Monthly Archives: September 2020

Symmetry, selection rules and reduction to a bare-bones model

When I was a graduate student, a group of us spent long hours learning group theory from scratch in effort to understand and interpret our experiments. One of our main goals back then was to understand Raman and infrared selection rules of phonons. We pored over the textbook and notes by the late Mildred Dresselhaus (the pdf can be found for free here). It is now difficult for me to remember what it was like looking at data without the vantage point of symmetry, such was the influence of the readings on my scientific outlook. Although sometimes hidden behind opaque mathematical formalism, when boiled down to their essence, the ideas are profound in how they simplify certain problems.

Simply stated, symmetry principles allow us to strip unnecessary complicating factors away from certain problems as long as the pertinent symmetries are retained. In this post, I will discuss Raman and IR selection rules in a very simple model that illustrates the essence of this way of thinking. Although this model is cartoonish, it contains the relevant symmetries that cut right to the physics of the problem.

To illustrate the concepts, I will be using the following harmonic oscillator-based model (what else would I use?!). Let’s consider the following setup, depicted below:

Figure 1: Two identical masses, two walls, three springs

It’s relatively intuitive to see that this system possesses two normal modes (see image below). One of these normal modes is inversion symmetric (i.e. maintains the symmetry about the dashed vertical line through the entirety of its oscillatory motion), while the other normal mode is manifestly inversion asymmetric (i.e. does not maintain symmetry about the dashed vertical line through the entirety of its oscillatory motion). In particular, this latter mode is anti-symmetric. These considerations lead us to label the symmetric mode “even” and the anti-symmetric mode “odd”. (I should mention that in group theoretical textbooks, even modes are often labelled gerade (German for even), while odd modes are labelled ungerade (German for odd), from which the u and g subscripts arise). Normal modes for the system of two oscillators are depicted below:

Figure 2: (Left) An even (gerade) normal mode. (Right) An odd (ungerade) normal mode.

These are the “natural” modes of the system, but our ability to observe them requires us to excite these modes in some way. How do we do this? This is where the analogy to IR and Raman spectroscopy comes in. I’ll describe the analogy in more detail below, but for now consider the possibility that we can move the walls in the oscillator system. Consider moving the walls in the following way. We can oscillate the walls back and forth, moving them closer and farther apart as depicted in the image below. Clearly, this oscillatory wall motion is also symmetric.

Figure 3: A symmetric perturbation that will excite the symmetric mode. (Left) Depiction of the wall motion at time t and (Right) at time t+T/2, where T is the period of oscillation.

This obviously isn’t the only way that we can move the walls. We could just as easily move them like this, which is anti-symmetric:

Figure 4: An anti-symmetric perturbation that will excite the anti-symmetric mode. (Left) Depiction of the wall motion at time t and (Right) at time t+T/2, where T is the period of oscillation.

While there are many other ways we could move the walls, it turns out that the above images essentially capture how Raman (gerade) and infrared (ungerade) modes are excited in a solid. Infrared modes are excited using an odd perturbation (proportional to the electric field \vec{E}), while Raman modes are excited with an even perturbation (proportional to two instances of the electric field \vec{E}\vec{E})^{**}. (Under an inversion operation, the electric field switches sign, thus the infrared perturbation is odd while the Raman perturbation is even). And that’s basically it — you can see from the images that an even (odd) perturbation will result in the excitation of the even (odd) normal mode!

While this model is unlikely to be taught in classrooms any time soon in reference to Raman and IR selection rules, it does capture the physical picture in a (what I would consider) meaningful way through the use of symmetry. You can even imagine changing the relative masses of the two blocks, and you would then start to see that the formerly IR and Raman modes start to “mix”. The normal modes would no longer be purely even and odd modes, and the perturbations would then excite linear combinations of these new modes (e.g. the even perturbation would excite both modes). The analogous spectroscopic statement would be that in a system that lacks inversion symmetry, normal modes are not exclusively Raman or IR active.

While pictures like this won’t give you a precise solution to most questions you’re trying to answer, they will often help you identify obviously wrong lines of reasoning. It’s been said that physics is not much more than the study of symmetry. While that’s not exactly true, it’s hard to overstate its importance.

^{**} Why is the Raman excitation even? There are many ways to explain this, but on a cartoon level, the first photon (the electric field vector) induces a dipole moment, and the second photon (the other electric field) interacts with the induced dipole. Because this is a two-photon process (i.e. photon-in, photon-out), the excitation is even under inversion. (I should mention that the strength of the induced dipole moment is related to the polarizability of system, which is often why folks talk about the polarizability in relation to Raman spectroscopy).

Why is the infrared excitation odd? Contrary to the Raman excitation, the infrared excitation requires the absorption of the incoming photon. Thus, infrared spectroscopy is a single photon process and requires only a single electric field vector to couple to a dipole moment. The excitation is thus odd under inversion.