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Next: 4. The Fundamental Problem of Reconstructing Up: A Non-Mathematical Introduction to X-ray Diffraction Previous: 2. The Problem of Focusing

3. The Essence of X-ray Diffraction

In principle one could say that the whole development of X-ray diffraction techniques really amounts to the development of alternatives to the focusing of X-ray images. The point is that the first stage of the imaging process, illustrated by the projector with no lens, can be performed but the crystallographer has no lens to put back in the projector and must try to make sense out of the diffuse patch in some other way.

If the problem were strictly analogous to this it is unlikely that any structures would ever have been solved. Fortunately there are two significant ways in which the X-ray crystallographerÆs case differs from that of the projectionist with no lens for his projector. First of all the projector uses white light with a broad frequency band which is also spatially incoherent and is produced from a large source. In the X-ray case it is usual (except under the special circumstances of Laue photographs with which we are not concerned here) to use monochromatic radiation which, as a result of travelling through a long, fine hole or slit has quite a high degree of spatial coherence. The second point is that the object usually exhibits some degree of regularity or crystallinity.

These two facts lead to the production of patterns which consist not of a diffuse patch but rather of a series of discrete spots (though there may well be diffuse spots and patches if the object is not sufficiently regular).

The parallel with the projector which most nearly matches the X-ray case would be a gas-phase laser beam falling on a regular grating (e.g. a finely woven handkerchief or a piece of gauze); the beam is scattered or diffracted into a number of well defined beams or spots arranged in a regular way. The optical diffractometer used in the preparation of the plates in the Atlas is merely a sophisticated development from this simple experiment.

The difference in the patterns of regular and irregular objects, all illuminated by monochromatic coherent radiation is illustrated in Plates 16 and 17 of the Atlas.

The process carried out by the lens of the projector or by the objective of the microscope and which needs to be carried out artificially by the X-ray crystallographer involves the mathematical operation of Fourier Synthesis. We shall however attempt to illustrate the process without resort to mathematics.

If the scattering (or diffraction pattern as it tends to be called if it consists of regular spots) is completely determined then it should, in principle, be possible to transform the beam into an image by the purely mathematical process of Fourier Synthesis. Unfortunately, however, it has so far proved quite impossible to record the relative phases: this immediately invalidates the direct mathematical process. The reason why the phase cannot be recorded becomes clear if one calculates the frequency of X-rays; determination of phase would, in effect, involve time measurements corresponding to a fraction of one period. If we assume X-rays of wavelength 1.5 Å (1.5 $\times$ 10- 10 m) the frequency is about 2 $\times$ 1018 and hence to measure a phase difference of (say) $\frac{1}{5}$th of a cycle would involve a time measurement of about 10-19 s which is certainly beyond our present resources. Perhaps one day a means of adding a coherent beam, as in optical-laser holography may become available and then the whole situation would change!

Let us first of all consider in a little more detail the relationships between an object and its scattering or diffraction pattern--regardless, for the moment, of whether we are dealing with light or with X-rays. Suppose that the object consists of two points only. Plate 1 of the Atlas shows that the result is a set of fringes whose spacing varies inversely with that of the points. These are the well known `Young's' or `Double-slit' fringes and can be shown to vary cosinusoidally in amplitude with alternate fringes $\pi$ out of phase with the rest. For the moment we will ignore the effects of the size of the scattering points and assume that they are mathematically small. The fringes will then, in principle, be of infinite extent and without the ring patterns superimposed as in Plate 1; the centre region will be the only one of interest. If we now add further pairs of points in different orientations, fringes of different orientation and spacing will be added and the resultant pattern becomes more complex (Plate 2). If the basic arrangement of points forming the object is repeated in any kind of regular way, further fringes are introduced and a two-dimensional `crystal' produces a pattern of regular spots (Plate 11). It will be clear from a study of plates 1, 2 and 11 that we can now separate two quite distinct variables. First the size and shape of the lattice (strictly the reciprocal lattice) in which the spots of the diffraction pattern are arranged depends solely on the size and shape of the lattice on which the groups of scatterers are arranged. And secondly the relative intensity of the spots depends on the arrangement of scatterers in each individual group. In the crystalline case the `individual group' is the unit cell contents.

next up previous
Next: 4. The Fundamental Problem of Reconstructing Up: A Non-Mathematical Introduction to X-ray Diffraction Previous: 2. The Problem of Focusing

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