💎Crystallography Unit 8 – Crystal Structure Determination

Fundamentals of Crystallography

• Crystallography studies the arrangement of atoms in crystalline solids
• Crystals have a periodic and ordered structure with repeating patterns in three dimensions
• The smallest repeating unit that represents the entire crystal structure is called the unit cell
• Unit cells are characterized by their lattice parameters (lengths a, b, c and angles α, β, γ)
• The arrangement of atoms within the unit cell determines the crystal's symmetry and properties
• Crystals can be classified into seven crystal systems based on their symmetry (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic)
• The concept of lattice planes and Miller indices (hkl) are used to describe the orientation and spacing of planes within a crystal

Crystal Systems and Symmetry

• The seven crystal systems are defined by the relationships between their lattice parameters and the presence of certain symmetry elements
  • Triclinic: a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°, no symmetry elements
  • Monoclinic: a ≠ b ≠ c, α = γ = 90° ≠ β, one two-fold rotation axis or mirror plane
  • Orthorhombic: a ≠ b ≠ c, α = β = γ = 90°, three perpendicular two-fold rotation axes or mirror planes
  • Tetragonal: a = b ≠ c, α = β = γ = 90°, one four-fold rotation axis
  • Trigonal: a = b = c, α = β = γ ≠ 90°, one three-fold rotation axis
  • Hexagonal: a = b ≠ c, α = β = 90°, γ = 120°, one six-fold rotation axis
  • Cubic: a = b = c, α = β = γ = 90°, four three-fold rotation axes
• Symmetry elements include rotation axes, mirror planes, inversion centers, and screw axes
• The combination of crystal systems and symmetry elements results in 32 crystallographic point groups
• Bravais lattices describe the 14 unique lattice types that can fill space without gaps (primitive, body-centered, face-centered, and base-centered)
• Space groups combine point group symmetry with translational symmetry, resulting in 230 unique space groups

X-ray Diffraction Basics

• X-ray diffraction (XRD) is the primary technique used to determine crystal structures
• X-rays interact with electrons in the crystal, causing them to scatter
• Constructive interference of scattered X-rays occurs when Bragg's law is satisfied: $nλ = 2dsinθ$
  • $n$: integer (order of reflection)
  • $λ$: wavelength of the X-rays
  • $d$: interplanar spacing
  • $θ$: angle between the incident X-rays and the lattice plane
• The resulting diffraction pattern consists of a series of spots (reflections) with varying intensities
• The positions of the reflections provide information about the unit cell dimensions and symmetry
• The intensities of the reflections are related to the arrangement of atoms within the unit cell
• The structure factor $F(hkl)$ is a complex number that represents the amplitude and phase of a diffracted wave from a particular set of lattice planes (hkl)

Data Collection Techniques

• Single-crystal X-ray diffraction is the most common method for collecting data on crystalline materials
• 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 data is recorded using a detector (e.g., CCD, CMOS, or pixel array detector)
• The data collection strategy aims to maximize completeness and redundancy while minimizing radiation damage
  • Completeness: the percentage of unique reflections measured
  • Redundancy: the average number of times each unique reflection is measured
• Data reduction involves integrating the intensities of the reflections, applying corrections (e.g., absorption, polarization), and scaling the data
• The quality of the data is assessed using various metrics, such as $R_{merge}$, $I/σ(I)$, and $CC_{1/2}$

Structure Factor Analysis

• The structure factor $F(hkl)$ is related to the electron density distribution in the unit cell by a Fourier transform
• The electron density $ρ(xyz)$ at a point $(x, y, z)$ in the unit cell is given by: $ρ(xyz) = \frac{1}{V} \sum_{hkl} F(hkl) e^{-2πi(hx+ky+lz)}$
  • $V$: volume of the unit cell
• The structure factor $F(hkl)$ is a complex number with amplitude $|F(hkl)|$ and phase $φ(hkl)$: $F(hkl) = |F(hkl)|e^{iφ(hkl)}$
• The measured intensities $I(hkl)$ are proportional to the square of the structure factor amplitudes: $I(hkl) ∝ |F(hkl)|^2$
• The structure factors depend on the positions $(x_j, y_j, z_j)$ and types of atoms in the unit cell: $F(hkl) = \sum_{j=1}^N f_j e^{2πi(hx_j+ky_j+lz_j)}$
  • $f_j$: atomic scattering factor of atom $j$
  • $N$: number of atoms in the unit cell
• The temperature factor (B-factor) is introduced to account for the effect of thermal motion on the atomic scattering factors

Phase Problem and Solutions

• The phase problem arises because the measured intensities only provide information about the amplitudes of the structure factors, not their phases
• Solving the phase problem is crucial for determining the electron density and, consequently, the crystal structure
• Several methods exist for obtaining phase information:
  • Direct methods: use statistical relationships between structure factor amplitudes to estimate phases (e.g., Karle-Hauptman method, Sayre's equation)
  • Patterson methods: use the Patterson function (a Fourier transform of the intensities) to locate heavy atoms and derive phases (e.g., heavy-atom method, isomorphous replacement)
  • Molecular replacement: uses the structure of a similar molecule as a starting model to estimate phases
  • Anomalous scattering: exploits the wavelength-dependent scattering of certain atoms to obtain phase information (e.g., multi-wavelength anomalous diffraction, single-wavelength anomalous diffraction)
• The choice of phasing method depends on the nature of the crystal and the available data

Refinement and Validation

• Once an initial model of the crystal structure is obtained, it is refined to improve its agreement with the observed data
• Refinement involves adjusting the atomic positions, temperature factors, and occupancies to minimize the difference between the calculated and observed structure factor amplitudes
• The refinement process is typically iterative, with alternating cycles of manual model building and computational refinement
• The quality of the refined model is assessed using various metrics, such as the R-factor and the free R-factor ($R_{free}$)
  • R-factor: measures the agreement between the observed and calculated structure factor amplitudes
  • $R_{free}$: calculated using a subset of reflections excluded from the refinement process to avoid overfitting
• Other validation tools include Ramachandran plots, clash scores, and geometry checks
• The final model should be chemically and physically reasonable and consistent with prior knowledge about the molecule or system

Advanced Methods and Applications

• Time-resolved crystallography allows the study of dynamic processes, such as enzyme catalysis or photochemical reactions, by collecting diffraction data at different time points
• Neutron diffraction can provide information about the positions of hydrogen atoms, which are difficult to locate using X-ray diffraction due to their low electron density
• Electron diffraction is useful for studying small crystals or thin films that are too small for X-ray diffraction
• Powder X-ray diffraction is used for polycrystalline or amorphous materials, where single crystals are not available
• Pair distribution function (PDF) analysis can provide information about local structure and disorder in materials
• Crystallography plays a crucial role in various fields, such as:
  • Drug discovery and design (e.g., structure-based drug design, fragment screening)
  • Materials science (e.g., characterization of new materials, understanding structure-property relationships)
  • Structural biology (e.g., protein structure determination, enzyme mechanism studies)
  • Mineralogy and geosciences (e.g., identification and characterization of minerals)