Visualising an alternative formulation of Planck's Radiation Law

[TheBigDig]

TL;DR Summary

Unable to find graph of alternate formulation of Planck's Law

I've come across this alternative formulation of Planck's Law which links the number density to energy gap

$$n(E) = \frac{2\pi}{c^2 h^3} \frac{E^2}{exp\big(\frac{E-\mu}{k_BT})-1}$$

I've tried visualising this relation and I imagine it will look similar to the spectral density relation but I'm just wondering if anyone has ever come across a graph of this.

 

[PeterDonis]

I've come across this alternative formulation of Planck's Law

Where? Please give a reference.

 

[vanhees71]

This is not Planck's Law. Since photons are massless bosons and also because there's no conservation law for photon number there cannot be a non-zero chemical potential for photons.

 

[Cthugha]

That is the generalized version of Planck's law extended towards more complex systems and non-thermal distribtions, which is valid e.g. for a photon gas in equilibrium with a set of electronic transitions, which are in turn in equilibrium with each other. To the best of my knowledge it was first given in:
P. Wurfel, Journal of Physics C, 15 (1982) ( https://iopscience.iop.org/article/10.1088/0022-3719/15/18/012 ).

In that article, he does not directly show a graph of the distribution, but figure 5 shows the luminescence intensity of a GaAs LED for a certain choice of chemical potentials, which is at least closely connected to the initial distribution. If you really need a plot of the initial distribution, it might help to check the papers, which cite the manuscript above.

 

[vanhees71]

I see. It's about photons in a cavity. Then at least a chemical potential is not mathematical suicide. I've to read the paper to make sense of it though, because which physical sense does it have if there's not a conservation law of some charge-like quantity (a "photon number" in some sense)?

 

[Cthugha]

Well, the focus is more on the emitter side. If you put some material with a band gap in a cavity, have it emit light and wait for equilibrium between photon emission and absorption to arise, the band gap energy will play an important role. If you pump the material somewhat more strongly, the lowest unoccupied state in the conduction band will not be exactly at the band gap energy but somewhat higher as more states become occupied. A similar thing goes on in the valence band. The difference between these two chemical potentials is the one that enters in this equation.

 

[vanhees71]

Yes, but all this is not an alternative formulation of Planck's law but entirely different physics. E.g., a laser can be seen as a material with "negative temperature", describing population inversion.

 

[Cthugha]

Sure, I fully agree. If I remember correctly, the author himself describes it as an "extension", which reduces to the real Planck's law in the limit of zero chemical potential. Adding any finite value there certainly means that one does not discuss a standard black body.

 

[TheBigDig]

Where? Please give a reference.

Sorry, it was from notes given by my lecturer. I looked up the reference material for the course but couldn't find any reference to it.

That is the generalized version of Planck's law extended towards more complex systems and non-thermal distribtions, which is valid e.g. for a photon gas in equilibrium with a set of electronic transitions, which are in turn in equilibrium with each other. To the best of my knowledge it was first given in:
P. Wurfel, Journal of Physics C, 15 (1982) ( https://iopscience.iop.org/article/10.1088/0022-3719/15/18/012 ).

In that article, he does not directly show a graph of the distribution, but figure 5 shows the luminescence intensity of a GaAs LED for a certain choice of chemical potentials, which is at least closely connected to the initial distribution. If you really need a plot of the initial distribution, it might help to check the papers, which cite the manuscript above.

Thanks for this discussion and explanation guys, really improved my understanding. I will take a look at this paper.