Planck formula and density of photons

In summary, the conversation discusses the computation of spontaneous and stimulated emission in a LASER system and compares two equations for the number of photons emitted by such a system in thermal equilibrium at a given temperature. One equation, from a document, has incorrect units and the other, derived by the speakers, has the correct units. The difference between the two equations is due to one being for number of photons per volume per unit frequency and the other being for number of photons per volume per unit energy. The refractive index must be included in the formula as it affects the density of states for photons in the material. This was derived for a solid material, not a vacuum.
  • #1
EmilyRuck
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6
Hello!
Let's consider again a system of atoms with only two permitted energy levels [itex]E_1[/itex] and [itex]E_2 > E_1[/itex]. When electrons decay from [itex]E_2[/itex] level to [itex]E_1[/itex], they generate a photon of energy [itex]E_{21} = E_2 - E_1 = h \nu[/itex]. The number of photons (per unit frequency, per unit volume) emitted by such a system in thermal equilibrium at a temperature [itex]T[/itex] can be determined dividing its black body radiation [itex]\rho (\nu)[/itex] by [itex]h \nu[/itex]:

[itex]
n_{ph} (E_{21}) = \rho (\nu) \displaystyle \frac{1}{h \nu} = \displaystyle \frac{8 \pi h \nu^3}{c^3 \left( e^{h \nu / (k_B T)} - 1 \right)} \frac{1}{h \nu} = \frac{8 \pi \nu^2}{c^3 \left( e^{h \nu / (k_B T)} - 1 \right)} = \frac{8 \pi E_{21}^2}{h^2 c^3 \left( e^{h \nu / (k_B T)} - 1 \right)}
[/itex]

where [itex]h[/itex] is the Planck constant, [itex]k_B[/itex] is the Boltzmann constant, [itex]c[/itex] is the speed of light. This computation is about spontaneous and stimulated emission in a LASER system.

In http://www.springer.com/cda/content/document/cda_downloaddocument/9784431551478-c2.pdf document (page 10, formula (2.13)), [itex]n_{ph} (E_{21})[/itex] is slightly different. It is

[itex]
n_{ph} (E_{21}) = \displaystyle \frac{8 \pi n_r^3 E_{21}^2}{h^3 c^3 \left( e^{h \nu / (k_B T)} - 1 \right)}
[/itex]

So:

1) If the above computation is correct, why in the book the denominator contains [itex]h^3[/itex] instead of [itex]h^2[/itex]?

2) What can the refractive index be related to? Maybe is it due to the fact that the material is a semiconductor and not the vacuum?

Thank you anyway!

Emily
 
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  • #2
I don't know if I can give you definitive answers here, but it's clear that the equation from the document has the wrong units, while the equation you derive has the correct units. So maybe the one in the book is a typo?

EmilyRuck said:
What can the refractive index be related to? Maybe is it due to the fact that the material is a semiconductor and not the vacuum?
This was my thought, given that the refractive index is raised to the same power as the speed of light. The derivation of the Planck formula involves counting the number of photon modes in the interval ##\nu +d\nu##. This assumes that these modes look something like ##A \sin (kx + \omega t)##, which implicitly gives the speed ##c =\frac {\omega }{k}##. For photons, this speed is generally assumed to be the speed of light, but if you're not in a vacuum, you need to correct the wave speed by the refractive index of the medium.

Caveat: this is my best guess. I didn't specifically go back through the whole derivation of Planck's law.
 
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  • #3
EmilyRuck,
Both formula are correct, just different quantities.
You derived number of photons per volume per unit frequency. The formula in the book is number of photons per volume per unit energy, thence the difference by a Planck's constant.

The refractive index does have to be included in the formula. It changes the density of states for photons in the material. Yes, the book formula is derived for a solid material, not vacuum.
 
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  • #4
TeethWhitener, I agree with you, the refractive index is at the same power as [itex]c[/itex] and in different media photons have different velocities.
Henryk, it was difficult because the book spoke about "density" without specifying anything else; so I thought it was per unit frequency.
Thank you both!
 

1. What is the Planck formula?

The Planck formula, also known as Planck's law, is an equation that describes the spectral energy density of blackbody radiation. It was developed by German physicist Max Planck in 1900 and is considered one of the fundamental equations in quantum mechanics.

2. How is the Planck formula related to the density of photons?

The Planck formula includes the term "photon density" which represents the number of photons present in a specific volume. This term is directly related to the density of photons, as the higher the photon density, the higher the number of photons present in a given volume.

3. How does the Planck formula explain the behavior of photons?

The Planck formula explains the behavior of photons by describing their spectral energy density, which is the amount of energy that a photon has at a specific wavelength. The formula also takes into account the frequency and temperature, which affect the behavior of photons in a given system.

4. What is the significance of the Planck formula in modern physics?

The Planck formula is significant in modern physics because it was one of the first equations to incorporate quantum mechanics and explain the behavior of photons and other particles at a microscopic level. It has been used in various fields such as astrophysics, cosmology, and quantum optics to understand the behavior of particles and their interactions with electromagnetic radiation.

5. Can the Planck formula be applied to other types of radiation besides blackbody radiation?

Yes, the Planck formula can be applied to other types of radiation such as thermal radiation from a heated object or even the cosmic microwave background radiation. However, it is important to note that the formula is specifically derived for blackbody radiation and may not accurately describe the behavior of other types of radiation in all scenarios.

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