J. C. KENDREW and M. F. PERUTZ 



to point peaks in the Patterson synthesis, and to examine a structure 

 containing only three such atoms in the unit cell (Figure 8). If the 

 coordinates of two atoms, for example A and B, differ by 2, 3, and 4, 

 say, the vector structure will show a peak at the distance \/2 2 +3 2 +4 2 

 from the origin. The vector joining the origin to this peak, drawn in 

 the figure as a wedged line, corresponds in length and direction to the 

 line joining the two atoms in the crystal structure on the left. The 

 same will apply to the lines joining any other pair of atoms — they will 

 all appear in the Patterson synthesis as lines radiating from a common 

 origin. It can be shown mathematically that if there are n atoms in 

 the unit cell of the crystal structure, the unit cell of the Patterson 

 synthesis will contain n(n — 1) vector peaks. In a real Patterson syn- 

 thesis the height of each peak will of course be equal to the product 

 of the electron densities at the centres of the corresponding pair of atoms. 



Figure 8. Left : Hypothetical structure showing three point 

 atoms A, B and C. Right : Corresponding vector structure. 

 The dot AB marks the vector peak corresponding to the line 

 joining the atoms A and B, and so on. O marks the origin. 



Let us now examine an actual experimental example of the relation- 

 ship between electron density map and Patterson synthesis in a simple 

 organic structure. The substance chosen is penta-erythritol which 

 crystallizes in the tetragonal system. Figure 9a shows this structure in 

 projection along the tetrad axis, with a quarter molecule at each corner 

 and a complete molecule at the centre of the unit cell (the latter lies 



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