Black-body radiation and the ultraviolet catastrophe

A black body is an ideal object that absorbs all electromagnetic radiation incident on it. It also emits radiation across a continuous range of frequencies. The distribution of this radiation depends on the black body’s temperature.

If emitted intensity is plotted against frequency, the observed curve rises to a maximum and then falls at higher frequencies. The maximum marks the frequency at which the emission is most intense. As the temperature increases, the spectrum changes: the peak moves to a higher frequency, and the emitted intensity increases.

The ultraviolet catastrophe

Classical physics predicted a different shape at high frequencies. Instead of falling after a maximum, the predicted intensity continued to increase without limit as frequency increased. This disagreement with the observed spectrum is called the ultraviolet catastrophe.

The prediction was not merely inaccurate at high frequencies. The total emitted power corresponds to the area under the intensity–frequency curve. Because the classical prediction increased without limit, it implied an infinite total emitted power. The observed curve falls at high frequencies and gives a finite total emitted power.

Planck's quantisation of energy

A black body emits thermal radiation over a range of frequencies. Its observed spectrum rises to a peak and then falls at high frequencies. Classical physics could not explain this fall: it predicted that the intensity would increase without limit as frequency increased.

Planck proposed that energy is exchanged between matter and radiation in discrete amounts, called quanta, rather than in arbitrary continuous amounts. For radiation of frequency $$f$$, the energy of one quantum is:

Here $$E$$ is the energy of one quantum and $$h$$ is Planck's constant. Energy is exchanged in whole-number multiples of $$hf$$, so a higher frequency means a larger energy quantum.

At a given temperature, exchanges involving these larger amounts of energy are less likely. High-frequency emission is therefore suppressed, causing the intensity to fall instead of rising without limit. Planck's quantisation explains the observed high-frequency fall in the black-body spectrum.

Einstein's explanation of the photoelectric effect

When electromagnetic radiation falls on a material, it can cause electrons to leave the surface. This is the photoelectric effect. The emitted electrons are called photoelectrons.

For a given material, emission depends on the radiation’s frequency:

• Below a minimum frequency, called the threshold frequency, no electrons are emitted, however intense the radiation.
• Above the threshold, electrons are emitted without a measurable delay.
• Increasing the frequency increases the maximum kinetic energy of the emitted electrons.
• Increasing the intensity at a fixed frequency above the threshold increases the number of electrons emitted, but not their maximum kinetic energy.

Why classical wave theory was insufficient

Classical wave theory treats the energy of radiation as continuously distributed across the wave. It therefore predicts that an electron should eventually collect enough energy to escape at any frequency, provided the radiation is intense enough or the exposure is long enough.

This does not explain why there is a threshold frequency or why emission occurs without a measurable delay above it. It also suggests that increasing intensity should increase the energy available to an electron, whereas the observed maximum kinetic energy depends on frequency.

Photons and the work function

Einstein proposed that radiation transfers energy in discrete packets called photons. A photon of frequency $$f$$ has energy $$hf$$, where $$h$$ is Planck’s constant. One photon transfers its energy to one electron.

An electron needs a minimum energy to escape from the material. This energy is the work function, $$\phi$$. If the photon’s energy is less than the work function, it cannot release the electron. If sufficient energy is transferred, the electron can escape; energy left after release becomes kinetic energy.

The greatest possible kinetic energy is given by:

Here, $$E_{\mathrm{k,max}}$$ is the maximum kinetic energy of an emitted electron. Some electrons have less kinetic energy because they lose energy before escaping.

At the threshold frequency $$f_0$$, a photon has just enough energy to supply the work function:

Below $$f_0$$, each photon has too little energy to release an electron. Above $$f_0$$, a photon can transfer enough energy in a single interaction, so emission does not require a measurable accumulation time. Increasing frequency increases the energy of each photon and therefore increases the maximum kinetic energy left after an electron escapes.

At a fixed frequency, greater intensity means more photons arrive per unit time. Above the threshold, these photons can release more electrons, but each photon still has energy $$hf$$. Increasing intensity therefore does not increase the maximum kinetic energy. Below the threshold, increasing intensity cannot make the individual photons energetic enough to release electrons.

The photoelectric effect provides evidence for the particle-like nature of electromagnetic radiation: energy is transferred in photons. Radiation also shows wave behaviour.