Radiation, the electron and X-rays
Classical physics predicted that a hot body would radiate infinite energy in the ultraviolet. To avoid that absurdity, Planck assumed that energy came in packets, and he himself considered the idea a mathematical trick.
01The ultraviolet catastrophe
Every hot body radiates. Nineteenth-century physics could calculate how: the radiation inside a cavity was treated as a set of modes, each one was assigned the average energy dictated by classical thermodynamics, and the results were summed. The outcome is the Rayleigh-Jeans law:
It was not an arithmetic mistake: it was the best physics available producing nonsense. Ehrenfest named it the ultraviolet catastrophe, and it was the first clear sign that something was wrong at the foundations.
In 1900 Planck found the formula that fit the data, and then looked for where it could come from. The only way he found was to suppose that the oscillators in the cavity could not have just any energy, but only multiples of a minimum packet proportional to the frequency:
The Planck curve and the classical prediction on the same graph. See where they coincide and where the classical one shoots up. The objects in the selector are real, with their temperatures.
He called it “an act of desperation” and spent years trying to derive his formula without the hypothesis of packets. He considered quantization a mathematical device to make the numbers work out, not a property of nature. It was Einstein, five years later, who took the idea seriously and applied it to light itself.
02The electron
In 1897 Thomson studied cathode rays by deflecting them with crossed electric and magnetic fields. By measuring the deflection he obtained the charge-to-mass ratio, and found something remarkable: it always came out the same, regardless of which metal the cathode was made of or what gas was in the tube.
That \( e/m \) was universal meant that this particle was a constituent of all matter, not a property of the metal: the first hint that the atom had an internal structure. In 1909 Millikan measured the charge separately with his oil-drop experiment, and with the two results the mass of the electron was determined: \( 9.109\times10^{-31} \) kg, about 1836 times smaller than that of the proton.
03X-rays
Röntgen discovered them in 1895 by accident: a fluorescent screen glowed several meters from a cathode-ray tube covered with black cardboard. In an X-ray tube the electrons are accelerated through tens of kilovolts and stopped abruptly against a metal target, and that braking produces two superimposed spectra.
| Component | Origin | What defines it |
|---|---|---|
| Continuous or bremsstrahlung | The braking of the electrons | The accelerating voltage |
| Characteristic | Inner electrons of the target that are knocked out and replaced | The target material |
The characteristic lines are the fingerprint of the element, and from them comes X-ray fluorescence, which identifies materials without destroying them. In electronics it is used to verify RoHS compliance —detecting lead in solder— and X-ray inspection is the only method for seeing the solder joints of a BGA package, which are hidden underneath the chip.
04In the lab
Power an incandescent lamp at different voltages, measuring the filament resistance to estimate its temperature. Observe the color shift and verify the trend predicted by the displacement law.
With a radiation sensor, measure the power emitted by a hot body at several temperatures. Plot it on a logarithmic scale against \( T \) and verify that the slope is four.
With a fine-beam tube and Helmholtz coils, measure the radius of the electron orbit as a function of the field and of the accelerating voltage. Solve for \( e/m \) and compare it with the tabulated value.
05Common mistakes
- Believing that Planck proposed that light is quantized. He quantized the oscillators in the wall; light was quantized by Einstein.
- Using Rayleigh-Jeans outside the far infrared. There it agrees with Planck; beyond that it diverges.
- Confusing the peak with the perceived color. The eye integrates the whole spectrum, not just the maximum.
- Forgetting emissivity. The laws hold for an ideal blackbody; a real body emits less.
- Confusing the continuous spectrum with the characteristic one. One depends on the voltage; the other, on the material.
- Thinking that X-rays are a different kind of particle. They are light, with a wavelength a thousand times shorter than visible light.
06Self-assessment
What is the ultraviolet catastrophe?
The classical prediction that the radiation from a hot body grows without limit as the wavelength gets shorter, with infinite total energy.
What did Planck assume to avoid it?
That the energy of the oscillators could only take multiples of \( h\nu \).
Where is the emission peak of the human body?
Near 9.4 µm, in the far infrared. That is why thermal cameras work between 8 and 14 µm.
What did Thomson measure?
The charge-to-mass ratio of the electron, and he found that it was the same with any cathode and any gas.
What does the Duane-Hunt limit depend on?
Only on the accelerating voltage: \( \lambda_{min} = 1240/V \) in nanometers with \( V \) in volts.
What are X-rays used for in electronics?
To inspect hidden solder joints of BGA packages and, through fluorescence, to verify RoHS compliance.
07Further reading
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica: átomos, moléculas, sólidos, núcleos y partículas”, Limusa). Chapter 1 develops thermal radiation and Planck's hypothesis with all the historical detail.
- Kenneth Krane. Modern Physics. 3rd ed., Wiley, 2019 (Spanish edition, “Física moderna”). A good treatment of the discovery of the electron, Millikan and the production of X-rays.
- Thomas Kuhn. Black-Body Theory and the Quantum Discontinuity, 1894-1912. University of Chicago Press, 1987. What Planck really thought when he introduced the quantum. A corrective to the textbook version.