Chapter 15. De Broglie’s Matter Waves

15.1 Compton scattering

German-Swiss-American physicist Albert Einstein’s theory of special relativity was first published in 1905[1] (discussed in Book I). This suggested that light has a momentum - which is classically equal to an object’s mass multiplied by its velocity - even if photons have no mass.

Special relativity shows that energy is related to mass via E2=p2c2+m2c4, where E refers to energy, p to momentum, c to the speed of light, and m to an object’s mass. An object with no mass, like a photon, will have an energy of E2=p2c2, or E=pc.

American physicist Arthur Compton proved that photons do have the momentum Einstein predicted in 1922.[2] Compton did this by firing X-rays at aluminium foil. When the X-rays hit the electrons in the outermost shell of the aluminium atoms, they transferred some of their angular momentum. The electrons gained enough energy to leave the atom, and the X-ray photon lost the same amount of energy. This process is now known as Compton scattering.

Diagram of Compton scattering, where a photon changes energy after colliding with a charged particle.

Figure 15.1
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Compton scattering.

When a photon collides with an electron and gains energy, the process is known as inverse Compton scattering. Compton scattering is now utilised in radiobiology,[3] and both Compton scattering and inverse Compton scattering are important in X-ray astronomy[4].

15.2 Electron waves

In 1924, French physicist Louis de Broglie used Einstein’s equations to show that electrons can act like waves, just as photons can act like particles.[5]

15.2.1 The wavelength of photons

The wavelength of a photon is calculated by combining Einstein’s equation for determining a photon’s energy with the Planck relation.[6] Using E = pc (which is the energy of photons according to special relativity) and E = hν (which is the energy of photons according to the Planck relation):

pc=hν (15.1)
Using c = λν
λ = h/p (15.2)

15.2.2 The wavelength of particles

De Broglie proposed that particles also have a wavelength, and that this can be calculated using the same equation, except here the particle’s momentum is equal to mv, where m is the particles’ mass, and v is its velocity:

λ = h/mv (15.3)

De Broglie realised that electrons must orbit the nucleus like a standing wave - a wave that is constrained at each end. This means that only a whole number of wavelengths fit exactly around each orbit:

λ = 2πr/n (15.4)

Here, the orbit is assumed to be circular. 2πr is the circumference of a circle of radius r, and n is the shell number. This means that the angular momentum of each electron is quantised, in agreement with Danish physicist Niels Bohr’s theory of the atom[7] (discussed in Chapter 10).

Bohr had previously shown that the angular momentum of each electron (L) is equal to the shell number multiplied by a constant:

h/mv = 2πr/n (15.5)
nh/2π = rmv
nh/2π = rp = L
Diagram showing that circular standing waves are only allowed if they fill the space they are contained in with a whole number of wavelengths.

Figure 15.2
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Allowed (left) and forbidden (right) standing waves.

Illustration showing how full wavelengths arrange themselves in a circle, with two, three, or four nodes.

Figure 15.3
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Standing waves for n = 1 to n = 5.

If an electron drops to a lower energy level, its orbit has a smaller radius. This means that fewer full wavelengths can fit, and so the frequency and energy are lower. The difference in energy between shells is always a quantised number, because E = hν. The lowest minimum energy is always more than zero because a full number of wavelengths must fit in each shell.

What is the wavelength of a person?

De Broglie’s theory can be extended to show that all matter exhibits the same wave-particle duality as light. This means that everything in the universe can act like a wave.[8]

λ = h/mv,   and   h=6.626×10-34 m2kg s-1.

This shows that an object’s wavelength gets smaller the more massive it is, and the faster it’s moving.

If a person has a mass of 75 kg, and is jogging at 8 km/h (which is about 2.2 m/s), then

λ = 6.626×10-34/7.5 × 2.2 = 4.016×10-36 m.

This is about 700 billion, billion times smaller than the classical electron radius, which is about 2.8×10-15 m.

Diffraction works best if the slit is about the same size as the wavelength, and so this explains why we do not notice wave-like behaviour in people.

De Broglie’s theory was greatly extended by German physicist Werner Heisenberg[9] (discussed in Chapter 16) and Austrian physicist Erwin Schrödinger[10] (discussed in Chapter 17) in 1925-1926. American physicists Clinton Davisson and Lester Germer proved that electrons have wave-like properties in 1927.[11] Davisson and Germer measured the wavelength of electrons by firing a beam of electrons at a nickel crystal, which acts like a diffraction grating, and then measuring the angles they were deflected by.

The wave-like nature of electrons meant that electron microscopes could be built in the 1930s.[12] The fact that electrons are more massive than photons means they have a smaller wavelength, and this is why electron microscopes have a better resolution than microscopes that use light.

In 1961, German physicist Claus Jönsson performed British natural philosopher Thomas Young’s double-slit experiment with electrons, and found that they behave the same way as photons.[13] This experiment has since been conducted on larger particles and molecules. In 2012, Thomas Juffmann and physicists from University of Vienna in Austria conducted the experiment on molecules containing over 100 atoms.[14]

By 1993, electron microscopes could be used to create images of individual atoms on metallic surfaces, known as quantum corrals[15] (discussed in Chapter 16). Electron waves could be seen in these images, and look like ripples on the metal’s surface.

A quantum corral, showing that electron waves can be produced by arranging atoms in a circle.

Figure 15.4
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A sculpture showing a quantum corral made from iron atoms (the raised points) on copper, the ripples on the surface are electron waves.

15.3 References

  1. Einstein, A. in The principle of relativity; original papers, The University of Calcutta, 1920 (1905).

  2. Compton, A. H., Physical Review 1923, 21, 483–502.

  3. Saha, G. B., Physics and Radiobiology of Nuclear Medicine, Springer Science & Business Media, 2013.

  4. Frank, J., King, A., Raine, D., Accretion Power in Astrophysics, Cambridge University Press, 2002.

  5. De Broglie, L., PhD thesis, University of Paris, 1924.

  6. Planck, M., Annalen der Physik 1901, 4, 90–100.

  7. Bohr, N., The London Edinburgh and Dublin Philosophical Magazine and Journal of Science 1913, 26, 1–25.

  8. Katz, D. M., Physics for Scientists and Engineers: Foundations and Connections, Extended Version with Modern, Cengage Learning, 2016.

  9. Heisenberg, W., Zeitschrift für Physik 1925, 33, 879–893.

  10. Schrödinger, E., Physical Review 1926, 28, 1049–1070.

  11. Davisson, C., Germer, L. H., Physical Review 1927, 30, 705–741.

  12. Palucka, T., Overview of Electron Microscopy, Caltech, 2002.

  13. Jönsson, C., Zeitschrift für Physik 1961, 161, 454–474.

  14. Juffmann, T., Milic, A., Müllneritsch, M., Asenbaum, P., Tsukernik, A., Tüxen, J., Mayor, M., Cheshnovsky, O., Arndt, M., Nature nanotechnology 2012, 7, 297–300.

  15. Crommie, M. F., Lutz, C. P., Eigler, D. M., Science 1993, 262, 218–220.

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