Patterson Map From a Crystal

The Patterson function P(u, v, w) includes all possible interatomic vectors in the crystallographic unit cell hence it must also contain the vector between every atom and itself. This vector always has components u = 0, v = 0, and w = 0. It is for this reason that the origin peak in a Patterson map is so large. It is the sum of the squares of the atomic numbers of all the atoms in the real unit cell. 

The Patterson map, or Patterson unit cell within the bounds of 0 to 1 along u, v, and w, must contain not only vectors from any atom x, y, z to x, y, z but also the vectors in the opposite directions from atoms x, y, z to jt, y, z. Hence every atomic relationship in real space is represented by a pair of centrosymmetrically related vectors u,v,w(x — x,y-y,z-z) and - u, —v, —w (x - x, y — y,z — z). This tells us that Patterson maps, like reciprocal space and the diffraction pattern, always contain a center of symmetry at their origin even when the real crystallographic unit cell does not. 

Let us now review what has been said and see how it would work in practice. The three-dimensional function P(u) = P(u, v, w), as deduced by Patterson, is the product, taken over the entire unit cell, of the electron density p(x, y, z) at each point (jc, y, z) and at the point related by vectors u = (u, v, w) for all vectors u that can be drawn in the unit cell. Formally, it may be written as 

A Patterson coordinate system is defined based on unit vectors u, v, and w, which are parallel with the axes of the real unit cell of the crystal. Each point (u, v, and w) in Patterson space defines the end point of a vector u having a unique direction and length from the origin of Patterson space to that point, that is, u = (u,v, w). Every point, or vector, in Patterson space (u, v, w) will have associated with it a value P(u). 

The process is now repeated for every vector u that can be drawn within the unit cell of the crystal until the value of P(u) for every point (u, v, w) in Patterson space has been calculated and entered. The collection of all points (u, v, w) with their associated value P(u, v, w) is the Patterson map of the crystallographic unit cell. The Patterson map yields, at least directly, no information regarding the absolute positions of scattering matter in the unit cell, atoms, but it does provide a map of all interatomic vectors in the crystal. The 

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