8.2 Fourier transforms in crystallography

Fourier transforms are the secret sauce of crystallography, turning complex crystal structures into manageable math. They let us jump between real space (where atoms live) and reciprocal space (where diffraction happens), making sense of X-ray data.

This magic trick helps us figure out where atoms are hiding in crystals. By playing with waves and frequencies, we can reconstruct electron clouds and see the invisible world of molecules.

Fourier Transforms and Reciprocal Space

Fundamental Concepts of Fourier Transforms

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• Fourier transform converts complex periodic functions from real space to frequency space
• Applies mathematical operation to decompose signals into constituent sinusoidal components
• Represents crystal structure as sum of periodic waves with different amplitudes and phases
• Enables analysis of crystal structures in reciprocal space, facilitating interpretation of diffraction data

Reciprocal Space and Its Significance

• Reciprocal space describes crystal lattice in terms of its periodicities rather than atomic positions
• Inverse relationship exists between real space and reciprocal space dimensions
• Reciprocal lattice points correspond to sets of lattice planes in real space
• Provides framework for understanding diffraction patterns and structure factor calculations
• Simplifies analysis of crystal symmetry and systematic absences in diffraction data

Applications of Convolution Theorem

• Convolution theorem states Fourier transform of convolution equals product of Fourier transforms
• Applies to crystallographic calculations involving electron density and structure factors
• Simplifies complex mathematical operations in reciprocal space
• Facilitates rapid computation of structure factors from atomic positions
• Enables efficient calculation of Patterson functions for structure solution methods

Inverse Fourier Transform in Crystallography

• Inverse Fourier transform converts reciprocal space data back to real space
• Reconstructs electron density distribution from measured structure factors
• Crucial for generating electron density maps from diffraction data
• Allows visualization of atomic arrangements in crystal structures
• Incorporates phase information, which must be determined separately in crystallographic experiments

Electron Density and Synthesis

Electron Density Distribution

• Electron density represents probability of finding electrons at specific locations in crystal
• Described by continuous function varying in three dimensions throughout unit cell
• Peaks in electron density correspond to atomic positions in crystal structure
• Influenced by atomic number, thermal motion, and bonding interactions
• Calculated from structure factors using inverse Fourier transform

Fourier Synthesis Techniques

• Fourier synthesis combines structure factors to reconstruct electron density distribution
• Utilizes measured diffraction intensities and calculated or experimental phases
• Generates electron density maps for structure visualization and refinement
• Improves with inclusion of higher-resolution reflections
• Affected by series termination errors due to limited experimental data

Fourier Difference Maps for Structure Refinement

• Fourier difference map highlights discrepancies between observed and calculated structure factors
• Calculated by subtracting calculated from observed structure factors before Fourier synthesis
• Reveals missing atoms, incorrect atomic positions, or unmodeled disorder in crystal structures
• Guides structural refinement process by indicating necessary model adjustments
• Particularly useful for locating hydrogen atoms and identifying solvent molecules

Resolution and Its Impact on Structure Determination

• Resolution determines level of detail observable in electron density maps
• Expressed as minimum d-spacing of reflections included in Fourier synthesis
• Higher resolution (smaller d-spacing) provides more accurate atomic positions and lower uncertainty
• Limited by experimental factors (crystal quality, data collection parameters)
• Affects ability to distinguish individual atoms and resolve fine structural details
• Typical protein structures require resolution better than 3 Å for reliable atomic model building