Stickybees said:
How do crystallographers go from a diffraction pattern to actually determining the structure of the crystal?
This is a fiendishly hard problem to solve. Hence the excitement when Bragg and others managed to crack it and to get crystal structure from X ray scattering photographs.
Start with the simplest example of an interference pattern - the two slits experiment. You can determine the spacing of the slits if you know the wavelength, the separation of the fringes and the distance to the screen (or, in fact, the angle of the resulting maxima and minima at a distance). Now take a grating, of many closely spaced wires - each of which will scatter some light, and shine light of just one wavelength on it. If you get the angle of the incident light just right, a ray will emerge (interfering sum of all the scattered contributions) as if it was reflected from a plane. There is no light scattered in other directions. Knowing the wavelength and the angle will tell you the spacing of the grating.
A crystal consists of many regularly spaced scattering points (in 3D, to make it more complicated). These points lie in planes (like the lines you can draw through a regular array of dots on graph paper) and each of these sets of planes are spaced regularly. The angle at which reflections occur will depend on the spacings and wavelength. Each of the dots on that picture (which is of just one 'perfect' crystal, I think) show the angles for which you get a reflection from a particular set of planes. The photograph will tell you the spacings and angles of all the planes that can be made up by joining scattering points.
'All that you have to do', now, is to decide, from the information of the spacings and angles of the planes, just how the points could be arranged inside the crystal, to form the planes.
It's very much like designing a directional TV transmitting array of many elements when you want a particular radiation pattern. The same sort of calculations are involved and, as Andy R. says, often involves the fact that the angular spacing of the reflections is related to the spatial arrangement of the scatterers by the Fourier relationship but a lot can also be gleaned about the symmetry of the crystal structure by looking at the symmetry of the spot pattern. I remember my Uni course about this and it was quite hard work, involving working in terms of 'k space'.
In general, it seems always to be easier to work from scattering array to radiated pattern than the other way round! But we usually know the radiated pattern and actually want information about the geometry of the scattering points.
Life gets even harder when you are dealing with 'powder' or Organic molecules, rather than one nice big crystal. The pictures become concentric circles and lines, rather than nice clear dots.
