Planck's law

"For a general introduction, see black body."

In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at temperature $T$. As a function of frequency $u$, Planck's law is written as: [harv|Rybicki|Lightman|1979|p=22]

:$I(\; u,T)\; =frac\{2\; h\; u^\{3\{c^2\}frac\{1\}\{\; e^\{frac\{h\; u\}\{kT-1\}.$

This function represents the emitted power per unit area of emitting surface, per unit solid angle, and per unit frequency. Sometimes, Planck's law is written as an expression $u(\; u,T)\; =\; pi\; I(\; u,T)$ for emitted power integrated over all solid angles. In other cases, it is written as $u(\; u,T)\; =\; 4pi\; I(\; u,T)/c$ for energy per unit volume.

The function $I(\; u,T)$ peaks for "h"$u$ = 2.82"kT". [Kittel, Thermal Physics p98] It falls off exponentially at higher frequencies and polynomially at lower.

As a function of wavelength λ, Planck's law written (for unit solid angle) as: [harv|Rybicki|Lightman|1979|p=22]

:$I(lambda,T)\; =frac\{2\; hc^2\}\{lambda^5\}frac\{1\}\{\; e^\{frac\{hc\}\{lambda\; kT-1\}.$

This function peaks for "hc" = 4.97λ"kT", a factor of 1.76 shorter in wavelength (higher in frequency) than the frequency peak. It is the more commonly used peak in Wien's displacement law.

The radiance emitted over a frequency range $[\; u\_1,\; u\_2]$ or a wavelength range $[lambda\_2,lambda\_1]\; =\; [c/\; u\_2,\; c/\; u\_1]$ can be obtained by integrating the respective functions.:$int\_\{\; u\_1\}^\{\; u\_2\}I(\; u,T),d\; u=int\_\{lambda\_2\}^\{lambda\_1\}I(lambda,T),dlambda.$The order of the integration limits is reversed because increasing frequencies correspond to decreasing wavelengths.

The following table provides the definition and SI units of measure for each symbol:

:

This is the radiation arriving at the top of the atmosphere. Radiation below 400 nm, or ultraviolet, is about 12%, while that above 700 nm, or infrared, starts at about the 49% point and so accounts for 51% of the total. The atmosphere shifts these percentages substantially in favor of visible light as it absorbs most of the ultraviolet and significant amounts of infrared.

History

Many popular science accounts of quantum theory, as well as some physics textbooks, contain some serious errors in their discussions of the history of Planck's Law. Although these errors were pointed out over forty years ago by historians of physics, they have proved to be difficult to eradicate. An article by Helge Kragh gives a lucid account of what actually happened. Kragh, Helge [http://www.physicsweb.org/articles/world/13/12/8/1 Max Planck: The reluctant revolutionary] Physics World, December 2000.]

Contrary to popular opinion, Planck did not quantize light; this is evident from his original 1901 paper Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901) [http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html Planck's original 1901 paper] .] and the references therein to his earlier work. He also plainly explains in his book "Theory of Heat Radiation" that his constant refers to Hertzian oscillators. The idea of quantization was developed by others into what we now know as quantum mechanics. The next step along this direction was made by Albert Einstein, who, by studying the photoelectric effect, proposed a model and equation whereby light was not only emitted but also absorbed in packets or photons. Then, in 1924, Satyendra Nath Bose developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck's law.

Contrary to another myth, Planck did not derive his law in an attempt to resolve the "ultraviolet catastrophe", the name given by Paul Ehrenfest to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is applied to black body radiation. Planck did not consider the equipartition theorem to be universally valid, so he never noticed any sort of "catastrophe" — it was only discovered some five years later by Einstein, Lord Rayleigh, and Sir James Jeans.

Appendix

A simple way to calculate the integral

:$J=int\_\{0\}^\{infty\}frac\{x^3\}\{e^x-1\},dx$

is to calculate the general case first and then compute the answer at the end. Consider the integral

:$int\_\{0\}^\{infty\}frac\{x^n\}\{e^x-1\},dx\; =\; int\_\{0\}^\{infty\}frac\{x^n\; e^\{-x\{1\; -\; e^\{-x,dx$

Since the denominator is always less than one, we can expand it in powers of $e^\{-x\}$ to get a convergent series

:$frac\{1\}\{1-e^\{-x\; =\; sum\_\{k=0\}^\{infty\}\; e^\{-kx\}.$

Here we have used the formula for the sum of a geometric series. The fraction on the left is the expression for the series indicated by the summation: $1\; +\; e^\{-x\}\; +\; e^\{-2x\}\; +\; e^\{-3x\}\; +\; cdots.$ The common multiplier is $e^\{-x\}$.

Then we have

:$int\_\{0\}^\{infty\}x^\{n\}\; e^\{-x\}\; sum\_\{k=0\}^\{infty\}\; e^\{-kx\},dx.$

Multiplication by the $e^\{-x\}$ on the left shifts our summation series one position to the right. That is, $e^\{-x\}\; +\; e^\{-2x\}\; +\; e^\{-3x\}\; +\; cdots$ becomes $e^\{-2x\}\; +\; e^\{-3x\}\; +\; e^\{-4x\}\; +\; cdots$. Therefore, we bump the index up by one and drop the $e^\{-x\}$:

:$int\_\{0\}^\{infty\}x^\{n\}\; sum\_\{k=1\}^\{infty\}\; e^\{-kx\},dx.$

By changing variables such that $u\; =\; kx$, thereby making $x^n\; =\; frac\{u^n\}\{k^n\}$ and $dx\; =\; frac\{du\}\{k\}$, we have

:$int\_\{0\}^\{infty\}frac\{u^n\}\{k^n\}\; sum\_\{k=1\}^\{infty\}\; e^\{-u\}frac\{du\}\{k\}$

or,

:$int\_\{0\}^\{infty\}u^n\; sum\_\{k=1\}^\{infty\}frac\{1\}\{k^\{n\; +\; 1\; e^\{-u\}du.$

Since each term in the sum represents a convergent integral, we can move the summation out from under the integral sign:

:$sum\_\{k=1\}^\{infty\}\; frac\{1\}\{k^\{n+1\; int\_\{0\}^\{infty\}u^\{n\}\; e^\{-u\},du.$

The summation on the left is the Riemann zeta function $zeta(n+1)$, while the integral on the right is the Gamma function $Gamma(n+1)$, and we are finally left with the general result

:$int\_\{0\}^\{infty\}frac\{x^\{n\{e^x-1\},dx\; =\; zeta(n+1)\; Gamma\{left(n+1\; ight)\}.$

or equivalently

:$int\_\{0\}^\{infty\}frac\{x^\{n-1\{e^x\; -\; 1\},dx\; =\; zeta\{left(n\; ight)\}\; Gamma\{left(n\; ight)\}.$

For our problem, the numerator contains $x^3$, leaving us with our specific result

:$J=zeta\{left(4\; ight)\}\; Gamma\{left(4\; ight)\}\; =\; frac\{pi^\{4\{90\}\; imes\; 6\; =\; frac\{pi^4\}\{15\}.$

Here we have used the fact that

:$sum\_\{n=1\}^\{infty\}frac\{1\}\{n^\{4$

is the Riemann zeta function evaluated for the argument 4, which is given by $pi^\{4\}/90$. (See "Finding Zeta(4)" at Wallis product for a simple though lengthy derivation of $zeta(4)$. This fact can also be proven by considering the following contour integral.)

:$oint\_\{C\_\{Rfrac\{picot(pi\; z)\}\{z^\{4.$

Where $C\_\{R\}$ is a contour of radius $R$ around the origin. In the limit, as $R$ approaches infinity, the integral approaches zero. Using the residue theorem the integral can also be written as a sum of residues at the poles of the integrand. The poles are at zero, the positive and negative integers. The sum of the residues yields precisely twice the desired summation plus the residue at zero. Because the integral approaches zero, the sum of all the residues must be zero. The summation must therefore equal minus one half times the residue at zero. From the series expansion of the cotangent function

:$cot(x)=frac\{1\}\{x\}\; -\; frac\; \{x\}\{3\}\; -\; frac\; \{x^3\}\; \{45\}\; +ldots,$

we see that the residue at zero is $-pi^\{4\}/45$ which yields the desired result. The evaluation of the Gamma function can be done by recognizing that for integral values of $n$, $Gamma(n+1)\; =\; n!$. In the appendix of the article Stefan-Boltzmann law we give a different derivation of this integral. (See also the integral of the Bose-Einstein distribution in the polylogarithm article.)

ee also

* Wien's displacement law

Notes

References

*citation
last1=Rybicki
first1=G. B.
first2=A. P.
last2=Lightman
year=1979
title=Radiative Processes in Astrophysics
publisher=John Wiley & Sons
location=New York
isbn=0-471-82759-2

*cite book
author= Thornton, Stephen T., Andrew Rex
year=2002
title=Modern Physics
publisher=Thomson Learning
location=USA
id=ISBN 0-03-006049-4

Further reading

*cite book | author=Peter C. Milonni | year=1994 | title=The Quantum Vacuum | publisher=Academic Press

External links

* [http://topex.ucsd.edu/rs/radiation.pdf Summary of Radiation]
* [http://www.vias.org/simulations/simusoft_blackbody.html Radiation of a Blackbody] - interactive simulation to play with Planck's law
* [http://scienceworld.wolfram.com/physics/PlanckLaw.html Scienceworld entry on Planck's Law]