29.6 The wave nature of matter  (Page 3/4)

Making connections: a submicroscopic diffraction grating

The wave nature of matter allows it to exhibit all the characteristics of other, more familiar, waves. Diffraction gratings, for example, produce diffraction patterns for light that depend on grating spacing and the wavelength of the light. This effect, as with most wave phenomena, is most pronounced when the wave interacts with objects having a size similar to its wavelength. For gratings, this is the spacing between multiple slits.) When electrons interact with a system having a spacing similar to the electron wavelength, they show the same types of interference patterns as light does for diffraction gratings, as shown at top left in [link] .

Atoms are spaced at regular intervals in a crystal as parallel planes, as shown in the bottom part of [link] . The spacings between these planes act like the openings in a diffraction grating. At certain incident angles, the paths of electrons scattering from successive planes differ by one wavelength and, thus, interfere constructively. At other angles, the path length differences are not an integral wavelength, and there is partial to total destructive interference. This type of scattering from a large crystal with well-defined lattice planes can produce dramatic interference patterns. It is called Bragg reflection , for the father-and-son team who first explored and analyzed it in some detail. The expanded view also shows the path-length differences and indicates how these depend on incident angle $\theta$ in a manner similar to the diffraction patterns for x rays reflecting from a crystal.

Let us take the spacing between parallel planes of atoms in the crystal to be $d$ . As mentioned, if the path length difference (PLD) for the electrons is a whole number of wavelengths, there will be constructive interference—that is, $\text{PLD}=\mathrm{n\lambda }(n=\mathrm{1,\; 2,\; 3,}\dots )$ . Because $\text{AB}=\text{BC}=d\text{sin}\theta \text{,}$ we have constructive interference when $\mathrm{n\lambda }=\text{2}d\text{sin}\mathrm{\theta .}$ This relationship is called the Bragg equation and applies not only to electrons but also to x rays.

The wavelength of matter is a submicroscopic characteristic that explains a macroscopic phenomenon such as Bragg reflection. Similarly, the wavelength of light is a submicroscopic characteristic that explains the macroscopic phenomenon of diffraction patterns.