J. C. KENDREW and M. F. PERUTZ 



origin in this Patterson projection correspond to the distance between 

 successive maxima of electron density in Figure 5. 



1st Order Term 

 2nd Order Term 

 '3rd Order Term 

 ■4th Order Term 



5th Order Term 



6th Order Term 



7th Order Term 



Sum cf Fourier Terms 



Figure 7. Graphical represen- 

 tation of Fourier summation for 

 Patterson projection. Each 

 sinusoidal wave represents one 

 Fourier component. The bottom 

 curve shows the linear Patterson 

 projection obtained by summing 

 all the components ; its vertical 

 j\/\/\/\/\/\/\ scale is half that of the other 



curves. The origin is marked by 

 a small circle for each curve. 



Onqin Peak 



PM 



" Peak 3 



Peak 

 Peak 2 



Following Patterson's original paper we can state the meaning of 

 the projection as follows. Consider an electron density distribution 

 p along a line z as in Figure 5, which may be written as p(z) ; the 

 Patterson equivalent of this function gives the distribution of a quantity 

 which we may call P(w), where the direction of w coincides with z 

 {Figure 7). It can be shown mathematically that P(w) is the integral 

 over the length of the unit cell of the products of the electron density 

 at any point z, multiplied by the electron density at another point 

 which is a distance w away from z. 



P(w) 



1 



c sin (3 



csin (3 

 J p (z) p (z + w) dvv 

 o 



In the example we have chosen, the peak (1) arises as the integral 

 of all the products of the electron density at any point z, multiplied 

 by the electron density at any point (z -f- 9). Since the electron density 

 projection contains four peaks which are spaced approximately 9 A 

 apart P(w) will contain a maximum at w = 9. On the other hand, the 

 curve contains no maximum at w = 6, since there is no pair of points 

 6 A apart in the electron density distribution where the electron density 

 is high at both points. The Patterson projection, then, has maxima 



172 



