Interstellar medium: Theory and models.

[Billy T]

In principle there could be transitions between substates of different angular momentum for high quantum numbers, but the point is that the energy difference between these is so small that the transition probability is practically zero: the energy difference for different l would only be caused by the spin-orbit interaction of the electron and this decreases like 1/r^3 with increasing distance from the nucleus. Since the orbital radius increases like n^2 with the principal quantum number n, this means that the energy difference betwwen diffeent l-values decreases like 1/r^6, which means in state n=50 it would be less than 1/10^8 compared to the difference in n=2 and the frequency of the transition would be correspondingly reduced as well (i.e. in the kHz rather than microwave range), which means that the transition probabilities between these states are practically zero (as I mentioned above, already the transition probability decreases with frequency v like 1/v^3, i.e. it would be more than 24 orders of magnitude smaller than transitions than transitions into lower n states). So one can say that in fact there are no transitions between different l values for such high n.
Also, the highest angular momentum values will hardly be occupied by recombination anyway. The maximum population is at small l and first decreases slowly and then very rapidly towards l=n-1 (at least that's what my own computations using the exact wave functions for hydrogen revealed).

OK - looks like no need to "correct" CBMR for this effect. But your last paragraph (I think) is ignoring where the electrons come from. - I amthinking that the typical free electron that gets captured (a recombination) has a lot of angular momentum and would very likely fall into a high l orbital. Is it not true that the continium photon released as it recombines can only carry away one unit of agular momentum? - Am I missing something or just too ignorant about all of this?

 

[Thomas2]

But your last paragraph (I think) is ignoring where the electrons come from. - I amthinking that the typical free electron that gets captured (a recombination) has a lot of angular momentum and would very likely fall into a high l orbital. Is it not true that the continium photon released as it recombines can only carry away one unit of agular momentum? - Am I missing something or just too ignorant about all of this?

You can't actually assign an angular momentum to a free electron because l can take on any integer number here (rather than l=0...n-1 for the bound state n). The angular momentum you have to assume for the free electron is merely determined by the angular momentum of the bound electron: if for instance a bound electron with l=3 is photoionized, the l-selection rule tells you that the resulting photoelectron must have l=2 or l=4. Reversely, you can assume that a free electron recombining into a bound state with l=3 must have had an angular momentum l=2 or l=4. But this is merely a formal rule in order to fix the corresponding free parameter of the wave function of the free electron when calculating the overlap integral for the transition. Apart from that it makes no sense to associate an angular momentum quantum number to a free electron as it does not have a discrete but a continuous energy spectrum. And as I said, even for a single electron with a fixed positive energy, the l-value for the wave function of a free electron can take on any integer number, i.e. the probability of having a particular l-value is statistically actually zero. This shows that the angular momentum quantum number for a free electron can only be defined through the angular momentum of the corresponding bound state involved in the transition (by means of the l-selection rule).

 

[Thomas2]

OK - looks like no need to "correct" CBMR for this effect?

I am not quite sure what you actually have in mind regarding the corrections for the CMB radiation:

Obviously, any contributions from within our galaxy will be highly unisotropic and thus can be easily subtracted (this is actually how WMAP or other experiments obtain the CMB background). Although some assumptions are being made here regarding the physical nature of the interstellar background emission (taking also spectral characteristics into account) this is as far as I am aware essentially an empirical correction (using sky maps obtained in other wavelength bands) which does not try to model the physics behind the production of the interstellar microwave background. So in this sense it does not really matter whether the interstellar microwave radiation is produced by atomic cascading between high levels or other mechanisms. But as I pointed out earlier, if cascading is responsible, then this component of the microwave radiation should be observed as a line spectrum (the lines will be broadened by the plasma, but because of its low density not sufficiently to make the spectrum appear continuous).

On the other hand, radiation from large intergalactic distances will be redshifted, so any microwave radiation would appear at much lower frequencies (and again you would have a discrete line spectrum). In fact, if you assume that the CMB has been redshifted by a factor of a few thousand, you find that this radiation was originally in the visible spectrum, which suggestes that it is redshifted radiation of distant stars, i.e. radiation produced by transitions between the lower atomic levels (because of the high plasma density in the photosphere of stars, the lines are broadened by such an amount here that the spectrum becomes continuous even in the visible region, not to mention the infrared or microwave region).

I have actually written a computer program years ago which calculates the radiation emitted by recombination and cascading in a plasma and takes all the issues that have been discussed in this thread into account. The problem is that, as I mentioned above already, for relatively low plasma densities much of the spectrum becomes a line spectrum and this leads to a very poor convergence of the numerical algorithm as the intensity at the chosen frequency points varies wildly (you can see this from this plot which holds for a plasma density of 10^14 cm^-3 ; for a plasma density of 10^3 cm^-3 (which would be about the highest plasma density encountered in the interstellar medium) the line region would fill actually the whole frequency range shown (in fact it would even extend two orders of magnitude further left to about 10^8 Hz). Since CMB observations are usually in the range 10^10 Hz - 10^11 Hz, these would be therefore definitely in the line region and the exact frequeny of the observations would then be very critical regards what is observed.

 

[Billy T]

You can't actually assign an angular momentum to a free electron because l can take on any integer number here (rather than l=0...n-1 for the bound state n). The angular momentum you have to assume for the free electron is merely determined by the angular momentum of the bound electron: if for instance a bound electron with l=3 is photoionized, the l-selection rule tells you that the resulting photoelectron must have l=2 or l=4. Reversely, you can assume that a free electron recombining into a bound state with l=3 must have had an angular momentum l=2 or l=4. But this is merely a formal rule in order to fix the corresponding free parameter of the wave function of the free electron when calculating the overlap integral for the transition. Apart from that it makes no sense to associate an angular momentum quantum number to a free electron as it does not have a discrete but a continuous energy spectrum. And as I said, even for a single electron with a fixed positive energy, the l-value for the wave function of a free electron can take on any integer number, i.e. the probability of having a particular l-value is statistically actually zero. This shows that the angular momentum quantum number for a free electron can only be defined through the angular momentum of the corresponding bound state involved in the transition (by means of the l-selection rule).

This seems valid to me provided that there is no concurrent interaction with a second electron that does not recombine. I know so little about the ISM that I can not guess if this is the case.

I just have the classical intuition that the typical free electron has velocity vector with relative large "impact parameter" relative to the proton that will capture it. (Lots of angular momentum to conserve.) Thus, I would expect it to fall into a high n level, which can also have a high l value. You have told me that the transition probably between close levels is small (Proportional to the cube of frequency of photon emits, if I remember what you said correctly.) but it is also true that delta l is only + or - 1 for radiative cascade down. Thus if free electron is captured into n= 50 & l = 47 level, as I think might be typical, I am confused as to how it could be quickly dropped into a low n level (as I think you have also told me.) Sorry if this problem is only due to my lack of time to go back and reread carefully what you have said.

 

[quantumdude]

Thomas,

Your userid's have been banned, and any future userid will be banned as well. The people on this site have had quite enough of you touting your misunderstandings of physics, and so your contribution is not welcome here anymore. There are many places on the internet at which you can spread your crackpot nonsense. This is not one of them.

Stop creating new accounts at Physics Forums.