Unit 2: Crystal Systems and Projection

Classification of Crystals into 7 Systems
Symmetry Elements
Description of Symmetry Elements of Normal Classes
Stereographic Projection
- Concepts and Principles
- Construction and Use of Wulff's Net

Classification of Crystals into 7 Systems

Crystals are classified into seven systems based on their crystallographic axes (lengths and angles) and their minimum required symmetry. All 32 possible crystal classes can be grouped into these seven systems. The "Normal Class" of each system is the class with the highest symmetry.

The 7 Crystal Systems
System	Axial Relationships (Lengths)	Axial Relationships (Angles)	Minimum Required Symmetry	Example Mineral
Cubic (Isometric)	a = b = c	α = β = γ = 90°	Four 3-fold axes of rotation	Pyrite, Garnet, Halite
Tetragonal	a = b ≠ c	α = β = γ = 90°	One 4-fold axis of rotation	Zircon, Rutile
Orthorhombic	a ≠ b ≠ c	α = β = γ = 90°	Three 2-fold axes or three mirror planes	Barite, Topaz, Staurolite
Hexagonal	a₁ = a₂ = a₃ ≠ c	γ = 120° (between a-axes), α = β = 90°	One 6-fold axis of rotation	Beryl, Apatite
Trigonal (Rhombohedral)	a₁ = a₂ = a₃ ≠ c or a = b = c	(Hexagonal axes) or α = β = γ ≠ 90°	One 3-fold axis of rotation	Calcite, Quartz, Tourmaline
Monoclinic	a ≠ b ≠ c	α = γ = 90°, β > 90°	One 2-fold axis or one mirror plane	Gypsum, Orthoclase, Muscovite
Triclinic	a ≠ b ≠ c	α ≠ β ≠ γ ≠ 90°	None (or just a center of symmetry)	Albite, Kyanite, Microcline

Mnemonic for Systems: A common way to remember the systems in order of decreasing symmetry is: C-T-O-H-T-M-T ("Can Tom Order Hot Tea? Maybe Tomorrow.").

Symmetry Elements

Symmetry operations are actions (like rotation or reflection) that, when performed on an object, result in an object that is indistinguishable from the original. The "element" is the geometric feature (a point, line, or plane) about which the operation is performed.

Plane of Symmetry (or Mirror Plane, 'm'): An imaginary plane that divides a crystal into two halves, where one half is the mirror image of the other.
Axis of Rotation (or Rotation Axis, 'A'): An imaginary line passing through the crystal. When the crystal is rotated around this line, it appears identical to its original position two or more times in a 360° rotation.
- 2-fold (A₂): Identical every 180° (2 times)
- 3-fold (A₃): Identical every 120° (3 times)
- 4-fold (A₄): Identical every 90° (4 times)
- 6-fold (A₆): Identical every 60° (6 times)
Note: 5-fold and >6-fold axes are not possible in crystals because they cannot fill 3D space without gaps (a "tessellation" problem).
Center of Symmetry (or Inversion, 'i'): An imaginary point in the center of the crystal. Any line drawn from a face or point on the surface, through the center, will find an identical face or point at an equal distance on the opposite side.
Axis of Roto-inversion ('Ā' or 'bar'): A compound symmetry operation. It involves two steps: (1) rotation around an axis, followed by (2) inversion through a center point.
- -1 (bar-1): 1-fold rotation (360°) + inversion = Same as a Center of Symmetry (i).
- -2 (bar-2): 2-fold rotation (180°) + inversion = Same as a Mirror Plane (m).
- -4 (bar-4): 4-fold rotation (90°) + inversion. This is a unique operation.

Description of Symmetry Elements of Normal Classes

The Normal Class (or Holohedral Class) is the class within each crystal system that possesses the highest possible symmetry for that system. When you define a system by its axes, the normal class is the one that has symmetry elements (planes, axes) aligned with all of those axes.

Cubic (Normal Class): 3 A₄, 4 A₃, 6 A₂, 9 m, i. (e.g., Galena, Halite)
- 3 four-fold axes (through the centers of opposite faces)
- 4 three-fold axes (through opposite corners/apices)
- 6 two-fold axes (through the centers of opposite edges)
- 9 mirror planes (3 parallel to faces, 6 diagonal)
- 1 center of symmetry
Tetragonal (Normal Class): 1 A₄, 4 A₂, 5 m, i. (e.g., Zircon)
- 1 four-fold axis (the 'c' axis)
- 4 two-fold axes (at 90° to the A₄)
- 5 mirror planes (1 horizontal, 4 vertical)
- 1 center of symmetry
Orthorhombic (Normal Class): 3 A₂, 3 m, i. (e.g., Barite)
- 3 two-fold axes (the 3 crystallographic axes a, b, c)
- 3 mirror planes (perpendicular to each axis)
- 1 center of symmetry
Hexagonal (Normal Class): 1 A₆, 6 A₂, 7 m, i. (e.g., Beryl)
- 1 six-fold axis (the 'c' axis)
- 6 two-fold axes (at 90° to the A₆)
- 7 mirror planes (1 horizontal, 6 vertical)
- 1 center of symmetry
Trigonal (Normal Class): 1 A₃, 3 A₂, 3 m, i. (e.g., Calcite)
- 1 three-fold axis (the 'c' axis)
- 3 two-fold axes (at 90° to the A₃)
- 3 mirror planes (vertical)
- 1 center of symmetry
Monoclinic (Normal Class): 1 A₂, 1 m, i. (e.g., Gypsum)
- 1 two-fold axis (the 'b' axis)
- 1 mirror plane (perpendicular to the 'b' axis)
- 1 center of symmetry
Triclinic (Normal Class): i (Just a center of symmetry). (e.g., Albite)
- This class has the lowest possible symmetry, with only a center. (The absolute lowest class has no symmetry at all).

Stereographic Projection

Concepts and Principles

A stereographic projection is a graphical method used to represent the 3D angular relationships of a crystal's faces and symmetry elements on a 2D piece of paper. It is an essential tool for crystallographers.

Principle of Projection:

Imagine the crystal is at the center of a hollow sphere (the "projection sphere").
From the center of the sphere, draw a perpendicular line (a "normal") from each crystal face outwards until it intersects the sphere's surface. This intersection point is called a pole (P).
All the angular information of the crystal is now represented by these poles on the sphere.
To get this onto 2D paper, we project these poles. Imagine the sphere is sitting on a flat plane (the "equatorial plane" or "primitive circle").
Poles on the upper hemisphere are projected by drawing a line from the South Pole (S), through the pole (P), until it hits the equatorial plane. These are marked with an open circle (○).
Poles on the lower hemisphere are projected by drawing a line from the North Pole (N), through the pole (P), until it hits the equatorial plane. These are marked with a solid dot (●) or a cross (x).

The resulting 2D diagram (a circle with dots and circles) is the stereogram. It accurately preserves all angular relationships, even though distances are distorted. Faces on the "equator" (the primitive circle) project onto the circle itself.

Construction and Use of Wulff's Net

A Wulff's Net is the "graph paper" used for stereographic projection. It is not a projection of a crystal itself, but a projection of the lines of latitude (small circles) and longitude (great circles) of the projection sphere.

Construction: The Wulff's net is a stereographic projection of a sphere's graticule (longitude and latitude lines), typically spaced at 2° or 5° intervals.

The straight horizontal line is the "equator."
The straight vertical line is a "great circle" passing through the poles.
The curved lines running from top to bottom are other great circles (representing longitude).
The curved lines running from side to side are small circles (representing latitude).

Use of the Wulff's Net:

Plotting Poles: If you know the angles of a crystal face (e.g., from a goniometer), you can use the Wulff's net to plot its pole accurately.
Measuring Angles: The primary use. To measure the angle between two poles (A and B) on your stereogram:
1. Place your stereogram (on tracing paper) over the Wulff's net.
2. Rotate the tracing paper until both poles A and B lie on the same great circle (one of the curved longitude lines).
3. Count the number of degrees along that great circle between A and B. This is the true interfacial angle.
Visualizing Symmetry: You can plot symmetry elements on the stereogram (e.g., mirror planes are shown as heavy lines, rotation axes as specific symbols) to visualize the crystal's symmetry class.