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.

List of actions

In the post before the previous one, I advocated for faculty and personnel in physics departments to reflect the make-up of the populace. Here is an excellent concrete list by Professor of Chemistry Kensha Marie Clark of the University of Memphis of actions that your department can undertake (click on the tweet for the whole list or on the unrolled tweet at the link). A lot of this list applies to other underrepresented folks as well:

https://threadreaderapp.com/thread/1268123251978403844.html

Meissner effect as amplified atomic diamagnetism

As you can probably tell from my previous post, I have found the recent activism inspiring and genuinely hopeful of it translating into some form of justice and meaningful action. At the end of this post I share a few videos that I found particularly poignant.

It’s hard to imagine the history of condensed matter physics without both the discovery and theory of superconductivity. Superconductivity has played and continues to play an outsized role in our field, and it is quite easy to understand why this is the case. All one has to do is to imagine what our world would look like with room temperature superconductivity. Besides the potential technological implications, it has also garnered attention because of the wealth of stunning effects associated with it. A few examples include the Josephson effect, flux quantization, persistent superconducting currents, vortex lattices and the Meissner effect.

Now, these effects occur for various reasons, but there are a couple of them that can be viewed to some extent as a microscopic effect on a macroscopic scale. To show what I mean by that, I am going to focus on the Meissner effect and talk about how we can view it as an amplification of atomic diamagnetism. One could also extend the this microscopic to macroscopic amplification picture to the relationship between a Josephson junction in a superconducting ring and the Aharonov-Bohm effect, but I’ll leave that discussion to another day.

To understand what I mean by amplification, let’s first look at atomic diamagnetism. Here we can use a similar logic that led to the Bohr model of the atom. Two conditions are important here — (i) the de Broglie relation \lambda = h/p and (ii) the Bohr quantization condition n\lambda = 2\pi r which states that only integer wavelengths are allowed in a closed loop (such as an atomic orbit). See the image below for a simple picture (click the image for the source).

We can use the classical relation for the momentum p=mv in addition to equations (i) and (ii) above to get mvr = n\hbar, which is what Bohr got in his atomic model. It’s worth noting here that when the atom is in its ground state (i.e. n=0), there is no “atomic current”, meaning that j = ev = 0. Without this current, however, it is not possible to have a diamagnetic response.

So how do we understand atomic diamagnetism? To do so, we need to incorporate the applied field into the deBroglie relation by using the canonical momentum. By making the “Peierls substitution”, we can write that p = mv+eA. Using the same logic as above, our quantization condition is now mvr = n\hbar - eAr. Now, however, something has changed; we do get a non-zero current in the ground state (i.e. j = ev = -e^2A/m for n=0). Qualitatively, this current circulates to screen out the field that is trying to “mess up” the integer-number-of-wavelengths-around-the-loop condition. Note also that we have a response that is strictly quantum mechanical in nature; the current is responding to the vector potential. (I realize that the relation is not strictly gauge invariant, but it makes sense in the “Coulomb gauge”, i.e. when \nabla\cdot A=0 or when the vector potential is strictly transverse). In some sense, we already knew that our answer must look obviously quantum mechanical because of the Bohr-van Leeuwen theorem.

If we examine the equation for the electromagnetic response to a superconductor, i.e. the London equation, we obtain a similar equation j = n_sev = -n_se^2A/m, where n_s is the superfluid density. The resemblance between the two equations is far from superficial. It is this London equation which allows us to understand the origin of the Meissner effect and the associated spectacular diamagnetism. Roughly speaking then, we can understand the Meissner effect as an amplification of an atomic effect that results in a non-zero ground state “screening” current.

I would also like to add that the Meissner effect is also visible in a multiply connected geometry (see below). This time, the magnetic field (for sufficiently small magnetic fields) is forbidden from going through the center of the ring.

What is particularly illuminating about this ring geometry is that you don’t have to have a magnetic field like in the image above. In fact, it is totally possible to have a superconducting ring under so-called Aharonov-Bohm conditions, where a solenoid passes through the center but the ring never sees the magnetic field. Instead, the superconducting ring “feels the vector potential”. In some sense, this latter experiment emphasizes the equation above where the current really responds (in a gauge-invariant way) to a vector potential and not just the magnetic field.

Understanding the Meissner effect in this way helps us divorce the Meissner effect from the at-first-sight similar effect of persistent currents in a superconducting ring. In the Meissner effect, as soon as the magnetic field is turned off, the current dies and goes back to zero. This is because through this entire process, the superconductor remains in its ground state. Only when the superconductor is excited to higher states (i.e. n=1,2,3…) does the current persist in a metastable fashion for a quasi-infinitely long time.

To me, understanding the Meissner effect in this way, which exposes the connection of the microscopic to the macroscopic, harks back to an old post I made about Frank Wilczek’s concept of upward inheritence. The Meissner effect somehow seems clearer through his lens.

Now as promised, here are the couple videos (if the videos don’t play, click on the panel to take you to the twitter website because these videos are worth watching!):

Listening

I am not Black. I am not American. I do not understand the many nuances of American and African-American culture. I do not understand the extra struggle African-American people have to go through each day. But there are some things that are easy to understand. It is easy to understand that the killing of yet another unarmed Black man is due to structural racism. It is easy to understand that the involved police officers did not view George Floyd as a man that was their equal. And it is easy to understand why people are incensed about this.

A lot needs fixing here, and it’s going to take a while for that to happen. But as I write this in Los Angeles with the sound of sirens going by my apartment every few minutes, it is hard not to think of the 1992 L.A. riots. It is hard not to think about what happened in the wake of the acquittal of the officers involved in the brutal beating of Rodney King. When looking at police violence against the Black community, it is easy to feel like very little has changed since then.

This time calls for some reflection about how all of us, in the institutions where we work or participate, can enact some change.

A few months ago, the physics and astronomy department at my new institution, UCLA, invited Sherard Robbins to come and speak about the demographics and minority representation in our department. He asked us to take a look around the room and to see if the representation in the room reflected that of the general population in Los Angeles. This was an embarrassing and shameful exercise. It is shameful because we do not have a single Black faculty member. It is also shameful because women are hugely underrepresented.

Representation matters. It particularly matters in positions of power. It matters because when you see people that look like you and are culturally similar to you in a position you thought was unattainable, you start to believe you can do it. It also matters because people tell stories, and stories are mediators of humanization. When you hear about your culturally different colleague’s weekend with their family at the beach, you see them as a parent, spouse, and human.

I am currently in a position of power. I have been an assistant professor now for almost a year, and because it is so new and fresh, it contrasts strongly with my previous position as a postdoc. I went from having almost no power and social responsibility to being thrust into a position where my words and actions do have an affect on undergraduates, graduate students, postdocs and other faculty members. I know that many of you who read this blog are or will be in similar positions in the future. So when you are afforded the privilege of such a position, it is your responsibility (just as it is now mine) to make sure that conduct in your department changes. It is your responsibility to make sure that the make-up of the department starts to reflect that of the greater population. It is your responsibility to ensure that traditionally underrepresented groups make it into positions of power. And it is your responsibility because if you don’t do it, no one else will.

The activists in the streets deserve tremendous credit for making their voices and anger heard. And it’s important that those in positions of power take actions that say “we hear you”.

I sign off with a rather profound quote adapted from the Talmud for the film Schindler’s List:

Whoever saves one life saves the world entire.

Slight detour

I am still planning to follow up my previous post on environmental negligence and will write a post about CFCs in the near future. However, I saw this YouTube video recently and found it harrowing. The British government had known about the consequences of acute radiation poisoning, but chose to perform these tests anyhow. In addition to the lives irreversibly changed, there is also the remarkable fact that these people were able to see live images of bones and blood vessels with their eyes. Does anyone have a good explanation as to how this would even be possible?