The UV catastrophe, Planck's quantisation, and the photoelectric effect

Black-body radiation and the ultraviolet catastrophe

• A black bodyAn idealised object that absorbs all incident electromagnetic radiation and, when hot, emits a continuous spectrum. The spectrum depends only on its temperature. absorbs all radiation falling on it and emits a continuous spectrum that depends only on its temperature.
• Experiments showed that the intensity of radiation increases with frequency to a peak, then decreases at higher frequencies.
• Classical wave theory (Rayleigh-Jeans law) predicted that intensity should increase without limit as frequency increases, heading towards infinity in the ultraviolet region. This absurd prediction was called the ultraviolet catastropheThe failure of classical wave theory to predict the observed spectrum of black-body radiation at short wavelengths, where it predicted infinite intensity..
• The classical prediction matched experiment at low frequencies but diverged catastrophically at high frequencies.

Planck's resolution (1900)

• Max Planck proposed that the oscillating atoms in a black body can only emit or absorb energy in discrete packets called quantaDiscrete packets of energy. A single quantum of electromagnetic energy is a photon..
• The energy of each quantum is:

• where $n$ is an integer, $h$ is Planck's constant ($6.63 \times 10^{-34}$ J s), and $f$ is the frequency.
• At high frequencies, each quantum has a large energy ($hf$). Very few oscillators have enough energy to emit even a single quantum at these frequencies, so the total intensity drops. This naturally limits the high-frequency emission and produces the correct spectrum.

The photoelectric effect and Einstein's photon theory

• When light shines on a metal surface, electrons are emitted if the light frequency exceeds a threshold frequencyThe minimum frequency of incident radiation required to release photoelectrons from a metal surface. Below this frequency, no electrons are emitted regardless of intensity. $f_0$.
• Classical wave theory predicted that any frequency of light should eventually eject electrons if the intensity is high enough. This was contradicted by experiment.
• In 1905, Einstein explained this by extending Planck's idea: light itself consists of particles (photons), each carrying energy $E = hf$.
• A single photon interacts with a single electron. If $hf$ is less than the work function $\phi$, no electron can escape, no matter how intense the light.
• Crucially, this was the first time light was shown to behave as particles in a measurable experiment, after decades of evidence for wave behaviour.