Dependence of Radiation Absorption on Refractive Index

[Hypatio]

I am reading the text 'thermal radiation heat transfer' by Howell et al. (2011). In it they describe normal spectral emission by an isothermal volume element dV as

$4\pi \kappa_\lambda I_{\lambda b}(S)dVd\lambda$

where $\kappa_{\lambda}$ is the absorption coefficient and $I_{\lambda b}$ is the black body spectrum, V is volume, and $\lambda$ is the wavelength.

And later state that if the refractive index $n$ is not equal to 1, it should be included as

$n_{\lambda}^2(S)\kappa_\lambda(S)I_{\lambda b}(S)dS$

Much earlier they define

$I_{\lambda b}=\frac{2\pi h c^2}{n^2\lambda^5\left[\exp\left(\frac{hc}{nk_B\lambda T}\right)-1\right]}$

It seems that this must mean that

$4\pi \kappa_\lambda \frac{2\pi h c^2}{\lambda^5\left[\exp\left(\frac{hc}{nk_B\lambda T}\right)-1\right]}(S)dVd\lambda$

is the correct form for calculating spectral power.

However, the authors continually jump around with and without the use of the index of refraction and I don't know what the correct form of $I_{\lambda b}$ is. This jumping around with definitions is all very confusing. So what is the correct form of the first equation? What is the actual dependnece of refractive index on emitted power?

Also, I have no idea why emission (a sub-atomic or atomic phenomenon) depends on the the index of refraction (an interatomic medium property).

Also, does this dependence on refractive index only occur in emission or does it also impact absorption? The authors always only talk about emission when referring to the dependence on the index of refraction.

 

[eljose79]

Thank you for bringing up your concerns regarding the text 'thermal radiation heat transfer' by Howell et al. (2011). I can understand your confusion with the various equations and definitions presented in the text. Allow me to clarify and provide some insights on the correct form of the first equation and the impact of refractive index on emission and absorption.

Firstly, the correct form of the first equation is indeed:

4\pi \kappa_\lambda \frac{2\pi h c^2}{\lambda^5\left[\exp\left(\frac{hc}{nk_B\lambda T}\right)-1\right]}(S)dVd\lambda

This is the spectral power emitted by an isothermal volume element dV at a specific wavelength \lambda, taking into account the absorption coefficient \kappa_\lambda and the black body spectrum I_{\lambda b}. The inclusion of the refractive index n in the equation is important because it affects the energy levels and transitions of the atoms or molecules within the isothermal volume element, thus impacting the emission of thermal radiation.

To understand this better, let's take a closer look at the black body spectrum I_{\lambda b}. This equation describes the spectral radiance of a black body at a specific temperature T, and it takes into account the refractive index n in the denominator. This is because the refractive index affects the energy levels and transitions of the atoms or molecules within the black body, thus impacting the emission of thermal radiation. Therefore, the inclusion of the refractive index in the equation is necessary to accurately determine the spectral power emitted by the isothermal volume element.

Additionally, the dependence of refractive index on emitted power is not limited to emission. It also impacts absorption, as the refractive index affects the energy levels and transitions of the atoms or molecules within the material, thus influencing the absorption of thermal radiation.

Lastly, it is important to note that the dependence of emission on refractive index is not limited to sub-atomic or atomic phenomena. It also applies to macroscopic objects, as the refractive index of a material can affect the emission of thermal radiation from its surface.

I hope this helps to clarify your doubts. If you have any further questions, please do not hesitate to ask.