Undergrad Two molecules with different polarizability in an EM field

[Salmone]

If I have two separated and non-interacting molecules with different constants polarizabilities $\alpha_1$ and $\alpha_2$ and I send an EM field of frequency $\omega$ first on the molecule no.$1$ and then on the molecule no.$2$ so that the two molecules will have a dipole moment $\vec{p_1}=\alpha_1\vec{E}$ and $\vec{p_2}=\alpha_2\vec{E}.$ What differences will there be in the two molecules?

1) Will they both oscillate with the same frequency as the EM field?

2) What does the difference in polarizability imply? That they will oscillate with the same $\omega$ but the oscillations of the molecule with the larger $\alpha$ will have larger amplitude?

3) Can one know by spectroscopy which of the two molecules has greater polarizability? I mean, we send EM field on the molecules and then the molecules begin to oscillate and generate a new EM field, in the domain of the frequencies we see the external EM field at frequency $\omega$ and the EM field produced by the molecules at the same $\omega$ (if my hypothesis are right), so is it possible to distinguish, by spectroscopy, two differents polarizability in a case like this one?

 

[Twigg]

1) Yes. You can think of it by analogy with a classical driven oscillator.

2) Classically, yes. Keep in mind that the classical model doesn't reflect reality very well for a single molecule. Another, more general consequence is that the potential energy of the two molecules will be different. For example, you can design an optical dipole trap at a "magic wavelength" $\lambda = \frac{2\pi c}{\omega}$ such that molecule #1 experiences a potential well but molecule #2 does not.

3) In principle it would be possible, but in practice the effect of the field produced by the oscillating dipole is tiny. Instead, spectroscopists usually measure the effect of the driving EM field on the molecules. The simplest technique is to measure the AC Stark shift on a molecular transition. This tells you the difference in polarizability between the initial and final states of the transition. Measuring the absolute polarizability of a single state is more a hassle, but there are ways. One of the more interesting ones is with atom interferometry. A much simpler method is to measure a related quantity, the transition dipole moment, and calculate the polarizability from that (see this paper for the math for atoms). This is how the labs I've worked in have done it. For molecules, the math is more complicated. Brown & Carrington's or Lefebvre-Brion & Field's spectroscopy bibles are the conventional place to start learning about molecular spectroscopy.

 

[Salmone]

2) Classically, yes.

Thanks, I am trying to get a schematic idea of the Raman effect and I wanted to figure out how to imagine an EM field encountering a molecule with a polarizability value that changes like a cosine, if the difference in polarizability is to change the amplitude of oscillation it is not clear how Stokes and anti Stokes lines can exist, the EM field finds a molecule with different polarizability and makes it oscillate at the same frequency but with oscillations of always different amplitude so it should always emit at the same frequency but perhaps the fact that it only classically behaves that way can explain the Raman effect.

 

[Twigg]

Sorry, there's a lot here I don't follow.

a molecule with a polarizability value that changes like a cosine

Can you elaborate? Are you saying that the polarizability of the molecule for an applied electric field that points at angle $\theta$ relative to the internuclear axis is proportional to $\cos \theta$?

if the difference in polarizability is to change the amplitude of oscillation

It can also change the direction ("polarizability tensor"), but I'm not sure where you're going with this.

it is not clear how Stokes and anti Stokes lines can exist, the EM field finds a molecule with different polarizability and makes it oscillate at the same frequency

"Lines" don't exist at all in a classical molecule. You only have lines when you have discrete states, and that only happens in a quantum mechanical theory of the molecule.

If we're talking about a semiclassical picture here, where the EM field is treated classically and the molecule is treated as a quantum bound system, then we can explain (coherent) Raman scattering. The EM field at frequency $\omega$ will cause Rabi flopping on both the carrier and (anti-)Stokes transitions. Even if the light is resonant with the carrier transition but not the (anti-)Stokes transition, there is some phase evolved on the (anti-)Stokes Rabi cycle, so there is some probability that the molecule will be found in the third state.

 

[Salmone]

I think I have to open a new thread in order to talk about Raman scattering, your first answer was clear about my first question, thank you.