Bragg's Law: X-Rays Meet a Crystal

A grating made by nature

To measure the wavelength of light, one uses a grating of fine lines ruled close together. Its lines must be about as far apart as the wavelength. X-rays have wavelengths near \(0.1\,\text{nm}\), the size of an atom. No ruling machine can space lines so finely. In 1912 Max von Laue saw that nature had already made such a grating: a crystal.

In a crystal of rock salt, sodium and chlorine atoms stand in a pattern that repeats in every direction. Through such a pattern one can draw many sets of parallel crystal planes, each full of atoms. The planes of one set are a fixed distance \(d\) apart. In rock salt, one important set has \(d = 0.282\,\text{nm}\).

Each plane of atoms acts like a very weak mirror. It sends back a tiny fraction of an X-ray beam, and lets the rest go on to the next plane. The reflected ray leaves at the same angle to the plane as the incoming ray. For X-rays the angle is measured from the plane itself. It is called the glancing angle \(\theta\).

A single plane reflects too little to notice. But a beam enters millions of planes, and every one of them reflects a little. Whether these small reflections add up depends on their paths. If they arrive in step, they reinforce into a strong beam. If not, they cancel, as two waves on a string do, and as light does in the chapter on interference.

The extra path

Look again at Fig 5.1. Ray 2 goes deeper than ray 1, by one plane. Drop a perpendicular from A onto ray 2 before it reflects, and another after it reflects. Up to the first perpendicular the two rays have gone equal distances, and after the second they go on side by side. The difference lies in the two thick pieces, CB and BD.

The two rays reinforce when this extra path is a whole number of wavelengths. Then the crests of ray 2 fall on the crests of ray 1, and the same holds for every deeper plane. This is the condition found by William Lawrence Bragg in 1912, Bragg's law.

Turn the angle slowly in the figure above. At most angles the extra path is some odd fraction of a wavelength, and the two waves partly cancel. With millions of planes, each a little further out of step, the reflections cancel almost completely. Only near \(15.8°\), \(33.1°\) and \(55.0°\) do all the planes agree, and the crystal sends back a strong beam.

How many orders?

The sine of an angle is never more than one. So \(n\lambda = 2d\sin\theta\) can be at most \(2d\). The orders that exist are the whole numbers \(n \le 2d/\lambda\). For copper Kα on rock salt, \(2d/\lambda = 0.564/0.154 = 3.66\), so there are three orders. A wave longer than \(2d\) gives no strong reflection at all.

The Bragg spectrometer

The Braggs, father and son, built an instrument to use the law. A narrow X-ray beam falls on a crystal mounted on a turntable. As the crystal turns through the glancing angle \(\theta\), a detector turns through \(2\theta\). So the detector always stands where a reflection would go. A graph of the count against \(\theta\) shows a sharp peak at every Bragg angle.

Check the three peaks in the figure above against Bragg's law. Their sines are \(0.273\), \(0.546\) and \(0.819\), in the ratio \(1 : 2 : 3\). That ratio tells the experimenter that the peaks are orders 1, 2 and 3 of one wavelength. The first gives \(\lambda = 2d\sin\theta = 0.564 \times 0.273 = 0.154\,\text{nm}\).

The spectrometer works in two directions. With X-rays of a known wavelength, the angles give the spacing \(d\) of a crystal's planes, and so the arrangement of its atoms. With a crystal of known \(d\), the angles give the wavelength of unknown X-rays. Moseley measured his Kα lines this way, with crystals of known spacing.

The same law works for any wave. In the chapter on Matter Waves, electrons of \(54\,\text{eV}\) reflected from a nickel crystal by exactly this rule. That is why a crystal proved that electrons are waves, and why it proves the same of X-rays.