What the effect is

The photoelectric effect is an emission process: electrons are released from a material surface when electromagnetic radiation of sufficiently high frequency is incident on it. The electrons that leave are called photoelectrons, which are electrons that have escaped from the surface after receiving energy from the incident radiation.

So a clean metal surface can turn light into an electric current. Shine suitable radiation on the surface, electrons leave, and if a second electrode collects them, charge flows through the external circuit. That’s the basic idea behind a photoelectric cell.

A photon is a quantum of electromagnetic radiation that transfers energy and momentum in discrete interactions. The energy of one photon is Eγ = h**f, where Eγ is the photon energy (J), h is the Planck constant (J s) and f is the frequency of the radiation (Hz). For light in vacuum, c = f**λ, where c is the speed of light in vacuum (m s⁻¹) and λ is the wavelength (m), so high frequency corresponds to short wavelength.

Why this is particle evidence

The main observations are awkward — genuinely awkward — for a purely classical wave model of light:

• Below a certain frequency, no electrons are emitted, however intense the radiation is.
• Above that frequency, emission begins without a measurable delay, even when the radiation is very weak.
• Increasing intensity increases the number of photoelectrons emitted per second, but it does not increase the maximum kinetic energy of each photoelectron.
• Increasing frequency increases the maximum kinetic energy of the photoelectrons.

In classical wave theory, light energy is spread continuously over the wavefront, and intensity is proportional to amplitude squared. A very weak beam should eventually give an electron enough energy if we wait long enough; a more intense beam should give electrons more energy. The experiment doesn’t show that. One photon interacts with one electron. If the photon has enough energy, the electron can leave at once.

This gave physics a clean case of falsification. The wave model of light was not thrown away — diffraction and interference still demand it — but the claim that light is only a wave could no longer survive. Einstein’s explanation put the particle description at the centre of the story.

The Bohr model does not rescue a wave-only explanation here. The Bohr model is an atomic model that describes electrons in quantized states of a hydrogen-like atom. The photoelectric effect concerns electrons near the surface of a metal, with bonding and conduction behaviour that the Bohr model is not built to describe. It is a useful historical model, not a general theory of metal surfaces.

The frequency cut-off

The threshold frequency is the lowest frequency of incident electromagnetic radiation that can release photoelectrons from a specified metal surface. It is written f₀, where f₀ is the threshold frequency (Hz).

The key word is “frequency”, not “intensity”. A weak ultraviolet source can release electrons from some metals; a very bright red source may release none. Examiners like this detail because it shows the photon model clearly, rather than the vague idea that “more light means more energy”.

Work function and threshold condition

The work function is the minimum energy needed for an electron to escape from a specified metal surface. It is written Φ, where Φ is the work function (J). At the threshold frequency,

h**f₀ = Φ.

If f < f₀, each photon has too little energy, so no photoelectrons are emitted. If f = f₀, the photon energy is just enough for escape, and the electron leaves with essentially no kinetic energy. If f > f₀, the extra photon energy becomes kinetic energy of the emitted electron.

Different metals have different threshold frequencies because their surface electrons are bound with different strengths. The state of the surface matters too: oxide layers, impurities and crystal structure can change the effective work function.

Energy accounting, one photon at a time

Einstein explained the photoelectric effect by applying conservation of energy to one photon and one electron. The photon is absorbed. Some of its energy frees the electron from the metal; the rest, if there is any, becomes kinetic energy of the photoelectron.

The maximum kinetic energy of the emitted photoelectrons is given by

Eₘₐₓ = h**f − Φ,

where Eₘₐₓ is the maximum kinetic energy of a photoelectron just after emission (J). “Maximum” is the key word here. Electrons do not all start from equally favourable places in the metal. Some lose energy on the way out, while the least hindered electrons can escape with Eₘₐₓ.

Since Φ = h**f₀, the same equation can be written as

Eₘₐₓ = h(f − f₀).

What the graph says

A graph of Eₘₐₓ against f gives a straight line. The gradient is h, the frequency-axis intercept is f₀, and the energy-axis intercept is −Φ. Changing the metal changes Φ, so the line shifts, but its gradient is still the Planck constant.

The graph also makes it clear why intensity does not appear in Einstein’s equation. At a fixed frequency above the threshold, higher intensity means more photons arriving each second, so the emission rate increases. The energy of each photon is unchanged, so Eₘₐₓ is unchanged too.

Stopping potential as a measurement

In a photoelectric cell, the collector can be made negative enough to repel even the fastest photoelectrons. The stopping potential is the potential difference that just brings the photoelectric current to zero. It is written Vₛ, where Vₛ is the stopping potential (V).

The maximum kinetic energy is then

Eₘₐₓ = e**Vₛ,

where e is the elementary charge (C). This is why the electronvolt is useful here: a photoelectron stopped by 1 V has lost 1 eV of kinetic energy.

Particle diffraction

Diffraction is the spreading and redistribution of a wave after it passes through an aperture or around an obstacle. You see it most clearly when the aperture or obstacle is about the same size as the wavelength. If particles produce a diffraction pattern, they’re acting like waves in that experiment.

The classic evidence comes from electron diffraction. In the Davisson–Germer experiment, electrons were fired at a nickel crystal, and the scattered intensity had strong maxima at particular angles. In school versions, electrons pass through a thin graphite film and make bright and dark rings on a fluorescent screen. Those rings are the clue. They’re not just a central bright patch; they show that the electrons have been scattered into directions fixed by wave interference.

A scattering experiment and minimum intensity

One clean way to imagine the measurement is this: send a collimated beam of electrons through a narrow slit or through a thin crystalline target, then move a detector across different scattering angles. If the electrons behaved only like tiny pellets, you’d expect a mainly geometrical spread. Instead, the detector measures intensity with maxima and minima.

For a single slit of width a, minima occur approximately when

a sin θₙ = n**λᵈᴮ,

where a is the slit width (m), θₙ is the angular position of the nth minimum from the central direction (rad or degrees), n is a positive integer order, and λᵈᴮ is the de Broglie wavelength of the particle (m). A detector placed at one of these θₙ positions records minimum intensity because probability amplitudes from different parts of the slit cancel.

That final idea links wave optics to quantum physics: the thing waving is not a tiny material ripple in space, but a probability amplitude. Its interference controls where particles are likely to be detected.

The dual description

Wave–particle duality is a quantum model in which something such as light or an electron shows wave-like behaviour in some experiments and particle-like behaviour in others. The object is not swapping between a tiny ball and a water wave. Rather, neither classical picture works fully by itself.

Matter looks particle-like when it is detected: an electron arrives at one point on a screen and transfers a localized amount of charge and energy. It looks wave-like as it travels through a diffraction or interference arrangement: after many electrons have been detected, their final distribution forms a pattern with maxima and minima.

What counts as wave evidence?

Waves are identified by behaviours such as superposition, diffraction and interference. Evidence for wave behaviour, then, is not just a beam spreading out. It is the appearance of systematic regions of high and low intensity, where reinforcement and cancellation have taken place.

Here is the unsettling part: the pattern still builds up when particles pass through the apparatus one at a time. Each electron is detected as a localized event, yet after many detections the same interference pattern appears. The wave model predicts the probability distribution; the particle detection gives the individual dots.

If you try to find out which slit an electron or photon passes through, you change the experiment. A measurement that can give path information disturbs the state enough to remove the two-source interference pattern. In quantum physics, observation is not always passive.

De Broglie’s proposal

The de Broglie wavelength is the wavelength linked to a moving particle because it has momentum. It is given by

λᵈᴮ = h / p,

where p is the magnitude of the particle momentum (kg m s⁻¹).

For a non-relativistic particle,

p = m**v,

where m is the particle mass (kg) and v is its speed (m s⁻¹). So for large, everyday objects, the de Broglie wavelength is absurdly tiny. A tennis ball, for example, does not diffract around a doorway in any observable way, because its wavelength is fantastically smaller than atomic and nuclear scales.

Electrons are the useful case

Electrons have a small mass, so their de Broglie wavelengths can be similar in size to atomic spacings. For an electron accelerated from rest through a potential difference V,

λᵈᴮ = h / √(2mₑeV),

where mₑ is the mass of an electron (kg) and V is the accelerating potential difference (V).

Increasing the accelerating potential difference gives the electron more momentum, so the de Broglie wavelength decreases. In an electron diffraction tube, the ring pattern contracts. Lower the potential difference, and the opposite happens: the wavelength increases and the rings spread out.

This gives a neat practical check on the idea. Change a particle quantity — the accelerating voltage, and therefore the momentum — and the wave pattern shifts just as the wavelength equation predicts.

The scattering idea

Compton scattering is an inelastic scattering interaction where a photon hits a loosely bound or free electron, then leaves in a new direction with lower energy. It gives extra evidence for the particle nature of light, since the result only makes sense when the photon is treated as something carrying energy and momentum.

For a photon, the momentum magnitude is

pγ = h / λ,

where pγ is the photon momentum (kg m s⁻¹).

In Compton’s experiment, X-ray photons were fired at a low atomic number target such as carbon. A crystal analyser and an ionization detector then measured scattered X-rays at different angles. The spectrum showed a wavelength-shifted peak, and the size of the shift depended on the scattering angle.

Why it is stronger than the photoelectric evidence

The photoelectric effect is strong evidence, but the incoming photon is absorbed and vanishes. Its particle nature has to be inferred from the electron emitted. With Compton scattering, the incoming and outgoing photon can both be studied: the photon isn’t just absorbed; it comes out with a measurable change in wavelength and direction.

That makes the conservation-law picture far more direct. You can compare energy and momentum before the interaction with energy and momentum after it, just like in a collision problem.

Similar to, but not the same as, solid-ball collisions

It resembles two solid balls colliding because total energy and total momentum are conserved, and the scattered energy depends on angle. The key difference is that a photon has zero rest mass, travels at speed c, and needs quantum and relativistic ideas to describe it. A solid ball collision involves many atoms and contact forces; Compton scattering is a single photon–electron interaction.

Increased wavelength means lost photon energy

In Compton scattering, the scattered photon leaves with a longer wavelength than the incident photon. Since Eγ = h**c / λ, that longer wavelength gives a smaller photon energy. The energy difference has gone to the recoil electron.

The electron gains momentum too, so it recoils at an angle to the original photon direction. So the interaction has to be treated as a two-dimensional conservation problem, not a simple straight-line rebound.

The two peaks in real data

Real Compton spectra often have two peaks. One sits at the original wavelength, caused by scattering from the atom as a whole or from tightly bound electrons; because the atom has a much larger mass, the wavelength change is too small to notice. The shifted peak is the Compton peak, from scattering by electrons that can be treated as free during the interaction.

As the photon scattering angle increases, the wavelength increase gets larger. In the straight-through direction, the shift is zero; at larger angles, the shifted peak moves to longer wavelength.