The Ultraviolet Catastrophe

THE ULTRAVIOLET CATASTROPHE

By Michelle Ng

Image source: https://jacksonjournal.news/amazing-star-wallpaper-hd/

THERMAL RADIATION

Light is often said to be a continuous electromagnetic wave. It is the oscillation of the electric and magnetic fields and can propagate through a vacuum. EM waves can have a continuous range of wavelengths and frequencies, forming the electromagnetic spectrum, ranging from long-wavelength, low-frequency radio waves to short-wavelength, high-frequency gamma rays. All EM waves propagate through a vacuum at a fixed speed c, where c = fλ.

Electric fields are regions around charges where another charge can experience a force, so when a charge accelerates, the electric field around it oscillates, creating an electromagnetic wave. [1] Since electrons are charged particles inside atoms, when electrons vibrate, EM waves are created and the atoms emit light.

You have probably heard that hot objects emit light, but actually, any body with a temperature above 0 Kelvin will emit EM waves. This is because all objects with T > 0 have internal energy, defined as the sum of kinetic and potential energies of the atoms inside a system. Atoms, and therefore electrons, in an object above 0K will always have some kinetic energy and so they will vibrate, emitting light. This emission of EM waves is known as thermal radiation, and is one of the ways in which energy is transferred via heating (alongside conduction and convection).

According to classical theory, because the electrons can have any kinetic energy, they can vibrate at any frequency, so the light emitted by electrons can have any frequency/wavelength.

In hot objects with a high internal energy, the atoms and electrons have more kinetic energy and vibrate more quickly and with greater amplitude, so higher intensity EM waves are emitted. This is quantified by the Stefan-Boltzmann relationship [2] (L = σAT4), where the luminosity, or total power output, of an object, is directly proportional to its surface area and the fourth power of its surface temperature (and temperature is directly proportional to the average kinetic energy of the molecules.) This relationship was established experimentally.

BLACKBODIES

A blackbody is a perfect absorber or emitter of EM radiation: it is a system that can absorb all EM radiation that is incident on it and can re-emit this energy exactly as well as it absorbed the energy. [3] The radiation it emits only depends on the temperature of the blackbody and nothing else. [4]

THE BLACKBODY CURVE

To study the radiation of blackbodies, scientists passed the light emitted by a hot blackbody through a diffraction grating to split the light into its constituent wavelengths. The principle behind this is that when light passes through the grating (a series of slits), each slit in the grating will act as a new source of light. The ‘new’ waves of light emanating from the slit will constructively and destructively interfere at different points on a screen, producing an interference pattern with light fringes (maxima) and dark fringes (minima). In the blackbody’s radiation, the different wavelengths of light will produce maxima and minima at different points on the screen (because the conditions for maxima and minima depend on wavelength), so the different wavelengths will split up on the screen, producing a spectrum. Scientists measured the intensity of each wavelength (which is proportional to the energy transferred for each wavelength.)

Through these experiments, physicists found that hot objects can radiate a range of wavelengths of light, but every wavelength is emitted at a different intensity—different wavelengths get different amounts of energy. Experiments also showed that at low frequencies and high frequencies, the intensity is very low. As frequency increases, intensity increases until the peak wavelength / frequency is reached before decreasing again. This curve is shown in Figure 1. This peak wavelength of emitted light is the wavelength of light that is emitted at maximum intensity.

As temperature increases, the EM waves emit light of higher frequencies overall and the peak wavelength of light is shifted to the shorter end of the spectrum. This relationship is described by Wien’s Law (λmaxT = 2.9x10-3 mK), which states that the temperature of an object is inversely proportional to the peak wavelength of light emitted (or directly proportional to peak frequency) [2]. For most objects on Earth, EM waves are emitted mainly in the infrared region of the spectrum, which explains why we cannot see thermal radiation of everyday objects; hotter objects are able to emit red or yellow light in the visible region, giving rise to the notion that hot objects glow. Even hotter objects can appear white or blue and that is because blue is at an even lower wavelength in the spectrum.




Figure 1 [3]: As temperature increases, the peak wavelength is shifted to the left. The intensity is also higher for higher temperatures. But for all temperatures, intensity is low for low frequencies and high frequencies.

THE RAYLEIGH-JEANS LAW AND THE ULTRAVIOLET CATASTROPHE

Classical theory had tried to describe the radiation of a blackbody, but classical physics contradicted the experimental findings described above. In more simple terms, classical physics predicts that if electrons can vibrate with any frequency and the EM waves can have any frequency, there should not be decreasing intensity as we move to higher frequencies (longer wavelengths): intensity should keep increasing beyond the peak frequency. [5] Examining this in more detail, we can consider the model of cavity modes.

A blackbody can be modelled by imagining a black box or ‘cavity’ with a small hole (figure 2), something that Kirchhoff thought of in 1859 [4]. Light is shone into the small hole and enters the cavity; because the hole is so small compared to the cavity itself, the chance of it escaping is very small, so the light will be repeatedly reflected off the inner walls until it is eventually absorbed, making the cavity a perfect absorber of radiation as it can be assumed that all light is absorbed.




Figure 2: Modelling a blackbody using a cavity [4]

The energy absorbed by the cavity can be modelled to be a series of standing waves inside the cavity; this energy can then be re-emitted. Inside the cavity, the waves are restricted by the condition that the electric field at the wall must be 0: the displacement of the EM wave oscillations must be 0 (at equilibrium) at the wall. For a given wavelength, there are only a limited number of ways in which this condition is satisfied, but intuitively you can see that as wavelength decreases and frequency increases, the number of ways to fulfil this restriction increases, as illustrated by figure 3. [3] These ‘ways’ are known as the possible modes of radiation in the cavity.


Figure 3 [3]

Using the cavity model, Lord Rayleigh tried to find an equation for how intensity changes with frequency. He calculated the number of modes that are possible as a function of frequency and found the relationship to be that the number of modes is proportional to frequency squared. [3] He also thought that the intensity of light for that frequency would be proportional to the number of possible modes for that frequency, because all modes have an equal chance of being produced. Therefore, intensity should be proportional to frequency squared, so the blackbody curve should have curved upwards, with intensity tending towards infinity as frequency increases. Evidently, this relationship did not agree with the actual experiments (Figure 4): the Rayleigh-Jeans law predicts that as frequency increases/wavelength decreases, intensity tends to infinity; the results show that as frequency increases past the peak frequency, intensity decreases and asymptotes to 0 again. This was known as the Ultraviolet Catastrophe, because at low frequencies the Rayleigh-Jeans law agrees with the data collected, but as you move to the UV end of the spectrum, the theory entirely breaks down. And since the area under the curve is related to energy, in classical theory, blackbody radiation appears to have infinite energy, which is impossible!



Figure 4 [4]

PLANCK’S SOLUTION

Since classical theory could not resolve these contradictions, Planck tried to explain this by using a new idea: that energy exists in discrete packets, or quanta, of energy, where the energy of a single quanta is related to its frequency—E = hf, where h is Planck’s constant. [5]

In classical theory, electrons can vibrate at any frequency, and there is an infinite number of energy values that can allow the electron to vibrate at that frequency. But Planck’s theory suggests that electrons that are vibrating at a given frequency f cannot actually have any value of energy: they can only have energy equal to integer multiples of hf. [6] And in order to be able to vibrate with that frequency f in the first place, the electron must have at least 1 quantum (1 hf) of energy since you cannot have half a quanta or a third of a quanta. As frequency increases, the energy of one quanta increases, so a smaller proportion of electrons in an object have an energy greater than at least one quanta, and thus a smaller proportion of electrons can actually vibrate with those frequencies and emit quanta of light at those frequencies.

Therefore, at higher frequencies, the chance that an electron has enough energy to emit light of that frequency actually decreases, so the intensity of that frequency decreases. At lower frequencies, although many electrons have enough energy to emit light at that frequency, the energy of one quantum is very small, so the total energy emitted is also very small, giving a small intensity. Planck’s equations successfully match up with the experimental observations.

LIGHT AS A PARTICLE

At the time, Planck did not really think of light as a particle, and it was only later when Einstein explained the photoelectric effect using Planck’s ideas and introduced the particle model of light. [5] Nonetheless the Ultraviolet Catastrophe led to the beginnings of quantum physics. In the case of the blackbody radiation, light is behaving more like a particle than a wave because only an integer number of quanta of energy of light can be emitted or absorbed, analogous to how you cannot really have half a particle, whereas a wave can propagate continuously.

This ‘particle’ of light, or a ‘packet’ of electromagnetic energy, is called a photon. The discovery of the particle nature of light led to the idea w wan of wave-particle duality and the principle of complementarity: the wave and particle ideas of light actually coexist, and both theories are needed to fully describe the behaviour of light. For example, the wave model of light can explain phenomena such as diffraction, but when considering the absorption and emission of light by objects, it is more useful to use the particle model of light. Later on, scientists also found, through experiments that showed that electrons can diffract, that electrons can also behave as waves. This was the beginning of quantum theory and the idea that anything can behave as a particle or a wave depending on the situation.

In fact, even we are, to some extent, ‘wave-like’. But the reason why we don’t observe this is because based on the De Broglie equation, λ = h/p (p = momentum, λ = wavelength), our wavelengths are extremely small (due to a large momentum). Diffraction effects caused by the ‘wave’ behaviour of our bodies are unnoticeable—diffraction only occurs significantly if the size of the aperture that the wave is passing through is comparable to the size of the wavelength. In comparison to the gap in a doorway, for example, our wavelengths are too tiny to have any effect.

BIBLIOGRAPHY

[1] Hu L., Delichatsios M. Thermal Radiation. Springer, Cham, 2019.

[2] "Radiation – The Physics Hypertextbook". The Physics Hypertextbook, 2021, https://physics.info/radiation/.

[3] "Blackbody Radiation". Hyperphysics.Phy-Astr.Gsu.Edu, 2021, http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html.

[4] "1.1: Blackbody Radiation Cannot Be Explained Classically". Chemistry Libretexts, 2021, https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/01%3A_The_Dawn_of_the_Quantum_Theory/1.01%3A_Blackbody_Radiation_Cannot_Be_Explained_Classically.

[5] Gribbin, John. In Search Of Schrodinger's Cat. Random House Publishing Group, 2011.

[6] "Three Failures Of Classical Physics". Physics.Weber.Edu, 2021, https://physics.weber.edu/carroll/honors/failure.htm.