What Is a Patterson Map

The Patterson synthesis was not derived by a mathematician and then put into use by an experimenter. On the contrary, it was calculated by a perceptive experimenter, who was initially uncertain of its meaning, who examined its features, and deduced its properties. Only some time later, and after considerable discussion among interested investigators was its actual physical basis deduced and its meaning mathematically established (see Glusker et al., 1987). It seems reasonable therefore to follow that same path and simply state how it is calculated, what it means in terms of atom distributions, and then later show how it may be used in structure determination. 

Remembering that Fflkl is simply the measured intensity / , a scaler quantity, then the expression can be recognized as simply the electron density equation (see Chapter 5) with squared coefficients and all phases 4 hkt set equal to zero. The normalization constant is here l/V2 because of the squared coefficient, where V is the volume of the unit cell. The units it implies for the function, something per volume squared, immediately indicates that P(u, v, w) is not electron density but some other spatial function. Because the equation yields something other than electron density, existing in some unique space, we cannot denote it by p(x, y, z) in jc, y, z (real space) we must designate it by P(u, v, w) in some alternative coordinate space whose variables are u, v, w. Otherwise, P(u, v, w) is the equation for aperiodic function in u, v, w space. The Patterson function, or Patterson wave 

Now what are these peaks, why do they have the values we observe, and what implications do their map positions ( , v, w) carry for the true distribution of atoms at x, y, z within the real crystallographic unit cell  

What Patterson theorized, and with the help of others eventually proved, was the following if we choose any point in the Patterson map at u, v, w, that point specifies a vector 

FIGURE 9.1 A section from a raw (directly from the computer) Patterson map calculated from the structure amplitudes recorded from a protein crystal. Contour lines have been drawn around areas of high Patterson function values to emphasize positions of Patterson peaks. Because of inaccuracies in the data and other sources of error, fluctuations in the background value of the Patterson function may be appreciable. 

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