💎Mathematical Crystallography Unit 10 – X-ray Diffraction: Techniques & Applications

Fundamental Principles of X-ray Diffraction

• X-ray diffraction (XRD) is a powerful non-destructive technique used to analyze the atomic and molecular structure of crystalline materials
• X-rays are electromagnetic waves with wavelengths ranging from 0.01 to 10 nanometers, comparable to the interatomic distances in crystals
• When X-rays interact with a crystalline sample, they are scattered by the electrons in the atoms, producing a diffraction pattern
• The diffraction pattern contains information about the crystal structure, including the arrangement of atoms, bond lengths, and angles
• Constructive interference occurs when the scattered X-rays are in phase, resulting in high-intensity peaks in the diffraction pattern
  • Constructive interference is governed by Bragg's law, which relates the wavelength of the X-rays, the interplanar spacing, and the angle of incidence
• Destructive interference occurs when the scattered X-rays are out of phase, leading to the cancellation of waves and low-intensity regions in the diffraction pattern
• The intensity of the diffracted X-rays depends on factors such as the atomic scattering factors, which are related to the number and distribution of electrons in the atoms

X-ray Sources and Instrumentation

• X-ray sources generate the X-rays used in diffraction experiments, with the most common types being X-ray tubes and synchrotron radiation sources
• X-ray tubes produce X-rays by accelerating electrons towards a metal target (anode), causing the emission of characteristic X-rays and a continuous spectrum (bremsstrahlung)
  • The choice of anode material (copper, molybdenum, etc.) determines the characteristic wavelengths of the X-rays produced
• Synchrotron radiation sources generate high-intensity, highly collimated X-rays by accelerating electrons to relativistic speeds in a circular storage ring
  • Synchrotron X-rays have a continuous spectrum and can be tuned to specific wavelengths using monochromators
• X-ray optics, such as mirrors and monochromators, are used to focus, collimate, and select specific wavelengths of X-rays for diffraction experiments
• Goniometers are devices that precisely control the orientation of the sample and detector relative to the incident X-ray beam, allowing for the measurement of diffraction patterns at different angles
• Detectors, such as photographic film, scintillation counters, and charge-coupled devices (CCDs), are used to record the diffracted X-rays and convert them into measurable signals
• Sample preparation is crucial for obtaining high-quality diffraction data, with factors such as particle size, sample thickness, and sample mounting affecting the results

Crystal Structure and Lattice Systems

• Crystals are solid materials with a regular, repeating arrangement of atoms or molecules in three-dimensional space
• The crystal structure describes the periodic arrangement of atoms in a crystal, including the positions of atoms, bond lengths, and bond angles
• Lattice systems are mathematical descriptions of the symmetry and periodicity of crystal structures, with seven unique lattice systems in three dimensions
  • The seven lattice systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal (rhombohedral), hexagonal, and cubic
• Each lattice system is characterized by its unit cell, which is the smallest repeating unit that contains all the symmetry and structural information of the crystal
  • Unit cell parameters include the lengths of the cell edges (a, b, c) and the angles between them (α, β, γ)
• Bravais lattices are the 14 unique ways of arranging points in three-dimensional space, with each lattice system having one or more associated Bravais lattices
• Miller indices (hkl) are used to describe the orientation of crystal planes and directions, with (hkl) denoting a specific set of parallel planes in the crystal
• Space groups are mathematical descriptions of the complete symmetry of a crystal structure, combining the lattice system with additional symmetry elements such as rotations, reflections, and translations

Bragg's Law and Diffraction Geometry

• Bragg's law is the fundamental equation that describes the conditions for constructive interference in X-ray diffraction
  • Bragg's law states that $nλ = 2d \sin θ$, where $n$ is an integer, $λ$ is the wavelength of the X-rays, $d$ is the interplanar spacing, and $θ$ is the angle of incidence
• Constructive interference occurs when the path difference between X-rays scattered from adjacent crystal planes is an integer multiple of the wavelength
  • This condition is satisfied when the angle of incidence equals the angle of reflection, known as the Bragg angle
• The interplanar spacing (d) is related to the Miller indices (hkl) and the unit cell parameters (a, b, c, α, β, γ) through the interplanar spacing formula, which varies depending on the lattice system
• The Ewald sphere is a geometric construction used to visualize the diffraction conditions in reciprocal space
  • The radius of the Ewald sphere is $1/λ$, and the origin of the reciprocal lattice is placed at the end of the incident X-ray vector
  • Diffraction occurs when a reciprocal lattice point intersects the Ewald sphere, satisfying Bragg's law
• The diffraction pattern is a two-dimensional projection of the reciprocal lattice, with each diffraction spot corresponding to a set of parallel crystal planes (hkl)
• The geometry of the diffraction pattern depends on the orientation of the crystal relative to the incident X-ray beam, which can be controlled using a goniometer

Data Collection and Processing Techniques

• Data collection involves measuring the intensities and positions of diffraction spots on a detector as a function of the sample orientation and X-ray wavelength
• The most common data collection methods are the rotating crystal method and the powder diffraction method
  • In the rotating crystal method, a single crystal is rotated about one or more axes while the diffraction pattern is recorded, allowing for the measurement of a complete set of diffraction data
  • In the powder diffraction method, a polycrystalline sample is used, and the diffraction pattern consists of concentric rings due to the random orientation of the crystallites
• Data processing involves converting the raw diffraction data into a usable format for structure determination and refinement
• Background subtraction removes the contribution of non-sample scattering (air, sample holder, etc.) from the diffraction pattern
• Peak search and indexing identify the positions and intensities of the diffraction spots and assign Miller indices (hkl) to each spot based on the unit cell parameters
• Intensity integration measures the total intensity of each diffraction spot, which is proportional to the square of the structure factor amplitude $|F_{hkl}|^2$
• Lorentz-polarization correction accounts for the geometric and polarization effects that influence the observed intensities of the diffraction spots
• Absorption correction compensates for the attenuation of X-rays as they pass through the sample, which depends on the sample composition and geometry
• Scaling and merging combine the intensity data from multiple measurements (e.g., different sample orientations or X-ray wavelengths) into a single, consistent dataset

Analysis Methods and Structure Determination

• Structure determination is the process of deriving the atomic positions, occupancies, and thermal parameters from the diffraction data
• The structure factor $F_{hkl}$ is a complex quantity that describes the amplitude and phase of the X-rays scattered by the atoms in the unit cell
  • The structure factor is related to the electron density distribution in the unit cell through a Fourier transform
• The electron density $ρ(xyz)$ at any point in the unit cell can be calculated from the structure factors using an inverse Fourier transform
  • $ρ(xyz) = \frac{1}{V} \sum_{hkl} F_{hkl} e^{-2πi(hx+ky+lz)}$, where $V$ is the volume of the unit cell
• The phase problem arises because the diffraction experiment only measures the amplitudes of the structure factors, not their phases
  • Solving the phase problem is a crucial step in structure determination, and various methods have been developed to estimate the phases
• Direct methods use statistical relationships between the structure factor amplitudes to estimate the phases, and are particularly effective for small molecules with high-resolution data
• Patterson methods calculate the Patterson function, which is a Fourier transform of the intensity data and represents the interatomic vector distribution in the crystal
  • The Patterson function can be used to locate heavy atoms and determine the initial phases for structure solution
• Charge flipping is an iterative algorithm that alternates between real and reciprocal space to solve the phase problem without prior knowledge of the structure
• Refinement is the process of optimizing the structural model to minimize the difference between the observed and calculated structure factor amplitudes
  • Least-squares refinement adjusts the atomic positions, occupancies, and thermal parameters to minimize the residual $\sum_{hkl} w(|F_{obs}| - |F_{calc}|)^2$, where $w$ is a weighting factor

Advanced X-ray Diffraction Techniques

• Synchrotron X-ray diffraction utilizes the high-intensity, highly collimated X-rays produced by synchrotron radiation sources to study a wide range of materials, including proteins, polymers, and nanostructures
  • Synchrotron X-rays enable high-resolution data collection, rapid data acquisition, and the study of small or weakly diffracting samples
• Anomalous scattering occurs when the X-ray energy is near an absorption edge of an element in the sample, leading to a change in the atomic scattering factors
  • Anomalous diffraction can be used to solve the phase problem by exploiting the differences in the diffraction intensities measured at different X-ray energies
• Time-resolved X-ray diffraction allows for the study of dynamic processes, such as chemical reactions, phase transitions, and structural changes, with high temporal resolution
  • Time-resolved experiments often employ pulsed X-ray sources, such as free-electron lasers or synchrotrons with fast detectors
• Grazing-incidence X-ray diffraction (GIXD) is a technique used to study thin films, surfaces, and interfaces
  • In GIXD, the incident X-ray beam makes a small angle with the sample surface, limiting the penetration depth and enhancing the surface sensitivity
• Pair distribution function (PDF) analysis is a technique that uses total scattering data (both Bragg and diffuse scattering) to study the local structure of materials, particularly in disordered or nanostructured systems
  • The PDF represents the probability of finding two atoms separated by a given distance, and can provide information about short-range order and local distortions in the structure
• Coherent X-ray diffraction imaging (CXDI) is a lensless imaging technique that uses the coherence properties of X-rays to reconstruct the three-dimensional structure of non-crystalline samples, such as nanoparticles or biological specimens
  • CXDI relies on iterative phase retrieval algorithms to reconstruct the real-space image from the diffraction pattern

Applications in Materials Science and Beyond

• X-ray diffraction is widely used in materials science to characterize the structure, composition, and properties of a variety of materials, including metals, ceramics, polymers, and composites
• Phase identification and quantification can be performed using X-ray diffraction, allowing for the determination of the types and relative amounts of phases present in a sample
  • Rietveld refinement is a powerful method for quantitative phase analysis, where the entire diffraction pattern is modeled using the crystal structures and relative abundances of the constituent phases
• Residual stress analysis uses X-ray diffraction to measure the strain in crystalline materials, which can arise from manufacturing processes, thermal treatments, or mechanical loading
  • The shift in the diffraction peak positions is related to the lattice strain, which can be converted to stress using the appropriate elastic constants
• Texture analysis involves measuring the orientation distribution of crystallites in a polycrystalline material, which can influence properties such as strength, ductility, and electrical conductivity
  • Pole figures and orientation distribution functions (ODFs) are used to represent the texture of a material
• Crystallite size and microstrain can be determined from the broadening of the diffraction peaks, using methods such as the Scherrer equation or Williamson-Hall analysis
  • Nanocrystalline materials often exhibit significant peak broadening due to their small crystallite size and the presence of lattice defects
• X-ray diffraction is also used in pharmaceutical research to study the crystal structure, polymorphism, and stability of drug compounds
  • Polymorphism refers to the ability of a substance to exist in multiple crystalline forms, which can have different solubilities, bioavailabilities, and mechanical properties
• In geology and mineralogy, X-ray diffraction is used to identify and characterize minerals, study the formation and transformation of rocks, and investigate the structure of Earth's interior
• X-ray diffraction is a crucial tool in the field of protein crystallography, where the three-dimensional structure of proteins is determined from single-crystal diffraction data
  • Understanding the structure of proteins is essential for elucidating their function, designing drugs, and developing new therapies for diseases