April 14, 2006

The Arangement of Atoms in Solid Matter

Max von Laue, Interference Phenomena with Röntgen Rays, 1912

Today we return to the world of the very small. While Rutherford was probing the structure of individual atoms, Max von Laue was single-handedly inventing the field of X-ray crystallography, the study of the positions of atoms in matter.

X-rays, which we now know are a very high frequency form of light, had only recently been discovered, and their properties were as yet poorly understood. von Laue believed they were waves rather than particles, and should therefore produce interference patterns. However, he took this line of thought an extra step further, which would win him a Nobel Prize just two years later. What, he asked, would an interference pattern caused by shining X-rays through a three-dimensional structure look like?

To simplify matters, von Laue looked at the mathematics of interference patterns caused by shining X-rays through crystals. It was already obvious that crystals had extremely simple structures at the atomic level, thanks to the way that they split into geometric forms when you knocked them apart. For example, even the smallest fragments of salt crystals appear to be cubic, or nearly so, which indicates to shrewd observers that the atoms of a salt crystal must be arranged in tiny cubes, which are then organized into larger cubes, and so on.

The regular pattern of atoms in a crystal means that a macroscopic beam of X-rays (say, 1mm in diameter) will not produce thousands of different overlapping interference patterns, but instead a single, constantly repeated pattern. With the right mathematics, you can work out from this pattern the relative positions of the atoms in a unit cell of the crystal, the smallest repeating pattern of atoms.

The core insight is that the arrangement of unit cells in a crystal causes rows, columns, and lines of atoms to form. The regularly spaced atoms act like a diffraction grating, so the final pattern is caused by the combination of three interference paterns: one from the rows, one from the columns, and one from the lines. The spots on the photographic plates are the places where all three patterns are simultaneously causing reinforcement of the waves of X-rays. By measuring the angles from the crystal to the spots, you can figure out the spacing of the atoms, and from that the structure of the unit cell.

This is not to say that the task is easy: it has been compared to reconstructing the positions of buoys in a river solely by studying the pattern of wakes at one point downstream. One difficulty von Laue delt with was that his beams of X-rays contained a range of different frequencies, which produced 5 different interference patterns on his assistants' photographic plates.

The importance of this work was immediately recognized. Beyond the matters of showing that X-rays do, in fact, behave like waves, and literally illuminating the structure of simple natural crystals, von Laue's methods (with some refinements) would later be used to determine the structure of complicated molecules such as DNA, Hemoglobin, and thousands more. You see, with a little work, practically any molecule can be crystalized.