Blackbody Math Help: Calculating from Maxwell & Boltzmann

[Hydr0matic]

http://www.phys.virginia.edu/classes/252/black_body_radiation.html

Have a look at this site, specifically the calculation of Planck's formula. I'm sure you've seen it many times before.

What I need help with is calculating a similar formula based on my own idea instead of the oven model. I'll try to explain what I'm thinking by referring to parts of the above site.

Making it very short... in the oven model there are a bunch of oscillators in the walls of the oven, all oscillating at different frequencies, ie those that "fit" ($\lambda = 2a, a, 2a / 3 ..$). Planck's "fix" was to quantize the energylevels and assume higher energies where less likely to be emitted.

First, I just want to say that I think the oven model is very lacking when it comes to reflecting reality. Though perfect blackbodies don't exist, the "almosts" - ie stars - are best represented by a simple chaotic (huge) quantity of oscillators, all oscillating in random directions. There are no walls and no "fits". Unlike the oven model, a perfect blackbody would produce a perfectly continuous spectrum.

The model we're going to focus on is the most realistic one, ie the gas analogy.

In fact, the general shape of the function $R_T(f)$ was very reminiscent of the energy distribution in a gas in thermal equilibrium, as analyzed by Maxwell and Boltzmann

According to the theorem of equipartition of energy the oscillators should have an average energy

$$K_\textrm{av} = \frac{mv^2}{2} = \frac{3k_{b}T}{2}$$

and thereby average velocity

$$v_\textrm{av} = \sqrt{\frac{3k_{b}T}{m}}$$

Similar to the oven model approach we must first figure out which wavelengths the oscillators are emitting. The main frequency $$f_0$$ is emitted perpendicular to the direction of oscillation and is thereby not doppler affected. $f_\textrm{max}$ and $f_\textrm{min}$ are emitted along the axis of oscillation since this is where most dopplereffect occurs.

ie

$$f_\textrm{max} = \left(f_0 \frac{c+v_\textrm{av}}{c-v_\textrm{av}}\right)$$
$$f_\textrm{min} = \left(f_0 \frac{c-v_\textrm{av}}{c+v_\textrm{av}}\right)$$

What I'm saying is that every single oscillator in the blackbody emitts all frequencies part of the blackbody's spectrum. And unlike the oven model, the mechanism controlling the energy- and frequency-distribution is the oscillation itself. Let me explain..

In the case with the oven the only radiation that matters is the one radiated perpendicular to the oscillators' direction of oscillation (due to the "fit"). So for an oscillator in the oven wall

$$f_0 = f_\textrm{max} = f_\textrm{min}$$

So at any given time, a single oscillator only radiates a single frequency.

However, this is not the case in our gas analogy. Every single oscillator radiates in all directions at all time, and so all angles must be accounted for. But why should this mean oscillators radiate more than one frequency ? This is where dopplereffect comes into play. Let me make an analogy with atoms and photons:
http://hydr0matic.insector.se/fysik/dopplershift.jpg

The analogy is very simple. While the atoms in a gas bump around and vibrate, they radiate photons. Depending on where the observer is, the atom radiating a specific photon might be moving towards or away from him. This causes the photon to shift it's frequency accordingly. If the atom is moving towards the observer while emitting the photon, the photon will be blueshifted. Now, if we would have a gas with an infinite number of atoms, there would be an infinite amount of relative angles to the observer, which of course would produce a continuous spectrum. This spectrum would stretch from it's most redshifted frequency, $f_\textrm{min}$, to it's most blueshifted one, $f_\textrm{max}$.

To this point the analogy with our blackbody gas model is excellent, but continuing will reveal some crucial differences.

The first difference is quantum related. A quantum atom will emit photons sporadically, but a classical oscillating charge will radiate continuously in every direction. This means that it would take an infinite amount of time for a quantum atom to emit the whole continuous spectrum, while it would only take a brief moment for the classical charge to do the same.

The most important difference from a blackbody perspective is the fact that photon intensity doesn't change with the relative angle. The observer will see as many unshifted photons as blueshifted and redshifted. If this was the case in our blackbody, the intensity wouldn't vary as a function of wavelength (As the famous Planck curve). But on the other hand, if the atoms were oscillating charges and the photons were electromagnetic waves, the intensity would vary very much indeed. In fact, Maxwell's equations tells us that the amplitude of an electromagnetic wave is a function of the angle between the oscillation and the radiated wave. This means that the intensity is highest perpendicular to the oscillation, and lowest (zero) in the direction of oscillation. This is why the Planck curve slopes down again at higher frequencies ! There are two relationships controlling the intensity curve - dopplershift and Maxwell's equations. These two are related as follows:

More dopplershift = less intensity // No dopplershift = highest intensity

$$\theta = \angle$$ radiation / oscillation

$$D_\textrm{amount}$$ = The amount of dopplereffect

$$D_\textrm{amount} \propto \cos{\theta}$$

Amplitude of EM field $$A \propto \sin{\theta}$$ (no radiation in direction of oscillation)


Anybody feel like helping me formulate this properly ? (mathematically)

Comments are appreciated...

 

[AndyW]

Hello there, thank you for sharing your interesting idea! I can definitely see where you're coming from and can understand the limitations of the oven model in representing a real blackbody. Your gas analogy is certainly a step towards a more realistic model. However, I would like to point out a few things that may help in formulating your idea more accurately.

First, I think it's important to clarify the difference between a classical oscillator and a quantum atom. In your analogy, you mention that a classical charge would radiate continuously in all directions, while a quantum atom would only emit photons sporadically. This is because a classical oscillator follows classical mechanics, where energy is continuous and can take on any value. In contrast, a quantum atom follows quantum mechanics, where energy is quantized and can only take on certain discrete values. This means that a classical oscillator would emit a continuous spectrum, while a quantum atom would emit discrete spectral lines.

Next, I think it would be helpful to define what you mean by "oscillator" in your gas analogy. Are you referring to individual atoms or molecules in the gas, or are you considering the gas as a whole to be the oscillator? This will affect how you calculate the average energy and velocity in your model.

Additionally, I think it would be important to consider the effects of collisions in your gas analogy. In a real gas, particles are constantly colliding with each other, which can affect their energy levels and therefore their emission spectra. This could potentially complicate your model, but it may also provide a more accurate representation of a blackbody.

In terms of formulating your idea mathematically, I would suggest starting with the basics of blackbody radiation and then incorporating your gas analogy into the calculations. This would involve using the Planck formula and the Stefan-Boltzmann law, as well as considering the effects of doppler shift and Maxwell's equations on the emission spectrum. It may also be helpful to look at existing models of blackbody radiation, such as the Bose-Einstein distribution, and see how you can modify them to fit your gas analogy.

Overall, I think your idea has potential and could lead to a more realistic model of blackbody radiation. I would suggest continuing to refine and develop your idea, and perhaps seeking input from other scientists in the field. Good luck!