Electron Diffraction

In 1927, George P. Thomson discovered that if a beam of electrons is made to pass through a thin gold foil then the regular atomic array in the foil acts as a sort of diffraction grating, so that, when a photographic film placed behind the foil is developed, an interference pattern is discernible. Independently, Clinton Davisson and Lester Germer found that electrons scattered by the surface of a nickel metal crystal display a diffraction pattern. Both these experimental results imply that electrons have wave-like properties. The electron wavelength, $\lambda$, or, alternatively, the wavenumber, $k =2\pi/\lambda$, can be deduced from the spacing of the maxima in the interference pattern. Thomson, Davisson, and Germer found that the momentum, ${\bf p}$, of an electron is related to its wavevector, ${\bf k}$, according to the following simple relation:

$${\bf p} = \hbar\,{\bf k}.$$ (4.5)

The associated wavelength, $\lambda =2\pi/k$, is known as the de Broglie wavelength, because the previous relation was first hypothesized by Louis de Broglie in 1926. (See Section 4.1.9.)

It turns out that wave-particle duality only manifests itself on lengthscales less than, or of order, the de Broglie wavelength. Under normal circumstances, this wavelength is fairly small. For instance, the de Broglie wavelength of an electron is

$$\lambda_e = 1.2\times 10^{-9}\,[E({\rm eV})]^{-1/2}\,{\rm m},$$ (4.6)

where the electron energy is conveniently measured in units of electron-volts (eV). (An electron accelerated from rest through a potential difference of $1000$ V acquires an energy of $1000$ eV, and so on. Electrons in atoms typically have energies in the range $10$ to $100$ eV.)