Unit 4: Twinning and X-ray Crystallography

Twinning: Concept, Elements, and Types

Concept of Twinning

Twinning is the symmetrical intergrowth of two or more crystals of the same substance. The twinned crystals (or "individuals") are related to each other by a specific symmetry operation that is not present in the original, untwinned crystal.

This is not a random growth; it is a specific, geometrically controlled relationship. The operation that relates the twin individuals is called the Twin Law.

Twinning Elements

The twin law can be described by one of three symmetry elements:

1. Twin Plane (m): The individuals are related by reflection across a shared plane (the twin plane). This is the most common element.
2. Twin Axis (A): The individuals are related by a rotation (usually 180°) around a common axis (the twin axis).
3. Twin Center (i): The individuals are related by inversion through a central point (the twin center). (This is rare).

The surface where the two individuals join is called the composition plane. Often, but not always, the twin plane and composition plane are the same.

Types of Twinning

• Contact Twins: The two individuals are joined at a single, well-defined composition plane. Example: Spinel Law, Japan Law (Quartz).
• Penetration Twins: The two individuals appear to grow through each other in an interpenetrating way. The composition surface is irregular. Example: Carlsbad Law (Feldspar), Staurolite's cross.
• Polysynthetic (or Repeated) Twinning: A series of contact twins are stacked parallel to each other. This often appears as fine parallel lines (striations) on a crystal face. Example: Albite Law (Plagioclase Feldspar).
• Cyclic Twinning: Successive twin planes are not parallel, causing the individuals to form a ring or star shape. Example: Aragonite, Rutile.

Laws of Twinning

A "Twin Law" is a specific, named relationship that defines the twin element and the crystal system it occurs in. There are many named twin laws.

Twinning in Feldspar, Quartz, and Staurolite

Twinning in Feldspar

Feldspar twinning is extremely common and is a key diagnostic feature.

• Orthoclase (K-Feldspar): Most commonly shows the Carlsbad Law, a simple penetration twin. You can often see a line down the middle of the crystal where the two individuals meet. Baveno and Manebach laws are also possible.
• Plagioclase Feldspar: Dominated by the Albite Law. These polysynthetic twins are visible to the naked eye (or with a hand lens) as fine, parallel striations on the (001) cleavage face. This is the #1 way to identify plagioclase.
• Microcline (K-Feldspar): Shows *both* Albite and Pericline twinning simultaneously. Because the two sets of twins are nearly at 90° to each other, they produce a "tartan" or "cross-hatch" pattern, which is best seen under a microscope.

Twinning in Quartz

Quartz twins are common, but often hard to see as the composition planes are not obvious. They are defined by the relationship between left-handed and right-handed quartz structures.

• Brazil Law: A penetration twin of a left-handed and right-handed crystal.
• Dauphiné Law: A penetration twin of two left-handed OR two right-handed crystals, related by a 180° rotation around the 'c' axis.
• Japan Law: A contact twin that forms a distinctive V-shape.

Twinning in Staurolite

Staurolite is famous for its penetration twins, which are often sold as "fairy crosses."

• St. Andrew's Cross: The two prismatic crystals interpenetrate at an angle of ~60°.
• Cruciform (Cross-shaped) Twin: The two individuals interpenetrate at an angle of ~90°.

X-ray Crystallography and Diffraction

X-ray Crystallography

X-ray Crystallography (XRC) is a powerful analytical technique used to determine the internal atomic structure of a crystal.

Why X-rays? The principle of diffraction requires that the wavelength of the waves used must be similar to the size of the gaps in the grating.

• The spacing between atoms in a crystal is ~1-3 Ångströms (Å).
• The wavelength of X-rays is also ~0.1-10 Å.

X-ray Diffraction (XRD)

Diffraction is the bending or scattering of waves as they pass around an obstacle or through an aperture. When multiple waves are scattered, they interfere with each other.

• Constructive Interference: If the scattered waves are "in-phase" (peaks align with peaks), they add up, and a strong beam is produced.
• Destructive Interference: If the waves are "out-of-phase" (peaks align with troughs), they cancel each other out, and no beam is detected.

In a crystal, X-rays are scattered by the planes of atoms. Only at specific, precise angles (where the path difference is just right) will the scattered waves be in-phase, leading to constructive interference. This creates a "diffracted beam." This relationship is described by Bragg's Law.

Bragg's Law

Formulated by W.L. Bragg and W.H. Bragg, this simple equation is the cornerstone of X-ray diffraction.

Bragg's Law: nλ = 2d sin(θ)

• n = an integer (1, 2, 3...), called the "order" of diffraction.
• λ (lambda) = the wavelength of the X-rays being used.
• d = the spacing between the atomic planes in the crystal (the "d-spacing").
• θ (theta) = the angle of incidence (the "Bragg angle") at which constructive interference occurs.

Derivation of Bragg's Law

1. Imagine two parallel X-ray beams (Ray 1 and Ray 2) hitting two parallel atomic planes (Plane 1 and Plane 2) separated by a distance d.
2. Both rays hit the planes at an angle θ.
3. Ray 1 hits Plane 1 and reflects off.
4. Ray 2 travels further, past Plane 1, hits Plane 2, and reflects off.
6. We need to find the extra path length (A-B-C) traveled by Ray 2.
   • Using trigonometry, the distance A-B is d sin(θ).
   • The distance B-C is also d sin(θ).
   • Total path difference = 2d sin(θ).
7. For the two reflected rays to be in-phase (constructive interference), this extra path length must be an integer multiple (n) of the X-ray's wavelength (λ).
8. Therefore: nλ = 2d sin(θ).

Powder and Single Crystal Methods

These are the two main experimental ways to use Bragg's Law to analyze crystals.

Powder Method (Debye-Scherrer Method)

• Sample: The crystal is ground into a fine powder. This creates millions of tiny crystals ("crystallites") in every possible random orientation.
• How it works: A monochromatic (single-wavelength) X-ray beam is shot at the powder.
  • Because the crystals are in every orientation, for any given set of 'd' planes (e.g., the (101) planes), there will be *some* crystals perfectly aligned at the correct Bragg angle (θ) to cause diffraction.
  • This diffracted beam exits the sample as a cone.
• Result: A detector measures these cones as a series of concentric rings (if on film) or as a graph of Intensity vs. Angle (2θ).
• Use: Mineral Identification. The resulting pattern (a list of 'd' and 'intensity' values) is a unique "fingerprint" that can be matched against a database (like the ICDD-PDF) to identify the unknown mineral(s).

Single Crystal Methods

• Sample: A single, well-formed crystal (less than 1mm in size).
• How it works: The crystal is mounted and precisely oriented in the X-ray beam.
  • Laue Method: Uses a stationary crystal and "white" (multi-wavelength) X-rays. The resulting pattern of spots reveals the crystal's point group symmetry.
  • Rotation/Precession Method: Uses a moving (rotating or precessing) crystal and monochromatic X-rays. As the crystal moves, different planes are brought into the correct orientation to satisfy Bragg's Law, producing a complex pattern of spots.
• Result: A 2D pattern of discrete spots. The position of the spots tells you the size and shape of the unit cell. The intensity of the spots tells you where the atoms are located within the unit cell.
• Use: Complete Structure Determination. This method is used to discover the *exact* 3D atomic arrangement of a new material or mineral.