De Broglie waves and electron diffraction: Dual Nature Of Radiation And Matter

3.1 The de Broglie hypothesis

De Broglie argued that if light waves can behave as particles, then particles should also show wave behaviour. He proposed that a moving particle has a wavelength inversely proportional to its momentum, using the same relation that holds for a photon. The constant of proportionality is Planck's constant. For a particle of mass m moving with speed v the wavelength is h divided by mv. The same particle accelerated from rest through a potential difference V has kinetic energy eV, so its wavelength can be written in terms of V, the mass and the charge. Heavier particles and slower particles have shorter wavelengths, which is why wave effects are easiest to see for electrons. ### The electron diffraction experiment

Davisson and Germer scattered a beam of electrons from the surface of a nickel crystal and recorded the intensity of scattered electrons at different angles. They found strong maxima at particular angles, exactly as a wave diffracted by the ordered atoms of the crystal would produce. The measured wavelengths agreed with the de Broglie relation. This experiment established that electrons have wave properties. Electron microscopes later used the short electron wavelength to resolve details far smaller than visible light can resolve. The same wave nature underlies the discrete energy levels of the hydrogen atom and the diffraction of neutrons and other particles. ### The photon picture and its consistency

The energy and momentum of a photon are connected by the same de Broglie relation: the photon momentum is h divided by the wavelength. The intensity of a light beam is the number of photons crossing unit area per unit time multiplied by the energy of one photon, which connects the wave picture with the particle picture. For an electron accelerated through V volts the de Broglie wavelength comes out of the order of a nanometre for a hundred volts, which is comparable to the spacing of atoms in a crystal. That is why crystals serve as diffraction gratings for electrons. ### Exercise coverage

The exercise set asks for the de Broglie wavelength of an electron accelerated through a given potential, the ratio of wavelengths for different particles with the same kinetic energy, the same speed or the same momentum, and the wavelength change when the kinetic energy is increased. Write the relation first, then identify which quantity is held fixed in the comparison. ## Worked de Broglie numericals

An electron accelerated through 100 V has a de Broglie wavelength of about 1.227 divided by the square root of 100 nanometres, which is 0.123 nm. A proton accelerated through the same potential has a much smaller wavelength because of its larger mass, by the square root of the mass ratio, which is about 43 times smaller. For the same kinetic energy the wavelength ratio is the inverse square root of the mass ratio. An alpha particle has about four times the proton mass, so its wavelength is half the proton wavelength at the same kinetic energy and one quarter at the same speed. These ratios are the standard exam comparisons. ## Exercise coverage detail

The textbook exercises pair each formula with a comparison: same potential, same energy, same momentum or same speed. Identify which quantity is held fixed, write the relation with that quantity constant, and take the ratio. A question on the Davisson-Germer experiment usually asks what was observed and how it supports the wave picture. ## Links to the rest of the book

The de Broglie relation explains why only certain orbits are allowed in the hydrogen atom: an orbit is allowed when its circumference contains a whole number of electron wavelengths. This idea is used in the next chapters on atoms and nuclei, so the relation is worth keeping sharp. ## More practice comparisons

For a proton and an alpha particle accelerated through the same potential, the kinetic energies are in the ratio of their charges, 1 to 2, so the momenta are in the ratio of the square roots of mass times charge. The wavelength ratio is therefore the inverse of that square root. Numbers can be computed once the ratio is written. For two particles with the same momentum the wavelengths are equal regardless of their masses. For the same kinetic energy the lighter particle has the longer wavelength. For the same speed the lighter particle again has the longer wavelength. These three cases, momentum, energy and speed, cover almost every exercise question in this section. The wavelength of a macroscopic object such as a cricket ball is far too small to detect, which is why wave behaviour is not observed in everyday life. The effect becomes measurable only for particles of very small mass and moderate speed, such as electrons. ## The Davisson and Germer data

The experiment used electrons accelerated through about 54 volts and found the strongest scattered beam at an angle of about 50 degrees. The spacing of the nickel crystal planes and the measured angle gave a wavelength of about 0.167 nm, close to the de Broglie value for that voltage. That agreement between the predicted and observed wavelengths is the evidence for matter waves. The same wave nature appears in electron microscopy, where a small accelerating voltage gives a wavelength thousands of times shorter than visible light. That is why electron microscopes can see objects far smaller than a light microscope can resolve.