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4. The Fundamental Problem of Reconstructing an X-ray Image

In the terms of this pamphlet we are now faced with the problem of deriving the arrangements shown on the left hand page of Plate 11, given the scattering patterns on the right hand of Plate 11.

Now let us consider the nature of the problem of recombining the scattered information to produce an image and we will start by once more considering the case of two point scatterers only. As we have seen, the diffraction or scattering pattern is a set of cosinusoidal fringes whose spacing is inversely related to that of the point. This looks very like a diffraction grating which if itself placed in a coherent monochromatic beam of light will give orders of diffraction. A good experiment at this point is to provide a few coarse diffraction gratings which can be placed in a laser beam and give a single row of regularly spaced sharp spots and it is easy to demonstrate the reciprocal relationship between the slit spacing of the grating and the space between the orders. If we now substitute for the ordinary diffraction grating (which has sharp transparent and opaque slits, i.e. has a square wave function) one with a cosinusoidal function determining its transparency variation we shall find only three orders: a bright centre one and a single weaker one on each side. Of course our grating really has a transparency distribution of (1 + $\cos\theta$) since we have not provided for negative transmission. If, using phase changing tricks with polarised light and mica, the details of which need not concern us here, we make a true representation of the cosine distribution with alternative strips in opposite phase, we arrive at a diffraction pattern with just two orders, one on either side of the original line of the beam. A logical development of this train of thought then is to see that if we were to place the patterns of (say) the right hand side of Plate 2 in a coherent monochromatic beam of light, then we should produce an image like the left hand page. In other words we can achieve the recombination trick merely by using a representation of the diffraction pattern itself as another diffracting object. The problem of representing the phase remains however and needs special consideration.

This reverse process is illustrated in Plate 29. On the left we have a series of pairs of points which build up in 29.8 Left to a representation of the complete scattering pattern in two dimensions of a crystal of Rhodium phthalocyanine derived using X-rays. The relative intensities of the spots are represented by varying the size of the holes. The diffraction pattern of this, 29.8 Right, is a reasonable reconstruction of an image of a Rhodium phthalocyanine molecule and one can see in the earlier figures on this page how the successive pairs of holes add further fringes to build up the pattern. This, as one might guess, is a very special case in which it just happens that the Rhodium atom at the centre is just sufficient to scatter enough coherent background over the whole pattern to bring the maximum negative regions to zero, and make corresponding enhancements of the positive regions (this principle is further explored in Plate 5). In this case therefore the phase problem does not cause difficulties. Such examples are, however, rare.

Following this line of argument with further examples we should be able to establish experimentally that the process of recombination is identical with that of scattering and that, under certain circumstances, the diffraction pattern of the diffraction pattern is an image of the object again. The mathematics could be introduced at this stage if it is desired but is not essential for non-specialists.

Copyright © 1997 International Union of Crystallography