Predicting Spectral Lines: Formula for All Elements

[prasannapakkiam]

Okay. I have made a reasonably accurate spectrometer. Is there a formula that can predict the lines for a given element with a given atomic number. I have done research. The best that I found, is the formula for 'hydrogen-like' elements. So is there a formula to predict all spectral lines of any ELEMENT?

 

[KingNothing]

As with most things, the more specific you get, the less an all-encompassing formula works.

 

[prasannapakkiam]

So there isn't one?

But with all this discovery on orbitals and quantum mechanics, how come nobody has derived a formula for this situation?

 

[Claude Bile]

There are equations, however they quickly become intractable once you start adding more and more electrons to the system - you need to make appropriate approximations (and a decent supercomputer) to predict an emission/absorption spectrum.

Claude.

 

[prasannapakkiam]

There are equations, however they quickly become intractable once you start adding more and more electrons to the system - you need to make appropriate approximations (and a decent supercomputer) to predict an emission/absorption spectrum.

Claude.

Don't think I have any of those lying around lately...

 

[KingNothing]

I don't think you need any supercomputers.

 

[prasannapakkiam]

Well, I discussed with my friends. If we know the first ionisation energies, surely this soud be enough t predict the lines. Are my friends and I just talking rubbish or is there some approximation or relationship to couple those 2 concepts.

 

[Claude Bile]

So is there a formula to predict all spectral lines of any ELEMENT?

Ah, now if you only want to know the values of specific transitions - and you have some energy values such as ionisation energies to aid you, then the problem becomes trivial.

If you know the ionisation energy from two different energy levels, then the energy corresponding to a transition between those two levels is simply the difference in the ionisation energy. Draw yourself an energy level diagram if you are not convinced - remember that the ionisation energy is the energy difference between the orbital of interest (negative) and a vacuum (zero).

Claude.

 

[prasannapakkiam]

thanks, I didn't know that it was quite so simple...

 

[damgo]

Just looking at ionization energies will cause you to miss most transitions, though, and include some that don't exist. Ionization energies are AFAIK given only for the energy levels occupied in the atom's ground state. ie for lithium you have three ionization energies corresponding to the three lowest energy levels. In reality, however, most transitions will involve higher energy levels... eg a transition of the valence electron 2s -> 2p or 2s->3s in lithium.

In addition there are various "selection rules" governing which transitions are possible. For example, photons have spin-1, so in single-photon transitions the total angular momentum must change by 0 or +-1.


In practice you would just look up the spectral lines in a book (eg the CRC) rather than trying to calculate themselves. In the early part of the 20th century there was a whole area of physics devoted to working out predictions (rather postdictions) of spectral lines. As Claude mentioned, it's just elementary quantum mechanics -- you just solve the Schrödinger equations for n interacting electrons in a Coulomb potential -- but this quickly becomes hairy and cleverness in choosing appriximation schemes is necessary. (Or you can just numerically solve the system using a computer.)