💎Mathematical Crystallography Unit 8 – Structure Factors & Fourier Transforms

Key Concepts and Definitions

• Structure factors quantify the amplitude and phase of a wave diffracted from crystal lattice planes
• Fourier transforms convert between real space (electron density) and reciprocal space (diffraction pattern)
• Bragg's law ($n\lambda = 2d\sin\theta$) relates the wavelength of incident X-rays to the spacing between lattice planes and the angle of the diffracted beam
  • $n$ represents the order of diffraction (integer)
  • $\lambda$ is the wavelength of the incident X-rays
  • $d$ is the spacing between lattice planes
  • $\theta$ is the angle between the incident beam and the lattice planes
• Reciprocal lattice is the Fourier transform of the real space lattice
  • Each point in the reciprocal lattice corresponds to a set of lattice planes in the real space lattice
• Atomic scattering factors describe the scattering amplitude of X-rays by individual atoms
  • Depends on the atomic number and the scattering angle
• Structure factor equation: $F_{hkl} = \sum_{j=1}^N f_j \exp[2\pi i(hx_j + ky_j + lz_j)]$
  • $F_{hkl}$ is the structure factor for the $(hkl)$ reflection
  • $f_j$ is the atomic scattering factor for the $j$-th atom
  • $x_j, y_j, z_j$ are the fractional coordinates of the $j$-th atom
  • $h, k, l$ are the Miller indices of the reflection

Mathematical Foundations

• Complex numbers are used to represent structure factors
  • Real part corresponds to the cosine term (in-phase component)
  • Imaginary part corresponds to the sine term (out-of-phase component)
• Euler's formula: $e^{ix} = \cos x + i\sin x$
  • Used to express the structure factor equation in exponential form
• Fourier series represent periodic functions as a sum of sinusoidal terms
  • Each term has a specific frequency, amplitude, and phase
• Fourier transforms extend Fourier series to non-periodic functions
  • Continuous Fourier transform: $F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ikx} dx$
  • Discrete Fourier transform (DFT): $F_k = \sum_{n=0}^{N-1} f_n e^{-2\pi ikn/N}$
• Convolution theorem relates the Fourier transform of a convolution to the product of the Fourier transforms
  • $(f * g)(x) \leftrightarrow F(k)G(k)$
• Sampling theorem (Nyquist-Shannon theorem) states that a band-limited signal can be perfectly reconstructed from its samples if the sampling rate is at least twice the maximum frequency present in the signal

Structure Factors Explained

• Structure factors are complex quantities that describe the amplitude and phase of X-rays scattered by a crystal
• The magnitude of the structure factor ($|F_{hkl}|$) is proportional to the square root of the measured intensity of the diffracted beam
• The phase of the structure factor ($\phi_{hkl}$) cannot be directly measured and must be determined indirectly (phase problem)
• Structure factors depend on the positions and types of atoms in the unit cell
  • Atoms with higher electron density (e.g., heavy atoms) contribute more to the structure factors
• Systematic absences occur when the structure factor is zero for certain reflections due to the presence of symmetry elements (e.g., screw axes, glide planes)
  • Used to determine the space group of the crystal
• Friedel's law states that the intensities of the $(hkl)$ and $(\bar{h}\bar{k}\bar{l})$ reflections are equal in the absence of anomalous scattering
  • Anomalous scattering can lead to differences in the intensities (Bijvoet differences)
• Temperature factors (B-factors) describe the effect of thermal motion on the structure factors
  • Higher temperature factors lead to a reduction in the scattered intensity, especially at high scattering angles

Introduction to Fourier Transforms

• Fourier transforms decompose a function into its constituent frequencies
• The Fourier transform of a function $f(x)$ is defined as: $F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ikx} dx$
  • $k$ represents the frequency domain variable
• The inverse Fourier transform recovers the original function from its Fourier transform: $f(x) = \int_{-\infty}^{\infty} F(k) e^{2\pi ikx} dk$
• Properties of Fourier transforms:
  • Linearity: $a f(x) + b g(x) \leftrightarrow a F(k) + b G(k)$
  • Scaling: $f(ax) \leftrightarrow \frac{1}{|a|} F(\frac{k}{a})$
  • Shift: $f(x-a) \leftrightarrow e^{-2\pi ika} F(k)$
  • Convolution: $(f * g)(x) \leftrightarrow F(k)G(k)$
• Discrete Fourier transform (DFT) is used for sampled data
  • Fast Fourier transform (FFT) is an efficient algorithm for computing the DFT
• Fourier transforms have wide applications in signal processing, image analysis, and crystallography

Applications in Crystallography

• Fourier transforms link the electron density in real space to the structure factors in reciprocal space
  • Electron density: $\rho(x,y,z) = \frac{1}{V} \sum_{h} \sum_{k} \sum_{l} F_{hkl} e^{-2\pi i(hx + ky + lz)}$
  • Structure factors: $F_{hkl} = \int_{V} \rho(x,y,z) e^{2\pi i(hx + ky + lz)} dV$
• Fourier synthesis is used to calculate the electron density from the structure factors
  • Requires both the amplitudes and phases of the structure factors
• Patterson function is the Fourier transform of the intensities (squared structure factor amplitudes)
  • Provides information about the interatomic vectors in the crystal
  • Used in methods such as heavy atom method and molecular replacement to solve the phase problem
• Fourier difference maps are used to locate missing atoms or to identify errors in the model
  • $\Delta\rho(x,y,z) = \frac{1}{V} \sum_{h} \sum_{k} \sum_{l} (F_{obs} - F_{calc}) e^{-2\pi i(hx + ky + lz)}$
• Fourier transforms are also used in powder diffraction to analyze the radial distribution function and pair distribution function

Practical Techniques and Methods

• Data collection: Measure the intensities of the diffracted X-rays using a detector (e.g., CCD, CMOS)
  • Optimize the data collection strategy to ensure complete and redundant data
  • Apply corrections for factors such as polarization, absorption, and Lorentz factor
• Data reduction: Convert the measured intensities to structure factor amplitudes
  • Perform data scaling and merging to combine multiple measurements of the same reflection
  • Assess the quality of the data using metrics such as $R_{merge}$, $R_{pim}$, and $I/\sigma(I)$
• Phasing: Determine the phases of the structure factors using methods such as:
  • Direct methods: Exploit statistical relationships between the structure factors
  • Patterson methods: Use the Patterson function to locate heavy atoms or known molecular fragments
  • Experimental phasing: Measure the phases directly using techniques such as anomalous scattering or isomorphous replacement
• Refinement: Improve the initial model by minimizing the difference between the observed and calculated structure factors
  • Least-squares refinement: Minimize the sum of the squared differences between $F_{obs}$ and $F_{calc}$
  • Maximum likelihood refinement: Maximize the probability of observing the measured data given the model
  • Apply constraints and restraints to maintain reasonable geometry and prevent overfitting

Common Challenges and Solutions

• Twinning: Occurs when multiple crystal domains are oriented in different ways within a single crystal
  • Can lead to overlapping reflections and complicate the data analysis
  • Detect twinning using statistical tests (e.g., L-test, H-test) and examine the intensity distribution
  • Refine the structure using specialized twinning algorithms or detwin the data if possible
• Disorder: Arises when atoms or molecules occupy multiple positions or conformations within the crystal
  • Can lead to diffuse scattering and reduced effective resolution
  • Model disorder using split occupancy, partial occupancy, or ensemble refinement
  • Apply non-crystallographic symmetry (NCS) restraints to improve the modeling of disordered regions
• Anisotropic data: Occurs when the diffraction quality varies significantly with the direction in reciprocal space
  • Can be caused by factors such as crystal shape, absorption, or thermal motion
  • Apply anisotropic scaling or use ellipsoidal truncation to correct for the anisotropy
  • Refine anisotropic displacement parameters (ADPs) to model the directional dependence of atomic motion
• Low resolution: Limits the level of detail that can be resolved in the electron density map
  • Typically encountered with large complexes, membrane proteins, or poorly diffracting crystals
  • Use low-resolution refinement techniques such as jelly-body refinement or group B-factor refinement
  • Apply external restraints derived from high-resolution structures or homology models

Real-World Examples and Case Studies

• Protein crystallography: Determining the three-dimensional structures of proteins
  • Example: Structure of the SARS-CoV-2 main protease (PDB ID: 6Y2F)
    • Key to understanding the viral replication mechanism and designing inhibitors
• Small molecule crystallography: Studying the structures of organic and inorganic compounds
  • Example: Structure of the superconductor YBa2Cu3O7 (YBCO)
    • Revealed the layered structure and the role of oxygen in high-temperature superconductivity
• Powder diffraction: Analyzing the structure of polycrystalline materials
  • Example: Identification of the mineral phases in Martian soil by the Curiosity rover
    • Provided insights into the geological history and potential habitability of Mars
• Electron crystallography: Using electron diffraction to study the structure of nanomaterials and biological specimens
  • Example: Structure of the membrane protein bacteriorhodopsin (PDB ID: 1BRD)
    • First high-resolution structure determined by electron crystallography
• Time-resolved crystallography: Capturing the structural changes during chemical reactions or biological processes
  • Example: Photocycle of the photoactive yellow protein (PYP)
    • Revealed the conformational changes and the role of the chromophore in the light-sensing mechanism