Physics Bootcamp

Subsection 56.2.1 X ray Diffraction from Single Crystals

X rays were discovered by William R\"ontgen in 1895 by the marks they left on photographic plates. They are produced when highly energetic electrons strike a target. X rays are very penetrating and can pass through human body and many other materials that are opaque to the visible light.

Are X rays particles or waves? Wilhelm Wien calculated that if X rays were waves their wavelength will be very small, of the order of \(10^{-11}\) m. Based on the calculation of the inter-atomic distances in crystals to be or the order of \(10^{-10}\) m, Max von Laue wondered if crystals could act as a diffraction grating for X rays. Two of his assistants, Walter Friedrich and Paul Knipping, set up the experiment to test Laue's ideas. First they used copper sulfate crystal since it was readily available in the lab. The crystal was fixed to a holder with wax which was placed in a lead box with a 3 mm hole in one side for the X rays to enter while the back end and the sides were lined with photographic plates. They found that, in addition to the central spot, there were many smaller dots on the screen in a regular pattern where diffracted beams had formed constructive interferences. These patterns are often referred to as Laue pattern. Von Laue received Nobel prize for his discovery in 1914.

Von Laue's patterns did not have all the diffraction spots expected from a source of continuous wavelengths. Von Laue took it to mean that the source he was using had only certain wavelengths. However, William Lawrence Bragg, the son of William Henry Bragg, the Cavendish Professor of Physics at the University of Leeds, thought differently. Lawrence Bragg was convinced that missing diffraction spots were due to some property of crystals themselves. He put forth a theory for diffraction of X rays based on their reflection from crystal planes.

Crystal planes are planes in space that are parallel to each other and spaced equally apart and every atom of the crystal belongs to one of these planes as illustrated for a square lattice in Figure 56.2.5. According to [William] Lawrence Bragg each crystal plane acts as a reflecting surface. The X rays reflected from these planes would interfere constructively if they were all in phase. For instance, the ray reflected from the surface plane will have traveled smaller distance than the ray that reflected off the plane a distance \(d\) from the surface just below as shown in Figure 56.2.4. The difference in the paths of the two rays will depend on the angle \(\theta\) and plane separation \(d\text{.}\)

If this distance is an integral multiple of the wavelength \(\lambda\text{,}\) then then reflected rays will interfere constructively in that direction.

The formula in Eq. (56.2.1) is called Bragg's law. This formula can be used to find \(\lambda\) of the X rays if we know \(d\) in a crystal and to find \(d\) between two crystal planes if we know \(\lambda\) of the X rays. By rotating the crystal, or changing the direction of the beam, or the direction of the detector, it is possible to figure out \(d\) corresponding to various crystal planes in the crystal as shown in Figure 56.2.5. From the data for \(d\) for various crystal planes, the arrangements of atoms in the crystal can be determined. The father/son team of Braggs worked on the crystal structure determination using this formula and were awarded Nobel prizes in 1915.

Subsection 56.2.2 The Powder X ray Diffraction

If the single crystal is not large enough, you can still gain the information about the crystalline structure from a powder of the material. If each particle of the powder itself is a tiny crystal, then each crystallite will diffract X rays independently of the way some other particle of the powder would. Since particle of the powder would be randomly oriented with respect to each other, we get circles at the photographic plate instead of sharp dots as illustrated in Figure 56.2.6. The rings are are called the Debye-Scherrer rings.

When the crystallites becomes smaller each peak spreads out over larger angles. Eventually, when there is no long range order, the material is called amorphous. The X ray diffraction of amorphous materials are broad and lack the sharp peaks.