Ultraviolet Catastrophe | Definition | Solution | 2 Important Laws


Contents : Ultraviolet Catastrophe

What is an ultraviolet catastrophe?

Ultraviolet catastrophe, also known as Rayleigh-Jeans catastrophe refers to the deviation from the statistical derivation of the Rayleigh-Jeans law at short wavelengths. According to Rayleigh-Jeans law, a blackbody at thermal equilibrium would radiate in all frequency range and would emit more energy as the wavelength decreases. In other words, it states that as frequency increases the blackbody starts to radiate an arbitrarily large amount of energy. However, this pattern is not seen physically. The error between the predicted energy radiation amount and the obtained energy radiation amount is much more pronounced at shorter wavelengths. Hence, it is termed the ultraviolet catastrophe.

What is ultraviolet light?

Ultraviolet light is electromagnetic radiation having a freqn in-between 8 × 1014 and 3 × 1016 Hz range and wavelength in-between 0.4 x 10-6 – 10-8 meter, so ultra violet light is not falling in the visible range of human eye-sights. The ultraviolet catastrophe is prominent in these wavelengths. Ultra Violet rays are extensively utilized in nullifying macrobacteria, sterilizing medical equipment’s, though overexposure to Ultra violet ray might not be good to human beings and could cause various skin infections. Ultraviolet (UV) sensors or detectors are used for sensing the UV radiation emitted during the time of ignition. UV flame sensors are capable of detecting fires and explosions within a timeframe of 3–4 milliseconds.

What is a blackbody?

In 1860, Gustav Kirchhoff gave the first idea of a blackbody. He stated that

..the supposition that bodies can be imagined which, for infinitely small thicknesses, completely absorb all incident rays, and neither reflect nor transmit any. I shall call such bodies perfectly black, or, more briefly, black bodies

“On the relation between the radiating and absorbing powers of different bodies for light and heat”

A blackbody is a material that is capable of absorbing and emitting light of every wavelength or frequency i.e. (e = a = 1). In nature, 100% blackbody cannot be found. A material known as carbon black is the closest to an actual blackbody on Earth. The sun is one of the main black bodies in the universe and emits light of all wavelengths. When a blackbody is in thermal equilibrium it emits the blackbody radiation. Blackbody radiation refers to the radiation emitted by a blackbody in every possible wavelength of light. It is also known as cavity radiation.

Statement of Rayleigh-jeans law

The British physicist Lord Rayleigh and Sir James Jeans has measured the spectral emission of a blackbody in the foundation of classical physics and multiple empirical factors, which could be stated as.

“A black body at thermal equilibrium will emits radiation in all freq. ranges and as the freq. increase the energy of emission of radiation increase”.

~Lord Rayleigh and Sir James Jeans

The mathematical equation of Rayleigh-jeans law is

Here, u(\nu ) is radiant energy density, \nu is frequency, T is the temperature in kelvins, c is the speed of light, K is Boltzmann constant. (ultraviolet catatrophe)

According to the expression shown above, radiant energy density is directly proportional to frequency, i.e. with increase in frequency, radiant energy should also increase diverging towards infinity as wavelength tends to zero. This law demonstrated a major error in the classical theory of physics.

Problem with the Rayleigh-Jeans Law

All the harmonic oscillator modes or degrees of freedom of a blackbody system at equilibrium should have average energy equal to KT as stated by the equipartition theorem of classical statistical mechanics. According to Rayleigh-Jeans law, radiant energy diverges towards infinity as wavelength tends to zero. This means radiated power is unlimited at a certain high-frequency range. This clearly violates the laws of physics that state that an object can never possess an infinite amount of power or energy, as proved by Albert Einstein.

Moreover, when measured physically the energy values obtained were much different from the predicted values. The error between the predicted energy radiation amount and the obtained energy radiation amount is much more pronounced at shorter wavelengths leading to the ultraviolet catastrophe.

The black curve denotes the values predicted by the Rayleigh-Jeans law, here, The blue, green, and red curves denote the values measured by Planck’s law at different wavelengths. Image source: Darth KuleBlack body, marked as public domain, more details on Wikimedia Commons (ultraviolet catastrophe.

This was a huge drawback of the Rayleigh-Jeans law. This issue of ultraviolet catastrophe was later solved by Max Planck and Albert Einstein by forming Planck’s law and Einstein’s equation respectively.

Statement of Planck’s Law

In the 1900s, Max Planck worked extensively on electromagnetic radiations and formulated one of the most important and controversial laws of the century. According to him, radiation energy came in tiny discrete packets called quanta that is proportional to the frequency of the radiation. His statement was:

The energy of electromagnetic radiation is confined to indivisible packets ( known as ‘quanta’), one of each packet has energies same as the product of the Planck constant and the frequency of the radiation.

~ Max Planck

The mathematical equation of Planck’s law of intensity spectral distribution function is

h is Planck’s constant, \nu is frequency, c is the speed of light, λ is the wavelength.

This lead to the formulation of the correct form of the spectral distribution functions (as shown below formulations) estimated by popular scientist Einstein (in the year 1905) and  Satyendra Nath Bose (in the year 1924). This distribution factor does not entirely depend in proportionate with frequency. The inverse proportionality with the exponential factor contributes to limiting the energy values obtained at shorter wavelengths or longer frequencies. This equation could actually predict the experimentally obtained values of radiation energy at a given wavelength eliminating the ultraviolet catastrophe.

Here, u(\nu ) is radiant energy density, \nu is frequency, T is the temperature in kelvins, c is the speed of light, K is Boltzmann constant, h is Planck’s constant.

Planck’s law also led to the formulation of the theory behind the photoelectric effect by Einstein. An electron present in a lower energy state tends to absorb external energy in the form of light (photons) to reach a higher energy state and occur only when the energy present in the photon is identical to the differences in energies in-between the two levels.

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About Sanchari Chakraborty

I am an eager learner, currently invested in the field of Applied Optics and Photonics. I am also an active member of SPIE (International society for optics and photonics) and OSI(Optical Society of India). My articles are aimed towards bringing quality science research topics to light in a simple yet informative way. Science has been evolving since time immemorial. So, I try my bit to tap into the evolution and present it to the readers.

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