This episode introduces an important phenomenon: wave – particle duality. In studying the photoelectric effect, students have learned that light, which we think of as waves, can sometimes behave as particles. Here they learn that electrons, which we think of as particles, can sometimes behave as waves.

Lesson Summary

• Demonstration: Diffraction of electrons (30 minutes)
• Discussion: de Broglie equation (15 minutes)
• Worked examples: Using the equation (15 minutes)
• Discussion: Summing up (10 minutes)
• Student questions: Practice calculations (30 minutes)

Demonstration: Diffraction of electrons

The diffraction of electrons was first shown by Davisson and Germer in the USA and G P Thomson in the UK, in 1927 and it can now be observed easily in schools with the correct apparatus.

Show electron diffraction. It will help if students have previously seen an electron-beam tube in use (e.g. the fine beam tube, or e/m tube).

Episode 506-1: Diffraction of electrons (Word, 69 KB)

Before giving an explanation, ask them to contemplate what they are seeing. It is not obvious that this is diffraction/interference, since students may not have seen diffraction through a polycrystalline material. (Note that the rigorous theory of crystal diffraction is not trivial – waves scattered off successive planes of atoms in the graphite give constructive interference if the path difference is a multiple of a wavelength, according to the Bragg equation. The scattered waves then appear to form a wave that appears to reflect off the planes of atoms, with the angle of incidence being equal to the angle of reflection. In the following we adopt a simplification to a 2D case – see Episode 506-1)

Qualitatively it can help to show a laser beam diffracted by two crossed diffraction gratings. Rotate the grating, and the pattern rotates. If you could rotate it fast enough, so that all orientations are present, you would see the array of spots trace out rings.

Episode 506-2: Diffraction of light (Word, 30 KB)

From their knowledge of diffraction, what can they say about the wavelength of the electrons? (It must be comparable to the separation of the carbon atoms in the graphite.) How does wavelength change as the accelerating voltage is increased? (The rings get bigger; wavelength must be getting smaller as the electrons move faster.)

Discussion: de Broglie equation

In 1923 Louis de Broglie proposed that a particle of momentum p would have a wavelength λ given by the equation:

• wavelength of particle λ  = hp
• where h is the Planck constant,
• or λ  = hmv for a particle of momentum mv.

The formula allows us to calculate the wavelength associated with a moving particle.

Worked examples: Using the equation

1. Find the wavelength of an electron of mass 9.00 × 10^-31 kg moving at 3.00 × 10⁷ m s^-1.
λ  = hp λ  = 6.63 × 10^-34 J s^-12.70 × 10^-23 kg m s^-1 λ  = 2.46 × 10^-11 m λ  = 0.025 nm This is comparable to atomic spacing, and explains why electrons can be diffracted by graphite.
2. Find the wavelength of a cricket ball of mass 0.15 kg moving at 30 m s^-1.
λ  = hp λ  = 6.63 × 10^-34 J s^-14.5 kg m s^-1 λ  = 1.47 × 10^-34 m This is a very small number, and explains why a cricket ball is not diffracted as it passes near to the stumps.
3. It is also desirable to be able to calculate the wavelength associated with an electron when the accelerating voltage is known. There are 3 steps in the calculation.

Calculate the wavelength of an electron accelerated through a potential difference of 10 kV.

1. Calculate the kinetic energy
E_k = e × V E_k = 1.6 × 10^-19 C × 10 × 10³ J C^-1 E_k = 1.6 × 10^-15 J
2. Calculate the momentum
E_k = p²2m p = 5.4 × 10^-23 kg m s^-1
3. Calculate wavelength
λ  = hp λ  = 6.63 × 10^-34 J s^-15.4 × 10^-23 kg m s^-1 λ  = 1.2 × 10^-11 m, or 0.012 nm.

Discussion: Summing up

You may come across a number of ways of trying to resolve the wave-particle dilemma. For example, some authors talk of wavicles. This is not very helpful.

Summarise by saying that particles and waves are phenomena that we observe in our macroscopic world. We cannot assume that they are appropriate at other scales.

Sometimes light behaves as waves (diffraction, interference effects), sometimes as particles (absorption and emission by atoms, photoelectric effect).

Sometimes electrons (and other matter) behave as particles (beta radiation etc), and sometimes as waves (electron diffraction).

It’s a matter of learning which description gives the right answer in a given situation.

The two situations are mutually exclusive. The wave model is use for radiation (i.e. anything transporting energy and momentum, e.g. a beam of light, a beam of electrons) getting from emission to absorption. The particle (or quantum) model is used to describe the actual processes of emission or absorption.

Student question: Interpreting electron diffraction patterns

Episode 506-3: Interpreting electron diffraction patterns (Word, 40 KB)

Episode 506-4: Electron diffraction question (Word, 25 KB)

You could extend the idea of electrons-as-waves further, to the realm of the atom.

Lesson Summary

• Demonstration: Melde’s experiment (20 minutes)
• Discussion: Electron waves in atoms (10 minutes)
• Demonstration: Standing waves on a loop (10 minutes)
• Student question: Electron standing waves (10 mins)

Demonstration: Melde’s experiment

Episode 507-1: Standing waves – for electrons? (Word, 161 KB)

In this section we are going to introduce the idea of standing waves within an atom. It is therefore useful first to demonstrate standing waves on a stretched elastic cord. This is known as Melde’s experiment.

(A very simple alternative to the vibration generator is an electric toothbrush.)

Show that there are only certain frequencies at which standing wave loops occur.

Discussion: Electron waves in atoms

The waves on the string are trapped between the two fixed points at the ends of the string and cannot escape.

If the electron has wave properties and it is also confined within an atom we could imagine a sort of standing wave pattern for these waves rather like the standing waves on a stretched string. The electrons are trapped within the atom rather like the waves being trapped on a stretched string. The boundaries of these electron waves would be the potential well formed within the atom.

This idea was introduced because the simple Rutherford model of the atom had one serious disadvantage concerning the stability of the orbits. Bohr showed that in such a model the electrons would spiral into the nucleus in about 1 × 10^-10 s, due to electrostatic attraction. He therefore proposed that the electrons could only exist in certain states, equivalent to the loops on the vibrating string.

If your students have met the idea of angular momentum, you could tell them that Bohr proposed that the angular momentum of the electrons in an atom is quantised, in line with Planck's quantum theory of radiation. He stated that the allowed values of the angular momentum of an electron would be integral multiples of h2 π . This implied a series of discrete orbits for the electron. We can imagine the electron as existing as a wave that fits round a given orbit an integral number of times.

Demonstration: Standing waves on a loop

The wire loop is a two dimensional analogy of electron waves in an atom. As it vibrates at the correct frequency, an integral number of waves fit round the orbit. These waves represent the electron waves in an atom.

Student question: Electron standing waves

de Broglie waves can be imagined as forming standing waves which fit into an atom.

Episode 507-2: Electron standing waves (Word, 323 KB)