Astrofest 2024

David: 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.

Ken,

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 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

Dominic Ford wrote:

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.

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.)