Bragg's equation and X-ray diffraction techniques for mineral studies

Bragg’s Law: The Theoretical Foundation

Bragg’s Law is mathematically expressed as: nλ = 2dsinθ, where:

• n is an integer representing the order of reflection (usually 1).
• λ is the wavelength of the incident X-rays.
• d is the interplanar spacing – the distance between parallel planes of atoms in the crystal lattice.
• θ is the angle of incidence (Bragg angle) – the angle between the incident X-ray beam and the crystal planes.

The derivation of Bragg’s Law is based on the concept of path difference. When X-rays are incident on a crystal, they are reflected by different atomic planes. Constructive interference occurs only when the path difference between the reflected rays is an integer multiple of the wavelength. This condition is satisfied by Bragg’s Law, allowing for the determination of ‘d’ spacing if λ and θ are known.

X-ray Diffraction Techniques for Mineral Studies

1. Powder X-ray Diffraction (PXRD)

PXRD is the most commonly used technique in mineralogy. A powdered sample containing numerous randomly oriented crystallites is irradiated with an X-ray beam. The diffraction pattern obtained is a series of peaks, each corresponding to a specific interplanar spacing (d-spacing) in the mineral.

• Procedure: The powder sample is placed in a sample holder and irradiated with monochromatic X-rays (typically Cu Kα radiation, λ = 1.5418 Å). The detector measures the intensity of the diffracted X-rays as a function of the diffraction angle (2θ).
• Applications:
  • Mineral Identification: Each mineral has a unique diffraction pattern, acting as a ‘fingerprint’. Comparing the obtained pattern with standard databases (e.g., ICDD Powder Diffraction File) allows for mineral identification.
  • Phase Analysis: Determining the presence and relative abundance of different phases in a mixture.
  • Crystallinity Determination: Assessing the degree of crystallinity of a sample.
  • Quantitative Analysis: Determining the amount of each phase present in a mixture using techniques like Rietveld refinement.

2. Single-Crystal X-ray Diffraction (SCXRD)

SCXRD provides a more detailed structural analysis than PXRD. It requires a single crystal of sufficient size and quality. The crystal is mounted on a diffractometer and rotated in the X-ray beam. The diffraction pattern is collected, and the data is used to determine the three-dimensional arrangement of atoms within the crystal.

• Procedure: A single crystal is mounted and rotated in the X-ray beam. The positions and intensities of the diffracted beams are measured.
• Applications:
  • Precise Structure Determination: Determining the exact atomic coordinates, bond lengths, and bond angles within the crystal structure.
  • Symmetry Determination: Identifying the crystal system and space group.
  • Anisotropy Studies: Investigating variations in physical properties with direction.

3. Other Techniques

Besides PXRD and SCXRD, other techniques like X-ray fluorescence (XRF) are often used in conjunction to determine the elemental composition of minerals, complementing the structural information obtained from diffraction techniques.

Factors Affecting Diffraction Patterns

Several factors can influence the quality and interpretation of X-ray diffraction patterns:

• Instrumental Factors: X-ray source stability, detector efficiency, and alignment.
• Sample Preparation: Particle size, homogeneity, and preferred orientation (in PXRD).
• Temperature: Thermal expansion can affect d-spacings.
• Solid Solution & Defects: Variations in composition and the presence of defects can broaden or shift diffraction peaks.

Technique	Sample Requirement	Information Obtained	Complexity
Powder XRD	Powdered sample	Mineral identification, phase analysis, crystallinity	Relatively simple
Single Crystal XRD	Single crystal	Precise structure determination, symmetry	Complex