Planck's law

In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Planck's law of black body radiation:

Black body spectrum as a function of wavelength

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

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

Symbol	Meaning	SI units of measure
$I\,$	spectral radiance, energy per unit time per unit surface area per unit solid angle per unit frequency	J·s-1·m-2·sr-1·Hz-1
$\nu \,$	frequency	hertz
$T\,$	temperature of the black body	kelvin
$h\,$	Planck's constant	joule per hertz
$c\,$	speed of light	meter per second
$e\,$	base of the natural logarithm, 2.718282...	dimensionless
$k\,$	Boltzmann's constant	joule per kelvin

The wavelength is related to the frequency by

$\lambda ={c \over \nu }.$

The law is sometimes written in terms of the spectral energy density

$u(\nu ,T)={4\pi \over c}I(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$

which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz).

The spectral energy density can also be expressed as a function of wavelength:

$u(\lambda ,T)={8\pi hc \over \lambda ^{5}}{1 \over e^{hc/\lambda kT}-1}$

as shown in the derivation below.

Max Planck originally produced this law in 1900 (published in 1901) in an attempt to improve upon an expression proposed by Wilhelm Wien which fit the experimental data at short wavelengths but deviated from it at long wavelengths. He found that the above function, Planck's function, fit the data for all wavelengths remarkably well. In constructing a derivation of this law, he considered the possible ways of distributing electromagnetic energy over the different modes of charged oscillators in matter. Planck's law emerges if it is assumed that these oscillators have energy proportional to frequency

$E=h\nu \,$.

Derivation (Statistical Mechanics)

(See also the gas in a box article for a general derivation.)

Consider a cube of side $L$. From the particle in a box article, the resonating modes of the electromagnetic radiation inside the box have wavelengths given by

${c \over \nu _{n}}={2L \over n}$

where $n$ is an integer, and the energy formulae for a photon

$E_{n}=h\nu _{n}={hc \over 2L}n$

This is in one dimension. In three dimensions the energy is

$E_{n}={hc \over 2L}{\sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}$

Let's now compute the total energy in the box

$U=\sum _{n=0}^{\infty }E_{n}\,{\bar {N}}(E_{n})$

where ${\bar {N}}(E_{n})$ is the number of photons in the box with energy $E_{n}$. In other words, the total energy is equal to the sum of energy multiplied by the number of photons with that energy (in one dimension). In 3 dimensions we have:

$U=\sum _{n_{x}=0}^{\infty }\sum _{n_{y}=0}^{\infty }\sum _{n_{z}=0}^{\infty }E_{n}\,{\bar {N}}(E_{n})$

In the Thomas Fermi approximation, we can assume that the $n_{i}$ are continuous variables, in which case the sum can be replaced with an integral

$U\approx \int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }E(n)\,{\bar {N}}\left(E(n)\right)dn_{x}dn_{y}dn_{z}$

In spherical coordinates:

$\ (n_{x},n_{y},n_{z})=(n\cos \phi \sin \theta ,n\sin \phi \sin \theta ,n\cos \theta )$

this expression is more easily handled:

$U=\int _{0}^{\pi /2}\int _{0}^{\pi /2}\int _{0}^{\infty }E(n)\,{\bar {N}}\left(E(n)\right)n^{2}\sin \theta \,d\theta d\phi dn$

So far, there is no mention of ${\bar {N}}(E)$, the number of photons with energy $E$. Photons obey Bose-Einstein statistics. Their distribution is given by the famous Bose-Einstein formula

$\langle N\rangle _{BE}={1 \over e^{E/kT}-1}$

Because a photon has two polarization states which do not affect its energy, the formula above must be multiplied by 2

${\bar {N}}(E)={2 \over e^{E/kT}-1}$

Substituting this into the energy integral yields

$U=\int _{0}^{\pi /2}\int _{0}^{\pi /2}\int _{0}^{\infty }{hcn \over 2L}{2 \over e^{hcn/2LkT}-1}n^{2}\sin \theta \,d\theta d\phi dn$

Changing the integration variable from $n$ to $\nu ={cn \over 2L}$ (frequency)

${U \over L^{3}}=\int _{0}^{\pi /2}\int _{0}^{\pi /2}\int _{0}^{\infty }{16 \over c^{3}}{h\nu \over e^{h\nu /kT}-1}\nu ^{2}\sin \theta \,d\theta d\phi d\nu$

Integrating over the angular variables:

${U \over L^{3}}=\int _{0}^{\infty }u(\nu ,T)d\nu$

where the spectral energy density $u(\nu ,T)$ is given by:

$u(\nu ,T)={8\pi h\nu ^{3} \over c^{3}}{1 \over e^{h\nu /kT}-1}$

$u(\nu ,T)$ is known as the black body spectrum. It is a spectral energy density function with units of energy per unit frequency per unit volume.

It can also be expressed in terms of $\lambda$, the wavelength, via the substitution $\nu =c/\lambda$, $d\nu =-c/\lambda ^{2}d\lambda$

${U \over L^{3}}=\int _{0}^{\infty }u(\lambda ,T)d\lambda$

where

$u(\lambda ,T)={8\pi hc \over \lambda ^{5}}{1 \over e^{hc/\lambda kT}-1}$

This is also a spectral energy density function with units of energy per unit wavelength per unit volume. This integral can be evaluated using polylogarithms, or Mathematica, to yield the total electromagnetic energy inside the box:

${U \over V}={8\pi ^{5}(kT)^{4} \over 15(hc)^{3}}$

where $V=L^{3}$ is the volume of the box. (Note - This is not the Stefan-Boltzmann law, which is the total energy radiated by a black body. See that article for an explanation.) Since the radiation is the same in all directions, and propagates at the speed of light (c), the spectral intensity (energy/time/area/solid angle/frequency) is

$I(\nu ,T)={\frac {u(\nu ,T)\,c}{4\pi }}$

which yields

$I(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}~{\frac {1}{e^{h\nu /kT}-1}}$

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. The article by Helge Kragh cited below gives a lucid account of what actually happened.

Contrary to popular opinion, Planck did not quantize light. This is evident from his original 1901 paper and the references therein to his earlier work. It is also plainly explained in his book "Theory of Heat Radiation," where he explains 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 road 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 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 equipartion 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.

External link and references

• Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901).
• Radiation of a Blackbody - interactive simulation to play with Planck's law
• Scienceworld entry on the Planck Law
• Kragh, Helge Max Planck: The reluctant revolutionary Physics World, December 2000