J. C. KENDREW and M. F. PERUTZ 



This is the heavy atom method ; another is the method of isomorphous 

 replacement, where a series of isomorphous crystals differing only in 

 the nature of one atom in the molecule (e.g. the alums) is examined. 

 Progressive changes in the amplitudes of corresponding reflexions in 

 successive members of the series may be attributed to the different 

 scattering contribution of the isomorphously-replaced atom, and hence 

 some signs may be deduced. 



Given the amplitudes and phase angles of all diffracted rays, the 

 positions of the atoms in the unit cell may be found by means of a 

 mathematical method known as a Fourier summation. The Fourier 

 method is the basis of all x-ray analysis of more complex types of 

 structure today, no matter whether it be that of a metal, a mineral or 

 an organic substance like penicillin or calciferol. One of its many 

 advantages is that it presents molecular structures in a realistic form 

 so that different types of atom and interatomic bonding are immediately 

 obvious to the eye (see Figure 6). The development of the Fourier 

 method in 1925 and 1926 by Compton, Havighurst and W. L. Bragg 

 was perhaps the most important advance in crystallography since 

 Bragg's original discovery of the structure of NaCl in 1913. The 

 nature of this method will now be explained. 



DETERMINATION OF STRUCTURE '. FOURIER SUMMATION 



By the methods of Fourier analysis it is possible to represent any 

 periodic function mathematically as the sum of a series of sinusoidally 

 varying quantities whose wave-lengths are 1, i, £,£,... 1/n of the 

 fundamental wave-length : each term in the series is fully determined 

 by its amplitude and phase. (The amplitude of a term of a Fourier 

 series is commonly referred to as its coefficient — a name which will 

 frequently recur below.) W. H. Bragg realized as early as 1915 that a 

 crystal structure in which a pattern repeats at regular intervals in three 

 dimensions could be represented mathematically by a Fourier series, 

 but at that time atomic theory was not sufficiently advanced for the 

 physical relationship between the observed diffracted rays and the 

 terms of the series to be fully understood. It was not discovered until 

 1925 that the amplitudes of the diffracted rays themselves formed the 

 coefficients of the Fourier terms, that the relative phases of the 

 diffracted rays formed the phase factors, and that the summation of 

 the series produced a periodic function in three dimensions which is 

 the electron density distribution in the crystal. 



It will be remembered that it is the electrons and not the atomic 

 nuclei which are responsible for the diffraction pattern, so that a 

 crystal structure will appear to the ' x-ray eye ' as a cloud of electrons 



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