284 H. K. SCHACHMAN AND E. C. WILLIAMS 



e. Patterson Vector Maps. One useful approach to the problem of deter- 

 mining the phases of the structure-amplitude terms was suggested by 

 Patterson (1934), who proposed that serious consideration be given to a 

 Fourier summation in which the squares of the Ff,ki terms would be the 

 coefficients and in which no phase-angle term would be present. Such a 

 summation can be evaluated from the observed intensity data, rather 

 readily so if only a two-dimensional projection is desired. Patterson, and 

 later Harker (1936), showed that this type of summation has a physical 

 significance. A simple example will suffice to show the interpretation of a 

 "Patterson summation". Suppose that there are three atoms: A, B, and C. 

 If we draw lines interconnecting the atoms, there will be three lines, but each 

 line will have two "directions": dependmg (for example) upon whether the 



line is directed A > B or B > A. There will thus be six "vectors", and, 



in general, there will be N(N — 1) such vectors for the case of N atoms. If 

 we now select an origin we may draw these six vectors from it, displacing 

 them as necessary but keeping their directions fixed. The end of each vector 

 will be given a weight (a so-called "vector density") corresponding to the 

 product of the number of electrons in the atoms at the two ends of the 

 vector. Thus, an analog of a true electron density map will be drawn. The 

 Patterson summation yields such a "vector density" map. In general it 

 cannot be analyzed to give the positions of the electron density peaks them- 

 selves. It is useful in that it gives general notions about what are most likely 

 the relative positions of the X-ray scattering centers; a pattern appearing 

 prominently in the Patterson projection sets limits upon the number of 

 guesses that can be taken about the positions of the true electron density 

 peaks. For complex molecules, at least, the most commonly appearing 

 contour maps are Patterson projections, since these can always be secured, 

 and they offer some shreds of information. 



/. Heavy Atom Replacement. While Patterson sununations are useful in 

 attempting to arrive at some useful notions as to electron density distribu- 

 tions, they have proved of only limited use in the case of complex molecules 

 such as the proteins and viruses. What is wanted, of course, is some method 

 whereby the relative phases of the Ff^j^i terms can be evaluated. Two some- 

 what similar methods exist whereby this problem can be directly approached: 

 (1) the method of heavy atom introduction, and (2) the method of isomor- 

 phous replacement (Green et at., 1954). In the former an atom of great 

 X-ray scattering power is introduced within the molecular structure, while 

 in the latter successive replacements of heavy atoms are attempted, the 

 crystalline structure remaining isomorphous during the replacements. In 

 both cases the hope is that the position of the heavy atom is the same within 

 each molecule of the crystal and that its position in the unit cell can be found. 

 The effect of the heavy atom is to modify the relative intensities of the 



