Comprehensive Introduction to Crystallography: Structures, Symmetry, and Systems

Learning Outcomes and General Overview

• Learning Objectives:

  • Define the nature of a crystal and distinguish it from amorphous substances.

  • Explain the concept of a Unit Cell.

  • Define and apply Steno’s Law of Interfacial Angles.

  • Explain and define the term Polymorphic.

  • Define and draw crystallographic axes.

  • Identify the three types of symmetry (Axis, Plane, Centre).

  • Categorize and describe the seven crystal systems, including their specific crystallographic axes and representative gemstones.

• Significance of Crystallography:

  • Crystals are minerals whose physical and optical properties vary with direction. These include:

    • Optical Properties: Refractive Index, Differential Selective Absorption of Light, and Dichroism.

    • Mechanical Properties: Hardness and Cohesion.

  • Gemstones possess elements of symmetry that dictate results observed through gemmological equipment.

  • Understanding crystallography is essential for interpreting results in a laboratory environment and serves as the foundation for most subsequent topics in gemmology.

Fundamental Definitions of Crystallography

• Crystal: A mineral characterized by three main traits:

  • An orderly and symmetrical atomic structure.

  • A definite external geometrical shape bounded by plane faces.

  • Physical properties which vary with direction.

• Crystalline: A subset of "Crystal" referring to a mineral that possesses an orderly and symmetrical atomic structure and physical properties that vary with direction, but does not possess a definite external shape.

• Amorphous (Non-crystalline):

  • Materials that lack an orderly and symmetrical atomic structure.

  • They do not have a definite external geometrical shape bounded by plane faces.

  • Their physical properties are the same in all directions (isotropic).

  • Example: "Swarovski Crystal" is actually a manufactured glass, which is non-crystalline or amorphous.

Comparative Analysis of Crystal Structures

• Chemical Composition and Arrangement:

  • Crystals are composed of atoms and rarely exist as pure elements (e.g., Diamond is a notable exception).

  • Minerals are described by chemical formulas. For example, Corundum is represented as $Al_2O_3$.

  • Atoms arrange themselves in a specific pattern unique to each mineral, which is repeated throughout the structure.

• Specific Mineral Examples:

  • Olivine (Peridot): Formulated as $(Mg, Fe)_2SiO_4$. It is a silicate comprising two atoms of Magnesium or Iron combined with isolated silicate tetrahedra ($SiO_4$). This molecule consists of one Silicon atom at the center of four Oxygen atoms arranged in a pyramid formation.

  • Diamond: Consists of five Carbon atoms linked by covalent bonds. Each atom serves as the geometric center of four other Carbon atoms. This structure results in more bonds in one specific direction than others, despite all bond lengths and strengths being identical.

The Unit Cell and Steno’s Law

• The Unit Cell:

  • Definition: The smallest possible spatial unit that can possess all the physical, chemical, and geometrical characteristics of the mineral as a whole.

  • Characteristics: This 3-dimensional pattern is unique to the mineral. Similar to "Lego blocks," the crystal is built by stacking these unit cells.

  • Atomic Count: The number of atoms in a unit cell is not necessarily the same as the number of atoms in the chemical formula; it often requires multiples of the formula to establish the repeating pattern.

  • Shapes: The unit cell for Diamond (Cubic system) is a cube, specifically a tetrahedral arrangement. The unit cell for Olivine/Fayalite (Orthorhombic system) is a rectangle, comparable to the shape of a breakfast cereal box.

• Steno’s Law of Interfacial Angles:

  • States that in crystals of the same substance, the angles between corresponding faces always have the same value when measured at the same temperature.

Geometrical Formations and Polymorphism

• Crystal Quality Terms:

  • Euhedral: A crystal that is perfectly formed, showing regular and symmetrical external forms with lustrous plane faces.

  • Anhedral: A crystal that is poorly formed and lacks a perfectly symmetrical external form.

• Polymorphism:

  • Definition: A group of two or more different minerals that share the same chemical composition but possess different physical and geometrical properties.

  • Example: Carbon ($C$):

    • Diamond: Cubic system, Hardness $10$, Dispersion $0.044$, Specific Gravity (SG) $3.52$, Refractive Index (RI) $2.417$.

    • Graphite: Hexagonal system, Hardness $1$ to $2$, Specific Gravity (SG) $2.09$ to $2.23$. Illustrative use includes miniature sculptures carved from pencil lead.

Elements of Symmetry

• Crystallographic Axis:

  • A set of imaginary lines that intersect at a point in space known as the Crystallographic Origin.

• Axis of Symmetry (Rotational Symmetry):

  • An imaginary line passing through the crystal center, about which the crystal can be rotated to show the same appearance multiple times during one full $360^\circ$ rotation.

  • Only $2$, $3$, $4$, or $6$-fold symmetry is possible in crystals.

  • Cubic System Example (13 Total Axes):

    • 3 Axes of 4-fold symmetry: Found by rotating between the centers of opposite faces. The same face appears every $90^\circ$.

    • 4 Axes of 3-fold symmetry: Found by rotating between opposite corners (points). The same pattern appears three times in a full circle.

    • 6 Axes of 2-fold symmetry: Found by rotating between the centers of opposite and parallel edges. The same pattern appears twice in a full circle.

• Plane of Symmetry:

  • An imaginary plane dividing a body into two halves where each is a mirror image of the other. A cube contains $9$ planes of symmetry.

• Centre of Symmetry:

  • Exists when an imaginary line can be passed from any point on the crystal surface through the center to find a similar point at an equal distance on the opposite side.

The Seven Crystal Systems

• 1. Cubic (Isometric):

  • Axes: Three axes ($a_1, a_2, a_3$) of equal length intersecting at $90^\circ$.

  • Examples: Fluorite, Spinel, Garnet, Cubic Zirconia, Diamond, Strontium Titanate, Pyrite.

• 2. Tetragonal:

  • Axes: Three axes intersecting at $90^\circ$. Two axes ($a_1, a_2$) are equal, but the vertical axis ($c$) is either longer or shorter (resembling a square milk carton).

  • Examples: Scapolite, Zircon, Rutile (rare as single specimens), Idocrase.

• 3. Trigonal and 4. Hexagonal:

  • Axes: Both systems utilize four axes. Three ($a_1, a_2, a_3$) intersect at $120^\circ$ in a horizontal plane, while the vertical $c$ axis intersects them at $90^\circ$. The $c$ axis is a different length than the $a$ axes.

  • Trigonal Examples: Calcite, Corundum (Sapphire and Ruby), Quartz, Tourmaline.

  • Hexagonal Examples: Beryl (Emerald, Aquamarine), Apatite, Synthetic Moissanite.

• 5. Orthorhombic:

  • Axes: Three axes ($a, b, c$) all of different lengths, but all intersecting at $90^\circ$ (resembling a rectangular box).

  • Examples: Iolite, Topaz, Kornerupine, Andalusite, Zoisite (Tanzanite), Danburite, Peridot.

• 6. Monoclinic:

  • Axes: Three axes of different lengths. Axes $a$ and $b$ intersect at $90^\circ$. Axis $c$ is at an incline to axis $a$, while intersecting axis $b$ at $90^\circ$.

  • Examples: Moonstone (Feldspar), Orthoclase (Feldspar), Jadeite, Nephrite, Spodumene (Kunzite), Diopside, Titanite (Sphene).

• 7. Triclinic:

  • Axes: Three axes of different lengths, all of which are inclined to each other (no right angles).

  • Examples: Kyanite, Feldspars (Microcline, Amazonite, Oligoclase, Sunstone, Labradorite).