X-Ray Diffraction

This page explains a little about how X-ray diffraction can help to find out the structure of DNA and other molecules. Because the answer is already known, it is easy to do simple calculations in reverse, to show the diffraction patterns of simple structures. This is far easier than making the original discovery, just as writing a good detective story, though far from easy, is completely different from solving a real case given only the available clues.

We have seen how the size of small objects has a profound effect on the behaviour of light waves. Two small slits produce interference patterns, while arrays of sources produce diffraction patterns. In all cases the patterns are dependent on the sizes and positions of the light sources. This knowledge was exploited by the two William Braggs to study crystal structure using X-rays. Because the spreading of a beam is inversely proportional to the distance between the atoms, the interpretation of the results is non-intuitive, and the required calculations can be very intricate. Before the age of computers the elucidation of structure was an art as well as a science.

Working out the structure of crystals and molecules from X-ray patterns is an example of in inverse problem, which is very much harder than going in the other direction. A simple example is the relationship between differentiating and integrating in maths. Another is given by convolution and deconvolution.

There are two basic ways of using the crystals. One large crystal can be used, or the a powder of tiny crystals can be used. In the first case, the required effects me be seen only for certain positions of the bean and the crystal. The X-ray image will be a series of spots on the film. The position and brightness give information about the crystal. In the second case, some crystals will always be in the right position. But for the same reason, the X-ray image will be a set of circles around the beam.

Because the positions of atoms in a large molecule are so various, the interpretation is far more difficult than it is for crystals. Dorothy Hodgkin achieved early result in this field, without the aid of computers. She worked out the structure of penicillin, vitamin B12, and insulin. A famous example of X-ray diffraction is the discovery by Crick and Watson of the structure of DNA, using data gathered by Franklin and Wilkins.

How does X-ray diffraction work? The basic idea is that the waves from all parts of the object combine to form the intensity in any direction. The intensity will vary depending on how the waves end to reinforce or cancel. As we have seen, the dispersion of the waves is bigger for smaller objects than for large, and so the diffraction pattern is not just an enlarged copy of the object.

A very simple example is seen when a camera is pointed at brilliant sources of light. Multi-pointed stars may be seen. These are caused by diffraction at the edges of the lens diaphragm. A square hole makes four lines, but a pentagonal hole makes ten. The star always has an even number of points because the diffraction pattern of an edge is symmetrical. These patterns are present in every photograph, but are generally too dim to see, though with tiny apertures they might produce a small blurring of the picture. If the sun, or its reflection, or bright lights, are in the picture, the star may become visible.

Here is an actual example, made with a camera which had a hexagonal lens aperture. The diffraction effect is sometimes exploited in photography and television by using a plastic sheet ruled with fine grooves. If we examine the diffraction pattern of a rectangular hole in isolation we find that it looks more like the yellow picture, simulating monochromatic light. In this and other pictures the brightness range has been compressed to make the side-lobes visible. Notice that the shorter dimension of the rectangle makes the wider diffraction pattern.

The next picture shows a helix and a double helix, viewed from the side and from an angle. The side views look like sinusoids in two dimensions. The second picture includes some blobs which represent atoms. There are ten atoms per cycle.

The other pictures above show calculated diffraction patterns of ten cycles of the structures, paired for comparison. These are - sine with double sine, sine with helix, double sine with double helix, and helix with double helix respectively. These structures are a long way from that of DNA, which is far more complex, but they illustrate the problem of working back from the diffraction pattern to the structure. In practice, of course, people had to work backwards from the X-ray patterns to find the structure. There was other information - a knowledge of the chemistry, for example, restricted the number of possible ideas that needed to be tried.

Unfortunately, there is a fatal flaw in this simple idea - the reflections from one molecule are impossible to detect. A vast number of molecules is needed. To ensure that they are all aligned in the same way they can be incorporated in a crystal. But now the dominant pattern is that crystal. From the fine details of that pattern the experimenter has to discover the structure of individual molecules.