Blackbody Radiation and the Quantum of Energy
A blackbody's thermal spectrum has a peak that moves to shorter wavelengths as temperature rises. The classical Rayleigh-Jeans model fails at short wavelengths. Planck explained the measured spectrum using discrete oscillator energy changes with spacing hf. Planck's constant h relates frequency to that energy spacing.
What a learner can do afterwards
- Describes how a blackbody's power-per-unit-wavelength spectrum changes with temperature and where the Rayleigh-Jeans model fails
- Explains how energy changes in multiples of hf reduce the high-frequency contribution and remove the short-wavelength divergence
- Uses Wien's displacement law with wavelength in metres and temperature in kelvin to estimate surface temperature
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You have met thermal emission and absorption. A blackbody is an ideal absorber of all incident electromagnetic radiation at every wavelength. At a given temperature, it has the maximum thermal emission per unit area. A small opening into a heated cavity closely approximates one. Many stars have broadly similar thermal spectra.
Plot emitted power per unit area per unit wavelength against wavelength. A blackbody curve rises to a peak and falls on either side. Heating increases total emission and moves this peak to shorter wavelengths. The spectrum remains broad. Visible glow colour is not a direct measurement of the peak wavelength.
The classical Rayleigh-Jeans model grows without limit as wavelength approaches zero, predicting infinite total power: the ultraviolet catastrophe. Planck used oscillator energy changes in multiples of hf. Here f is frequency and h is Planck's constant. High-frequency steps require more energy, so those modes are less often excited at a fixed temperature. This gives the measured finite spectrum.
Wien's displacement law is lambda_max T = b, where b = 2.898 x 10^-3 m K. The peak wavelength lambda_max refers to the power-per-unit-wavelength spectrum and must be in metres. T is in kelvin. Rearrange to T = b / lambda_max. One nanometre is 10^-9 m. For an approximately blackbody star, the peak gives an estimated surface temperature, not its age or distance.
A blackbody spectrum peaks at 1.20 x 10^-6 m. Its temperature is T = (2.898 x 10^-3) / (1.20 x 10^-6) = 2415 K. Divide the constant by the wavelength. A shorter peak wavelength gives a higher kelvin temperature. When comparing two bodies, T_B / T_A = lambda_A / lambda_B.
The Stefan-Boltzmann law gives total emitted power: P = sigma A T^4. A is the blackbody's area, T is in kelvin and sigma is a constant. Dividing by area gives P/A = sigma T^4. This power includes all wavelengths. It differs from apparent brightness at a detector, which also depends on distance, and from net cooling, which counts incoming energy too.
A blackbody has a broad thermal spectrum. Wien's law locates its peak. Energy changes in multiples of hf explain the measured spectrum.
This lesson, its questions and flashcards were written for this dataset. Scientific source records were used for facts and teaching design; their wording is not reproduced.
The readings and flashcards were written for this dataset. Blackbody radiation is a linked Khan Academy video. The video remains subject to publisher and YouTube terms; no reuse licence is asserted here.
More to read
What a thermal spectrum's axes tell you
A spectrum shows how emission is distributed. For the wavelength form used here, the horizontal axis is wavelength. The vertical quantity is emitted power per unit area per unit wavelength. The area under that curve gives total emitted power per unit area.
The peak identifies the wavelength with the largest value of this vertical quantity. It does not mean that the body emits only that wavelength. Nor is the peak's height the same as the total emission.
Wien's law uses this wavelength-spectrum peak: lambda_max T = 2.898 x 10^-3 m K. Convert wavelength to metres before dividing the constant by it. The result is temperature in kelvin. A spectrum expressed per unit frequency has a different peak convention, so check the axes.
The classical Rayleigh-Jeans model agrees at long wavelengths but diverges toward short wavelengths. Measurements instead show a finite peak followed by falling emission. That disagreement required a different physical model.
How energy spacing changes thermal emission
Planck's model describes energy exchange between radiation in a cavity and oscillators in its walls. An oscillator of frequency f changes energy in integer multiples of hf. The units explain the role of h: joule seconds multiplied by frequency in inverse seconds gives joules. The constant h alone is not an energy packet.
At fixed temperature, large excitation energies are less likely than small ones. A high frequency gives a large hf spacing. Those modes therefore contribute less than the classical equal-energy treatment predicts. High-frequency radiation remains possible; quantisation does not impose a sharp cutoff.
The resulting distribution has finite total power and matches the observed blackbody spectrum. It also leads to Wien's law and the fourth-power relationship for total emission.
Equal spacing belongs to this oscillator model. Other quantum systems can have unequal energy gaps. In an atom, the energy of an emitted photon equals the particular difference between the two levels involved.
Testing photon energy with emitted electrons
In the usual single-photon photoelectric experiment, an absorbed photon can supply energy to an electron in a metal. The work function phi is the minimum energy needed to free an electron from that surface. Emission requires sufficient photon energy, and the maximum electron kinetic energy follows K_max = hf - phi.
At a fixed frequency above threshold, increasing light intensity supplies more photons each second. It can increase the electron emission rate without increasing the maximum kinetic energy. Increasing frequency changes the energy of each photon and raises that maximum. Different surfaces can have different thresholds.
Einstein's explanation treated light itself as packets carrying hf. This differs from Planck's original model of quantised energy exchange by oscillators in a thermal radiator.
Photoelectric measurements and blackbody spectra are separate tests. Both support the frequency-energy relationship, but a photoelectric threshold is not the wavelength peak of a thermal spectrum.
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