Crystallography

X-rays were discovered in 1895, and for years no one knew whether they were particles or waves. In 1912 Max von Laue in Munich reasoned that if they were very short waves, the regular rows of atoms in a crystal should scatter them the way closely ruled lines scatter light. Walter Friedrich and Paul Knipping tried it with a crystal of copper sulphate and found a pattern of spots. X-rays were waves, and crystals were lattices, as the theory of electromagnetism and the crystallographers had both supposed.

Laue's own analysis of the spots was complicated. That autumn William Lawrence Bragg, a 22-year-old student at Cambridge, found a far simpler picture. Each plane of atoms reflects a little of the beam, and the reflections from many parallel planes add up only at particular angles. His father, William Henry Bragg, built an instrument to measure those angles precisely. In 1913 the Braggs solved rock salt and diamond. In salt, each sodium atom sits among six chlorine atoms and each chlorine among six sodium atoms, with nothing that could be called a molecule of sodium chloride. Some chemists found this hard to accept. They shared the 1915 Nobel prize, although the son felt for years that the public credited his father with his law.

The method spread from minerals to chemistry. In 1929 Kathleen Lonsdale showed that the benzene ring, the backbone of organic chemistry, is flat. Solving larger molecules was a matter of heavy calculation, since the diffraction spots record how strongly each set of planes reflects but not the timing, or phase, of the reflected waves. Crystallographers found ways round this, such as adding a heavy atom as a marker.

Dorothy Hodgkin pushed the method furthest. She solved penicillin in 1945, which showed chemists the unusual ring at the heart of the drug, then vitamin B12 in 1956, with the help of some of the first electronic computers. Insulin took her 34 years and was finished in 1969. Meanwhile X-ray photographs of DNA fibres helped reveal the double helix, and Max Perutz and John Kendrew solved the first protein structures, the start of molecular biology as a science of shapes.

In 1982 the rules themselves were broken. Dan Shechtman saw a sharp diffraction pattern with tenfold symmetry in a rapidly cooled alloy of aluminium and manganese. The symmetry was forbidden for any repeating lattice. By his own account, his group leader asked him to leave, and Linus Pauling is reported to have said that there were no quasicrystals, only quasi-scientists. But others reproduced the result, and theorists explained it as order without repetition, like a Penrose tiling. In 1992 the definition of a crystal was changed to fit.

X-rays reflected from two neighbouring planes of atoms, a distance $d$ apart, travel paths that differ by $2d\sin\theta$, where $\theta$ is the angle between the beam and the planes. The reflections reinforce only when that difference is a whole number of wavelengths:

The Braggs could measure angles, but to get distances they needed one length to start from. They took it from the density of salt. A cube of rock salt with side $a$ holds four sodium and four chlorine atoms. Salt's density is 2.165 g/cm³ and a mole of NaCl weighs 58.44 g, so

One ångström, Å, is $10^{-10}$ m. With the spacing known, the angles give the X-ray wavelength, and after that any crystal can be measured.

Take the X-rays used in most laboratories today, from copper, with $\lambda = 1.5406$ Å. The planes in salt that hold alternate layers of sodium and chlorine are $a/2 = 2.82$ Å apart. Other sets of planes cut the cube at slants:

For the (200) planes, the second and third orders, $n = 2$ and $n = 3$, appear at 33.11° and 55.03°. A fourth order would need $\sin\theta > 1$, so it does not exist.

The same arithmetic shows why visible light cannot do this. Green light has $\lambda \approx 5000$ Å, and reflecting it from planes 2.82 Å apart would need $\sin\theta \approx 890$. Only waves shorter than $2d = 5.64$ Å can be reflected at all. X-rays, with wavelengths near one ångström, are the right size for atoms.

Crystallography grew from a branch of mineralogy into the basic tool for seeing matter at the scale of atoms. It showed physicists the lattices through which electrons move, the starting point of solid-state physics. It gave chemists the shapes of molecules and biologists the shapes of proteins. Neutrons and electrons are now diffracted as well as X-rays, and electron microscopes can image single frozen proteins. Yet the simplest question, which crystal a given molecule will form, still cannot be answered reliably in advance.