Draft:Kirchhoff-Clausius's Law

In thermal radiation using geometrical optics, the Kirchhoff-Clausius law was named after Gustav Kirchhoff and Rudolf Clausius, who published their initial findings in 1862. and 1863.

The Kirchhoff-Clausius law states that:

"The  emissive power  of  perfectly  black  bodies  is  directly proportional  to  the  square  of  the  index  of refraction of the surrounding medium (Kirchoff), and therefore inversely proportional to the squares of the velocities of propagation in the surrounding medium (Clausius)."

With the formula:

$I=n^{2}I^{0}$

(where $I$ = emissive power as radiance and n = index of refraction, all in the surrounding medium, and $I^{0}$ = emissive power of a perfectly black body in a vacuum; one assigns the index zero for all quantities relating to the vacuum).

In principle, this means that an object's emissive power increases when it is embedded inside a medium, which is more refractive than a vacuum. But if you write it in a monochromatic form in wavelength $\lambda$, with $n={\frac {\lambda ^{0}}{\lambda }}$ it gives

$I_{\lambda }\lambda ^{2}=I_{\lambda }^{0}(\lambda ^{0})^{2}$

So, you see there that the energy flux in a wavelength-squared surface is independent of the medium. In fact, when the electromagnetic wave slows down in a medium, its wavelength, whose energy is constant, decreases, thus obtaining a greater energy density (from the emissive power) while maintaining the same energy flow. Then passing into the vacuum, the emissive power decreases again.

Demonstration

One gives the following definitions:

One names the frequency $\nu$, the wavelength $\lambda$ and the speed of light c.

The quantity “$I$” is the energy intensity (radiance) of isotropic radiation at thermal equilibrium whose one gives the spectral decomposition by the following integral:

$I=\int _{0}^{\infty }I_{\nu }\cdot d\nu$

 It is now a matter of calculating the energy intensity at thermal equilibrium in a homogeneous and isotropic transparent medium of refractive index n. This equilibrium radiation is established in a closed cavity entirely occupied by the medium considered and whose walls are at a constant temperature. The same radiation will exist in the medium, which occupies only part of the cavity. Consider a cavity with a given medium for one part and the vacuum in the remaining. The equilibrium radiation in the medium and vacuum depends neither on the shape nor the properties of the separation surface between the medium and the vacuum. One can assume that this separation surface is flat and perfectly smooth.

Figure 1.

Figure 1.

 One considers that the energy exchange between the medium and the vacuum will only result from reflections and refractions of the radiation on the separation surface. This exchange of energy applies the principle of detailed balance, and this exchange cannot alter the state of equilibrium between the radiation in the medium and the vacuum. From there, one can establish a relationship between the radiation intensity $I^{0}$ in a vacuum and the same quantities $I$ in the medium.

 As one verifies the principle of detailed balance, it is sufficient to consider only a part of the total radiation, including the frequencies between $\nu$ and $\nu +d\nu$. The flux per unit time from the vacuum, falling on the unit area of ​​the separation surface, and contained in the solid angle cone $d\Omega _{0}$ where $\varphi _{0}$ is the angle of incidence (Figure 1), is:

$\Phi _{1}=I_{\nu }^{0}d\Omega _{0}d\nu \cos(\varphi _{0})$

According to the principle of detailed balance, an equal flow of energy must propagate in the opposite direction. That consists of two flows: the first flow results from the reflection of the flow $\Phi _{2}$ and has the value:

$(1-A_{\nu })I_{\nu }^{0}d\Omega _{0}d\nu \cos(\varphi _{0})$

The second results from refracting the flow $\Phi _{3}$ from the medium ($A_{\nu }$ being the absorbing power coefficient for frequency $\nu$). According to Fresnel's formulas, the reflection coefficients on the separation surface of rays propagating in opposite directions are equal; the second flow is, therefore, equal to:

$A_{\nu }I_{\nu }d\Omega d\nu \cos(\varphi _{0})$

$\varphi$ being the angle of refraction and $d\Omega$ the solid angle in the medium, which, after refraction, becomes equal to $d\Omega _{0}$. After division by $d\nu$, one writes the detailed balance condition as follows:

$(1-A_{\nu })I_{\nu }^{0}\cos(\varphi _{0})d\Omega _{0}+A_{\nu }I_{\nu }\cos(\varphi )d\Omega =I_{\nu }^{0}\cos(\varphi _{0})d\Omega _{0}$

It gives:

$I_{\nu }\cos(\varphi )d\Omega =I_{\nu }^{0}\cos(\varphi _{0})d\Omega _{0}$

Let's take for $d\Omega _{0}$ the solid angle (not shown in Figure 1) delimited by the cones whose generators make the angles with the normal to the separation surface $\varphi _{0}$ and $\varphi _{0}+d\varphi _{0}$; it results in:

$d\Omega _{0}=2\pi \sin(\varphi _{0})d\varphi _{0}$

Likewise:

$d\Omega =2\pi \sin(\varphi )d\varphi$

Equality (2) is now writen:

$I_{\nu }\cos(\varphi )\sin(\varphi )d\varphi =I_{\nu }^{0}\nu \cos(\varphi _{0})\sin(\varphi _{0})d\varphi _{0}$

According to the law of refraction, $\sin(\varphi _{0})=n\sin(\varphi )$, and subsequently:

$\cos(\varphi _{0})d\varphi _{0}=n\cos(\varphi )d\varphi$

Therefore:

$I_{\nu }\sin(\varphi )=nI_{\nu }^{0}\sin(\varphi _{0})$

Hence, by the same law of refraction:

$I_{\nu }=n^{2}I_{\nu }^{0}$

Kirchhoff found this solution in 1860, and Clausius, who knew it, found it in 1863 by another route and with another form:

$I_{\nu }c_{n}^{2}=I_{\nu }^{0}c^{2}$

It is called the Kirchhoff-Clausius law.

As $c=\lambda \nu$ and $\nu$ is invariant with the medium, you can also write:

$I_{\nu }(\lambda _{\nu })^{2}=I_{\nu }^{0}(\lambda _{\nu }^{0})^{2}$

This means that in an equilibrium radiation the energy fluxes passing through a rectangular area of ​​side equal to the wavelength $\lambda$ (in the medium) are the same for all media. This is valid both for the total flux and for its monochromatic components.

History

In 1849, Foucault noticed that bright lines occur where the Fraunhofer double line D of the solar spectrum is found and that this dark line D is produced or made more intense when the rays of the sun, or those from one of the incandescent carbon poles, are passed through the luminous arc. Gustav Kirchhoff discovered the law of thermal radiation in 1859 while collaborating with Robert Bunsen at the University of Heidelberg, where they developed the modern spectroscope. He proved it in 1861 and then, in 1862, defined the perfect black body. The same year, as he had noticed, like Foucault, that the spectrum of sunlight was amplified in the flame of the Bunsen burner, he found a theoretical explanation in geometric optics with a formula that gave the amplification coefficient with the square of the refractive index ($I=n^{2}I^{0}$) for a new law, which will become the Kirchhoff-Clausius law.

In 1863, Rudolf Clausius revisited Kirchhoff's study in the spirit of the second law of thermodynamics. For this, he considered two perfect black bodies (a and c) side by side and at the same temperature, immersed in two different adjacent media, such as water and air, and radiating towards each other at different speeds. So, to respect the second law, the mutual thermal radiation between these two black bodies must be equal, and he obtains a different form of the formula:$I_{a}\cdot v_{a}^{2}=I_{c}\cdot v_{c}^{2}$
(where $I_{a}$ and $I_{c}$= emissive power, and $v_{a}$ and $v_{c}$ = light speed, in each surrounding medium ). Hence $I_{a}\div I_{c}=v_{c}^{2}\div v_{a}^{2}$ and $I_{a}\div I_{c}=n_{a}^{2}\div n_{c}^{2}$.
As written Clausius, Kirchhoff used only one black body in a vacuum and radiated it in another media, so it had a vacuum refractive index of one. So, to simplify, you obtain the Kirchhoff form: $I_{c}=n_{c}^{2}\cdot I_{a}$

Afterward, this law was mainly used in astrophysics, maybe first by Georges MESLIN in 1872.

Marian Smoluchowski de Smolan also studied it in 1896 in Paris

Above all, it became a crucial point in Planck's demonstration of the law of black body radiation in 1901. With the Kirchhoff-Clausius law, he demonstrated that the energy density emitted by a black body is the same in any media and is a universal function of its temperature and frequency. If you replace $I$ with the energy density $U$, the Kirchoff-Clausius law becomes $U_{c}=n_{c}^{3}\cdot U_{a}$. Then, as $n_{c}=\lambda _{a}/\lambda _{c}$ ( $\lambda$ for wavelength), you obtain $\lambda _{c}^{3}U_{c}=\lambda _{a}^{3}U_{a}$. So, the energy of an equilibrium radiation localized in a cube with an edge equal to the wavelength $\lambda$ is the same in any media. In addition, as it also led to the Planck-Einstein relationship, it became indirectly a key point in Albert Einstein's demonstration of the photoelectric effect in 1905.

In 1902, Rudolf Straubel extended this law to the plane parallel to the radiation, which is why it is sometimes known as the Kirchhoff-Clausius-Straubel law.

Wolfgang Pauli also demonstrated the law.

E. Schoenberg, in 1929, in his article on "Theoretical Photometry," used the law.

MOLCHANOV, A.P., in 1966, applied the law in his "Physics of the Solar System" course. He applied it for small volumes in local thermodynamic equilibrium.

Sivoukhine D., in his "General Physics Course" in 1982, resumes the demonstration of Planck's law for black body radiation, detailing both the Kirchhoff-Clausius law and the other laws, which is rare.

In 1993, a Ukrainian Academy of Sciences team briefly cited the law in its introduction.

In 2014, a research article used the law on the subject of disinfection of water by UV.

The same year, a research article by a team working on the theory of "Variable speed of light", used the law.

You can also find some biography of Gustav Kirchhoff citing the law.