How We Came to Know the Cosmos: Light & Matter

# Chapter 15. De Broglie’s Matter Waves

## 15.1 Compton scattering

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.

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.

 Figure 15.1Image credit 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, 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 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
 Figure 15.2Image credit Allowed (left) and forbidden (right) standing waves.
 Figure 15.3Image credit 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 Werner Heisenberg[9] (discussed in Chapter 16) and Erwin Schrödinger[10] (discussed in Chapter 17) in 1925-1926. The 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, the German physicist Claus Jönsson performed 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 the University of Vienna 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.

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