Draft:Kirchhoff-Clausius's Law

The Kirchhoff-Clausius law, also known as the Clausius's law or the n-squared law for radiance, defines the relationship between the radiance of a black body thermal radiation and the medium through which the emitted light passes. It establishes that the radiance (total power radiated per unit of surface and unit of solid angle, expressed in watts per square meter per steradian) is directly proportional to the square of the refractive index of the medium (Kirchhoff) and, therefore, inversely proportional to the square of the speed of propagation of the radiation in the medium (Clausius):

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

(where $L$ = emissive power as radiance and n = index of refraction, all in the surrounding medium, and $L^{0}$ = emissive power of a perfectly black body in a vacuum; one assigns the index zero for all quantities relating to the vacuum).
This law is a theoretical discovery based on geometrical optics. Its name is in tribute to Gustav Kirchhoff and Rudolf Clausius, who published their initial findings in 1862. and 1864.
The law was proved experimentally by Marian Smoluchowski de Smolan in 1896 , comparing the emission of hot, blackened copper in carbon dioxide and hydrogen.

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)."
This is one of the 4 classical laws on which Planck's law is based, in order of discovery:

• Kirchhoff's law on thermal radiation.
• Kirchhoff-Clausius law.
• Stefan-Boltzmann law.
• Wien's displacement law.

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 (and previously Foucault) had noticed that the spectrum of sunlight was amplified in the flame of the Bunsen burner, Kirchoff 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 would become known as 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 for example, 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 $I_{a}\div I_{c}=n_{a}^{2}\div n_{c}^{2}$.
Clausius and Kirchhoff wrote this form using only one black body in a vacuum and radiating it in another media, so it had a vacuum refractive index of one. So, to simplify, you obtain the Kirchhoff form: $I_{c}=n_{c}^{2}\cdot I_{a}$

Marian Smoluchowski de Smolan also studied it and validated it experimentally in 1896 during his stay in Paris, by comparing the emission of hot and blackened copper in carbon dioxide and hydrogen.

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.

The law will be demonstrated again by Wolfgang Pauli, then by D. Sivoukhin under the supervision of V. Ginzburg.

Later, a link will be established with electromagnetism for radiative transfert.

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.

 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 written:

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

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.

Notable applications

Radiative transfer