💎Crystallography Unit 9 – Electron and Neutron Diffraction

Fundamentals of Diffraction

• Diffraction occurs when waves encounter an obstacle or aperture and bend around it, resulting in an interference pattern
• Constructive interference happens when waves are in phase and amplitudes add up, leading to bright spots in the diffraction pattern
• Destructive interference occurs when waves are out of phase and cancel each other out, resulting in dark spots in the pattern
• Bragg's law ($n\lambda = 2d\sin\theta$) relates the wavelength ($\lambda$), interplanar spacing ($d$), and scattering angle ($\theta$) for constructive interference
  • $n$ represents the order of diffraction (integer)
• Reciprocal lattice is a Fourier transform of the crystal lattice and provides a convenient way to analyze diffraction patterns
• Ewald sphere is a geometric construction used to visualize the conditions for diffraction in reciprocal space
• Structure factor ($F_{hkl}$) is a complex quantity that describes the amplitude and phase of the scattered wave from a set of lattice planes ($hkl$)

Wave Properties of Electrons and Neutrons

• De Broglie relationship ($\lambda = h/p$) connects the wavelength ($\lambda$) and momentum ($p$) of a particle, where $h$ is Planck's constant
• Electrons and neutrons exhibit wave-particle duality, allowing them to be used for diffraction experiments
• Electron wavelength depends on the accelerating voltage ($\lambda = h/\sqrt{2meV}$), where $m$ is the electron mass and $e$ is the electron charge
  • Typical electron wavelengths range from 0.01 to 0.1 Å
• Neutron wavelength is determined by the neutron velocity ($\lambda = h/mv$), which can be controlled by a moderator
  • Common neutron wavelengths are around 1-2 Å
• Coherence length is the distance over which the wave maintains a constant phase relationship and affects the quality of the diffraction pattern
• Electrons interact strongly with matter via Coulomb forces, leading to multiple scattering and a short penetration depth
• Neutrons interact weakly with matter via nuclear forces and have a high penetration depth, making them suitable for bulk analysis

Electron Diffraction Techniques

• Transmission Electron Microscopy (TEM) is a powerful technique that uses a high-energy electron beam to probe the structure of thin samples
  • Selected Area Electron Diffraction (SAED) is performed in TEM by using an aperture to select a specific region for diffraction analysis
• Convergent Beam Electron Diffraction (CBED) uses a focused electron beam to obtain local structural information and study defects
• Reflection High-Energy Electron Diffraction (RHEED) is a surface-sensitive technique that uses grazing incidence electrons to study thin films and surfaces
  • RHEED patterns consist of streaks and spots, providing information about surface roughness and crystallinity
• Low-Energy Electron Diffraction (LEED) is another surface-sensitive technique that uses low-energy electrons (20-500 eV) to probe the surface structure
• Electron Backscatter Diffraction (EBSD) is a scanning electron microscopy (SEM) based technique used to study the microstructure and orientation of crystalline materials
• Precession Electron Diffraction (PED) reduces dynamical effects by precessing the electron beam, improving the quality of the diffraction patterns
• Electron diffraction patterns are affected by multiple scattering, which can be reduced by using thinner samples or higher electron energies

Neutron Diffraction Methods

• Single crystal neutron diffraction is used to determine the atomic and magnetic structure of materials
  • Large single crystals (several mm³) are required due to the low flux of neutron sources
• Powder neutron diffraction is a technique for studying polycrystalline materials, where the sample is a fine powder with randomly oriented crystallites
  • Rietveld refinement is a method for analyzing powder diffraction data by fitting a theoretical model to the observed pattern
• Time-of-Flight (TOF) neutron diffraction uses pulsed neutron sources and measures the time taken by neutrons to travel from the source to the detector
  • TOF allows for a wide range of wavelengths to be used simultaneously, improving the efficiency of data collection
• Neutron Laue diffraction is a single crystal technique that uses a polychromatic neutron beam to collect a large number of reflections simultaneously
• Small-Angle Neutron Scattering (SANS) probes the structure of materials on the nanoscale (1-100 nm) by measuring the scattering at small angles
• Neutron reflectometry is a technique for studying the structure and composition of thin films and interfaces by measuring the specular reflection of neutrons
• Polarized neutron diffraction is used to study the magnetic structure of materials by exploiting the interaction between the neutron spin and the sample magnetization

Scattering and Intensity Factors

• Atomic scattering factor ($f$) describes the scattering amplitude of an atom as a function of the scattering angle and wavelength
  • For electrons, $f$ is proportional to the atomic number ($Z$)
  • For neutrons, $f$ is replaced by the scattering length ($b$), which varies irregularly with atomic number
• Structure factor ($F_{hkl}$) is the sum of the atomic scattering factors multiplied by a phase factor, taking into account the positions of atoms in the unit cell
  • $F_{hkl} = \sum_{j=1}^N f_j \exp[2\pi i(hx_j + ky_j + lz_j)]$, where $x_j, y_j, z_j$ are the fractional coordinates of atom $j$
• Intensity of the diffracted beam is proportional to the square of the structure factor ($I \propto |F_{hkl}|^2$)
• Multiplicity factor accounts for the number of symmetrically equivalent reflections contributing to the same diffraction peak
• Lorentz factor corrects for the variation in the time a reciprocal lattice point spends in the vicinity of the Ewald sphere
• Polarization factor accounts for the change in the intensity of the scattered beam due to the polarization of the incident beam
• Absorption factor corrects for the attenuation of the beam as it passes through the sample, which depends on the sample geometry and composition
• Temperature factor (Debye-Waller factor) describes the reduction in the scattered intensity due to thermal vibrations of atoms

Structural Analysis from Diffraction Patterns

• Indexing is the process of assigning Miller indices ($hkl$) to the observed diffraction peaks based on their positions and the unit cell parameters
  • Methods for indexing include the Rietveld method, the Werner algorithm, and the Ito method
• Unit cell determination involves finding the lattice parameters ($a, b, c, \alpha, \beta, \gamma$) that best fit the observed diffraction pattern
  • Least-squares refinement is commonly used to optimize the unit cell parameters
• Space group determination is the process of identifying the symmetry elements present in the crystal structure based on the systematic absences in the diffraction pattern
  • Systematic absences are missing reflections due to destructive interference caused by the presence of certain symmetry elements (screw axes, glide planes)
• Fourier synthesis is a method for calculating the electron density or nuclear density distribution in the unit cell from the measured structure factors
  • $\rho(xyz) = \frac{1}{V} \sum_{hkl} F_{hkl} \exp[-2\pi i(hx + ky + lz)]$, where $V$ is the unit cell volume
• Patterson function is a Fourier synthesis that uses the squared structure factors ($|F_{hkl}|^2$) and provides information about the interatomic vectors in the crystal structure
• Direct methods are a set of techniques for solving the phase problem in crystallography by exploiting statistical relationships between the structure factors
  • Examples of direct methods include the tangent formula, the Sayre equation, and the maximum entropy method
• Rietveld refinement is a method for refining the crystal structure by minimizing the difference between the observed and calculated diffraction patterns
  • The method involves optimizing parameters such as atomic positions, occupancies, thermal factors, and background coefficients

Applications in Materials Science

• Phase identification and quantification in complex mixtures using powder diffraction and Rietveld analysis
  • Quantitative phase analysis (QPA) determines the relative amounts of different phases in a sample
• Strain and stress analysis in materials using diffraction peak shifts and broadening
  • Williamson-Hall plot is a method for separating the contributions of size and strain to peak broadening
• Texture and orientation analysis in polycrystalline materials using pole figures and orientation distribution functions (ODFs)
  • Pole figures represent the distribution of crystal orientations relative to the sample reference frame
• Residual stress measurement in engineered components using neutron or synchrotron diffraction
  • Depth-resolved residual stress profiles can be obtained by measuring the peak shifts at different depths
• In-situ studies of phase transformations, chemical reactions, and mechanical deformation using diffraction techniques
  • Examples include the study of martensitic transformations, hydride formation, and crack propagation
• Characterization of nanostructured materials, thin films, and interfaces using specialized diffraction techniques (GISAXS, XRR, RHEED)
  • Grazing-Incidence Small-Angle X-ray Scattering (GISAXS) probes the morphology and spatial arrangement of nanostructures on surfaces
• Investigation of the magnetic structure and spin dynamics in materials using neutron diffraction and inelastic neutron scattering
  • Magnetic structure determination involves measuring the intensities of magnetic Bragg peaks and fitting them to a model
• Pair Distribution Function (PDF) analysis for studying the local structure of amorphous and disordered materials
  • PDF is obtained by Fourier transforming the total scattering data, including both Bragg and diffuse scattering

Limitations and Challenges

• Sample preparation can be challenging, especially for air-sensitive or reactive materials
  • Special sample environments (vacuum, inert atmosphere, cryostats) may be required
• Preferred orientation (texture) in the sample can lead to inaccurate intensity measurements and affect the structural analysis
  • Strategies to mitigate texture effects include sample rotation, using a side-loaded sample holder, or applying a correction during data analysis
• Extinction and multiple scattering effects can cause deviations from kinematical diffraction theory and affect the measured intensities
  • Extinction arises from the attenuation of the incident and diffracted beams within a perfect crystal
  • Multiple scattering occurs when the diffracted beam is re-scattered by other crystallites, leading to a redistribution of intensity
• Peak overlap in powder diffraction patterns can make it difficult to extract accurate intensities and perform structural refinements
  • Using high-resolution diffractometers, synchrotron radiation, or neutron TOF techniques can help resolve overlapping peaks
• Disorder and defects in the crystal structure can lead to diffuse scattering and complicate the interpretation of diffraction patterns
  • Modeling disorder requires specialized techniques such as the Pair Distribution Function (PDF) analysis or the 3D-ΔPDF method
• Limited access to advanced diffraction facilities (synchrotrons, neutron sources) can hinder the study of complex materials systems
  • Collaborations and proposal-based access to these facilities are essential for many research projects
• Data interpretation and analysis can be complex and require specialized software and expertise
  • Examples of widely used software packages include GSAS, FullProf, TOPAS, and JANA
• Complementary techniques (spectroscopy, microscopy, simulations) are often necessary to obtain a comprehensive understanding of the material's structure and properties
  • Combining diffraction with techniques such as X-ray absorption spectroscopy (XAS), Raman spectroscopy, or Density Functional Theory (DFT) calculations can provide valuable insights