X-Ray Crystallography of Biological Macromolecules 



halfway up the unit cell above the plane of projection). This picture 

 is derived from the electron density projection of Figure 9b where each 

 atom in the unit cell appears as a peak on a contour map ; it will be 

 seen that the peaks corresponding to oxygen atoms are slightly higher 

 than those corresponding to carbon atoms. Figure 9c is a section of 

 the three-dimensional Patterson synthesis through the origin. Unlike 

 the electron density projection of Figure 9b which shows all the atoms 

 in the unit cell, this section shows only those vectors which lie in the 

 basal plane passing through the origin. This means that only those 

 pairs of atoms in Figure 9b which are situated at the same level in the 

 unit cell, so that the line joining the pair (and hence its corresponding 

 vector) is parallel to the plane of projection, will give rise to a peak in 

 this Patterson section. In Figure 9b three such pairs are marked, one 

 by a full, one by a dashed and one by a dot-dashed line ; the corre- 

 sponding vectors are seen radiating from the origin in Figure 9c, each 

 giving rise to a different type of peak. As all the other peaks in the 

 section are related to these three peaks by symmetry, we have already 

 accounted for all the peaks which occur. In the actual analysis of the 

 structure the reasoning would of course be reversed and the orientation 

 of the penta-erythritol molecules would be found from the position 

 of the peaks on the Patterson section. 



Projection of Structure 



Fourier Projection p (xy) 



Potter son Section P(xyO) 



Figure 9. (a) Unit cell of penta-erythritol projected on the face normal 

 to the c axis, (b) Electron density projection from which (a) was derived. 

 Atoms lying at equal levels above the plane of projection are connected 

 by lines, (c) Section through the origin of the three-dimensional vector 

 structure. O marks the origin. Vectors correspond to lines in (b). 

 (Reproduced from Llewellyn, Cox and Goodwyn 9 .) 



A more complex structure would contain a far more intricate system 

 of vector peaks. In fact, since the number of vector peaks in the unit 

 cell increases almost as the square of the number of atoms, the vector 

 equivalent of any but the simplest crystal structure would at first sight 

 appear to be of forbidding complexity. There are, however, special 

 cases where a certain resemblance arises between the real structure of 



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