💎Mathematical Crystallography Unit 7 – Diffraction Theory Fundamentals

Key Concepts and Definitions

• Diffraction phenomenon where waves bend around obstacles or spread out after passing through an opening
• Crystalline materials have regular, repeating structures that can diffract waves (X-rays, electrons, neutrons)
• Bragg's law ($n\lambda = 2d\sin\theta$) relates the wavelength of incident radiation ($\lambda$), the spacing between crystal planes ($d$), and the angle of incidence ($\theta$)
  • $n$ represents an integer known as the order of diffraction
  • Constructive interference occurs when Bragg's law is satisfied, resulting in diffraction peaks
• Reciprocal lattice mathematical construct representing the Fourier transform of the crystal lattice
  • Each point in the reciprocal lattice corresponds to a set of parallel planes in the crystal
  • Distances in reciprocal space are inversely related to distances in real space
• Structure factor ($F_{hkl}$) complex quantity describing the amplitude and phase of a diffracted wave from a set of crystal planes ($hkl$)
  • Depends on the positions and types of atoms in the unit cell
  • Fourier transform of the electron density distribution in the crystal
• Intensity of diffracted beams proportional to the square of the structure factor magnitude ($I_{hkl} \propto |F_{hkl}|^2$)

Historical Context and Development

• Discovery of X-ray diffraction by Max von Laue in 1912 demonstrated the wave nature of X-rays and the periodic structure of crystals
• William Henry Bragg and William Lawrence Bragg developed Bragg's law in 1913, enabling the determination of crystal structures
• Early X-ray diffraction experiments focused on simple inorganic crystals (NaCl, diamond)
• Development of the Patterson function in 1934 by Arthur Lindo Patterson facilitated the determination of heavy atom positions in crystals
• Advances in computing power and algorithms (direct methods, Fourier synthesis) in the mid-20th century revolutionized structure determination
• Synchrotron radiation sources and free-electron lasers have greatly enhanced the capabilities of diffraction techniques in recent decades

Mathematical Foundations

• Fourier transforms play a central role in diffraction theory, connecting real space and reciprocal space
  • Crystal structure can be represented as a Fourier series, with coefficients given by the structure factors
  • Electron density distribution can be obtained by an inverse Fourier transform of the structure factors
• Convolution theorem states that the Fourier transform of a convolution is the product of the Fourier transforms of the individual functions
  • Allows the calculation of diffraction patterns from the convolution of the lattice and the motif (unit cell contents)
• Ewald sphere geometric construction relating the incident and diffracted wave vectors in reciprocal space
  • Diffraction occurs when the Ewald sphere intersects points in the reciprocal lattice
  • Radius of the Ewald sphere is inversely proportional to the wavelength of the incident radiation
• Scattering factors describe the scattering power of individual atoms as a function of the scattering angle
  • Depend on the electron density distribution of the atom
  • Decrease with increasing scattering angle due to destructive interference of waves scattered from different parts of the atom

Types of Diffraction

• X-ray diffraction most common technique, using X-rays with wavelengths comparable to interatomic distances in crystals (0.5-2.5 Å)
  • Scattered by electrons in the crystal
  • Suitable for a wide range of materials, including inorganic and organic compounds, proteins, and nucleic acids
• Electron diffraction uses a beam of electrons with wavelengths much shorter than X-rays (< 0.1 Å)
  • Stronger interaction with matter compared to X-rays, allowing the study of thin films, surfaces, and nanostructures
  • Multiple scattering effects can complicate data interpretation
• Neutron diffraction employs neutrons with wavelengths similar to X-rays
  • Scattered by atomic nuclei and magnetic moments
  • Particularly useful for studying light elements (hydrogen), magnetic structures, and isotope effects
• Powder diffraction used for polycrystalline or powdered samples with randomly oriented crystallites
  • Produces cone-shaped diffraction patterns instead of discrete spots
  • Requires specialized data analysis techniques (Rietveld refinement) to extract structural information

Experimental Techniques and Setups

• Single-crystal X-ray diffraction most common setup, using a monochromatic X-ray source and a single crystal sample
  • Crystal is rotated to collect a complete dataset of diffraction intensities
  • Requires high-quality, sufficiently large crystals (typically > 50 μm)
• Powder X-ray diffraction (PXRD) uses a polycrystalline sample and a monochromatic X-ray source
  • Sample is usually packed into a capillary or flat plate and rotated during data collection
  • Provides average structural information over many crystallites
• Laue diffraction employs a polychromatic (white) X-ray source and a stationary crystal
  • Produces a pattern of diffraction spots that can be used for crystal orientation and symmetry determination
  • Limited application in structure determination due to overlapping reflections
• Time-resolved diffraction techniques use pulsed X-ray sources (synchrotrons, free-electron lasers) to study dynamic processes in crystals
  • Can probe structural changes on timescales ranging from femtoseconds to seconds
  • Requires specialized sample delivery methods (liquid jets, microfluidic devices) and data analysis techniques

Applications in Crystallography

• Determination of crystal structures the primary application of diffraction techniques in crystallography
  • Involves measuring diffraction intensities, solving the phase problem, and constructing an atomic model
  • Provides detailed information on the positions, types, and connectivity of atoms in the crystal
• Study of phase transitions and structural changes as a function of temperature, pressure, or composition
  • Diffraction patterns can reveal the appearance or disappearance of peaks, changes in peak positions or intensities, and symmetry changes
• Characterization of disorder, defects, and strain in crystals
  • Diffuse scattering, peak broadening, and peak splitting can indicate the presence of disorder, defects, or strain
• Determination of particle size and shape using peak broadening analysis (Scherrer equation)
• Identification of unknown crystalline phases by comparing diffraction patterns with databases (powder diffraction file, PDF)

Challenges and Limitations

• Phase problem the fundamental challenge in crystallography, as diffraction intensities only provide information about the amplitudes of structure factors, not their phases
  • Solved using methods such as Patterson synthesis, direct methods, molecular replacement, or anomalous scattering
  • Requires additional experimental data (heavy atom derivatives, anomalous scatterers) or prior structural knowledge
• Radiation damage caused by the interaction of X-rays or electrons with the sample, leading to structural changes or sample degradation
  • Particularly problematic for sensitive materials (proteins, organic compounds)
  • Can be mitigated by using cryogenic temperatures, shorter exposure times, or advanced sample delivery methods
• Sample preparation and quality critical for obtaining high-quality diffraction data
  • Single crystals must be sufficiently large, well-ordered, and free of twinning or other defects
  • Powder samples must be homogeneous, finely ground, and properly packed to minimize preferred orientation effects
• Overlapping reflections in powder diffraction patterns can complicate data analysis and structure determination
  • Requires advanced profile fitting and refinement techniques (Rietveld method)
  • May limit the complexity of structures that can be solved using powder diffraction alone

Advanced Topics and Future Directions

• Coherent diffraction imaging (CDI) technique that uses the coherence properties of X-ray sources to image non-crystalline samples
  • Relies on oversampling the diffraction pattern and iterative phase retrieval algorithms
  • Enables the study of single particles, nanostructures, and biological systems without the need for crystallization
• Serial femtosecond crystallography (SFX) uses ultrashort X-ray pulses from free-electron lasers to collect diffraction data from a stream of microcrystals
  • Overcomes radiation damage limitations by using pulses shorter than the timescale of damage processes
  • Allows the study of small, delicate crystals and time-resolved structural changes
• Electron crystallography combines electron diffraction and imaging techniques to study the structure of thin films, surfaces, and nanostructures
  • Offers higher spatial resolution than X-ray diffraction due to the shorter wavelength of electrons
  • Requires specialized sample preparation and data analysis methods to account for multiple scattering effects
• Integration of diffraction techniques with other methods (spectroscopy, microscopy, computational modeling) to provide a more comprehensive understanding of structure-property relationships in materials
  • Enables the correlation of structural information with electronic, optical, or mechanical properties
  • Facilitates the rational design of materials with tailored functionalities