Grating Spectroscopy Theory for Undergraduate Lab Assignment

[Sojourner01]

OK, I have a lab assignment on absorption spectra. The nuts and bolts of generating a spectrum are a) pretty clear to me and b) nothing to do with quantum mechanics, but a large part of the assignment is theory on why the spectrum looks the way it does. The sample is vapour of molecular iodine illuminated with a tungsten lamp - so pretty low-tech, but this is an undergraduate lab.

So far I've come up with the Franck-Condon principle for explaining the numerous spectral lines. Wiki has plenty to say on the subject, and so far I've gathered that:

- The very prominent absorption lines are due to electron transitions into different principal quantum numbers i.e. different values of n.
- When one of these transitions occurs, the vibrational mode of the molecule can also change instantaneously provided that the initial and final equilibrium positions of the nuclei are identical.
- The likelihood of transitioning into a particular combination of n and vibrational mode is related to the overlap of the initial and final states' wavefunctions.
- The numerous, less prominent absorbtions in a spectrum are due to vibrational mode transition

However, I'm very fuzzy on the formalism. The wiki article talks about a 'vibrational quantum number' - now this isn't one of the 4 I know of - principal, magnetic, angular momentum and spin - unless someone's using bad terminology, which annoys me no end.

Plus, I don't understand bra-ket notation because our course doesn't teach it, and pretty much all quantum mechanics articles on wiki are expressed in bra-ket.

 

[OlderDan]

OK, I have a lab assignment on absorption spectra. The nuts and bolts of generating a spectrum are a) pretty clear to me and b) nothing to do with quantum mechanics, but a large part of the assignment is theory on why the spectrum looks the way it does. The sample is vapour of molecular iodine illuminated with a tungsten lamp - so pretty low-tech, but this is an undergraduate lab.

So far I've come up with the Franck-Condon principle for explaining the numerous spectral lines. Wiki has plenty to say on the subject, and so far I've gathered that:

- The very prominent absorption lines are due to electron transitions into different principal quantum numbers i.e. different values of n.
- When one of these transitions occurs, the vibrational mode of the molecule can also change instantaneously provided that the initial and final equilibrium positions of the nuclei are identical.
- The likelihood of transitioning into a particular combination of n and vibrational mode is related to the overlap of the initial and final states' wavefunctions.
- The numerous, less prominent absorbtions in a spectrum are due to vibrational mode transition

However, I'm very fuzzy on the formalism. The wiki article talks about a 'vibrational quantum number' - now this isn't one of the 4 I know of - principal, magnetic, angular momentum and spin - unless someone's using bad terminology, which annoys me no end.

Plus, I don't understand bra-ket notation because our course doesn't teach it, and pretty much all quantum mechanics articles on wiki are expressed in bra-ket.

This is my first encounter with the Franck-Condon Principle, so I did a quick google myself and found this article

http://www.life.uiuc.edu/govindjee/biochem494/Abs.html

I think this is a pretty good overview without resorting to the mathematics of state transition probabilities. The quantization of vibrational states and associated quantum numbers is not someone's bad terminology. One of the problems you solve in introductory QM is the harmonic oscillator. A quick refresher on that is here

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

What the first article is saying is that the electron energy levels associated with the quantum numbers you are already familiar with are altered by the quantized vibrational states of the molecules. The vibrational levels are associated with variations in the internuclear distances of the molecule, and the nuclear separation affects the energy levels of the states you are familiar with. This results in the precise levels associated with the other three quantum numbers being split into many closely spaced levels associated with the vibrational states of the molecule. The Franck-Condon Principle is that the state transitions of the electrons are too fast to immediately change the relative momentum and positions of the heavy nuclei, so the absorption transitions between what you are accustomed to thinking of as the ground and excited electron states occur at a particular internuclear separation that often involves transitions from a lower to a higher vibrational energy state.

It's pretty involved stuff, but the bottom line is that the vibrational states add a bit of energy to the states you are already familiar with, creating many more electron energy states available for absorption transitions.

Some others might be able to do a better job of explaining the details, but quantized vibrational states and associated quantum numbers are not someone's fiction.

In solids, the vibrational states are waves within the solid. The energy states of these waves are quantized and there are wave functions associated with the positions and vibrating modes of the atoms. These vibrations are referred to as phonons, and they play an important role in determining the thermal and electrical properties of materials.

 

[Sojourner01]

Thanks for that, glad my understanding is accurate.

I was confused about the vibrational quantum number thing because we haven't touched molecular quantum mechanics at all yet, only single atoms. I come to expect mistakes from wikipedia, hence my scepticism.

I'm still not entirely clear on why vibrational states must be quantised, but I'll accept it for now.

I'm probably getting ahead of myself a little here, but for the formal paper we're expected to produce I'll probably have to derive the full iodine spectrum from vibronic transitions. This is pretty heavy stuff as so far in QM classes we've done: (i) the harmonic oscillator (ii) the hydrogen atom, and little else.

 

[OlderDan]

I'm still not entirely clear on why vibrational states must be quantised, but I'll accept it for now.

I'm probably getting ahead of myself a little here, but for the formal paper we're expected to produce I'll probably have to derive the full iodine spectrum from vibronic transitions. This is pretty heavy stuff as so far in QM classes we've done: (i) the harmonic oscillator (ii) the hydrogen atom, and little else.

You should be no more surprised that vibrational states are quantized than you are that light and atomic electronic states are quantized. Nobody even suspected the microscopic world behaved the way it does until forced to thinking in new ways to explain phenomena that could not be understood in terms of classical physics. Now that you know that things on the atomic level are quantized, what would be shocking would be to find something that was not.

You are being asked to do a lot if you are expected to derive the states corresponding to the spectra you observe. The good news is that the usual approximation, which seems to work well, is that the wave functions for the vibrating molecule are separable into an electronic part and a vibrational part. You will recall from the hydrogen atom that being able to separate the variables makes the problem much easier to deal with. Even so, the presence of a second nucleus in the neighborhood of the electron adds considerable complexity to the electronic part of the wave function.

If you think you need to apply your knowledge of the Schroedinger equation approach used for the hydrogen atom to the more complex problem of diatomic molecules, you might take a look at this

http://www.ubishops.ca/ccc/div/sci/chem/quantum/HAMMOL.html

I tend to doubt you are expected to master something this complex after just learning the hydrogen atom and harmonic oscillator