X-Ray Crystallography of Biological Macromolecules 



of varying density. The cloud will be densest near the centre of an 

 atom and thinnest halfway between two atoms. Given then the 

 amplitudes and phases of all the reflexions in the diffraction pattern, 

 and applying the process of Fourier summation, the electron density 

 can be calculated throughout the unit cell and the positions of all the 

 atoms can thus be determined. The kind of crystal whose structure 

 interests people today produces a very large number of reflexions ; 

 hence the number of data will be large and the work of computation 

 great. However, various devices have been constructed to shorten this 

 labour and at the present time a number of projects are on foot to 

 design electronic computers for this purpose. 



Often such a calculation of the electron density in three dimensions 

 may be impossible, because the phase factors cannot be found ; but 

 much useful information can be derived by calculating the electron 

 density projected on to a plane, or even on to a line passing through 

 the unit cell. We should like to dwell in detail on examples of the 

 latter kind because they afford convenient and simple illustrations of 

 the Fourier method. 



Figure 4. Hypothetical unit 

 cell containing three atoms. 

 The curve on the left is a linear 

 electron density projection on 

 the c axis, obtained by pro- 

 jecting all the electrons in the 

 unit cell onto c as indicated. 



A linear Fourier projection can be visualized in the following way. 

 Consider the array of atoms in the hypothetical unit cell of Figure 4. 

 If all the electrons in these atoms are projected on to the line marked 

 c, along directions which are perpendicular to c, we shall obtain a 

 varying distribution of projected electron density along this line. We 

 may now plot this in the form of a curve which will actually show the 

 height of each atom above the base of the unit cell. To calculate the 

 electron density distribution in the unit cell projected on to this line, 

 the intensities of the different order diffracted rays from the set of 

 lattice planes normal to that line would have to be measured. Their 

 amplitudes would form the coefficients of the terms of the series, and 

 provided the phase factors could be found, the series could be summed. 



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