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

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

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

{| border="1" cellspacing="0" cellpadding="5"

!Symbol Meaning SI units of measure $I\; \backslash ,$ 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 $\backslash nu\; \backslash ,$ frequency hertz $T\; \backslash ,$ temperature of the black body kelvin $h\; \backslash ,$ Planck's constant joule per hertz $c\; \backslash ,$ speed of light meter per second $e\; \backslash ,$ base of the natural logarithm, 2.718282... dimensionless $k\; \backslash ,$ Boltzmann's constant joule per kelvin

The wavelength is related to the frequency by

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

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

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

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:

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

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 emerged when he assumed that the energy of these oscillators was limited to a set of discrete, integer multiples of a fundamental unit of energy, E, proportional to the oscillation frequency ν:

$E=h\backslash nu\backslash ,$.

Planck made this quantization assumption five years before Albert Einstein hypothesized the existence of photons as a means of explaining the photoelectric effect. At the time, Planck believed that the quantization applied only to the tiny oscillators that were thought to exist in the walls of the cavity (what we now know to be atoms), and made no assumption that light itself propagates in discrete bundles or packets of energy. Moreover, Planck did not attribute any physical significance to this assumption, but rather believed that it was merely a mathematical device that enabled him to derive a single expression for the black body spectrum that matched the empirical data at all wavelengths.

Ultimately, Planck's assumption of energy quantization and Einstein's photon hypothesis became the fundamental basis for the later development of Quantum Mechanics. Both scientists would eventually receive (separate) Nobel prizes in recognition of these major contributions to the advancement of physics.

Derivation (Statistical Mechanics)

---

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

Consider a cube of side $L$ with conducting walls filled with electromagnetic radiation. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The wavelength $\backslash lambda\_\{i\}$ in the three directions $i=1\backslash ldots\; 3$ orthogonal to the walls can be:

$\backslash lambda\_\{i\}\; =\; \backslash frac\{2L\}\{n\_\{i\}\}$

where the $n\_\{i\}$ are integers. For each set of integers $n\_\{i\}$ there are two linear independent solutions (modes). According to quantum theory, the energy levels of a mode are given by:

$E\_\{n\_\{1\},n\_\{2\},n\_\{3\}\}\backslash left(r\backslash right)=\backslash left(r+\backslash frac\{1\}\{2\}\backslash right)\backslash frac\{hc\}\{2L\}\backslash sqrt\{n\_\{1\}^\{2\}+n\_\{2\}^\{2\}+n\_\{3\}^\{2\}\}\backslash mbox\{\; (1)\}$

The quantum number $r$ can be interpreted as the number of photons in the mode. The two modes for each set of $n\_\{i\}$ correspond to the two polarization states of the photon which has a spin of 1. Note that for $r=0$ the energy of the mode is not zero. This vacuum energy of the electromagnetic field is responsible for the Casimir effect. In the following we will calculate the internal energy of the box at temperature $T$ relative to the vacuum energy.

According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by:

$P\_\{r\}=\backslash frac\{\backslash exp\backslash left(-\backslash beta\; E\backslash left(r\backslash right)\backslash right)\}\{Z\backslash left(\backslash beta\backslash right)\}$

Here

$\backslash beta\backslash equiv\; 1/\backslash left(kT\backslash right)$.

The denominator $Z\backslash left(\backslash beta\backslash right)$, is the partition function of a single mode and makes $P\_\{r\}$ properly normalized:

$Z\backslash left(\backslash beta\backslash right)=\backslash sum\_\{r=0\}^\{\backslash infty\}\backslash exp\backslash leftE\backslash left(r\backslash right)\backslash right=\backslash frac\{1\}\{1-\backslash exp\backslash left*\}$

Here we have defined

$\backslash varepsilon\backslash equiv\backslash frac\{hc\}\{2L\}\backslash sqrt\{n\_\{1\}^\{2\}+n\_\{2\}^\{2\}+n\_\{3\}^\{2\}\}$

which is the energy of a single photon. As explained here, the average energy in a mode can be expressed in terms of the partition function:

$\backslash left\backslash langle\; E\backslash right\backslash rangle=-\backslash frac\{d\backslash log\backslash left(Z\backslash right)\}\{d\backslash beta\}=\backslash frac\{\backslash varepsilon\}\{\backslash exp\backslash left(\backslash beta\backslash varepsilon\backslash right)-1\}$

This formula is a special case of the general formula for particles obeying Bose-Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero.

The total energy in the box now follows by summing $\backslash left\backslash langle\; E\backslash right\backslash rangle$ over all allowed single photon states. This can be done exactly in the thermodynamic limit $L\backslash rightarrow\backslash infty$. In this limit, $\backslash varepsilon$ becomes continuous and we can then integrate $\backslash left\backslash langle\; E\backslash right\backslash rangle$ over this parameter. To calculate the energy in the box in this way, we need to evaluate how many photon states there are in a given energy range. If we write the total number of single photon states with energies between $\backslash varepsilon$ and $\backslash varepsilon\; +\; d\backslash varepsilon$ as $g\backslash left(\backslash varepsilon\backslash right)d\backslash epsilon$, where $g\backslash left(\backslash varepsilon\backslash right)$ is the density of states which we'll evaluate in a moment, then we can write:

$U\; =\; \backslash int\_\{0\}^\{\backslash infty\}\backslash frac\{\backslash varepsilon\}\{\backslash exp\backslash left(\backslash beta\backslash varepsilon\backslash right)-1\}g\backslash left(\backslash varepsilon\backslash right)d\backslash varepsilon\; \backslash mbox\{\; (2)\}$

To calculate the density of states we rewrite equation (1) as follows:

$\backslash varepsilon\backslash equiv\backslash frac\{hc\}\{2L\}n$

where $n$ is the norm of the vector $\backslash vec\{n\}=\backslash left(n\_\{1\},n\_\{2\},n\_\{3\}\backslash right)$:

$n=\backslash sqrt\{n\_\{1\}^\{2\}+n\_\{2\}^\{2\}+n\_\{3\}^\{2\}\}$

For every vector $n$ with integer components larger or equal than zero there are two photon states. This means that the number of photon states in a certain region of n-space is twice the volume of that region. An energy range of $d\backslash varepsilon$ corresponds to shell of thickness $dn=\; \backslash frac\{2L\}\{hc\}d\backslash varepsilon$ in n-space. Because the components of $\backslash vec\{n\}$ have to be positive, this shell spans an octant of a sphere. The number of photon states $g\backslash left(\backslash varepsilon\backslash right)d\backslash epsilon$ in an energy range $d\backslash varepsilon$ is thus given by:

$g\backslash left(\backslash varepsilon\backslash right)d\backslash epsilon=2\backslash frac\{1\}\{8\}4\backslash pi\; n^\{2\}dn=\backslash frac\{8\backslash pi\; L^\{3\}\}\{h^\{3\}c^\{3\}\}\backslash varepsilon^\{2\}d\backslash varepsilon$

Inserting this in Eq. (2) gives:

$U\; =L^\{3\}\backslash frac\{8\backslash pi\}\{h^\{3\}c^\{3\}\}\backslash int\_\{0\}^\{\backslash infty\}\backslash frac\{\backslash varepsilon^\{3\}\}\{\backslash exp\backslash left(\backslash beta\backslash varepsilon\backslash right)-1\}d\backslash varepsilon\backslash mbox\{\; (3)\}$

From this equation one easily derives the spectral energy density as a function of frequency $u(\backslash nu,T)$ and as a function of wavelength $u(\backslash lambda,T)$:

$\backslash frac\{U\}\{L^3\}\; =\; \backslash int\_0^\{\backslash infty\}u(\backslash nu,T)\; d\backslash nu$

where:

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

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

And:

$\backslash frac\{U\}\{L^3\}\; =\; \backslash int\_0^\backslash infty\; u(\backslash lambda,T)\; d\backslash lambda$

where

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

This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithms. In this case, however, it is possible to calculate the integral in closed form. Let's first make the integration variable in Eq. (3) dimensionless by substituting $\backslash varepsilon\; =\; k\; T\; x$:

$u(T)\; =\backslash frac\{8\backslash pi\; (kT)^\{4\}\}\{(hc)^\{3\}\}\; J$

Here $J$ is given by:

$J=\backslash int\_\{0\}^\{\backslash infty\}\backslash frac\{x^\{3\}\}\{\backslash exp\backslash left(x\backslash right)-1\}dx\; =\; \backslash frac\{\backslash pi^\{4\}\}\{15\}$

We prove this result in the Appendix below. The total electromagnetic energy inside the box is thus given by:

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

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

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

which yields

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

Derivation (Thermodynamics)

---

The fact that the energy density of the box containing radiation is proportional to $T^\{4\}$ was derived by Ludwig Boltzmann in 1884 using thermodynamics. It follows from classical electrodynamics that the radiation pressure $P$ is related to the internal energy density:

$P=\backslash frac\{u\}\{3\}$

The total internal energy of the box containing radiation can thus be written as:

$U=3PV$

Inserting this in the fundamental law of thermodynamics

$dU=T\; dS\; -\; P\; dV$

yields the equation:

$dS=4\backslash frac\{P\}\{T\}dV\; +\; 3\backslash frac\{V\}\{T\}dP$

We can now use this equation to derive a Maxwell relation. We read off that:

$\backslash left.\backslash frac\{\backslash partial\; S\}\{\backslash partial\; V\}\backslash right|\_\{P\}=4\backslash frac\{P\}\{T\}$

And

$\backslash left.\backslash frac\{\backslash partial\; S\}\{\backslash partial\; P\}\backslash right|\_\{V\}=3\backslash frac\{V\}\{T\}$

The symmetry of second derivatives of $S$ w.r.t. $P$ and $V$ then implies:

$4\backslash left.\backslash frac\{\backslash partial\; \backslash left(\backslash frac\{P\}\{T\}\backslash right)\}\{\backslash partial\; P\}\backslash right|\_\{V\}=\; 3\backslash left.\backslash frac\{\backslash partial\; \backslash left(\backslash frac\{V\}\{T\}\backslash right)\}\{\backslash partial\; V\}\backslash right|\_\{P\}$

Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume. In the derivative on the r.h.s. the temperature is thus a constant. Evaluating the derivatives gives the differential equation:

$\backslash frac\{1\}\{P\}\backslash frac\{dP\}\{dT\}=\backslash frac\{4\}\{T\}$

This implies that $u=3P\; \backslash propto\; T^\{4\}$

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.

Appendix

---

A simple way to calculate the integral

$J=\backslash int\_\{0\}^\{\backslash infty\}\backslash frac\{x^\{3\}\}\{\backslash exp\backslash left(x\backslash right)-1\}dx$

is as follows. After multiplying the numerator and denominator of the integrand we can expand the integrand in powers of $\backslash exp(-x)$.

$J=\backslash int\_\{0\}^\{\backslash infty\}\backslash frac\{x^\{3\}\backslash exp\backslash left(-x\backslash right)\}\{1-\backslash exp\backslash left(-x\backslash right)\}dx=\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash int\_\{0\}^\{\backslash infty\}x^\{3\}\backslash exp\backslash left(-n\; x\backslash right)dx=6\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{1\}\{n^\{4\}\}=\backslash frac\{\backslash pi^\{4\}\}\{15\}$

Here we have used that $\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{1\}\{n^\{4\}\}$ is the Riemann zeta function evaluated for the argument 4, which is given by $\backslash frac\{\backslash pi^\{4\}\}\{90\}$. This fact can be proven by considering the contour integral

$\backslash oint\_\{C\_\{R\}\}\backslash frac\{\backslash pi\backslash cot(\backslash pi\; z)\}\{z^\{4\}\}$

Where $C\_\{R\}$ is a contour of radius $R$ around the origin. In the limit $R\backslash rightarrow\backslash infty$ 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. This means that $\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{1\}\{n^\{4\}\}$ equals minus $\backslash frac\{\backslash pi^\{4\}\}\{2\}$ times the coefficient of $x^3$ of the series expansion of the series expansion of the cotangent function.

External link and references

---

• Planck's 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).
• 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

Statistical mechanics | Foundational quantum physics | Eponymous laws

Закон на Планк | Planckův vyzařovací zákon | Plancksches Strahlungsgesetz | Ley de Planck | Loi de Planck | Wet van Planck | Plancks strålingslov | Planckin laki mustan kappaleen säteilystä