The 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 , 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 between atoms in a crystal structure
represents the convolution of with its inverse
appear at the origin of the Patterson map, corresponding to atoms with themselves
occur between different atoms, providing information about interatomic distances
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 and
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 on specific planes
Mathematical Formulation and Properties
Patterson function defined as P(u,v,w)=V1∑h,k,l∣Fhkl∣2cos[2π(hu+kv+lw)]
Patterson map exhibits , regardless of the crystal's symmetry
Self-vectors contribute to the origin peak with intensity proportional to ∑jZj2
Cross-vectors appear at positions corresponding to interatomic vectors rj−ri
Harker sections occur at specific coordinates determined by the symmetry (u = 2x, v = 2y, w = 2z for P2₁2₁2₁)
Heavy Atom Methods
Principles of Heavy Atom Method
utilizes atoms with high atomic numbers to solve crystal structures
Isomorphous replacement involves introducing heavy atoms without changing crystal structure
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
Key Terms to Review (15)
Atomic positions: Atomic positions refer to the specific locations of atoms within a crystal lattice, defined by their coordinates in three-dimensional space. Understanding these positions is essential for interpreting crystal structures and plays a crucial role in methods like the Patterson function and heavy atom techniques, which are used to solve the phases of crystal diffraction patterns.
Centrosymmetry: Centrosymmetry refers to a property of a crystal structure where every point in the crystal has an equivalent point at an equal distance in the opposite direction from a central point, making it symmetric about that center. This concept is important in crystallography because it helps in understanding the arrangement of atoms and how they interact, particularly when analyzing diffraction patterns and interpreting the Patterson function, which relies on the symmetry of the crystal to identify the positions of atoms.
Cross-vectors: Cross-vectors are mathematical constructs that arise in vector analysis, specifically relating to the concept of the cross product. They are essential in crystallography for representing relationships between different vectors, particularly when analyzing three-dimensional crystal structures and their symmetry properties.
Electron density: Electron density refers to the probability of finding electrons in a specific region of space around an atom or a molecule. This concept is crucial for understanding how atoms bond and interact, influencing the overall structure and properties of crystalline materials. By analyzing electron density, scientists can derive valuable information about atomic positions and arrangements in a crystal, which is essential for techniques like structure factor calculations and Patterson functions.
Harker Sections: Harker sections are graphical representations used in crystallography to analyze the data from the Patterson function, helping to determine the arrangement of atoms within a crystal. These sections are obtained by slicing through a three-dimensional Patterson map along specific planes, allowing researchers to visualize the distribution of heavy atom positions and enhance the understanding of molecular structures. They are especially important in solving structures using heavy atom methods, as they simplify the complexity of the Patterson function by providing more accessible two-dimensional data.
Heavy atom method: The heavy atom method is a technique used in crystallography to enhance the determination of molecular structures by incorporating heavy atoms, which scatter X-rays more strongly than lighter atoms. This method allows for improved resolution of the electron density map, facilitating phase determination that is crucial for interpreting crystallographic data. By introducing heavy atoms into the crystal structure, researchers can take advantage of their scattering properties to derive more accurate structural information.
Interatomic Vectors: Interatomic vectors are directional vectors that represent the distance and direction between two atoms in a crystal lattice. They are crucial for understanding the arrangement of atoms within a crystal structure and play a vital role in techniques used to analyze crystal symmetry and electron density distributions, especially in methods like the Patterson function and heavy atom techniques.
Isomorphous Replacement: Isomorphous replacement is a method used in crystallography to determine the structure of a molecule by replacing one or more of its atoms with similar atoms that have different scattering properties. This technique relies on the principle that the crystal lattices of the original and modified structures are nearly identical, allowing for comparison of the diffraction patterns to derive information about the original structure. By analyzing the differences in the diffraction data caused by these heavy atoms, researchers can solve the phase problem, which is crucial for accurate structure determination.
Patterson Function: The Patterson function is a mathematical tool used in crystallography to derive information about the arrangement of atoms in a crystal structure. It calculates a three-dimensional distribution of vectors between pairs of scattering centers, helping to identify the relative positions of these centers without needing phase information. This function is particularly significant in heavy atom methods, where it aids in solving complex crystal structures by exploiting the presence of heavier atoms.
Patterson map: A Patterson map is a mathematical tool used in crystallography to visualize the distribution of electron density within a crystal structure. By analyzing the intensities of the scattered X-rays from a crystal, the Patterson function generates a map that reveals the positions of atoms relative to one another, providing critical information for determining molecular structures, especially in heavy atom methods.
Patterson Superposition: Patterson superposition is a technique used in crystallography to determine the three-dimensional arrangement of atoms in a crystal by exploiting the Patterson function. This method allows researchers to resolve the positions of atoms by comparing the calculated Patterson map from known heavy atom derivatives with experimental data, thereby facilitating the phase determination crucial for crystal structure analysis.
Self-vectors: Self-vectors, also known as eigenvectors, are specific vectors associated with a linear transformation represented by a matrix, where the transformation merely scales the vector by a corresponding eigenvalue. They are critical in analyzing symmetry and periodicity in crystallography, especially in understanding molecular structures and their interactions. Their significance extends to computational methods used in crystallography, including Patterson functions and heavy atom techniques, where self-vectors help identify atomic positions and refine structural models.
Space Group: A space group is a mathematical classification that describes the symmetry of a crystal structure, incorporating both translational and point symmetry operations. It combines the various symmetry elements such as rotations, reflections, and translations that define how a crystal's lattice points are arranged in three-dimensional space. Understanding space groups is essential for analyzing the geometric and physical properties of crystalline materials.
Symmetry-related peaks: Symmetry-related peaks refer to the reflections in a crystallographic dataset that arise due to the symmetrical nature of the crystal structure. These peaks appear at positions in reciprocal space that are related by symmetry operations defined by the crystal's space group. Understanding these peaks is crucial for analyzing the Patterson function and applying heavy atom methods effectively.
Vector distribution: Vector distribution refers to the arrangement of vectors in a crystallographic context, representing the spatial relationship of atoms within a crystal lattice. This concept plays a crucial role in analyzing the symmetry and periodicity of crystal structures, providing insights into their properties and behaviors. In mathematical crystallography, vector distributions are essential for understanding diffraction patterns and the interpretation of the Patterson function and heavy atom methods.