Photoelectric Effect & Wave–Particle Duality
Light as particles, and particles as waves.
Photons
Light arrives in packets.
Light makes interference fringes, so it is a wave. Yet it also knocks electrons out of metals one at a time, so it comes in particles. Depending on the experiment, one face or the other shows.
The photon
Interference needs waves; the photoelectric effect (next lesson) can only be understood with particles. These particles of light are photons. Their properties:
- A photon always travels at c = 299 792 458 m/s ≈ 3.0 × 10⁸ m/s in vacuum, in every frame of reference.
- Its rest mass is zero (Newtonian mass isn’t meaningful for it).
- Each photon has a definite energy and a definite momentum, tied to the frequency and wavelength the same light shows as a wave:
- In a collision with a particle, total energy and momentum are conserved; photons can be absorbed or created, so their number need not be conserved.
- Brighter light of the same wavelength means more photons per second, not more energetic ones.
For a given wavelength, light energy can change only in whole steps of hc/λ. That is why photon theory is also called the quantum theory of light.
A parallel beam of 500 nm light has intensity 50 W/m². Find each photon’s energy and momentum, and how many cross 1 cm² each second.
How many photons does a 3.0 mW green laser (532 nm) emit each second?
A 100 W monochromatic source emits 3.0 × 10²⁰ photons per second. Find its wavelength.
Photons push
A hydrogen atom (1.67 × 10⁻²⁷ kg) absorbs a 103 nm photon coming head-on and stops dead. How fast was it moving?
A 5.0 W beam of 600 nm light falls normally on a perfectly absorbing surface. Find the photons absorbed per second and the force on the surface.
- 1Light comes in photons: speed c, zero rest mass.
- 2E = hν = hc/λ, p = h/λ = E/c (42.1); hc ≈ 1240 eV nm.
- 3Intensity sets the number of photons; wavelength sets each photon's energy.
- 4Photons carry momentum: absorbed light pushes with force P/c.
Check yourself
The Photoelectric Effect
Single photons free single electrons.
Shine short-wavelength light on a metal and electrons fly out. The details of how they do so can’t be explained by light waves at all, and they won Einstein his Nobel Prize.
Einstein’s photoelectric equation
Light falling on a metal can eject electrons: the photoelectric effect, and the ejected electrons are photoelectrons. The metal’s free electrons need at least the work function φ to escape.
A photon striking a free electron may give it all its energy E = hν. If E > φ the electron may get out, but it can also lose energy in collisions inside the metal (or fail to escape at all). An electron near the surface that uses its energy most economically leaves with E − φ; others leave with less. So photoelectrons have kinetic energies from zero up to
| Metal | φ (eV) | Metal | φ (eV) |
|---|---|---|---|
| Cesium | 1.9 | Calcium | 3.2 |
| Potassium | 2.2 | Copper | 4.5 |
| Sodium | 2.3 | Silver | 4.7 |
| Lithium | 2.5 | Platinum | 5.6 |
Threshold
Kmax = 0 when λ = λ₀ = hc/φ. For longer wavelengths each photon has less energy than φ, and no electrons come out however bright the light. λ₀ is the threshold wavelength and ν₀ = φ/h the threshold frequency; with φ = hν₀,
Find the longest wavelength that can eject electrons from sodium (φ = 2.3 eV).
The experiment
Two plates sealed in a vacuum: the emitter (cathode) C, lit through a window, and the collector (anode) A. Batteries, a rheostat and a reversing switch set A’s potential, positive or negative, and an ammeter reads the photocurrent.
- With A positive, more electrons are pulled across against the space charge; at high enough voltage every emitted electron is collected and the current reaches its saturation value.
- With A negative, electrons are turned back. At the stopping potential V₀ even the fastest ones fail: eV₀ = Kmax.
Light of 400 nm falls on potassium (φ = 2.2 eV). Find the maximum kinetic energy of the photoelectrons and the stopping potential.
Stopping potential against frequency
Plotting V₀ against 1/λ (or ν) gives a straight line for each metal. The slope hc/e (or h/e) is the same for every metal; the line meets the axis at the threshold.
Summary of the observations
- Short-wavelength light ejects electrons almost instantly.
- Each metal has a threshold wavelength; longer wavelengths eject nothing.
- Photoelectron kinetic energies range from 0 to Kmax = hc/λ − φ.
- A negative collector voltage stops the current; the stopping potential is proportional to Kmax.
- The stopping potential, and so Kmax, does not depend on intensity.
- It does depend on wavelength.
- Higher intensity gives a larger photocurrent.
Why waves can’t explain it
- A wave spreads its energy over the whole surface. In dim light an electron would need a long time to soak up φ, yet emission is immediate.
- Brighter waves carry more energy, so Kmax should rise with intensity. It doesn’t.
- With bright enough light of any colour, waves should eventually free electrons: there should be no threshold. There is one.
Photons explain all three: dim light means fewer photons, but any electron that is hit gets a photon’s whole energy at once, and that energy depends only on λ.
A 2.0 mW point source is 3.0 m from a metal (φ = 2.3 eV). If light were a wave and an electron could gather all the energy falling on a circle of radius 1.0 × 10⁻⁹ m around it, how long would it take to collect enough to escape?
- 1K_max = hν − φ (42.2) = h(ν − ν₀); threshold λ₀ = hc/φ.
- 2Stopping potential eV₀ = K_max: depends on λ and the metal, not on intensity.
- 3Saturation current grows with intensity.
- 4V₀ against ν: same slope h/e for every metal.
- 5No delay, no intensity dependence of K_max and a threshold all rule out the wave picture here.
Check yourself
Matter Waves
Electrons diffract too.
If light, a wave, can behave like particles, can particles behave like waves? De Broglie said yes, and experiments agreed: electrons diffract and interfere just as light does.
Particles as waves
Electrons usually behave as particles, but many experiments now show them interfering and making fringes like waves; the electron microscope depends on it. Protons, neutrons and larger particles have the same wave nature, if the experiment is set up to reveal it.
Louis de Broglie proposed, in his PhD thesis (Nobel Prize 1929), that a particle of momentum p has wavelength
For an electron accelerated from rest through a potential difference V, p = √(2meV), so
Find the de Broglie wavelength of electrons accelerated through 100 V.
Compare the de Broglie wavelengths of a proton moving at 1.0 × 10⁵ m/s and a 160 g cricket ball at 40 m/s.
Where Newton stops working
Since electrons are both waves and particles, Newton’s laws can’t describe them in every situation. Classical mechanics works well for objects larger than about 10⁻⁴ cm; for smaller ones we need quantum mechanics, which builds in this dual nature.
- 1Particles have wave properties: λ = h/p (42.4).
- 2Electron through V volts: λ ≈ 1.23/√V nm.
- 3Wave effects matter only when λ is comparable to the sizes involved; quantum mechanics is needed for sub-microscopic particles.