1.6 X-ray diffraction

X-ray diffraction is a powerful tool in condensed matter physics for studying atomic structures. It uses X-rays to probe materials, revealing crystal structures, atomic positions, and bond lengths. This technique is crucial for understanding how microscopic structures relate to macroscopic properties.

The process involves X-rays interacting with electrons in atoms, creating diffraction patterns. Bragg's law describes the conditions for constructive interference, allowing researchers to determine atomic arrangements and analyze crystal structures in various materials.

Principles of X-ray diffraction

• X-ray diffraction serves as a powerful tool in condensed matter physics for probing atomic and molecular structures
• Enables researchers to determine crystal structures, atomic positions, and bond lengths in solid materials
• Fundamental to understanding the relationship between microscopic structure and macroscopic properties in condensed matter systems

X-ray properties

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• Electromagnetic waves with wavelengths ranging from 0.01 to 10 nanometers
• Generated by accelerating electrons in a metal target (copper, molybdenum)
• Possess high penetration depth allows for non-destructive analysis of materials
• Coherent scattering of X-rays by electrons in atoms forms the basis of diffraction patterns
  • Intensity of scattered X-rays depends on the atomic number of the element

Interaction with matter

• X-rays interact with electrons in atoms through various mechanisms
• Elastic scattering preserves the energy and wavelength of incident X-rays
• Photoelectric absorption results in ejection of core electrons
• Compton scattering involves inelastic collisions with loosely bound electrons
• Pair production occurs at high energies (above 1.02 MeV) creating electron-positron pairs

Bragg's law

• Describes the conditions for constructive interference of scattered X-rays
• Expressed mathematically as $n\lambda = 2d\sin\theta$
• $n$ represents the order of diffraction (integer)
• $\lambda$ denotes the wavelength of incident X-rays
• $d$ signifies the interplanar spacing in the crystal lattice
• $\theta$ refers to the angle between the incident beam and the lattice plane
• Predicts the angles at which intense diffracted beams will occur

Crystal structure analysis

• Enables determination of atomic arrangements and bond lengths in crystalline materials
• Crucial for understanding structure-property relationships in condensed matter systems
• Provides insights into material symmetry, phase transitions, and defect structures

Reciprocal lattice concept

• Mathematical construct representing the Fourier transform of the real space lattice
• Points in reciprocal space correspond to sets of lattice planes in real space
• Reciprocal lattice vectors defined as $\mathbf{b}_1 = 2\pi\frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}$
• Simplifies the description of diffraction phenomena and crystal symmetry
• Ewald sphere construction in reciprocal space visualizes diffraction conditions

Structure factor

• Complex quantity describing the amplitude and phase of scattered X-rays
• Depends on atomic positions and scattering factors of atoms in the unit cell
• Expressed as $F_{hkl} = \sum_j f_j e^{2\pi i(hx_j + ky_j + lz_j)}$
• $f_j$ represents the atomic scattering factor of atom j
• $(x_j, y_j, z_j)$ denote fractional coordinates of atom j in the unit cell
• $(h, k, l)$ are Miller indices of the diffracting plane

Systematic absences

• Occur when structure factor becomes zero for certain combinations of Miller indices
• Result from specific symmetry elements in the crystal structure (glide planes, screw axes)
• Help determine space group and crystal system of the material
• Absence of (h00) reflections for h = odd in body-centered cubic structures
• (0k0) reflections absent for k = odd in structures with b-glide perpendicular to a-axis

Experimental techniques

• X-ray diffraction experiments encompass various methods tailored to different sample types
• Choice of technique depends on sample characteristics, available instrumentation, and research objectives
• Advances in X-ray sources and detectors continually expand the capabilities of diffraction analysis

Powder diffraction

• Utilizes polycrystalline samples with randomly oriented crystallites
• Produces characteristic ring patterns on 2D detectors or intensity vs. 2θ plots
• Suitable for phase identification, quantitative analysis, and structure refinement
• Advantages include simple sample preparation and ability to analyze mixtures
• Limitations include peak overlap and loss of some 3D structural information

Single crystal methods

• Employs individual crystals to obtain detailed 3D structural information
• Rotation method collects diffraction patterns at various crystal orientations
• Laue method uses polychromatic X-rays to obtain diffraction patterns without rotation
• Provides precise atomic positions, bond lengths, and angles
• Challenges include growing suitable crystals and handling radiation-sensitive samples

Synchrotron radiation

• Utilizes high-intensity X-rays produced by electron acceleration in storage rings
• Offers tunable wavelengths, high brilliance, and pulsed time structure
• Enables time-resolved studies, high-pressure experiments, and microbeam analysis
• Facilitates investigation of weakly scattering materials and subtle structural changes
• Applications include protein crystallography and in-situ studies of material transformations

Diffraction patterns

• Contain rich information about crystal structure, symmetry, and material properties
• Interpretation requires understanding of reciprocal space and diffraction physics
• Advanced software tools assist in pattern analysis and structure solution

Interpretation basics

• Diffraction peaks correspond to specific lattice planes in the crystal structure
• Peak positions relate to interplanar spacings through Bragg's law
• Peak intensities depend on structure factor and multiplicity of reflections
• Peak shapes influenced by instrumental factors and sample characteristics
• Background arises from various sources (air scattering, fluorescence, amorphous content)

Miller indices

• Three-integer notation (hkl) describing lattice planes and directions in crystals
• Represent the reciprocal of the fractional intercepts of planes with crystal axes
• Determine the orientation of crystal planes relative to the unit cell edges
• Low-index planes (small h, k, l values) often have higher diffraction intensities
• Systematic absences in Miller indices reveal information about crystal symmetry

Intensity analysis

• Peak intensities provide information about atomic positions and occupancies
• Integrated intensities proportional to the square of the structure factor magnitude
• Lorentz-polarization factor corrects for geometric and physical effects on intensity
• Temperature factor (Debye-Waller factor) accounts for thermal vibrations of atoms
• Extinction and absorption effects may require additional corrections in some cases

Advanced applications

• X-ray diffraction techniques extend beyond basic structure determination
• Enable investigation of complex materials systems and in-situ processes
• Provide insights into material behavior under various conditions and treatments

Thin films vs bulk materials

• Thin film diffraction employs grazing incidence geometry to enhance surface sensitivity
• Reveals information about film thickness, texture, and epitaxial relationships
• Bulk diffraction probes the entire sample volume, suitable for homogeneous materials
• Thin films often exhibit strain effects and preferred orientation not present in bulk
• Reflectivity measurements provide information on film density and interface roughness

Strain and texture analysis

• Lattice strain manifests as peak shifts in diffraction patterns
• Williamson-Hall analysis separates strain and size effects on peak broadening
• Texture (preferred orientation) alters relative peak intensities
• Pole figure measurements map the distribution of crystallite orientations
• Residual stress analysis uses sin²ψ method to determine stress state in materials

In-situ diffraction studies

• Monitor structural changes during heating, cooling, or application of external fields
• High-temperature XRD reveals phase transitions and thermal expansion behavior
• Pressure-dependent studies explore structural changes at extreme conditions
• Electrochemical cells enable investigation of battery materials during cycling
• Time-resolved experiments capture transient structures in chemical reactions

Data analysis methods

• Extracting meaningful information from diffraction data requires sophisticated analysis techniques
• Combines mathematical modeling with physical understanding of diffraction phenomena
• Continuous development of analysis methods enhances the power of X-ray diffraction

Peak fitting

• Determines precise peak positions, intensities, and shapes
• Employs mathematical functions (Gaussian, Lorentzian, Pseudo-Voigt) to model peak profiles
• Least-squares refinement optimizes fit parameters to match observed data
• Deconvolution techniques separate overlapping peaks
• Provides quantitative information for further structural analysis and refinement

Rietveld refinement

• Whole-pattern fitting method for powder diffraction data
• Refines structural model to match calculated and observed diffraction patterns
• Adjusts parameters including atomic positions, occupancies, and thermal factors
• Accounts for instrumental effects and sample characteristics
• Yields detailed structural information and quantitative phase analysis

Phase identification

• Compares observed diffraction patterns with database of known structures
• Search-match algorithms identify potential phases based on peak positions and intensities
• Considers chemical composition and sample history to narrow down possibilities
• Handles multi-phase mixtures through peak deconvolution and quantitative analysis
• Challenges arise with unknown phases, solid solutions, and highly textured samples

Limitations and challenges

• Understanding limitations of X-ray diffraction ensures proper interpretation of results
• Awareness of challenges guides experimental design and data analysis strategies
• Ongoing research addresses these limitations through improved techniques and instrumentation

Peak overlap

• Occurs when diffraction peaks from different lattice planes coincide
• More pronounced in low-symmetry structures and complex mixtures
• Complicates phase identification and quantitative analysis
• Addressed through high-resolution instruments and advanced peak deconvolution methods
• Complementary techniques (neutron diffraction) may provide additional peak separation

Preferred orientation

• Non-random distribution of crystallite orientations in polycrystalline samples
• Alters relative peak intensities, affecting structure refinement and quantitative analysis
• Common in materials with anisotropic crystal shapes (platelets, needles)
• Minimized through careful sample preparation (spray-drying, backloading)
• Modeled in Rietveld refinement using spherical harmonics or March-Dollase functions

Sample preparation effects

• Improper sample preparation can introduce artifacts and errors in diffraction data
• Grinding may induce strain or phase transformations in sensitive materials
• Surface roughness affects peak intensities and backgrounds in reflection geometry
• Insufficient particle statistics in coarse-grained samples lead to poor reproducibility
• Absorption effects in highly absorbing or inhomogeneous samples require correction

Complementary techniques

• X-ray diffraction often used in conjunction with other analytical methods
• Complementary techniques provide additional structural and compositional information
• Combining multiple approaches enables comprehensive characterization of complex materials

Neutron diffraction vs X-ray

• Neutrons interact with atomic nuclei rather than electron clouds
• Provides better contrast for light elements and neighboring elements in periodic table
• Deeper penetration allows for bulk analysis of larger samples
• Magnetic scattering reveals magnetic structure and ordering in materials
• X-rays offer higher resolution and are more readily available in laboratory settings

Electron diffraction

• Utilizes high-energy electrons instead of X-rays for diffraction
• Enables analysis of very small sample volumes (nanocrystals, thin films)
• Provides both real-space imaging and reciprocal space diffraction information
• Rotation electron diffraction (RED) allows 3D structure determination of nanocrystals
• Challenges include multiple scattering effects and beam damage to sensitive samples

Spectroscopic methods

• X-ray absorption spectroscopy (XAS) probes local atomic environment and oxidation states
• X-ray photoelectron spectroscopy (XPS) analyzes surface composition and chemical states
• Raman spectroscopy provides complementary information on molecular vibrations and symmetry
• Nuclear magnetic resonance (NMR) reveals local atomic environments in both crystalline and amorphous materials
• Combining diffraction with spectroscopic techniques enables comprehensive structural characterization