Physics · Atomic and quantum physics · Grade 11

Photon momentum and the Compton effect

Can sunlight push a sail, and what happens when a photon hits an electron?

The situation

A spacecraft that moves through space without fuel — a solar sail — is made of mirror film only micrometres thick. In 2010 the Japanese probe IKAROS flew past Venus with such a sail. Light has no mass, yet it pushes the sail — so a photon carries momentum. The same momentum is why an X-ray photon that strikes an electron sets it in motion, while the wavelength of the scattered radiation increases. How does light “give a push”, and how can this be measured?

Topic

Momentum of a photon, the Compton effect. Solar sail and the scattering of X-rays by an electron

Key ideas

Photon momentum p = h/λ
Radiation pressure
Compton shift

Formulas

E = h ν, p = \frac{h}{λ} = \frac{E}{c}

Photon energy and momentum: the momentum of a 550 nm photon is $1.2 \cdot 10^{- 27}$ kg·m/s.

p_{light} = \frac{S}{c} (1 + R) \cos^{2} α

Radiation pressure: $S$ is the intensity, $R$ the reflection coefficient, $α$ the angle of incidence. At 1 AU ( $S = 1361$ W/m²) the pressure is 4.54 µPa on an absorbing surface and 9.08 µPa on an ideal mirror.

\vec{S} = \vec{E} \times \vec{H}, ⟨ S ⟩ = \frac{1}{2} c ε_{0} E_{0}^{2}, g = \frac{S}{c^{2}}

The Umov–Poynting vector is the energy flux density; $g$ is the volume density of the momentum of an electromagnetic wave.

Δ λ = λ^{'} - λ = \frac{h}{m_{e} c} (1 - \cos θ)

Compton formula: $h / (m_{e} c) = 2.426$ pm is the Compton wavelength of the electron.

E^{'} = \frac{E}{1 + \frac{E}{m_{e} c^{2}} (1 - \cos θ)}, T_{e} = E - E^{'}

Energies of the scattered photon and the recoil electron ( $m_{e} c^{2} = 511$ keV).

Theory

Photon momentum

Light carries energy in portions called photons, each with $E = h ν$ . A photon has zero rest mass, but it does have momentum: $p = E / c = h / λ$ . The momentum of a visible photon is very small ( $\approx 10^{- 27}$ kg·m/s), but the number of photons is enormous: in Earth's orbit the Sun sends ≈ $6 \cdot 10^{21}$ photons per second onto every square metre. If a surface absorbs a photon, it receives its momentum ( $p$ ); if a mirror reflects the photon straight back, it receives twice as much ( $2 p$ ), because the photon's momentum is reversed. That is why the pressure on a mirror is 2 times greater than on a black surface. More about photons — in the “The photoelectric effect” experiment.

Radiation pressure and the Umov–Poynting vector

According to Maxwell's theory, an electromagnetic wave carries an energy flux (the Umov–Poynting vector $\vec{S} = \vec{E} \times \vec{H}$ ) and momentum: the momentum per unit volume is $S / c^{2}$ , and the pressure on a surface is $S / c$ if the wave is absorbed and $2 S / c$ if it is reflected. At 1 AU the amplitude of the field of sunlight is ≈ 1 kV/m, while the pressure is only ≈ 4.5–9 µPa — $10^{10}$ times smaller than atmospheric pressure. Even so, a light, large sail accumulates it for months and reaches a noticeable speed. If sunlight strikes a mirror sail at an angle of incidence $α$ , the force is directed along the sail's normal and is proportional to $\cos^{2} α$ ; its component along the orbit is $\cos^{2} α \sin α$ — this is greatest at $\tan α = 1 / \sqrt{2}$ , i.e. at $α \approx 35.3 °$ . With this force the craft spirals away from the Sun.

Lebedev's experiment

Radiation pressure was first measured by Pyotr Lebedev in 1899–1900. Thin vanes hang on a fine thread inside an evacuated glass vessel; the radiation pressure twists the thread, and the twist is measured with a mirror and a scale (an optical lever): $φ = M / κ$ . The main difficulty is radiometric forces: the illuminated side of a vane heats up, the remaining gas molecules rebound from it more strongly, and this force can be many times greater than the radiation pressure. The vanes of the famous Crookes radiometer turn precisely because of the gas, not the radiation pressure. Lebedev reduced these errors by using a high vacuum, thin metal vanes and alternating the illumination; the result agreed with Maxwell's theory. In 1910 he also measured the pressure of light on gases.

Compton effect

In 1923 Arthur Compton found that in molybdenum X-rays scattered by graphite there is, besides the original $λ$ , a longer wavelength $λ^{'}$ ; the shift $Δ λ$ depends neither on the target material nor on $λ$ , only on the scattering angle. Classical wave theory cannot explain this. Compton treated it as an elastic collision of a photon with a free electron: conservation of energy and momentum gives $Δ λ = \frac{h}{m_{e} c} (1 - \cos θ)$ . The unshifted line comes from scattering by inner electrons tightly bound to the atom — then the whole atom takes the recoil, and its mass is very large. For visible light $Δ λ / λ \sim 10^{- 5}$ — unnoticeable; for X-rays it is a few per cent.

Interaction of gamma rays with matter

A photon interacts with matter in three main ways: the photoelectric effect (dominant at low energies and in heavy elements), Compton scattering (dominant in light materials roughly between 0.1 and 10 MeV) and electron–positron pair production ( $E > 2 m_{e} c^{2} = 1.022$ MeV). A 662 keV gamma quantum from Cs-137 scattered through 180° rebounds with an energy of 184 keV, and the electron receives at most 477 keV (the Compton edge) — these limits can be seen in the spectrum of a scintillation detector. If the photoelectric effect leaves a vacancy in an inner shell, the atom settles down by emitting an X-ray quantum or an Auger electron. In technology, the photoelectric effect is used in photocells, solar panels and camera sensors. For more, see the “The photoelectric effect” experiment and the experiments on radiation detectors.

About the model

Model: the sail is flat, reflects like a mirror and absorbs the rest; its own thermal radiation and the size of the Sun's disc are ignored; the orbit lies in the plane of the ecliptic (the Sun's attraction + radiation pressure). The parameters of the Lebedev balance (thread stiffness $κ = 10^{- 10}$ N·m/rad, arm 1 cm, vane Ø 5 mm, scale 1 m) and the radiometric force formula are approximate. The widths of the peaks in the Compton spectrum (electron motion — Doppler broadening) and their heights are approximate; the probabilities for a free electron are taken from the Klein–Nishina formula.

Did you know?

In 1619 Johannes Kepler noticed that a comet's tail always points away from the Sun and suggested that sunlight pushes it. Today we know that it is radiation pressure that shapes the dust tail of a comet, while the gas tail is shaped by the solar wind.
In 1923 Arthur Compton discovered that the wavelength of X-rays scattered by graphite increases, and in 1927 he was awarded the Nobel Prize. This experiment showed directly that the photon has momentum.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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