Unit 1: Fundamentals of Crystallography

Parts of Crystals

A crystal is a 3D solid bounded by flat surfaces. These geometric components are the basic "parts" of a crystal.

Face, Edge, Apex (Vertex)

• Face: A flat, planar surface on the exterior of a crystal. Each face represents a plane of atoms with a specific orientation in the crystal's internal lattice.
• Edge: The sharp, linear intersection where two adjacent crystal faces meet.
• Apex (or Vertex): A point or corner where three or more edges (and faces) intersect.

Solid Angle

This is simply another term for an apex or vertex. It is the "corner" formed by the intersection of three or more faces, enclosing a three-dimensional angle.

Interfacial Angle

This is one of the most fundamental measurements in crystallography.

Interfacial Angle: The angle between the perpendicular lines (called "normals") drawn to two adjacent crystal faces.

Important: It is not the internal or external angle between the faces themselves. This measurement is taken using an instrument called a goniometer. The constancy of this angle is the basis for the first law of crystallography.

Zone

A Zone is a group or set of crystal faces whose intersection edges are all mutually parallel.

• Zone Axis: An imaginary line passing through the center of the crystal that is parallel to all the intersection edges in that zone.
• For example, all the vertical faces on a hexagonal prism (like in quartz) belong to the same zone, and the zone axis is the vertical 'c' crystallographic axis.

Crystal Form and Habit

• Crystal Form: A form is a group of all crystal faces that have the same relationship to the internal symmetry of the crystal. For example, the 6 identical faces of a cube all belong to the "cube form."
  • Open Form: A set of faces that does not completely enclose space (e.g., a prism, which is open on the ends). It must be combined with another form (like a pinacoid, or "cap") to make a full crystal.
  • Closed Form: A set of faces that does completely enclose space (e.g., a cube or an octahedron). A crystal can consist of just one closed form.
• Crystal Habit: The habit is the general, characteristic shape of a crystal. It describes the overall appearance, which is a result of the *relative* development of different forms. Habit is heavily influenced by the growth environment (temperature, pressure, impurities).
  • Example: Both Ruby and Sapphire are the mineral Corundum. Ruby often has a tabular (flat, tablet-like) habit, while Sapphire often has a prismatic or bipyramidal habit.
  • Common Habits: Acicular (needle-like), Prismatic (elongated), Tabular (flat), Bladed, Equant (equal-sided), Fibrous.

Laws of Crystallography

First Law: The Law of Constancy of Interfacial Angles (Steno's Law)

Steno's Law (1669): All crystals of the same substance have constant interfacial angles between their corresponding faces.

This means a tiny, perfectly-formed quartz crystal and a large, distorted quartz crystal will have the exact same angle between their corresponding faces. This law proves that the external shape is a reflection of a fixed, ordered internal structure.

Second Law: The Law of Rational Indices (Haüy's Law)

Haüy's Law (1784): The intercepts that any crystal face makes with the crystallographic axes can be expressed as simple whole-number ratios of the unit intercepts.

This law established the concept of a unit cell—a fundamental repeating block. It means faces are not random; they must align with planes of atoms in the lattice. This law is the basis for the Miller Indices system.

Parameters and Indices

These are notation systems used to describe the orientation of a crystal face relative to the crystallographic axes (imaginary lines labeled a, b, and c).

Weiss Parameters

An older, clunky system. It describes a face by its direct intercepts on the axes, relative to a "unit face."

• A face intercepts the 'a' axis at 1 unit, the 'b' axis at 2 units, and is parallel to 'c'.
• Its Weiss Parameters are: 1a : 2b : ∞c.

The use of infinity (∞) makes this system difficult for calculations.

Miller Indices (hkl)

The modern, standard notation. It solves the "infinity problem" by using reciprocals.

How to find the Miller Indices (hkl):

1. Step 1: Find the intercepts (Weiss Parameters).
   • Example: 1a : 2b : 3c
2. Step 2: Take the reciprocals of the intercepts.
   • Example: 1/1 : 1/2 : 1/3
3. Step 3: Clear the fractions (multiply by a common denominator).
   • Example: Multiply by 6: (1/1)*6 : (1/2)*6 : (1/3)*6 → 6 : 3 : 2
4. Step 4: Write the indices in parentheses, no commas.
   • Example: (632)

Classification of Crystals (7 Systems)

All crystals are classified into 7 systems based on the lengths of their crystallographic axes (a, b, c) and the angles between them (α, β, γ).

Symmetry Elements

This classification is based on symmetry. The main symmetry elements are:

• Plane of Symmetry (m): A mirror plane that divides the crystal into two identical, reflected halves.
• Axis of Rotation (A): An imaginary line; when the crystal is rotated around it, it looks identical multiple times in a 360° turn (e.g., 2-fold, 3-fold, 4-fold, 6-fold).
• Center of Symmetry (i): A central point; any line drawn through it finds identical features at equal distances on opposite sides.
• Axis of Roto-inversion (Ā or "bar"): A compound operation of rotation + inversion.

The 7 Systems and their Normal Classes

The "Normal Class" (or Holohedral Class) is the class within each system that has the highest possible symmetry.