions and orientation in relation to the crystal's symmetry. Understanding the b-axis is essential for interpreting both Hermann-Mauguin notation and stereographic projections, as well as characterizing the properties of different crystal systems.
C-axis: The c-axis is one of the principal axes in a crystal system, specifically referring to the vertical axis in a three-dimensional lattice structure. It plays a crucial role in defining the geometry and symmetry of the crystal, affecting how other axes are oriented and measured. Understanding the c-axis is essential for determining crystal parameters, which are represented in Hermann-Mauguin notation, as well as for categorizing crystals into one of the seven distinct crystal systems based on their symmetry properties.
Classification of minerals: The classification of minerals is the systematic arrangement of minerals into groups based on their chemical composition and crystal structure. This classification helps in understanding the properties, uses, and origins of different minerals, facilitating their identification and study in fields like geology and material science. The use of specific notations and projections is essential in visualizing and analyzing these classifications effectively.
Cubic: Cubic refers to a specific geometric shape characterized by having equal edge lengths and right angles between all adjacent edges, often associated with crystal systems and lattices. This term plays a crucial role in understanding crystal structures, as cubic symmetry influences various physical properties and behaviors of materials across many fields.
Glide Plane: A glide plane is a type of symmetry operation in crystallography that combines translation along a direction with reflection across a plane. This operation results in the movement of points in the crystal structure, which can significantly influence the arrangement of atoms and the properties of the crystal. Glide planes are essential in defining the symmetry elements of a crystal system and play a critical role in Hermann-Mauguin notation and stereographic projection.
Great Circle: A great circle is the largest circle that can be drawn on the surface of a sphere, representing the shortest path between two points on that sphere. In the context of crystallography, great circles are essential in understanding the orientation of crystals and their properties, especially when using Hermann-Mauguin notation and stereographic projection to visualize symmetry and orientations in three-dimensional space.
Hermann-Mauguin notation: Hermann-Mauguin notation is a system used to describe the symmetry of crystalline structures using specific symbols. This notation helps in identifying the various symmetry elements present in crystals, including rotation axes, mirror planes, and inversion centers. By translating these symmetry features into a standardized format, Hermann-Mauguin notation allows for a clearer understanding and communication of the three-dimensional symmetry of different crystal classes.
Hexagonal: Hexagonal refers to a crystal system characterized by a six-fold rotational symmetry and a lattice structure defined by three equal axes in a plane, intersecting at 120-degree angles, and a fourth axis that is perpendicular to this plane. This unique arrangement leads to various properties and behaviors in crystalline materials, making it essential in understanding their significance in multiple fields.
Identifying Crystal Classes: Identifying crystal classes involves categorizing crystals based on their symmetry properties and the arrangement of their atoms, which helps in understanding their behavior and characteristics. This classification is crucial in crystallography as it links the geometric aspects of a crystal structure to its physical properties. By using specific notations and visualizations, one can efficiently communicate and analyze different crystal systems and their related forms.
Inversion: Inversion refers to a symmetry operation that transforms a point in a crystal to its equivalent position through a central point, effectively flipping the structure through that point. This operation is significant in understanding the overall symmetry of crystals, allowing for the classification and analysis of crystal structures by how they behave under various transformations. Inversion helps to identify point groups and can influence the Hermann-Mauguin notation used to describe symmetry in crystals.
Lattice Parameters: Lattice parameters are the measurements that define the size and shape of the unit cell in a crystal lattice. They are crucial for understanding the geometric arrangement of atoms in a crystal, and include parameters such as the lengths of the cell edges and the angles between them. These parameters not only help in describing different crystal systems but also connect to concepts such as symmetry and the properties of various crystal structures.
Mirror Plane: A mirror plane is a symmetry element in which one half of an object is the mirror image of the other half when divided by that plane. This concept highlights the reflective symmetry of a crystal structure, indicating that if you were to fold the object along the mirror plane, both halves would align perfectly. Understanding mirror planes is essential for identifying point groups and recognizing how symmetry operations can characterize the geometric properties of crystals.
Point Group: A point group is a set of symmetry operations that leave at least one point unchanged in a molecule or crystal structure. This collection of operations includes rotations, reflections, and inversions that describe the symmetrical properties of the structure. Understanding point groups is essential for analyzing crystal structures and their symmetry in various contexts, from determining the Bravais lattices to employing techniques like X-ray diffraction.
Projection Plane: A projection plane is an imaginary or defined surface onto which a three-dimensional object is projected to create a two-dimensional representation of that object. In the context of crystallography, this concept is essential for visualizing crystal structures and understanding the orientation of planes within a crystal, particularly when using Hermann-Mauguin notation and stereographic projections.
Reflection: Reflection is a symmetry operation that involves flipping a structure across a plane, resulting in a mirror image. This operation plays a crucial role in determining the symmetry of crystalline structures and is essential for understanding point groups and their associated symmetry operations. Reflection helps in visualizing how certain features of a crystal can be transformed, aiding in the classification and analysis of crystalline materials.
Rotation: Rotation is a symmetry operation that involves turning a crystal or geometric object around a fixed point or axis by a certain angle. This operation is critical in understanding how shapes and structures maintain their integrity when subjected to changes in orientation, especially in the study of crystal systems. Recognizing the significance of rotation helps in identifying point groups and their associated symmetry operations, as well as in applying Hermann-Mauguin notation for crystallographic descriptions.
Rotation Axis: A rotation axis is an imaginary line around which a crystal structure can be rotated by a certain angle to produce an equivalent arrangement of its components. This concept is essential in understanding symmetry operations in crystallography, particularly how Hermann-Mauguin notation classifies various rotational symmetries and how these symmetries can be represented in stereographic projections for visualizing crystal orientations.
Schoenflies notation: Schoenflies notation is a systematic way to describe the symmetry of molecular and crystalline structures by categorizing them into point groups. Each point group represents a specific set of symmetry operations, such as rotations and reflections, that can be performed on a molecule or crystal without changing its overall shape. This notation allows for clear communication about the symmetry properties of various structures in a concise manner.
Space group: A space group is a mathematical description of the symmetry of a crystal structure, combining both translational and rotational symmetry elements. It provides critical information for understanding how atoms are arranged in a crystal and how they interact with each other, which is essential for analyzing various crystal forms, including the arrangement of points in a Bravais lattice and the properties of specific materials.
Stereographic Pole: A stereographic pole is a point on a stereographic projection that represents the orientation of a crystal face or a direction in three-dimensional space. It is essential for visualizing how crystal structures relate to one another and for understanding symmetry in crystallography. By mapping these poles onto a two-dimensional plane, one can easily analyze and interpret the crystallographic relationships and orientations of different planes and axes.
Tetragonal: Tetragonal is one of the seven crystal systems characterized by a unit cell that has two equal axes and one unique axis, resulting in a rectangular prism shape. This symmetry and dimensionality relate closely to symmetry operations, lattice parameters, and crystal structures, making it essential for understanding crystal behavior and properties in materials science.
Unit Cell Dimensions: Unit cell dimensions refer to the specific measurements that define the size and shape of the smallest repeating unit in a crystal lattice. These dimensions, typically represented as lengths of the edges and angles between them, are critical for characterizing the geometry of various crystal structures, influencing their properties and how they are represented in different notations.