5.2 Interaction of X-rays with matter (absorption, scattering)

X-rays interact with matter through absorption and scattering. Absorption transfers energy to electrons, following the Beer-Lambert law. Scattering changes X-ray direction, with elastic and inelastic types playing crucial roles in crystallography.

Atomic properties and X-ray characteristics influence absorption, while scattering processes vary in complexity. The atomic scattering factor, a key concept in crystallography, describes how atoms scatter X-rays and is essential for structure determination.

X-ray Absorption and Scattering

Absorption Mechanisms and Processes

• X-ray absorption transfers incident X-ray photon energy to an atom's electron, ejecting or exciting it to a higher energy level
• Photoelectric effect serves as primary absorption mechanism, completely transferring X-ray energy to inner-shell electron
• X-ray intensity decreases exponentially through matter, following Beer-Lambert law: $I = I₀e^(-μx)$ (μ represents linear attenuation coefficient)
• X-ray fluorescence emits characteristic X-rays as electrons transition between energy levels after absorption excitation
  • Occurs in elements like calcium and iron
  • Used in X-ray fluorescence spectroscopy for elemental analysis

Scattering Types and Characteristics

• X-ray scattering changes X-ray direction upon interaction with matter
  • Elastic scattering maintains energy
  • Inelastic scattering alters energy
• Compton scattering transfers part of X-ray photon energy to electron
  • Results in scattered photon with lower energy and longer wavelength
  • Significant in medical imaging and radiation therapy
• Elastic scattering (coherent or Rayleigh scattering) preserves X-ray wavelength
  • No energy transfer between X-ray photon and scattering electron
  • Crucial for X-ray diffraction experiments
• Thomson scattering classically describes elastic scattering by free electrons
  • Scattered intensity proportional to $(1 + cos²θ)/2$ (θ represents scattering angle)
  • Forms basis for understanding X-ray scattering in crystallography

Factors Influencing X-ray Absorption

Atomic Properties and X-ray Characteristics

• Atomic number (Z) strongly influences X-ray absorption
  • Higher Z elements exhibit greater absorption due to increased electron density
  • Lead (Z=82) absorbs X-rays more effectively than aluminum (Z=13)
• X-ray wavelength affects absorption
  • Shorter wavelengths (higher energy) penetrate matter more easily than longer wavelengths
  • Hard X-rays (wavelength < 0.1 nm) penetrate deeper than soft X-rays (wavelength > 0.1 nm)
• Mass attenuation coefficient (μ/ρ) describes X-ray absorption effectiveness per unit mass of material
  • Varies with atomic number and X-ray energy
  • Used in radiation shielding calculations

Energy-dependent Absorption Phenomena

• Absorption edges occur at specific energies corresponding to electron binding energies
  • Cause abrupt changes in absorption characteristics
  • K-edge, L-edge, M-edge represent absorption edges for different electron shells
• Photoelectric effect dominates absorption at lower X-ray energies
  • More prominent for heavy elements and lower energy X-rays
• Compton scattering becomes more significant at higher energies
  • Increases with atomic number but less rapidly than photoelectric absorption
• Heavy atoms in a sample significantly alter X-ray absorption patterns
  • Exploited in anomalous X-ray scattering techniques
  • Used in multi-wavelength anomalous diffraction (MAD) for phase determination

Types of X-ray Scattering

Elastic and Inelastic Scattering Processes

• Elastic scattering (coherent or Rayleigh scattering) involves no energy transfer
  • Preserves X-ray wavelength
  • Dominant mechanism in X-ray diffraction experiments
• Inelastic scattering (Compton scattering) results in energy transfer
  • Changes X-ray wavelength
  • Contributes to background noise in diffraction patterns
• Differential cross-section for Compton scattering described by Klein-Nishina formula
  • Accounts for quantum mechanical effects in electron-photon interactions
  • Predicts scattering probability as function of scattering angle and photon energy

Complex Scattering Phenomena

• Resonant (anomalous) scattering occurs when X-ray energy approaches absorption edge of an atom
  • Causes significant changes in scattering amplitude and phase
  • Utilized in multi-wavelength anomalous diffraction (MAD) for phase determination
• Multiple scattering events occur in thick or densely packed samples
  • Complicates analysis of diffraction patterns
  • Requires advanced computational methods for interpretation (dynamical diffraction theory)
• Thermal diffuse scattering results from atomic thermal vibrations
  • Contributes to diffuse background in diffraction patterns
  • Provides information about lattice dynamics and phonon modes

Atomic Scattering Factor in Crystallography

Fundamental Concepts and Dependencies

• Atomic scattering factor (f) describes how an atom scatters X-rays
  • Fundamental quantity in determining intensity of diffracted X-rays
  • Depends on atomic number, X-ray wavelength, and scattering angle
• Forward scattering (θ = 0) atomic scattering factor approximately equals atomic number Z
  • Reflects total number of electrons in atom
  • Decreases with increasing scattering angle
• Atomic scattering factor expressed as complex quantity: $f = f₀ + f' + if"$
  • f₀ represents normal scattering factor
  • f' and f" represent real and imaginary components of anomalous scattering

Applications and Significance in Structure Determination

• Atomic form factor describes variation of f with scattering angle
  • Fourier transform of electron density distribution in atom
  • Crucial for accurate structure factor calculations
• Accurate knowledge of atomic scattering factors essential for structure determination
  • Directly influences structure factor calculations and electron density maps
  • Critical for interpreting X-ray diffraction data
• International Tables for Crystallography provide tabulated values and analytical approximations
  • Essential reference for crystallographic analysis
  • Include scattering factors for different elements and ions
• Anomalous scattering components (f' and f") enable phase determination techniques
  • Single-wavelength anomalous diffraction (SAD)
  • Multi-wavelength anomalous diffraction (MAD)