Structure factors are crucial in crystallography, representing how atoms scatter X-rays in a crystal. They combine atomic positions and scattering powers to determine diffraction intensities. Understanding structure factors is key to solving crystal structures.

Calculating structure factors involves summing contributions from all atoms in the unit cell. This process links real space (atomic positions) to reciprocal space (diffraction patterns). Analyzing structure factors reveals crystal symmetry and helps determine molecular structures.

Structure Factor and Components

Understanding Structure Factors and Atomic Scattering

• Structure factor represents the collective scattering of X-rays by all atoms in a unit cell
• Atomic scattering factor measures how effectively an atom scatters X-rays
  • Depends on the atomic number and scattering angle
  • Decreases with increasing scattering angle
• Form factor describes the scattering power of an atom relative to that of a single electron
  • Varies with the type of atom and scattering angle
  • Larger atoms generally have higher form factors
• Amplitude of the structure factor indicates the intensity of the diffracted X-ray beam
  • Proportional to the square root of the measured intensity
  • Affected by the atomic positions and types of atoms in the unit cell
• Phase angle represents the phase shift of the scattered wave relative to a reference wave
  • Crucial for determining the electron density distribution in the crystal
  • Cannot be directly measured in X-ray diffraction experiments (crystallographic phase problem)

Calculating and Interpreting Structure Factors

• Structure factor calculation involves summing contributions from all atoms in the unit cell
• Express structure factor as a complex number with amplitude and phase components
• Use Fourier transform to convert between real space (electron density) and reciprocal space (structure factors)
• Analyze structure factor patterns to deduce crystal symmetry and atomic arrangements
• Apply structure factor calculations in protein crystallography to determine molecular structures
  • Combine experimental diffraction data with phase information from various methods (heavy atom, molecular replacement)

Systematic Absences and Friedel's Law

Systematic Absences in Diffraction Patterns

• Systematic absences occur when certain reflections are missing from the diffraction pattern
• Result from specific symmetry elements in the crystal structure
  • Screw axes, glide planes, and centering translations
• Help determine the space group of the crystal
• Examples of systematic absences:
  • Body-centered cubic lattice: h + k + l must be even for non-zero intensity
  • Face-centered cubic lattice: h, k, and l must be all odd or all even for non-zero intensity
• Analyze systematic absences to narrow down possible space groups
• Use International Tables for Crystallography to identify space groups based on observed absences

Friedel's Law and Its Applications

• Friedel's law states that the intensities of (hkl) and (-h-k-l) reflections are equal
• Applies to centrosymmetric crystals and non-centrosymmetric crystals without anomalous scattering
• Breaks down in the presence of anomalous scattering
  • Occurs when the incident X-ray energy is close to an absorption edge of an atom in the crystal
• Use violations of Friedel's law to determine absolute structure
  • Important in pharmaceutical industry for identifying chiral molecules
• Apply Friedel pair analysis in protein crystallography for phase determination
  • Single-wavelength anomalous diffraction (SAD) and multi-wavelength anomalous diffraction (MAD) methods

Structure Factor Equation and Phase Problem

Structure Factor Equation and Its Components

• Structure factor equation: $F(hkl) = \sum_{j=1}^N f_j e^{2\pi i(hx_j + ky_j + lz_j)}$
• Summation over all atoms (N) in the unit cell
• $f_j$ represents the atomic scattering factor of atom j
• $(x_j, y_j, z_j)$ are the fractional coordinates of atom j in the unit cell
• $(h, k, l)$ are the Miller indices of the reflection
• Exponential term accounts for the phase shift due to atom position
• Calculate structure factors for different $(hkl)$ values to generate the complete diffraction pattern
• Use structure factor equation in reverse to determine electron density from measured intensities

The Phase Problem and Solution Strategies

• Phase problem arises because only intensities can be measured in X-ray diffraction experiments
• Intensity is proportional to the square of the structure factor amplitude
• Phase information is lost during the measurement process
• Solving the phase problem is crucial for determining the crystal structure
• Various methods to overcome the phase problem:
  • Direct methods: use statistical relationships between structure factor amplitudes
  • Patterson methods: exploit heavy atom positions in the unit cell
  • Isomorphous replacement: compare diffraction patterns of native and heavy-atom derivatives
  • Anomalous scattering: utilize wavelength-dependent scattering effects
  • Molecular replacement: use known structures as initial phase models
• Combine phase estimates with measured amplitudes to calculate electron density maps
• Iterative refinement of phases and model building to improve structure solution