Wave–particle duality and the photoelectric effect

Electron diffraction and the de Broglie equation, photons, the photoelectric effect and atomic energy levels.

Understand how diffraction experiments provide evidence for the wave nature of electrons

Be able to use the de Broglie equation $\lambda = \frac{h}{p}$

Understand how the behaviour of electromagnetic radiation can be described in terms of a wave model and a photon model, and how these models developed over time

Be able to use the equation $E = hf$ , that relates the photon energy to the wave frequency

Understand that the absorption of a photon can result in the emission of a photoelectron

Understand the terms threshold frequency and work function and be able to use the equation $hf = \phi + \frac{1}{2}mv_{max}^2$

Be able to use the electronvolt (eV) to express small energies

Understand how the photoelectric effect provides evidence for the particle nature of electromagnetic radiation

Understand atomic line spectra in terms of transitions between discrete energy levels and understand how to calculate the frequency of radiation that could be emitted or absorbed in a transition between energy levels

Light of wavelength $400\,\text{nm}$ falls on a sodium surface. The work function of sodium is $2.28\,\text{eV}$ .

(a) Calculate the energy of one photon in eV.

(b) Calculate the maximum kinetic energy of the emitted electrons in joules.

$E = hc/\lambda = 6.63 \times 10^{-34} \times 3.00 \times 10^{8}/400 \times 10^{-9} = 4.97 \times 10^{-19}\,\text{J}$ (1)
$E = 4.97 \times 10^{-19}/1.60 \times 10^{-19} = 3.11\,\text{eV}$ (1)
$KE_\text{max} = 3.11 - 2.28 = 0.83\,\text{eV}$ (1)
$KE_\text{max} = 0.83 \times 1.60 \times 10^{-19} = 1.3 \times 10^{-19}\,\text{J}$ (1)

In a hydrogen atom, an electron moves from the energy level at $-3.40\,\text{eV}$ to the ground state at $-13.6\,\text{eV}$ .

Calculate the wavelength of the photon emitted.

Energy difference $= 13.6 - 3.40 = 10.2\,\text{eV}$ (1)

Electrons are accelerated from rest through a potential difference of $150\,\text{V}$ and then directed at a thin graphite film.

(a) Calculate the de Broglie wavelength of the electrons.

(b) Explain why the electrons produce a diffraction pattern with graphite.

$KE = eV = 1.60 \times 10^{-19} \times 150 = 2.4 \times 10^{-17}\,\text{J}$ (1)

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