💎Crystallography Unit 6 – X–ray Diffraction Techniques

Fundamentals of X-ray Diffraction

• X-ray diffraction (XRD) is a powerful analytical technique used to identify and characterize crystalline materials
• Utilizes the interaction between X-rays and the regular arrangement of atoms in a crystal lattice
• X-rays have wavelengths comparable to interatomic distances in crystals (angstroms)
• Constructive interference of scattered X-rays occurs at specific angles, producing a diffraction pattern
• Diffraction patterns provide information about the crystal structure, including lattice parameters, symmetry, and atomic positions
• Intensity of diffracted X-rays depends on the type and arrangement of atoms in the crystal
• XRD is non-destructive and can be applied to a wide range of materials (powders, single crystals, thin films)

X-ray Sources and Production

• X-rays are electromagnetic radiation with wavelengths between 0.01 and 10 nanometers
• Generated by accelerating electrons to high energies and colliding them with a metal target (anode)
• Common anode materials include copper, molybdenum, and chromium
• Characteristic X-rays are produced when electrons from higher energy levels fill vacancies in inner shells
  • Vacancies created by the ejection of inner shell electrons due to electron bombardment
  • Characteristic X-rays have specific wavelengths determined by the energy difference between the shells
• Continuous (white) X-rays are generated by the deceleration of electrons as they interact with the anode
• Synchrotron radiation sources produce high-intensity, tunable X-rays for advanced diffraction experiments

Crystal Structure Basics

• Crystals are solid materials with a regular, repeating arrangement of atoms in three dimensions
• The smallest repeating unit of a crystal is called the unit cell
• Unit cells are characterized by their lattice parameters (a, b, c, α, β, γ)
• There are seven crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, cubic)
• 14 Bravais lattices describe the possible arrangements of lattice points in three dimensions
• Atoms, ions, or molecules occupy specific positions within the unit cell (lattice points, interstitial sites)
• Miller indices (hkl) are used to describe the orientation of crystal planes and directions

Bragg's Law and Diffraction Geometry

• Bragg's law describes the conditions for constructive interference of X-rays scattered by a crystal
• $nλ = 2d \sin θ$, where $n$ is an integer, $λ$ is the X-ray wavelength, $d$ is the interplanar spacing, and $θ$ is the scattering angle
• Diffraction occurs when the path difference between X-rays scattered from parallel planes is an integer multiple of the wavelength
• The scattering angle (2θ) is the angle between the incident and diffracted X-ray beams
• Reciprocal lattice is a mathematical construct used to represent the diffraction pattern of a crystal
• Ewald sphere is a geometric construction that relates the incident X-ray wavelength, crystal orientation, and diffraction angles

X-ray Diffraction Techniques and Instruments

• Powder XRD is used for polycrystalline samples, where crystals are randomly oriented
  • Produces a one-dimensional diffraction pattern (intensity vs. 2θ)
  • Identifies phases, determines lattice parameters, and estimates crystallite size
• Single crystal XRD is used for large, high-quality single crystals
  • Produces a three-dimensional diffraction pattern
  • Determines the complete crystal structure, including atomic positions and thermal parameters
• Laue diffraction uses polychromatic (white) X-rays to rapidly determine crystal orientation
• Diffractometers consist of an X-ray source, sample stage, and detector
  • Common geometries include Bragg-Brentano, Debye-Scherrer, and Guinier
• Synchrotron-based diffraction techniques offer high resolution, fast data collection, and in situ capabilities

Data Collection and Processing

• Diffraction data is collected by measuring the intensity of diffracted X-rays as a function of scattering angle (2θ)
• Factors affecting data quality include X-ray wavelength, beam size, sample preparation, and instrumental resolution
• Background subtraction removes contributions from non-crystalline components and instrumental noise
• Peak identification and indexing assign Miller indices (hkl) to each diffraction peak
• Intensity integration determines the total intensity of each diffraction peak
• Absorption and polarization corrections account for the attenuation and polarization of X-rays by the sample and instrument
• Data reduction converts the raw diffraction data into a format suitable for structure determination

Structure Determination Methods

• Patterson methods use the Fourier transform of the diffraction intensities to determine the interatomic vectors in a crystal
  • Useful for structures with heavy atoms or known molecular fragments
• Direct methods estimate the phases of diffraction peaks based on statistical relationships among the intensities
  • Applicable to small to medium-sized structures with atoms of similar scattering power
• Charge flipping is an iterative algorithm that alternates between real and reciprocal space to solve the phase problem
• Rietveld refinement is a full-profile fitting method that refines the crystal structure by minimizing the difference between the observed and calculated diffraction patterns
• Maximum entropy methods incorporate prior knowledge and minimize assumptions in the structure determination process

Applications in Materials Science and Beyond

• Phase identification and quantification in complex mixtures (alloys, ceramics, minerals)
• Determination of lattice parameters, strain, and stress in materials
• Characterization of thin films, multilayers, and epitaxial structures
• Study of phase transitions, thermal expansion, and temperature-dependent phenomena
• Investigation of nanostructured materials, including nanoparticles, nanowires, and nanocomposites
• Structural analysis of proteins, nucleic acids, and other biological macromolecules
• Forensic science applications, such as the identification of illicit drugs and explosives
• Cultural heritage studies, including the analysis of pigments, ceramics, and archaeological artifacts