Sine-Patterson maps

The intensity differences between the Bragg reflections hkl and hkl suggested to Yoshi H. Okaya and Raymond Pepinsky their use in a sine function (rather than the cosine function of the normal Patterson map) to give an analogue of the Patterson map. This sine-Patterson map has some interesting properties (see Figure 14.27). Peaks in the map represent vectors from the anomalously scattering atom, A, to all other atoms. 

X, but now the vectors between atoms have a distinguishable direction. When peaks in the sine-Patterson map are positive they represent A- X vectors, while the holes (troughs) in the map represent X- A vectors. 

FIGURE 14.27. (a) The cosine (normal) and (b) the sine Patterson maps of potassium isocitrate (Ref. 13). Peaks in these maps are marked with filled circles if they are positive and open circles if they are negative. Note that the high K- -K peaks at u = 0.25, V — 0.50 and u = 0.50, v = 0.0 (present in the cosine Patterson map), are missing in the sine Patterson map. Note that each vector in the sine Patterson map has a sign (positive or negative). 

Thus, by comparing peaks in a normal Patterson map with those in the sine-Patterson function, a sign for each peak can be found that directly gives the absolute structure of the crystal. This is shown in the example in Figure 14.27. 

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