Rayleigh-Jeans Law & The Ultraviolet Catastrophe

Illustration of vibrations of a drum.
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Lord Rayleigh made an attempt to explain the energy distribution in black body radiation, which was completed by Jeans in 1900. The results obtained by then are known as Rayleigh-Jeans’ Rules on Black Body Radiation. The law covering these rules is called Rayleigh Jean’s Law.

The black body emits radiation of continuously variable wavelengths right from zero to infinite. This radiation can be imagined as broken up into monochromatic waves. These monochromatic waves originate as a result of a different mode of vibration of the medium, which at that time was supposed to be an electromagnetic sensitive medium called ‘ETHER’.

According to well-known result of statistical mechanics, the number of such modes of vibration lying between the wavelength range $ \lambda $ and $ \lambda+d\lambda $ is equal to $ 8\pi{\lambda}^{-4}d\lambda $ per unit volume. And also according to the theorem of equipartition of energy, the total energy of a system for each mode of vibration (or degree of freedom) is equal to $ kT $ , where $ k $ is the Boltzmann constant and $ T $ is the temperature of the system in Kelvin. Hence, the total energy of the radiation lying between the wavelength range $ \lambda$ and $ \lambda + d\lambda $ per unit volume is
$ u_{\lambda}d\lambda$ =number of mode of vibration $ \times kT $
or, $ u_{\lambda}d\lambda=8\pi kT{\lambda}^{-4}d\lambda $

This is Rayleigh Jean Law.

About Rayleigh-Jean’s Law

Lord Rayleigh, a British physicist, derived the λ-4 dependence of the Rayleigh-Jeans law based on empirical facts and classical physical arguments. In 1905, Rayleigh and Sir James Jeans put forth a more complete derivation that included the proportionality constant. The Rayleigh-Jeans law notably exposed a crucial flaw in the theory of physics of that time; it predicted an energy output that diverges towards infinity as wavelength approaches zero and frequency tends to infinity.

Scientists later measured the spectral emission of actual black bodies and discovered that the emission was in line with the Rayleigh-Jeans Law at low frequencies (large wavelengths) below 105 GHz. However, it diverged at higher frequencies (shorter wavelengths) and reached a maximum before falling with frequency. Thus, in the end, the total energy emitted is finite. This inconsistency between observations and predictions of classical physics is often referred to as the “ultraviolet catastrophe”, which I’ve discussed in detail down below.

The Ultraviolet Catastrophe

The Rayleigh-Jeans law led to the “ultraviolet catastrophe” because it predicted that black bodies at thermal equilibrium would keep getting brighter and brighter at higher frequencies of radiation, and that the total power radiated per unit area (corresponding to ultraviolet light) of the black body would be infinite. The term “ultraviolet catastrophe” was first used by Paul Ehrenfest in 1911.

As we’ve discussed before, the Rayleigh-Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical physical arguments.

For a given wavelength λ , it is:

Bλ(T) = 2ckbT/ λ4

where Bλ is the spectral radiance (the power emitted per unit emitting area, per steradian, per unit wavelength), c is the speed of light, kb is the Boltzmann constant, and T is the temperature in Kelvin.

For a given frequency ν, the above expression would become:

Bν(T) = 2ν2kbT/c2

We get this formula from the equipartition theorem of classical statistical mechanics which states that all harmonic oscillator modes (that is, degrees of freedom) of a system at equilibrium possess an average energy of kbT.

In other words, the ultraviolet catastrophe is essentially a way of stating that the above formula goes haywire at higher frequencies, or Bν(T) → ∞ when ν→∞.

The problem of the ultraviolet catastrophe

Let us consider an example provided in Mason’s A History of the Sciences that effectively describes the phenomenon of multi-mode vibration using a simple piece of string. Being a natural vibrator, the string will oscillate with particular modes (that is, standing waves generated in a string during harmonic resonance) that are dependent on the length of the string.

When it comes to classical physics, an object radiating energy will act as a natural vibrator. Since every mode possesses the same energy, the bulk of the energy in a natural vibrator is present in higher frequencies and smaller wavelengths (which have the highest number of modes).

Speaking of classical electromagnetism, the number of electromagnetic modes in a three-dimensional cavity, per unit frequency, is directly proportional to the square of the given frequency. Thus, we may construe in turn that the radiated power per unit frequency is proportional to the frequency squared.

Thus, you can see that both the power at a certain frequency and the total radiated power are unlimited as we approach higher frequencies. However, the total radiated power of a cavity is not found to be infinite upon observation, implying that this conclusion not in accordance to physics. Both Einstein and Lord Rayleigh and Sir James Jeans independently made this point in 1905.

What is the solution to the ultraviolet catastrophe?

Earlier in 1900, Max Planck made some rather unconventional (for the time) assumptions to derive the correct form for the intensity spectral distribution function. Notably, he assumed that electromagnetic radiation can be absorbed or emitted only in discrete packets of energy known as “quanta”.

Equanta = hν = hc/λ

where h is Planck’s constant, c is the speed of light, λ is the wavelength of light, and ν is the frequency of light. With the help of Planck’s assumptions, scientists derived the correct form of the spectral distribution functions:

Bλ(λ,T) = 2hc25 × 1/ehc/(λkbT) -1

In 1905 and 1924 respectively, Albert Einstein and Satyendra Nath Bose managed to solve the problem of the ultraviolet catastrophe by asserting that Planck’s quanta were actual physical particles (known today as photons) and not just imaginary mathematical entities. They subsequently modified statistical mechanics in the style of Boltzmann to a group of photons.

A photon, as described by Einstein, possessed energy directly proportional to its frequency. Moreover, he also described Stokes’ unpublished law and the world-famous phenomenon of photoelectric effect. The Nobel Prize in Physics committee particularly cited this published postulate in their decision to honor Einstein with the prize in 1921.

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