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Is there anyone going to Astrofest 2024 in London who is familiar with the concept and determination of “oscillator strength” and is prepared to discuss the topic with me over lunch? I am attending on both days.
Regards
Ken WhightDavid: I am supposed to know what “oscillator strength” means in the context of time dependent quantum mechanics but my memory has rotted in the last 40 years or so and can no longer describe it particularly coherently. Time I ran a refresh cycle on my RAM.
Mark me as recently mystified.
Beside which, I will not be Astrofesting this year.
Sorry if my post was a little short of information. My request is directly linked to my other topic raised in this spectroscopy section i.e. Spectral Line Modelling.
When I got interested in astronomical spectroscopy I decided to start a project to see what information I could extract from the spectra I was collecting. I was expecting to find, in the literature, a ready made simple global thermodynamic equilibrium analysis but couldn’t find one and so started to develop one myself.
It was a much more difficult task than I anticipated (I am more of a solid state physicist/mathematician/numerical modeller) and my project took about 10 (often frustrating!) years to complete but in the end I was quite successful (gaining interesting insights on the way) at determining the pressure and thickness of the Sun’s photosphere and made predictions for other stars.
Trouble is my model uses thermal equilibrium photon capture cross-sections, calculated from thermal equilibrium functions (Boltzmann & Planck) and Einstein A coefficients rather than conventional “oscillator strengths” which seem to require an additional somewhat mysterious quantum mechanical calculation.
I was hoping to meet someone who was familiar with the conventional approach to discuss how the two approaches fit together.
My sources were:-
“Astronomical Spectroscopy for Amateurs” Ken M. Harrison, Springer
“Spectroscopy: The Key to the Stars” Keith Robinson, Springer
“Atomic Astrophysics and Spectroscopy” Anil K. Pradhan and Sultana N. Nahar, Cambridge University PressKen,
I fear this may be a rather trickier task then you think.
The first problem is that stellar atmospheres are not in thermal equilibrium. There’s convection going on, dredging up hot gas to the surface, which then cools and sinks back down. The cooling gas is not in thermal equilibrium.
Furthermore, stellar atmospheres are translucent: light can penetrate a certain depth into the atmosphere. Temperature and pressure change with depth. This means that to model a stellar spectrum you need to do ray-tracing through a finite depth of partially opaque medium, modelling absorption, emission, and photon scattering. This is the branch of astrophysics called radiative transfer.
And to make things worse, the quantum mechanical equations for atomic line spectra are not soluble for anything more complicated than hydrogen. Computational models of bigger atoms exist, but they’re notoriously inaccurate. To be sure of what an atom’s spectrum looks like, you really need to measure it empirically in a laboratory. But that’s hard, because it involves recreating the conditions at the surface of the Sun. Atoms need to be exceedingly hot to reach the ionisation states they attain in the Sun. So you need a very large laser, some very hot and very pure samples of each chemical element, and a very fast camera that can measure the spectrum of the plasma formed after the laser fires.
To compound the challenge, there are > 100 elements in the periodic table, each of which can have N-1 ionisation states, leading to thousands of atomic states, each with distinct spectra. And that’s just the atoms – stellar atmospheres have molecules too.
I have professional experience working with a couple of codes which attempt to model all of the above, and each is the culmination of several PhDs worth of work. Turbospectrum (Bertrand Plez et al. 2012) is simpler and open source. PySME (Nikolai Piskunov et al. 2018) is more sophisticated in its modelling of non-equilibrium effects, and is available in binary form online but not open source. Both rely on the VALD list of atomic lines – maintained at Uppsala University – which is essentially a synopsis of a vast number of laboratory studies. The oscillator strengths in there are mostly not calculated by quantum mechanics; they’re fitted to empirical data. Quantum mechanics may allow you to write down equations for oscillator strengths, but that’s not much good if they can’t be solved.
I hope that’s vaguely helpful.
Best wishes,
Dominic
Hi Dominic, Thanks for your interesting reply. In my career within Philips research I learned that modelling silicon devices was difficult enough even with the luxury of having them on the bench infront of you! However assuming thermal equilibrium got you a long way. We did go to a “hydrodynamic” model where the silicon lattice, electrons and holes could be at different temperatures but I couldn’t claim that that was a major advance in terms of designing better devices. I also know that “Monte Carlo” methods were necessary for modelling GaAs devices but I never had to get into that thankfully!
Yes stellar photospheres are not in a global thermal equilibrium but I was hoping that estimates of photosphere pressure and thickness would be “reasonable” for a fair number of stellar types (my estimates for the Sun seemed OK – see the attached file and others under my “Spectral Line Modelling topic). Having developed a global equilibrium model it could fairly easily be extended to a local equilibrium model (but not by me!) to cover more (main sequence?) stars.
However, the particular “insight” that I wanted to discuss was that whilst detailed balance under thermal equilibrium allows you to calculate the Einstein B coefficients in terms of a cross-section area (if you express the Planck function as a photon flux). These cross-sections infuriatingly only apply in the monochromatic case. In the polychromatic case they proved useless as each atom now has a choice of photon to capture. I think this is what necessitated the additional calculation of “oscillator strengths”. However if you use the properties of thermal equilibrium you can relate the polychromatic capture cross-sections simply to the Einstein A coefficients of any elemental spectral series as demonstrated in my attached paper Europa.pdf (equation A.4.12) and I’m hoping to test this out on the Sodium principle series for the Sun if I can get the necessary data, sadly my experimental setup is somewhat moribund (offers of data anyone?).Hi Ken,
Interesting that you worked at Philips Research. Our paths may well have crossed in the early 2000s, when I did three summer internships at PRL. At the time I was torn between a career in astronomy versus joining Philips, but the decision was made for me when PRL closed down. Just in case the world wasn’t already small enough – I’m guessing you worked in Alan Knapp’s group? His wife taught me chemistry at school…
As you say – you can get a long way by assuming local thermal equilibrium. How far is an interestingly controversial question. Without any independent way of measuring the physical conditions and composition of a star, it’s hard to verify exactly how accurate models are.
It’s a very long time since I’ve looked at these kinds of calculation, but I think the jigsaw piece you’re missing is Kirchoff’s Law. From memory, this has the consequence that any plasma that is in equilibrium for the polychromatic case is also in equilibrium with regard to emission and absorption at every monochromatic wavelength of light. The result is that you never need to solve the polychromatic case. You solve the equilibrium equations monochromatically for every wavelength you’re interested in. As I recall, if you’re interested in solving for the equilibrium occupation probabilities of the quantum states, your monochromatic equations give you a bunch of (thousands of) simultaneous equations that you can solve with a big (sparse) matrix inversion operation. You should end up with something resembling a Boltzmann distribution.
The oscillator strengths reflect the fact that transitions are more likely between quantum mechanical states with similar wavefunctions – which give rise to strong lines – versus those with very dissimilar wavefunctions – which give rise to weak “forbidden” lines. But calculating wavefunctions is somewhere between difficult and impossible, and numerical approximation often don’t seem to resemble reality particularly well. Hence the tendency to use empirical lab measurements.
Best wishes,
Dominic
Hi Dominic, Amazing! I transferred to Alan Knapp’s group in 2002 (or thereabouts) in the twilight of my career with Philips, avoiding redundancy in the process. I eventually took redundancy and early retirement when what was left of the lab moved to Cambridge in 2008. A sad end to what was part of a prestigeous institution that was up there (almost) with Bell labs. I had many interesting projects and conference/business trips over the years since I joined, in 1973, what was then the Mullard Research Laboratory. My happiest time was through the 1980’s to early 90’s modelling silicon power devices, computing power was increasing according to Moore’s law and the models could therefore become more and more detailed so I’m very familiar with solving thousands of sparse profiled matrix equations. I still have the software to do this (if you’re interested). It can solve (in principle) via LU decomposition any number of equations in any number of dimensions to any level of “fill in” all the way up to a direct solver using various iteration methods developed in the 1980’s (conjugate gradients, bi-conjugate gradients etc). I posted some of my work on the “legacy page” of my website http://www.thewhightstuff.co.uk.
Back to spectroscopy: I think you are referring to the principle of detailed balance in your last reply and that is what I used to determine the Einstein B coefficients but they vary too strongly between the Balmer series lines to reproduce my measurements. They are also temperature independent whereas my measurements, over a number of stars, definitley show a temperature dependance hence the reasoning leading to my equation A.4.12. Another pleasing feature of my model is that all calculated internal parameters have believeable values, the impact parameter for pressure calculation is approximately 8 Bohr radii and photon cross-sections are of the order of 10 Bohr areas.
I suppose I am looking for someone who would look at my model and say “yes this is how a ball of gas in thermal equilibrium would look spectroscopically” or “no it isn’t because….”, whether it’s a good model of any particular star is a separate question though the Sun looks to be well modelled in it’s gross features. I realise this is a big imposition on anyone so if it is of interest to you please continue this discussion via my email address ken.whight@btinternet.com and if you still live in the South East or are attending Astrofest it would be great to meet up. if it’s not of interest then thank you very much for taking the trouble to comment.
Regards
Ken WhightThe oscillator strengths reflect the fact that transitions are more likely between quantum mechanical states with similar wavefunctions – which give rise to strong lines – versus those with very dissimilar wavefunctions – which give rise to weak “forbidden” lines. But calculating wavefunctions is somewhere between difficult and impossible, and numerical approximation often don’t seem to resemble reality particularly well. Hence the tendency to use empirical lab measurements.
This is bringing back memories.
I am but a humble experimental chemist and know rather little quantitative about atomic spectroscopy, other than hydrogenic systems in the Schrödinger approximation. My DPhil research, was on robvibronic structure in the electronic spectra of Cu2 and CeO in the gas phase. More recently I helped the Exomol team with AlH. A beauty of that field of work is there is a plethora of rotational lines, by and large, and frequently a good number of vibrational bands. Fairly easy to measure the temperature also. With all those measurements fitting a potential curve is not entirely trivial, especially near dissociation, and very difficult when two states perturb each other. However, very accurate results are possible and it is (usually) straightforward to reverse engineer properties such as dipole moment, polarizability and so on. In particular, oscillator strengths — which is what this thread has done to refresh my RAM.
Thank you both.
(BTW, One state in AlH is barely bound and I failed to get a good enough approximatiom to the potential energy curve. Believe it or not, an ab initio calculation was the key to solving this one. It gave a fairly good PEC but a not particularly good absolute energy. The latter was known very precisely from the spectra and putting the two together gave excellent predictions for the oscillator strengths.)
Hi Paul, I’m glad the discussion revived your grey cells! If you are interested in understanding my work in more detail then you’re very welcome to contact me at my email address above. I suppose I should try to get it reviewed by submitting a paper to a journal, the trouble is that it falls between two stools i.e. to basic for professional journals and a bit maths heavy for amateur journals. I would have thought though that the approach would be good for teaching the basics of stellar spectroscopy at undergraduate level in a similar way to being taught the Bohr model of an atom.
Regards
Ken
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