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Black-body radiation and Planck’s quanta
Black bodies and their radiation curves, Wien's displacement law, the failure of classical physics known as the ultraviolet catastrophe, and Planck's hypothesis that energy is quantised in units of E = hf.
Heat a steel rod in a forge and watch it change. First you feel the heat but see nothing. Then it glows a dull red, then orange, then yellow-white. Every blacksmith knew this sequence, and by 1900 physicists could measure it precisely. Yet when they used the best physics of the day to predict the colours, the answer was nonsense: any hot object should blaze out infinite energy in ultraviolet light. Fixing that one curve forced Max Planck into an idea he himself called "an act of desperation", and it started quantum physics.
What is a black body?
A black body is an ideal object that absorbs all radiation falling on it, reflecting none. Because it is also a perfect emitter, the radiation it gives out depends only on its temperature, not on what it is made of. This is called black-body radiation.
A good practical model is a hollow box with a tiny hole. Any light entering the hole bounces around inside until it is absorbed, so the hole looks perfectly black. When the box is heated, the radiation escaping from the hole is black-body radiation. Stars, glowing filaments and the cosmic microwave background are all close to black bodies.
Black-body curves
Measurements in the 1890s by Lummer and Pringsheim in Berlin mapped these curves accurately. Three features stand out:
- Each curve has a peak, and falls to zero at both very short and very long wavelengths.
- Hotter bodies emit more at every wavelength: the total power rises steeply with temperature.
- The peak moves to shorter wavelengths as temperature rises: Wien's displacement law (Wilhelm Wien, 1893):
$$\lambda_{max} = \frac{b}{T}, \qquad b = 2.898 \times 10^{-3} \text{ m K}$$
Temperature must be in kelvin. Objects at room temperature peak in the far infrared, which is why thermal cameras detect them. Around 1000 K the short-wavelength tail reaches the red end of the visible spectrum and objects glow dull red. The Sun, at 5772 K, peaks in the visible.
Human skin is at about 34 °C. At what wavelength does it emit most strongly, and what kind of detector is needed?
The ultraviolet catastrophe
Classical physics treated the radiation inside the hot cavity as standing electromagnetic waves. Two well-tested classical ideas then gave a prediction (the Rayleigh–Jeans law, 1900–1905):
- The number of possible standing waves in the cavity increases enormously at shorter wavelengths, because more short waves fit.
- Energy is shared equally, so each standing wave has the same average energy, $kT$, whatever its wavelength.
More waves at short wavelength, each with the same energy, means intensity should rise without limit as wavelength decreases. Integrated over all wavelengths, the total energy would be infinite. The prediction matched experiment at long wavelengths but failed completely at short ones, where real curves fall to zero. This failure was later called the ultraviolet catastrophe (Paul Ehrenfest's phrase, 1911).
Planck's quantum hypothesis
In December 1900, Max Planck found a formula that fitted the measured curves exactly. To derive it he had to assume that the oscillating charges in the walls of the cavity could only have energies that were whole-number multiples of a basic amount:
$$E = nhf, \qquad n = 1, 2, 3, \dots$$
so energy is exchanged with the radiation in packets, or quanta, of size
$$E = hf$$
where $f$ is the frequency and $h$ is Planck's constant, $6.626 \times 10^{-34}$ J s. (Since 2019, $h$ has been fixed at exactly $6.626\,070\,15 \times 10^{-34}$ J s, and the kilogram is defined from it.)
Why quantisation fixes the curve. At long wavelengths (low $f$), one quantum $hf$ is much smaller than the typical thermal energy $kT$, so energy can be shared out smoothly and the classical answer is fine. At short wavelengths (high $f$), one quantum is much larger than $kT$. The thermal jostling rarely has enough energy to supply even one quantum, so those high-frequency waves are almost never excited. Their average energy drops towards zero, and the curve turns over instead of rising forever.
Planck saw this as a mathematical trick and spent years trying to avoid it. In 1905 Einstein took it seriously, proposing that light itself is made of quanta (7I). Planck received the Nobel Prize in 1918.
Calculate the energy of a quantum of (a) green light of wavelength 530 nm and (b) FM radio at 100 MHz. Comment on the difference.
A 5.0 mW laser pointer emits light of wavelength 650 nm. How many quanta does it emit each second?
- Start with the Sun. Check $\lambda_{max}$ = 502 nm with Wien's law, and the photon energy at the peak with $E = hc/\lambda$.
- Compare the dashed classical curve with Planck's curve. Where do they agree, and where do they disagree?
- Read "Classical ÷ Planck at 300 nm". By what factor does classical physics overestimate the ultraviolet? Try the toaster. Why is the factor so much larger?
- Look at the energy bars. In which band does each wave get its full classical share of $kT$? Which bands are "frozen out", and why?
- Use the slider to find the temperature at which the peak is at 1.00 µm. Check with $T = b/\lambda_{max}$.
Test yourself
A star's surface temperature doubles. Its peak wavelength
$\lambda_{max} \propto 1/T$.
The "ultraviolet catastrophe" refers to
Equal energy $kT$ for ever more short-wavelength modes gave infinite energy.
Planck's key assumption was that
Planck quantised the energy of the emitters. Light quanta came from Einstein.
The energy of a quantum of ultraviolet light of frequency $1.5 \times 10^{15}$ Hz is
$E = hf = 6.626 \times 10^{-34} \times 1.5 \times 10^{15} = 9.9 \times 10^{-19}$ J.
Label the black-body graph
Order these objects by the wavelength at which they emit most strongly, shortest first.
A black body all radiation that falls on it. As temperature rises, the peak wavelength becomes . Classical physics predicted unlimited energy in the . In 1900 proposed that energy is exchanged in .
Explain how the study of black-body radiation led to the idea that energy is quantised. Refer to experimental observations, the classical prediction and Planck's hypothesis.
Experiments in the 1890s showed that a black body emits a continuous spectrum with a peak that shifts to shorter wavelengths as temperature increases ($\lambda_{max} = b/T$), and that intensity falls to zero at both very long and very short wavelengths.
Classical physics treated the radiation as standing waves in a cavity. More short-wavelength waves fit in the cavity, and each was expected to carry the same average energy $kT$. This Rayleigh–Jeans prediction agreed at long wavelengths but predicted intensity increasing without limit at short wavelengths, and an infinite total energy: the ultraviolet catastrophe. Classical physics clearly could not explain the observations.
In 1900 Planck derived a formula that matched the curves exactly by assuming that the oscillators emitting the radiation can only have energies in whole-number multiples of $hf$, so energy is exchanged in quanta $E = hf$. At high frequencies a single quantum is much larger than the available thermal energy $kT$, so those waves are rarely excited and the intensity falls. Because only quantisation could explain the measured spectrum, black-body radiation provided the first evidence that energy is quantised.