8.3 Patterson function and heavy atom methods

The Patterson function is a powerful tool in crystallography, helping solve crystal structures without phase info. It calculates vector distributions between atoms, creating a map that shows interatomic distances. This method is crucial for unraveling complex structures.

Heavy atom methods take advantage of atoms with high atomic numbers to simplify structure determination. By introducing these atoms or using isomorphous replacement, researchers can more easily locate key positions and solve crystal structures.

Patterson Function and Map

Understanding the Patterson Function and Map

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• Patterson function calculates the vector distribution between atoms in a crystal structure
• Patterson map represents the convolution of electron density with its inverse
• Self-vectors appear at the origin of the Patterson map, corresponding to atoms with themselves
• Cross-vectors occur between different atoms, providing information about interatomic distances
• Harker sections contain peaks corresponding to symmetry-related atoms, simplifying structure determination

Applications and Interpretation of Patterson Analysis

• Patterson function helps solve crystal structures without phase information
• Patterson map interpretation reveals atomic positions and interatomic vectors
• Self-vectors create a large peak at the origin, proportional to the sum of squared atomic numbers
• Cross-vectors appear as smaller peaks, representing distances between different atoms
• Harker sections simplify structure solution by concentrating symmetry-related peaks on specific planes

Mathematical Formulation and Properties

• Patterson function defined as $P(u,v,w) = \frac{1}{V} \sum_{h,k,l} |F_{hkl}|^2 \cos[2\pi(hu+kv+lw)]$
• Patterson map exhibits centrosymmetry, regardless of the crystal's symmetry
• Self-vectors contribute to the origin peak with intensity proportional to $\sum_{j} Z_j^2$
• Cross-vectors appear at positions corresponding to interatomic vectors $r_j - r_i$
• Harker sections occur at specific coordinates determined by the space group symmetry (u = 2x, v = 2y, w = 2z for P2₁2₁2₁)

Heavy Atom Methods

Principles of Heavy Atom Method

• Heavy atom method utilizes atoms with high atomic numbers to solve crystal structures
• Isomorphous replacement involves introducing heavy atoms without changing crystal structure
• Patterson superposition combines multiple Patterson maps to locate heavy atom positions

Implementing Heavy Atom Techniques

• Heavy atom method exploits the strong scattering power of atoms with high atomic numbers (mercury, platinum)
• Isomorphous replacement requires preparing crystals with and without heavy atoms (native and derivative)
• Patterson superposition overlays multiple Patterson maps to identify common features and atom positions

Advantages and Challenges of Heavy Atom Approaches

• Heavy atom method simplifies phase determination by dominating the diffraction pattern
• Isomorphous replacement provides phase information through intensity differences between native and derivative crystals
• Patterson superposition reduces noise and enhances peaks corresponding to heavy atom positions
• Challenges include finding suitable heavy atoms and maintaining isomorphism in crystal structures
• Multiple isomorphous replacement (MIR) uses several heavy atom derivatives to improve phase accuracy