## Limits of resolution; X-ray diffraction

8-3-00

### Resolving power

The resolving power of an optical instrument, such as your eye, or a telescope, is its ability to separate far-away objects that are close together into individual images, as opposed to a single merged image. If you look at two stars in the sky, for example, you can tell they are two stars if they're separated by a large enough angle. Some stars, however, are so close together that they look like one star. You can only see that they are two stars by looking at them through a telescope. So, why does the telescope resolve the stars into separate objects while your eye can not? It's all because of diffraction.

If you look at a far-away object, the image of the object will form a diffraction pattern on your retina. For two far-away objects separated by a small angle, the diffraction patterns will overlap. You are able to resolve the two objects as long as the central peaks in the two diffraction patterns don't overlap. The limit is when one central peak falls at the position of the first dark fringe for the second diffraction pattern. This is known as the Rayleigh criterion. Once the two central peaks start to overlap, in other words, the two objects look like one.

The size of the central peak in the diffraction pattern depends on the size of the aperture (the opening you look through). For your eye, this is your pupil. A telescope, or even a camera, has a much larger aperture, and therefore more resolving power. The minimum angular separation is given by:

The factor of 1.22 applies to circular apertures like your pupil, a telescope, or a camera lens.

The closer you are to two objects, the greater the angular separation between them. Up close, then, two objects are easily resolved. As you get further from the objects, however, they will eventually merge to become one.

### X-ray diffraction

Things that look a lot like diffraction gratings, orderly arrays of equally-spaced objects, are found in nature; these are crystals. Many solid materials (salt, diamond, graphite, metals, etc.) have a crystal structure, in which the atoms are arranged in a repeating, orderly, 3-dimensional pattern. This is a lot like a diffraction grating, only a three-dimensional grating. Atoms in a typical solid are separated by an angstrom or a few angstroms; . This is much smaller than the wavelength of visible light, but x-rays have wavelengths of about this size. X-rays interact with crystals, then, in a way very similar to the way light interacts with a grating.

X-ray diffraction is a very powerful tool used to study crystal structure. By examining the x-ray diffraction pattern, the type of crystal structure (i.e., the pattern in which the atoms are arranged) can be identified, and the spacing between atoms can be determined.

The two diagrams below can help to understand how x-ray diffraction works. Each represents atoms arranged in a particular crystal structure.

You can think of the diffraction pattern like this. When x-rays come in at a particular angle, they reflect off the different planes of atoms as if they were plane mirrors. However, for a particular set of planes, the reflected waves interfere with each other. A reflected x-ray signal is only observed if the conditions are right for constructive interference. If d is the distance between planes, reflected x-rays are only observed under these conditions:

That's known as Bragg's law. The important thing to notice is that the angles at which you see reflected x-rays are related to the spacing between planes of atoms. By measuring the angles at which you see reflected x-rays, you can deduce the spacing between planes and determine the structure of the crystal.