Use of Projection Diagrams to Represent Crystal Symmetry | Geology Optional Notes for UPSC PDF Download

Crystal Projections

• Cystal projection is a method to represent a 3D crystal on a 2D surface.
• Various projections have specific rules to ensure a reproducible relationship to the crystal.
• This method showcases the symmetry present in crystals, providing detailed graphical representations of point groups.

Principles of Crystallography

• In crystallography, it's essential to depict crystal planes and directions in two dimensions for discussion and measurement purposes.
• Angular relationships in three dimensions of the crystal should be accurately represented in a two-dimensional diagram.

Conformal Projection in Crystallography

• A conformal projection faithfully reproduces angular relationships from three dimensions to two dimensions.
• The primary conformal projection used in crystallography is the stereographic projection.

Origin of Stereographic Projection

• The stereographic projection dates back to the second century A.D., credited to Claudius Ptolemy for representing stars.
• It was used to symbolize the stars on the celestial sphere, and its early documentation comes from a sixteenth-century Latin translation.

Purpose of Stereographic Projection

• The stereographic projection aids in accurately representing angular relationships from 3D to 2D dimensions.
• This technique is crucial in crystallography for visualizing crystal symmetry effectively.

Stereographic Projection in Crystallography

• Introduction to Stereographic Projection

  • Stereographic projection, introduced to crystallography by F. E. Neumann and further developed by W. H. Miller, is a technique used to visualize crystal structures.

• Visualizing Crystal Planes

  • Imagine a crystal positioned with its center at the center of a sphere known as the sphere of projection.
  • Normals to crystal planes are drawn through the center of the sphere, intersecting the sphere's surface at points like P, known as poles.
  • A direction is represented by a point on the sphere's surface where a line parallel to the given direction meets the sphere's surface.
  • Crystal planes can be represented by extending parallel planes through the sphere's center until they intersect the sphere's surface.

• Representation of Directions in Crystals

  • Directions in a crystal, such as normals to lattice planes or lattice directions, are represented by points (poles) on the surface of the sphere.

• Understanding Great Circles

  • When a plane passes through the center of the sphere, it is a diametral plane, and the line of intersection with the sphere forms a great circle.
  • A great circle is a circle on the sphere's surface with a radius equal to the sphere's radius.

Crystal Projections: Understanding Spherical Projection of Crystals

• Spherical Projection Concept

  Spherical projection in crystallography refers to representing crystal structures on a two-dimensional surface. The angle between two planes, defined by their normals, is equivalent to the angle created at the center of the sphere of projection by the arc of the great circle passing through the poles.

• Projection Process

  To simplify the visualization of crystal structures, poles are projected onto a suitable two-dimensional plane, such as a piece of paper. This allows for the preservation of angular relationships in a two-dimensional drawing.

• Types of Projections

  Various projection methods exist, such as orthographic projection, where a pole is projected from a point at infinity onto a plane parallel to the equatorial plane to create a projection point.

• Spherical Projection Analogies

  Imagine spherical projection similar to a terrestrial globe. Analogous to the Earth's north and south poles, we define poles N and S on our projection. The equatorial plane, perpendicular to the NS line, intersects the sphere to form the equator, a great circle.

• Illustrative Example

  Consider a globe where you have a north pole, a south pole, and an equator. Translating this to crystallography, we use spherical projection to map crystal structures onto a flat surface for easier study and analysis.

Crystal Projections Overview

• Orthographic Projection:
  • The orthographic projection involves projecting the pole P onto a plane parallel to the equatorial plane passing through N.
  • It is particularly useful in crystallography for visualizing crystal shapes.
• Gnomonic Projection:
  • In the gnomonic projection, the point of projection is the center of the sphere, projecting the pole at P'G on a plane parallel to the equatorial plane passing through N.
  • It is relevant for labeling electron back-scattered electron diffraction patterns in scanning electron microscopes.
  • Angles are distorted in this projection as it is not conformal.
• Stereographic Projection:
  • In the stereographic projection, the pole P is projected from a point S on the sphere's surface to a plane normal to OS.
  • The equatorial plane (no SO) is commonly used for this projection.
  • The point P' produced on this plane is defined as the stereographic projection of P.
  • The plane of projection intersects the sphere at a great circle known as the primitive circle.

Visualizing Projections

• True Projection Concept:
  • Illustrated in Fig. 3(a), where a pole P₁ in the northern hemisphere projects to P'₁ inside the primitive circle.
• Projection Method:
  • Fig. 3(b) demonstrates how a small circle projects as a circle in crystallography and mineralogy contexts.

Crystal Projections in Geology

Stereographic Projection

• Points on a sphere are projected onto a plane.
• Poles in the northern hemisphere project inside the primitive, while those in the southern hemisphere project outside.
• To simplify working with projected poles outside the primitive, poles in the southern hemisphere can be projected from the north pole.
• Projected poles are marked with rings to differentiate them from true projections.
• The stereographic projection preserves angles and projects circles on the sphere as circles on the plane.

Stereographic Net (Wulff Net)

• A template of the projected coordinate system used for measuring and plotting angles.
• Also known as the Wulff net after G. V. Wulff, a Russian Crystallographer.
• Facilitates the visualization of crystal projections on a 2D surface.
• Helps in understanding the orientation of crystals in crystallography.

Crystallography and Mineralogy Concepts

• Stereographic Projection Overview

  • Primitive Circle: It is the outer circle that encloses the stereonet.
  • Great Circles: These are curved lines connecting the N and S points on the stereonet, including the E-W and N-S axes. Angular measurements between points are possible only along Great Circles.
  • Small Circles: These are highly curved lines that curve both upward and downward on the stereonet.

• Crystal Face Plotting Rules

  • Crystal Face Representation: All crystal faces are represented as poles.
  • Q Angle: The q angle is measured from the b axis in a clockwise sense in the equatorial plane. For instance, the (010) face has a q angle of 0°.
  • P Angle and O Angle: The p angle is the angle between the c axis and the pole to the crystal face, measured downward from the North pole of the sphere. The o angle is measured in the horizontal equatorial plane.
  • Stereographic Projection Device: A stereographic net or stereonet is utilized to facilitate the plotting of stereographic projections.

Crystallography and Mineralogy

Crystal Projections Module

• Crystallographic Axis: The b crystallographic axis serves as the reference point, plotted at p = 0° and p = 90°.
• Measurement of Angles:
  • Positive φ angles are measured clockwise on the stereonet, while negative φ angles are measured counterclockwise.
  • To plot a face, first measure the q angle along the outermost great circle and mark it on tracing paper. Rotate the paper so the mark aligns with the E-W axis of the stereonet. Measure the p angle from the center of the stereonet along the E-W axis.
  • Angles are measured along great circles like the primitive circle, and the E-W and N-S axes of the stereonet.
• Zones: Faces on the same great circle are in the same zone. Zone axes are determined by setting two faces on the same great circle and counting 90° away from their intersection along the E-W axis.
• Plotting Crystal Faces:
  • Crystal faces above the crystal (p < 90) are represented as open circles, while faces below (p > 90) are marked as "⛶" signs.
• Geological Application: Place tracing paper on the stereonet, trace the outermost great circle, and make a reference mark. Follow specific procedures for accurate plotting.
• Illustration: Use stereographic nets for measurements. For instance, if p = 60° and 30°, plot the pole to the face on the stereonet by aligning N of the tracing paper with the N of the net and following the specified steps.

Stereographic Projection in Crystallography

• Overview of Stereographic Projection: Stereographic projection is a technique used in crystallography to represent crystal planes and directions in a two-dimensional format. It allows for the measurement of angles between different crystallographic features.
• Key Concept: In stereographic projection, the crystal is envisioned at the center of a sphere known as the stereographic sphere. The normals to the crystal faces extend out from the center, intersecting the sphere to form points that represent crystal faces or planes, each labeled with the appropriate Miller index.
• Angle Preservation: Stereographic projection maintains the angular relationships between different crystallographic directions while disregarding linear distances. This technique helps in visualizing the symmetry relationships between crystal faces.
• Symmetry Representation: The symmetry of the arrangement of points on the stereographic sphere reflects the symmetry of the crystal structure. By analyzing these points, one can understand the symmetry elements present in the crystal lattice.
• Spherical Projection: In spherical projection, the crystal is placed within a reference sphere or sphere of projection. Crystal planes and directions are then projected onto the surface of this sphere for visualization and analysis.