Black Body radiation in a cavity.

How Black Body Problem Led to Birth of Revolutionary Quantum Physics


Introduction

A Black Body is an ideal physical body that absorbs all the radiation falling on it. The term Black Body was introduced by the German physicist Gustav Kirchhoff in 1860. The thermal radiation emitted by heated solid objects is found to be of a continuous distribution of frequencies.

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An approximate realization of a black body as a tiny hole in an insulated enclosure. Radiation incident on the hole will pass into the cavity, and is very unlikely to be re-emitted if the cavity is large.
An approximate realization of a black body as a tiny hole in an insulated enclosure. Radiation incident on the hole will pass into the cavity, and is very unlikely to be re-emitted if the cavity is large. MikeRunCC BY-SA 4.0Source

Black Body Spectrum

The radiation emitted by heated gases has a discrete distribution spectrum of frequencies consisting of few colored lines with no light in between.

A body with a temperature higher than its surroundings will have its rate of emission greater than its rate of absorption until a thermal equilibrium is reached between the body and its surroundings, at which the rates of absorption and emission are equal.

Explaining the continuous energy distribution of solid objects was one of the greatest unsolved problems of Physics in the 19th Century.


Black Body Problem

The experimental data obtained showed that in the black body, after obtaining thermal equilibrium, the radiation emitted has a well-defined, continuous energy distribution.

To each frequency, there corresponded an energy density that depended only on the temperature of the cavity walls, not on the chemical composition of the object or its shape.

There was a maximum energy density for a given frequency, and this increased with temperature.

Black body curves of Planck for various temperatures and comparison with the classical theory of Rayleigh-Jeans.
Black body curves of Planck for various temperatures and comparison with the classical theory of Rayleigh-Jeans.

What were the attempts made to explain the Black Body Radiation?

What does Stefan-Boltzmann Law for the black body stand for?

The temperature of the Sun was estimated near to the accurate value for the first time with the Stefan-Boltzmann law for Black Body radiation.
The temperature of the Sun was estimated near to the accurate value for the first time with the Stefan-Boltzmann law.

“The total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body’s temperature”.

Stephen-boltzmann law

Where σ=5.67×10-8Wm-2K-4 is the Stefan-Boltzmann Constant, and a≤1, for black body a=1.

It was an empirical law based on experimental observations. The temperature of the Sun was estimated approximately for the first time by using this law.


What is Wein’s Displacement Law? Why is it used for black body radiation?

The color of a star is determined by its temperature, according to Wien’s law. The difference is clearly visible in the Orion constellation. The stars Betelgeuse (T ≈ 3300 K, upper left), Rigel (T = 12100 K, bottom right), Bellatrix (T = 22000 K, upper right), and Mintaka (T = 31800 K, rightmost of the 3 “belt stars” in the middle). Credits – Mouser • CC-BY-SA-3.0Source

“Black Body energy distribution for different temperatures will peak at different wavelengths that are inversely proportional to the temperature”.

Wein’s Displacement Law

b is the Wein’s Displacement Constant whose value is b=2.898×10-3mK. We can find the effective temperature of the Sun which happens to be 5778 K.


Wein’s Energy Density Distribution for a Black Body

Computed black body spectrum vs. frequency υ for T = 5800K according to Planck’s law. Wien’s and Rayleigh-Jeans approximations are shown additionally as dashed lines. Credits – Geek3 • CC BY-SA 4.0Source

It was derived by the German physicist Wilhelm Wein in 1896.

Wein assumed that the radiation inside a hollow enclosure is produced by a resonator of molecular dimensions.

He also assumed that the emitted radiation is proportional to the kinetic energy of the resonators and that the intensity of radiation of any particular wavelength is proportional to the number of resonators.

The energy density per unit frequency according to Wein’s distribution is

where U(ν, T) is the amount of energy per unit surface area per unit time per unit solid angle per unit frequency emitted at a frequency ν, c is the speed of light, T is the temperature of the Black Body, h is the Planck Constant, k is the Boltzmann Constant.

In terms of wavelength λ, the energy distribution can be written as

Wein’s distribution fits well with the Black Body radiation distribution only at high frequencies, at low frequencies it fails badly.


What is Rayleigh-Jean’s Law? Does it explain Black Body radiation?

John William Strutt (Lord Rayleigh).
Credits – Unknown • Public Domain • Source

British physicist Lord Rayleigh assumed that the radiation inside the cavity consists of standing waves with nodes at the metallic surfaces using the classical electromagnetic theory.

The standing waves are considered to be harmonic oscillators as these result from the harmonic oscillations of a large number of electrical charges that are present in the walls of the cavity.

A geometrical argument is made to find out the number of standing waves in the frequency interval and how this number depends on the frequency.

When the system is in thermal equilibrium, the classical kinetic theory is used to calculate the average total energy of the waves, which depends only on the temperature T. Using this, the energy density of the black body can be found.

The number of modes of the radiation in the frequency interval ν to ν+dν is

The electromagnetic energy density in the frequency range ν to ν+dν is given by

The Rayleigh-Jeans Law explains the black body spectrum at large wavelengths but doesn’t hold well at small wavelengths.

The integration of the energy density equation over all frequencies diverges, which implies that the cavity has an infinite amount of energy. This result is famously called the Ultraviolet catastrophe. This was one of the major failures of Classical physics, as this was based on the assumption of the energy and matter being continuous, but the energy is only allowed in discrete packets in the quantum theory which later perfectly explained the black body radiation.


What is Planck’s Law? Why is it so important? Did it perfectly explain Black Body radiation?

Max Planck.
Credits – Unknown • Public Domain • Source

German physicist Max Planck found that the classical theory gave wrong results with the experimental data for the black body radiation, considered violating the law of equipartition of energy.

Planck first produced an empirical law as an improvement to Wein’s energy distribution of the black body, which fitted extremely well with the experimental data, and then tried to find a theoretical explanation for the law.

Planck considered a cavity that consisted of resonant oscillatory bodies, with perfectly reflective walls.

Boltzmann, in his law of equipartition of energy, had assumed the energy was continuous and tends to zero.

Planck assumed that in the several oscillators of each of the finitely many characteristic frequencies, the total energy was distributed to each in an integer multiple of a definite physical unit of energy, which is a characteristic of the respective characteristic frequency.

So, the smallest amount of energy possible for a particular frequency ν is E=hν, where h=6.626×10-34Js is the fundamental constant of nature called the Planck constant.

The energy density per unit frequency of radiation is given by

In terms of wavelength


References

  1. Eisberg, R. M. (2006). QUANTUM PHYSICS: OF ATOMS, MOLECULES, SOLIDS, NUCLEI AND PARTICLES. Wiley.
  2. Feynman, R. P., Leighton, R. B., & Sands, M. (1989). The Feynman Lectures on Physics: Commemorative Issue Vol 1 (Commemorative Issue ed.). Addison Wesley.
  3. Planck, M. (1900a). Entropie und Temperatur strahlender Wärme. Annalen Der Physik, 306(4), 719–737. https://doi.org/10.1002/andp.19003060410
  4. Planck, M. (1900b). Über eine Verbesserung der Wien’schen Spectralgleichung. Verhandlungen Der Deutschen Physikalischen Gesellschaft, 2, 202–204. https://archive.org/stream/verhandlungende01goog#page/n212/mode/2up
  5. Planck, M. (1900c). Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum. Verhandlungen Der Deutschen Physikalischen Gesellschaft, 2, 237–245. https://archive.org/stream/verhandlungende01goog#page/n247/mode/2up
  6. Planck, M. (1991). The Theory of Heat Radiation. Dover Publications.
  7. Stefan, J. (1879). Über die Beziehung zwischen der Wärmestrahlung und der Temperatur. Sitzungsberichte Der Mathematisch-Naturwissenschaftlichen Classe Der Kaiserlichen Akademie Der Wissenschaften, 79, 391–428. http://www.ing-buero-ebel.de/strahlung/Original/Stefan1879.pdf
  8. Wien, W. (1896). Ueber die Energievertheilung im Emissionsspectrum eines schwarzen Körpers. Annalen Der Physik Und Chemie, 294(8), 662–669. https://doi.org/10.1002/andp.18962940803
  9. Zettili, N. (2009). Quantum Mechanics. Wiley.

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