15:4/ X-ray Analyses of Proteins and Nucleic Acids 285 



The relative intensity will be proportional to |/v fc ;| 2 . In a typical 

 experiment, these are measured and the types of atoms present (and 

 hence, the values of /„) are known. Thus, the assumed values of u n , 

 v n , w n can be used to compute \F hkl \ 2 . It remains to adjust u n , v n , w n 

 for each atom until the final structure agrees both with chemical data 

 and also with the X-ray diffraction pattern. 



Complex crystals or even simple crystals of complex molecules give 

 rise to complex diffraction patterns. The number of points necessary 

 increases both as the number of atoms per molecule and also as the size 

 of the unit cell of the crystal increase. In order to obtain useful informa- 

 tion from these complicated diffraction patterns, it is necessary to know 

 the relative intensities of the various maxima as well as their direction. 



In interpreting diffraction by complex molecules, it is more convenient 

 to deal with electron densities than with atomic positions. After the 

 electron density has been mapped, the atoms may be located at the 

 center of the density maxima. From the preceding paragraphs, it may 

 be seen that the crystal structure factor F hkl can be defined by an 

 absolute value |/^ fc/ |, and a phase angle a hkh as 



, „ , amplitude of the wave scattered by all atoms in the unit cell 

 amplitude of the wave scattered by an electron 



a hki = phase difference between the wave scattered by the unit cell 

 and the wave scattered by an electron at the origin 



Adding the contribution of each electron as before 



F hkl = jjj P (u, v, w) e 2n « hu+kv +lw > du dv dw (3) 



where 



p = electron density at u, v, w 



Readers familiar with Fourier series will recognize that Equation 3 has 

 the form of the coefficients of a Fourier series. Accordingly, one may 

 rewrite it as 



p{u, V,W) =222 \ F hkl\ COS {lirihu + kv + Iw) + a hkl ) 

 h k I 



(4) 



Thus, if one can guess the values of a hkl and can measure a sufficient 

 number of intensities \F hkl \ 2 , then one can map the electron density p 

 and hence, locate all the atoms. This is called a Fourier synthesis. 



The problem of correctly guessing the phases either in Equation 2 or 

 in Equations 3 and 4 has intrigued mathematically minded crystallog- 

 raphers. The general procedure is to guess phase values and then 

 keep adjusting these to give sharper and sharper electron-density 

 contours. If one gets on the right track, these contours define atoms 



