Mod 7 UV Catastrophe Question (1 Viewer)

[askit]

Hey guys, could anyone explain to me how classical theory of light assuming that every oscillator having the same kinetic energy at a given temperature is an incorrect assumption leads to the ultraviolet catastrophe. Also, I get how quantising energy abides by the law of conservation of energy (original theory of infinite emission of energy doesn't make sense) but I still don't really understand how this relates to the intensity wavelength graph. Any help much appreciated

 

[wizzkids]

Firstly, let's clear up your description of the "classical" intensity distribution function. The "classical" intensity distribution function was derived by Lord Rayleigh and Sir James Jeans around 1905 - 1910. This function plots the intensity (watts per square metre per steradian) versus frequency (or its inverse, wavelength). Here is an outline how this model was developed. A classical oscillator can store energy in two forms - kinetic and potential. Each form will acquire (on average) 1/2kT of energy, so that makes kT energy per oscillator (where k is the Boltzmann Constant and T is the absolute temperature in Kelvin). Every oscillator can spend this energy "budget" in a mode that has a particular frequency or wavelength, and there are no excluded values. All frequencies and all wavelengths are permitted. If the energy is equally distributed across all frequencies, then when the data is plotted with wavelength on the horizontal axis, the energy density tends to go to infinity at short wavelengths because there is an infinite number of frequencies at short wavelengths. This was called the "ultraviolet catastrophe" (it was a catastrophe for the theory, not for the real world). Now you might be thinking that the Rayleigh-Jeans Law was useless, but it did give a reasonable fit to the blackbody radiation at long wavelengths, as shown below so it had some validity.

Prof. Max Planck introduced a "correction" to the Rayleigh-Jeans Law, which avoided the "ultraviolet catastrophe", and brought the radiation function into agreement with experiments.
Planck's correction postulated that all radiation is emitted and absorbed in quanta of energy given by E = hf.
He still assumed that all oscillators would have average energy "budget" of kT. However there would be some frequencies or wavelengths that would now be impossible or excluded because the average "budget" of kT was smaller than the minimum quantum hf required to emit that radiation, so the radiation intensity function would naturally drop off sharply for frequencies (energies) above hf/kT.
I hope that helps.

 

[askit]

Thankyou so much for your response, I've got a few questions from your explanation

Every oscillator can spend this energy "budget" in a mode that has a particular frequency or wavelength, and there are no excluded values - given that on average they all have kT energy ie. all oscillators have the same energy, what causes them to spend their "budget" on different frequencies?

energy density tends to go to infinity at short wavelengths because there is an infinite number of frequencies at short wavelengths -confused with this part, can't we same the same for any point on the graph eg. theres an infinite amount of frequencies for long wavelengths?

 

[wizzkids]

Thankyou so much for your response, I've got a few questions from your explanation

Every oscillator can spend this energy "budget" in a mode that has a particular frequency or wavelength, and there are no excluded values - given that on average they all have kT energy ie. all oscillators have the same energy, what causes them to spend their "budget" on different frequencies?

energy density tends to go to infinity at short wavelengths because there is an infinite number of frequencies at short wavelengths -confused with this part, can't we same the same for any point on the graph eg. theres an infinite amount of frequencies for long wavelengths?

Good questions - this shows you are really engaging.
(i) Something I didn't mention before it is extremely unlikely that all the oscillators will have exactly the average energy kT. Entropy will make sure that this doesn't happen. There is a spread of energies or a distribution function for these energies. This distribution function was figured out by our old friend, Ludwig Boltzmann way back in 1868. This creates a "natural" spread to their energies.
(ii) No, there is not an infinite amount of frequencies for long wavelengths. I will borrow the model used by Lord Rayleigh. Consider a box or cavity that is at thermal equilibrium confining the radiant energy, and the waves form standing waves when bouncing backwards and forwards between the ends of the box. The shortest standing wave that will fit between the ends of the box is half λ equal to the length of the box. This fixes a lower limit to the frequency of the radiation. The next standing wave is when λ is equal to the length of the box, then 3/2 λ and so on, however there is no upper limit to the standing waves that can fit between the ends of the box. You can keep increasing the frequency to infinity and they can still form standing waves between the ends of the box. So there is a lower limit to the allowed frequency but no upper limit to the allowed frequency.
As for the shape of the Intensity distribution curve, if you start by plotting intensity versus frequency, and then you transform the horizontal axis by the reciprocal function (inverse transform function λ = c/f) to turn it into intensity versus wavelength, this distorts the horizontal axis, it stretches out the intensity for long wavelengths (low frequencies) and squeezes the function for short wavelengths (high frequencies) so you get this concentration of the intensity towards short wavelengths. Now, we know in the real world this doesn't happen, so there must be another mechanism that cuts off the intensity sharply when hf > kT , and this was Max Planck's hypothesis. He then had to empirically work out what the constant "h" needed to be to fit the data, and he came up with h=6 x 10^-34 Joule.seconds

 

[ilovemangos]

in the hsc/school exam, how much detail would you include for question that asks how Planck's quantisation of energy resolves the UV catastrophe? would you just talk about the probability of certain frequencies being lower

 

[askit]

in the hsc/school exam, how much detail would you include for question that asks how Planck's quantisation of energy resolves the UV catastrophe? would you just talk about the probability of certain frequencies being lower

From what I've seen so far, yea its good enough to just talk about E=hf, f increases, E increases thus probability decreases, but I really don't like how HSC physics is structured that way. I keep getting confused because half the time it just wants me to accept things

 

[wizzkids]

I think you could say, "Planck showed that there was a decreasing intensity of radiation for frequency f > kT/h because radiant energy was quantized according to E = hf and if E >>kT then it could not be emitted."

 

[MaccaFacta2]

The responses and questions above are excellent. I'd like to add a perspective that I have found to be helpful with my own HSC Physics students.

If you are asked a question about energy quantisation, use the visible emission spectra of the hydrogen atom as your example (or absorption spectrum - it is the same physics in reverse). The reason we get the red, blue-green and two violet lines from a hydrogen discharge tube is because hydrogen's electron can only move around the hydrogen nucleus in quantised energy levels (or orbitals if you are studying HSC Chemistry). These wavelengths (656.3 nm, 486.1 nm etc) are specific because the electron can only form a standing wave at particular radii from the nucleus.

On the "oscillator" question - a useful analogy is to consider "twanging" a ruler with different distances hanging over the edge of your desk. The shorter the length, the higher the frequency of the "twang". At long overhangs, there is (to a very good approximation) a continuum of lengths that can be twanged. Rayleigh and Jeans developed their equation at a point where atoms were speculative (Ernst Mach once famously declared, after an 1897 lecture by Ludwig Boltzmann at the Imperial Academy of Science in Vienna, "I don't believe that atoms exist!"). So there was no need to postulate a lower bound on the length of the ruler. These days we would figure that a 10 atom overhang would oscillate at a different frequency from, say, an 11 atom overhang. A 10.5 atom overhang could be imagined as an impossibility.

Both of the examples I have used have deadly flaws in them because I'm trying to map non-quantum analogies onto quantum systems. Be aware that HSC Physics is an introduction to important ideas - it is certainly not the last word on them.