Translation:On the Distribution of Energy in the Emission Spectrum of a Blackbody

by Wilhelm Wien

Even though the shift in the radiation of a blackbody with temperature, and the distribution over all wavelengths can be derived on the basis of the electromagnetic theory of light in a purely thermodynamic way without the help of any special hypotheses, It has not yet been possible to determine the distribution of the energy itself. And yet it is in the nature of things that the dependence of the intensity on the wavelength should also be completely determinable through the properties of the radiation itself, because it depends only on the temperature and not on the special properties of individual bodies.

The radiation of a blackbody corresponds to the state of thermal equilibrium and consequently to a maximum of entropy. For example, If a process were known by which the wavelengths could be changed in a manner such that the entropy increases without an expenditure of work and without absorption, then the distribution of energy in the spectrum of a blackbody would be completely determined from the condition of maximum entropy. As I have shown in an earlier work, it is always possible to state the entropy of radiation of known intensity and color, but for the time being there are no known physical processes that would cause such a change in color. It is therefore not possible to determine the energy distribution without hypotheses.

An attempt to base a complete law of radiation on certain assumptions was made by E.v. Lommel ^[1] and W. Michelson ^[2]. The latter makes the following prerequisites:

1. Maxwell's law of the distribution of velocities for a large number of molecules is also valid for solid bodies.

2. The period of oscillation $\tau$ that a molecule exites depends on the velocity of the molecule, by the equation:

$\tau ={\frac {4\varrho }{v}}$

where $\varrho$ is a constant. (This assumption results from a certain idea about the modes excitation of the radiation.)

3. The intensity of the radiation emitted by a molecule is proportional to the number of molecules having the same period of oscillation. It is an indefinite function of temperature, and also an unknown function of kinetic energy, which, by a further assumption, is then restricted to a power of $v^{2}$.

The law which Michelson obtains from these assumptions for a given wavelength $\lambda _{m}$ at maximum energy is:

$\lambda _{m}={\frac {\mathrm {const} }{\sqrt {\vartheta }}}$

Where $\vartheta$ denotes the absolute temperature. Other than that this law leaves the total emission as a function of temperature undetermined.

I have now endeavored to utilize Michelson's fortunate idea of using Maxwell's law of the distribution of velocities as the basis of the law of radiation, but reduce the number of hypotheses which have been put forward, because our complete ignorance of the cause of the radiation makes these particularly uncertain, by using the results obtained by Boltzmann and myself in a purely thermodynamic way.

The remaining hypotheses still leave some uncertainty in their theoretical justification, but offer the advantage that their results can be compared directly with experiment, for a wide range of values. The confirmation or refutation by experiment will therefore also decide whether or not this hypothesis is correct and in this respect whether or not it will be useful for further development.

The law that an empty cavity surrounded by walls of the same temperature contains blackbody radiation, also applies if the radiation emanates from gases are sealed off from the cavity by means of transparent walls and from the outside byreflecting walls. However, the gases must have a finite absorptivity for all wavelengths. There is no doubt that there are gases such as carbonic acid and water vapor which emit heat rays simply by increasing their temperature. ^[3]. Highly superheated vapors can be treated as gases, and by appropriately mixing different substances one can always imagine a gas mixture being produced which has a finite absorptivity for all wavelengths. But one must not think here of the radiation which the gases emit under the influence of electrical or chemical processes.

If one assumes a gas as a radiating body, Maxwell's law of the distribution of velocities will apply if we take as our basis the kinetic theory of gases. The absolute temperature will be proportional to the mean kinetic energy of the gas molecules. This assumption has acquired a high degree of probability through the work of Clausius ^[4] and Boltzmann ^[5] and is supported by the investigations of Helmholtz ^[6] on monocyclic systems, according to which both the kinetic energy and the absolute temperature have the property of being the integrating denominator of the differential of the energy supplied.

In order to avoid unnecessary verbosity which would arise from considering different components of mixtures of gases, let us imagine a mixture such that the homogeneous radiation under consideration is emitted preferentially by one constituent of the gaseous mixture.

The number of molecules having velocity between $v$ and $v+dv$ is proportional to the quantity

$v^{2}e^{-{\frac {v^{2}}{\alpha ^{2}}}}dv$ ,

where $\alpha$ is a constant that can be deduced from the average velocity ${\bar {v}}$ through the equation

${\bar {v}}^{2}={\frac {3}{2}}\alpha ^{2}$

The absolute temperature is therefore proportional to $\alpha ^{2}$.

Now for radiation emitted by a molecule, of velocity $v$, it is completely unknown how the radiation depends on the state of the molecule. The view that the electric charges in the molecules can excite electromagnetic waves is nowadays generally accepted.

We hypothesize that each molecule emits radiation with a wavelength that depends only on the velocity and has intensity that a function of that velocity.

This conclusion can be reached by various special assumptions about the process of radiation, but since such assumptions are completely arbitrary for the time being, it seems to me safest at first to make the necessary hypothesis as simple and general as possible.

Since the wavelength $\lambda$ of the radiation emitted by a molecule is a function of $v$, $v$ should also be a function of $\lambda$

The intensity of radiation whose wavelength is between $\lambda$ and $\lambda +d\lambda$ is proportional to

1. The number of molecules that emit radiation in this range.

2. A function of the velocity $v$, and therefore also a function of $\lambda$.

That means,

$\varphi _{\lambda }=F(\lambda )e^{-{\frac {f(\lambda )}{\vartheta }}}$

where $F$ and $f$ are two unknown functions and $\vartheta$ is the absolute temperature.

Now the change of radiation with temperature, according to the theory given by Boltzmann ^[7] and myself ^[8] consists of an increase of the total energy in proportion to the fourth power of the absolute temperature and a change in wavelength for each quantum of energy enclosed between $\lambda$ and $\lambda +d\lambda$, in the sense that the associated wavelength changes in inverse proportion to the absolute temperature. If one then imagines the energy at a temperature to be plotted as a function of wavelength, this curve would remain unchanged, with changing temperature, if the scale of the graph were changed so that the ordinates were proportionally diminished and the abscissas proportionally increased. The latter is for our value of $\varphi _{\lambda }$ only possible if $\lambda$ and $\vartheta$ occur in the exponent and only if they do so as the product $\lambda \vartheta$

If $c$ is a constant, then

${\frac {f(\lambda )}{\vartheta }}={\frac {c}{\lambda \vartheta }}$

The increase in the total energy determines the value of $F(\lambda )$. Because it has to be

$\int \limits _{0}^{\infty }F(\lambda )e^{-{\frac {c}{\vartheta \lambda }}}d\lambda =\mathrm {const.} \,\vartheta ^{4}$.

$F(\lambda )$ can be determined using the method of undetermined coefficients.

We imagine $F(\lambda )$ expanded as a series and set $\lambda ={\frac {c}{y\vartheta }}$, so that

${\begin{aligned}F(\lambda )=F\left({\frac {c}{y\vartheta }}\right)=a_{0}&+a_{+1}{\frac {\vartheta y}{c}}+a_{+2}{\frac {\vartheta ^{2}y^{2}}{c^{2}}}+\ldots +a_{n}{\frac {\vartheta ^{n}y^{n}}{c^{n}}}+\ldots \\&+a_{-1}{\frac {c}{\vartheta y}}+a_{-2}{\frac {c^{2}}{\vartheta ^{2}y^{2}}}+\ldots +a_{-n}{\frac {\vartheta ^{-n}y^{-n}}{c^{-n}}}.\end{aligned}}$

Integrating this results in

$\int \limits _{0}^{\infty }F(\lambda )e^{-{\frac {c}{\vartheta \lambda }}}d\lambda ={\frac {c}{\vartheta }}\int \limits _{0}^{\infty }F\left({\frac {c}{y\vartheta }}\right)e^{-y}{\frac {dy}{y^{2}}}=\sum _{n}a_{n}{\frac {\vartheta ^{n-1}}{c^{n-1}}}\int \limits _{0}^{\infty }e^{-y}y^{n-2}dy$.

It therefore follows that

$\mathrm {const.} \vartheta ^{4}=\sum _{n}a_{n}{\frac {\vartheta ^{n-1}}{c^{n-1}}}\Gamma (n-1)$

So all but one of the coefficients will vanish, and the result will be

$\vartheta ^{n-1}=\vartheta ^{4}$,

And therefore $n=5$

From this, it therefore follows that

$F(\lambda )={\frac {\mathrm {const.} }{\lambda ^{5}}}$.

The equation for $\varphi _{\lambda }$ now becomes

$\varphi _{\lambda }={\frac {C}{\lambda ^{5}}}e^{-{\frac {c}{\lambda \vartheta }}}$.

From this it follows that

${\begin{aligned}{\frac {d\varphi }{d\lambda }}&=-{\frac {Ce^{-{\frac {c}{\lambda \vartheta }}}}{\lambda ^{6}}}\left(5-{\frac {c}{\lambda \vartheta }}\right),\\{\frac {d^{2}\varphi }{d\lambda ^{2}}}&={\frac {Ce^{-{\frac {c}{\lambda \vartheta }}}}{\lambda ^{7}}}\left(30-{\frac {12c}{\lambda \vartheta }}+{\frac {c^{2}}{\lambda ^{2}\vartheta ^{2}}}\right);\end{aligned}}$

for

$\lambda ={\frac {c}{5\vartheta }}$ we have ${\frac {d\varphi }{d\lambda }}=0$,

${\frac {d^{2}\varphi }{d\lambda ^{2}}}=-{\frac {5Ce^{-5}}{\lambda ^{7}}}$;

${\frac {d^{2}\varphi }{d\lambda ^{2}}}$ is negative, so the value corresponds to a maximum. We shall refer to this value as $\lambda _{m}$. The associated value of $\varphi$ is

$\varphi _{m}={\frac {C}{\lambda _{m}^{5}}}e^{-5}$.

Since both $\varphi$ and ${\frac {d\varphi }{d\lambda }}$ vanish for $\lambda =\infty$, then the curve is an asymptote to the $\lambda$-axis. Furthermore ${\frac {d^{2}\varphi }{d\lambda ^{2}}}=0$ for the roots of the equation

$30\lambda ^{2}\vartheta ^{2}-12c\lambda \vartheta +c^{2}=0$,

therefore for

$\lambda =\lambda _{m}\left(1\pm {\sqrt {\tfrac {1}{6}}}\right)$.

For these two points, the curve has inflection points. We will set $\lambda =\lambda _{m}(1+\varepsilon )$, so that

$\varphi _{\lambda }={\frac {Ce^{-{\frac {c}{\lambda _{m}(1+\varepsilon )\vartheta }}}}{\lambda _{m}^{5}(1+\varepsilon )^{5}}}={\frac {Ce^{-{\frac {5}{1+\varepsilon }}}}{\lambda _{m}^{5}(1+\varepsilon )^{5}}}$,

therefore

$\log {\frac {\varphi }{\varphi _{m}}}=-5\left(\log(1+\varepsilon )-{\frac {\varepsilon }{1+\varepsilon }}\right)=-5\left({\tfrac {1}{2}}\varepsilon ^{2}-{\tfrac {2}{3}}\varepsilon ^{3}+{\tfrac {3}{4}}\varepsilon ^{4}\ldots \right)$

We will set $-\varepsilon$ for $\varepsilon$ , so that

$\log {\frac {\varphi }{\varphi _{m}}}=-5\left({\tfrac {1}{2}}\varepsilon ^{2}+{\tfrac {2}{3}}\varepsilon ^{3}+{\tfrac {3}{4}}\varepsilon ^{4}\ldots \right)$

Here the absolute value of the series is larger, i.e ${\frac {\phi }{\phi _{m}}}$ smaller than with positive $\varepsilon$ . So long as $\varepsilon <1$ the ordinates at the same distance from the maximum are smaller on the short wavelength side.

In an earlier work ^[9] I showed that the energy curves of black bodies at different temperatures cannot intersect. From this it could be further deduced that the curve must drop more slowly on the side of the long waves than the curve

${\frac {\mathrm {const.} }{\lambda ^{5}}}.$

Now this is actually the case with our curve; ${\frac {d\varphi _{\lambda }}{d\lambda }}$ is always smaller than the absolute value ${\frac {5C}{\lambda ^{6}}}$ and only reaches this limit for $\vartheta =\infty$. For infinitely increasing temperature would mean $\varphi _{\lambda }={\frac {C}{\lambda ^{5}}}$ and the maximum energy will approach zero wavelength indefinitely.

Just as I had derived the formula for $\varphi _{\lambda }$ from the aforementioned theoretical considerations, Prof Paschen had independently found the formula

$\varphi _{\lambda }={\frac {C}{\lambda ^{\alpha }}}e^{-{\frac {c}{\lambda \vartheta }}}$

(Where $\alpha$ a constant), to represent his observations the best, and he was kind enough to let me know of it, and to grant permission to communicate his formula here.

Prof. Paschen intends to determine the value of the constant $\alpha$ from the complete calculation and comparison of his observations. If $\alpha$ is not $5$, the total emission would not follow Stefan's law.

Charlottenburg , June 1896.

1. E.v. Lommel, Wied. Ann. 3. p. 251. 1877.
2. W. Michelson, Journ. de phys. (2) 6. 1887.
3. Paschen, Wied. Ann. 50 p.409 1893.
4. Clausius, Pogg. Ann. 142. p. 195. 1866.
5. Boltzmann, Wien. Ber. (2) 53. p. 195.
6. Helmholtz, Ges. Abh. 3. p.119
7. Boltzmann, Wied. Ann. 22. p.291. 1884.
8. W. Wien, Ber. d. Berl. Akad. 9. Febr. 1893.
9. W. Wien, Wied. Ann. 52. p. 159. 1894.