💎Mathematical Crystallography Unit 12 – Crystal Structure Determination & Refinement

Fundamentals of Crystal Structures

• Crystals are solid materials with a highly ordered microscopic structure consisting of a repeating pattern of atoms, ions, or molecules
• The smallest repeating unit that shows the full symmetry of the crystal structure is called the unit cell
• Unit cells are characterized by their lattice parameters: lengths (a, b, c) and angles (α, β, γ)
• There are seven crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic) based on the symmetry of the unit cell
  • Each crystal system has specific constraints on the lattice parameters (e.g., cubic: a = b = c, α = β = γ = 90°)
• Bravais lattices are the 14 unique lattice types that describe the translational symmetry of crystal structures
• Point groups describe the symmetry operations (rotations, reflections, inversions) that leave at least one point in the crystal fixed
• Space groups combine the symmetry elements of point groups with translational symmetry, resulting in 230 unique space groups

X-ray Diffraction Theory

• X-ray diffraction (XRD) is a powerful technique for determining the atomic structure of crystals
• X-rays interact with the electron density in a crystal, causing the X-rays to scatter
• Constructive interference of scattered X-rays occurs when Bragg's law is satisfied: $2d\sin\theta = n\lambda$
  • $d$ is the interplanar spacing, $\theta$ is the incident angle, $n$ is an integer, and $\lambda$ is the X-ray wavelength
• The resulting diffraction pattern is a Fourier transform of the electron density in the crystal
• The intensity of each diffraction spot (hkl) is proportional to the square of the structure factor $F_{hkl}$
• The structure factor is a complex number that describes the amplitude and phase of the scattered X-rays from a particular set of lattice planes (hkl)
• The phase information is lost during the measurement process, leading to the "phase problem" in crystallography

Data Collection Techniques

• Single-crystal X-ray diffraction is the most common method for collecting data to determine crystal structures
• A single crystal is mounted on a goniometer and rotated in the X-ray beam to collect a complete dataset
• The crystal is typically cooled to low temperatures (e.g., 100 K) to reduce thermal motion and improve data quality
• Monochromatic X-rays are used to avoid complications from wavelength-dependent effects
• The diffraction pattern is recorded using a detector (e.g., CCD, CMOS, or pixel array detector)
• The intensities of the diffraction spots are integrated and corrected for various factors (absorption, polarization, etc.)
• Data quality is assessed using metrics such as resolution, completeness, and merging statistics (Rmerge, Rpim)
• High-pressure X-ray diffraction can be used to study crystal structures under extreme conditions

Structure Factor Calculations

• The structure factor $F_{hkl}$ is a complex number that represents the resultant of all waves scattered by atoms in the unit cell for a given reflection (hkl)
• It is calculated as a sum over all atoms in the unit cell: $F_{hkl} = \sum_{j=1}^N f_j \exp[2\pi i(hx_j + ky_j + lz_j)]$
  • $f_j$ is the atomic scattering factor for atom $j$, and $x_j, y_j, z_j$ are its fractional coordinates
• The atomic scattering factor depends on the element and the scattering angle, and it decreases with increasing angle
• The structure factor equation can be simplified using symmetry operations of the space group
• Systematic absences in the diffraction pattern arise from destructive interference due to certain symmetry elements (e.g., screw axes, glide planes)
• The temperature factor (B-factor) is included in the structure factor calculation to account for the effect of thermal motion on the atomic scattering factors
• Anomalous scattering can be used to help solve the phase problem by introducing a known phase shift in the structure factors

Phase Problem and Solutions

• The phase problem arises because the measured intensities in X-ray diffraction experiments only provide the amplitudes of the structure factors, not their phases
• Solving the phase problem is crucial for determining the electron density and, consequently, the atomic positions in the crystal structure
• There are several methods to overcome the phase problem, including:
  • Direct methods: Exploit statistical relationships between structure factor amplitudes to estimate phases
  • Patterson methods: Use the Patterson function, a Fourier transform of the intensities, to locate heavy atoms or known fragments
  • Molecular replacement: Use the structure of a similar molecule as a starting model for phase calculation
  • Anomalous scattering: Exploit the differences in scattering factors for certain elements at specific wavelengths to obtain phase information
• Density modification techniques, such as solvent flattening and histogram matching, can be used to improve and refine the initial phase estimates
• Phase extension methods, like non-crystallographic symmetry averaging, can be employed to extend phases to higher resolution

Direct Methods in Structure Determination

• Direct methods are a class of algorithms that attempt to solve the phase problem by exploiting statistical relationships between structure factor amplitudes
• The main principle behind direct methods is the positivity and atomicity of the electron density
• Sayre's equation relates the structure factors of certain reflections: $F_{hkl} \approx \sum_{h'k'l'} F_{h'k'l'} F_{h-h',k-k',l-l'}$
• The tangent formula is used to estimate the phase of a reflection based on the phases of other reflections: $\tan(\phi_{hkl}) \approx \frac{\sum_{h'k'l'} |E_{h'k'l'}E_{h-h',k-k',l-l'}|\sin(\phi_{h'k'l'} + \phi_{h-h',k-k',l-l'})}{\sum_{h'k'l'} |E_{h'k'l'}E_{h-h',k-k',l-l'}|\cos(\phi_{h'k'l'} + \phi_{h-h',k-k',l-l'})}$
• Normalized structure factors (E-values) are used in direct methods to give equal weight to all reflections
• The triplet relationship states that the product of the signs of three structure factors, related by $h_1 + h_2 = h_3$, tends to be positive
• Figures of merit, such as the residual and the combined figure of merit (CFOM), are used to assess the quality of the phase sets generated by direct methods

Patterson Methods

• The Patterson function is a Fourier transform of the intensities, rather than the structure factors
• It represents a map of interatomic vectors in the crystal structure
• The Patterson function is calculated as: $P(uvw) = \frac{1}{V} \sum_{hkl} |F_{hkl}|^2 \cos[2\pi(hu + kv + lw)]$
• Peaks in the Patterson map correspond to vectors between atoms in the structure, with the peak height proportional to the product of the atomic numbers
• Patterson methods are particularly useful when the structure contains a few heavy atoms (e.g., metals) among many light atoms
• The positions of the heavy atoms can be determined from the Patterson map and used to calculate initial phases for the structure factors
• Difference Patterson maps can be used to locate additional atoms or to identify structural changes between related structures
• Superposition methods, like SHELXS, use the Patterson function to orient and position known molecular fragments in the unit cell

Structure Refinement Techniques

• Structure refinement is the process of optimizing the atomic model to best fit the experimental data
• The goal is to minimize the difference between the observed and calculated structure factor amplitudes
• Least-squares refinement is the most common method, where the quantity $\sum_{hkl} w(|F_{obs}| - |F_{calc}|)^2$ is minimized
  • $w$ is a weighting factor that accounts for the precision of each measurement
• The refinement process typically involves adjusting atomic positions, thermal parameters (B-factors), occupancies, and other model parameters
• Constraints and restraints can be applied to the model to maintain chemically reasonable geometry and to stabilize the refinement
• The quality of the refinement is assessed using the R-factor and the free R-factor (Rfree), which measure the agreement between the model and the data
• Rfree is calculated using a small subset of reflections (5-10%) that are excluded from the refinement process to avoid overfitting
• Difference Fourier maps (Fo-Fc and 2Fo-Fc) are used to identify areas of the model that require improvement or to locate missing atoms
• Anisotropic displacement parameters (ADPs) can be refined for high-resolution data to better model the thermal motion of atoms
• Occupancy refinement is used for structures with disorder or partial occupancy of atomic sites

Advanced Topics and Special Cases

• Twinning occurs when multiple crystal domains are oriented in different ways within a single crystal
  • Merohedral twinning: domains are related by a symmetry operation of the crystal system
  • Non-merohedral twinning: domains are related by a transformation that is not a symmetry operation
• Twinned data can be identified by abnormal intensity statistics and treated using specialized refinement methods
• Modulated structures have a periodic distortion of the basic crystal structure, requiring higher-dimensional crystallography for their description and refinement
• Quasicrystals have ordered but non-periodic structures, characterized by forbidden rotational symmetries (e.g., 5-fold, 10-fold)
• Incommensurate structures have multiple periodic components with periods that are not rationally related, requiring specialized methods for data collection and refinement
• Disorder in crystal structures can be modeled using split atomic positions, partial occupancies, or anisotropic displacement parameters
• Absolute structure determination is important for chiral molecules and requires the use of anomalous scattering or reference compounds of known chirality
• Charge density studies aim to experimentally determine the electron density distribution in crystals, providing insights into bonding and chemical properties