X-Ray Crystallography of Biological Macromolecules 



intensity of scattering at any point in a crystal is a function of the 

 electron density {i.e. the number of electrons per unit volume) at that 

 point. The amplitudes and phases of the diffracted beams of x-rays 

 will be a function of the distribution of electron density within each 

 repeat scattering unit, in this case the unit cell. 



THE ANALYSIS OF X-RAY DIFFRACTION PATTERNS 



In the grating it would be possible, by measuring the amplitudes and 

 phases of all the diffracted rays, to deduce the shape of the lines in the 

 grating. Actually calculation is unnecessary, since by a suitable optical 

 arrangement the rays can be recombined to form an image of the 

 original grating, the process of recombination automatically taking 

 care of both amplitudes and phases ; this, of course, is the basis of 

 Abbe's treatment of the optical microscope, where the object is con- 

 ceived as replaced by a small ideal grating 1 . 



Unfortunately there are no such things as x-ray lenses (in other 

 words we do not know how to refract x-rays appreciably), so it is not 

 possible to construct an x-ray microscope which would recombine the 

 reflexions of the diffraction pattern to form an enlarged image of the 

 unit cell of the crystal. Furthermore, it is not generally possible to 

 measure the phase of a reflected x-ray beam ; all that can be recorded 

 in the x-ray camera is the intensity of the reflexion, the square root 

 of which is its amplitude. The x-ray diffraction pattern in itself, there- 

 fore, does not provide sufficient data for a direct calculation of the 

 electron density in the unit cell which produced it. This is the funda- 

 mental difficulty of all x-ray analysis, but it is encountered in its most 

 acute form in the analysis of very large molecules. 



Progress can be made in simple cases where it is possible, by making 

 certain assumptions, to deduce or guess the phases of some or all of 

 the reflexions. For example, one simplifying factor may be that the 

 symmetry of the crystal is such that the phase angles between all the 

 diffracted rays and the incident ray must be or n — in other words 

 the diffracted rays are all exactly in or exactly out of phase with the 

 incident ray ; the problem is then reduced to one of finding out merely 

 the signs of the amplitudes of each reflexion. 



In such cases it may be possible to apply various tricks to find out 

 the signs of some reflexions. For example, if one atom in the unit 

 cell has very much larger atomic number than any of the others, its 

 contribution to each reflexion will also be very large by virtue of its 

 high electron density, and will effectively determine the sign of all but 

 the weakest reflexions ; once the position of the heavy atom is known 

 direct calculation will often give the signs of enough strong reflexions 

 to enable a first approximation to the crystal structure to be computed. 



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