Visualising an alternative formulation of Planck's Radiation Law

Click For Summary

Discussion Overview

The discussion centers around an alternative formulation of Planck's Radiation Law that incorporates a chemical potential, linking the number density of photons to energy gaps in a system. Participants explore the implications of this formulation, its validity, and its relation to traditional Planck's Law, with a focus on theoretical and conceptual aspects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents an alternative formulation of Planck's Law involving a chemical potential, questioning its visualization and relevance.
  • Another participant challenges the formulation, stating that a non-zero chemical potential for photons is not valid due to their massless nature and lack of a conservation law.
  • A different viewpoint suggests that the formulation is a generalized version of Planck's Law applicable to complex systems and non-thermal distributions, referencing a specific paper by P. Wurfel.
  • Some participants discuss the implications of introducing a chemical potential, questioning its physical significance in the absence of a conservation law for photons.
  • There is a mention of the role of band gap energy in photon emission and absorption within a cavity, highlighting the dynamics of chemical potentials in this context.
  • One participant argues that the alternative formulation represents fundamentally different physics, such as in the case of lasers with population inversion.
  • Another participant agrees that the formulation is an extension of Planck's Law, which reverts to the original law under specific conditions.
  • There is a request for references to the alternative formulation, with some participants indicating difficulty in finding relevant literature.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity and implications of the alternative formulation of Planck's Law. Multiple competing views are presented, and the discussion remains unresolved regarding the physical interpretation of the chemical potential in this context.

Contextual Notes

Participants note that the alternative formulation may apply to specific systems and conditions, such as photon gases in equilibrium with electronic transitions, but the assumptions and limitations of this formulation are not fully explored.

TheBigDig
Messages
65
Reaction score
2
TL;DR
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.
 
Physics news on Phys.org
TheBigDig said:
I've come across this alternative formulation of Planck's Law

Where? Please give a reference.
 
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.
 
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.
 
  • Like
Likes   Reactions: PeterDonis and TheBigDig
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)?
 
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.
 
  • Like
Likes   Reactions: 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.
 
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.
 
  • Like
Likes   Reactions: TheBigDig and vanhees71
PeterDonis said:
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.

Cthugha said:
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.
 

Similar threads

  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 18 ·
Replies
18
Views
2K
  • · Replies 9 ·
Replies
9
Views
4K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 23 ·
Replies
23
Views
3K
  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 2 ·
Replies
2
Views
9K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K