PHYSICS

Quantum Fluorescence.

Understanding the universe’s most popular particle

Image by FLY:D on Unsplash

Next time you see a face, watch your favourite YouTube video, or enjoy a nice sunset, try to remember that you’re taking invisible disturbances in electromagnetic fields and turning them into a world of colour.

Red, yellow, blue, everything in between — bouncing, reflecting, combining into rainbows, pouring out of headlamps, bursting out of our windows just when we needed that extra five minutes of sleep. It’s easy to forget that his entire sense of vision that defines our experience so much is just our minds trying to make sense of a million flavours of the same particle.

In our lives, few things are more ubiquitous than light. Yet, in all of physics, we probably understand it the least.

What is light? Where does it come from? Why does it seem to interact differently with different things? And what about fireflies, or things that glow in the dark, and everything else that seems to defy any intuitive understanding of light?

These are the some of the great questions of luminescence. Some of them still don’t have answers — many never will. But by understanding what light does at the subatomic level, we can get closer to the truth.

In this article, I won’t attempt to cover the entire domain at once (nor do I think I’m the right person to do that), but at the very least, let’s try to tackle the first piece of the puzzle in some depth: fluorescence.

Welcome to Quantum City

While the standard Bohr-Rutherford model of the atom does hold some value in helping us describe chemical reactions and larger-scale behaviours in biology, it starts to break down when photons enter the scene.

This is because electrons are quantum objects that don’t behave like life-sized yellow balls (as tempting as that scene is to imagine). So, to try and avoid as much confusion as possible with fluorescence, our standard model is going to need a few upgrades.

TL;DR: In the quantum world, electrons don’t necessarily orbit their respective nuclei as planets orbit around stars. Rather, they can be described as having a certain probability of being found around the nucleus when we choose to observe them.

These regions of high electron probability are called orbitals, and they’re mathematically defined as the solutions to a fundamental physical formula known as Schrödinger’s wavefunction. And, as dictated by these solutions, it turns out that the vast majority of them form shapes that are far a lot more exotic than spheres:

Some orbital solutions to Schrödinger’s wavefunction. The red contours represent regions of high electron probability (take them with a grain of salt since they’re not to scale). In reality, the orbitals shown here should keep getting bigger and further from the nucleus as you go to the right.

We can conveniently organize all the electrons in an atom into relatively discrete groups called energy shells. As we’ll cover soon enough, these electrons all have a similar amount of energy, which loosely correlates to their average distance from the nucleus.

Shells can be further subdivided into subshells, which are smaller groupings of electrons that do have the same amount of energy. Finally, subshells can be subdivided into orbitals, each always containing two electrons. Once again, the electrons here have the same energy, but a different angular momentum (although this really isn’t necessary for what we’ll be talking about).

For illustration, I’ve found that it’s always helped to think of an atom’s electrons as a city. Different energy shells correspond to the different neighbourhoods: each one holds a lot of people (electrons) with a general way of life (energy) that only others in the same community could really understand.

Within neighbourhoods, we have blocks (subshells). Although the residents of different blocks have quite a lot in common and agree at a high level, they still differ in more nuanced ways and often choose to keep to their little social circles.

A so-called Jabłoński diagram of electrons at different shells and orbitals next to our neighbourhood analogy of electron configurations. Any electrons within the same house (or orbital) have the same energy, while electrons in different houses. Why? That’s up next.

Finally, we reach the smallest unit of a neighbourhood: the house. I know the average American family size is about 3.5 or something, but in Quantum city, it’s exactly 2. Always. And since it’s harder to get any closer than family, no energy differences here either.

Note: This explanation might be a bit redundant for those of you that already have a basic idea of orbital theory and electron configuration. If you don’t have any idea what this means, this article isn’t the best place to start since this explanation is just meant to be a refresher. To quickly learn about these concepts, watch these videos from Khan Academy or Crash Course.

Energy and Luminescence

So, most atoms have more than one electron, and the distance or position of each electron from the nucleus somehow determines its energy? Let’s define this more scientifically.

When we refer to energy (at least as it relates to electrons), what we’re really talking about is one specific kind of energy known as electrical potential energy.

It’s common knowledge that electrons are attracted to the nucleus because of the oppositely charged protons it contains. More technically, this occurs because their interaction warps the electromagnetic field, creating a force that tends to pull them together.

With this, it stands to reason that when left alone, an electron and a proton will spontaneously maximize their attraction by minimizing their distance. If we could assign free will to these star-crossed particles, it’s almost as if they “want” to stay as closer to each other as possible.

For a better understanding of electrical potential energy, consider this. An elastic band has a certain resting state, in the sense that you don’t need to put any effort to hold it there. This would be the point at which the band has its lowest elastic potential energy.

The main idea here is that neither elastics nor subatomic particles will stay in a place that’s energetically unfavourable to them. You need energy to hold them there.

To increase its potential energy — that is, stretch it out — you’d need to do work (transfer your energy to it). The more you do this, the more potential the elastic has to move back to its default state. And that’s exactly what happens. As soon as you let go, all the energy you put into that elastic gets converted into kinetic energy as it snaps right into regular conformation.

Similarly, any electron held within the attractive grasp of a nucleus has a default position termed its ground state. This is an orbital that the electron can exist in without any need for external pushing or pulling, and corresponds to a state of relatively low electric potential energy.

Using that same logic, pulling an electron away from the nucleus (against their electromagnetic attraction) will increase its electrical potential since it has just as much room to move back toward the nucleus when you set it free.

The magnitude of this attraction can be mathematically predicted via Coulomb’s law, which relates the electromagnetic force between any two charged particles to their charge and distance from each other.

Fₑ represents the electrostatic force, while k is a constant, and q1 and q2 represent the two charges involved. It may seem contradictory that the orbital with the bigger radius has a higher PE, but keep in mind that Coulomb’s law solves for force. To even get an electron this far away from its default state (near a proton), you’d first have to continually apply this force to get to the same radius as the smaller orbital — and then some.

Consequently, it makes sense that different shells and subshells have different amounts of energy, since the electrons in each one are more likely to be found closer or further away from the nucleus. But this is just the beginning.

Bridging the Gap

While the exact geometry and trajectory that orbitals trace out are fixed, the positions of the electrons themselves certainly aren’t.

When an electron receives energy from extra heat, a collision with a neighbouring electron, or, in the case of luminescence, a photon, it gains the ability to move to a higher energy orbital in a process called excitation.

Not for long, though. As in our elastic band analogy, this process will only last as long as there’s energy to continue keeping the electron further away from the nucleus than it would like to be. Almost as soon as that energy is gone, so is the electron — back to its ground state.

But here’s where our convenient analogy begins to break down. You can energize an elastic as much or as little as you’d like, with the only real limit being how far it can stretch before breaking.

On the contrary, quantum mechanics is based on the principle of quantization. At such a small, small, small scale, electrons can only exist in these fixed orbital regions. Never in between.

As long as you provide the exact amount of energy needed for the electron to get from shell one to shell two, or subshell A to subshell B, it will make the transition.

But if that quanta of energy is even slightly less or more than what’s required to get it to its next fixed orbital, it will most likely reject it.

In optics, we use a surprisingly brief formula to calculate the amount of energy a photon of light can carry, simply based on its wavelength or frequency:

E=ℏv relates the frequency of a photon (γ) to the amount of energy it carries. represents the Planck constant, which is pretty big deal in quantum mechanics. When a photon meets an electron and has enough energy to help it bridge a shell (e.g. n1-n2), excitation can occur. Then, the electron re-emits all that energy. When E is lesser than the energy required to bridge a shell, electrons don’t absorb the photon or go anywhere.

Overall, this maps out the general process of all luminescence. An electron at ground state can absorb a photon, use its energy to transiently stay in a higher energy orbital, and then release it all back out by emitting another photon before it returns to its ground state. What goes in must come out.

Stairwells and Fluorescence

We know that after being exposed to the exact quanta of energy needed to jump to a new shell or subshell, electrons will eventually release all that energy back out. But the process of returning to ground state can play out in a lot of ways.

Let’s try drawing a parallel to a piece of architecture we’re all familiar with: the common staircase. Naturally, the top would represent the highest energy point an electron can occupy, and also the furthest from the nucleus.

The bottom would be something like a ground state, where the electrical potential energy is about as low as it can be. That would make you the personification of an electron.

Somehow a photon has given you enough energy to get to the very top. Now, how do you go about getting to the bottom?

It’s an odd-looking staircase in the background, but a staircase nonetheless…

Well, you could jump down the thing all at once. Or, you could go one step at a time. Or, two at a time. Or, two steps, then three, then four.

While some ways might be less…traditional (and painful) than others, it all works out as long as you eventually get to the bottom, where your attraction to the nucleus is highest.

When you emit a concentrated pulse of light at something, this is what’s happening behind the scenes. You’re using a stream of photons to excite its atoms’ electrons. And as it turns out, excited electrons can take any one of these approaches (relaxation pathways) to descend back to ground state.

For more on relaxation pathways — and some forbidden pathways (yes, you read that correctly) — feel free to read this article.

The vast majority of photons, upon making contact with an atom’s electron, will excite it to an entirely new energy shell. Almost immediately afterward (typically within 10⁻¹⁵s) it will return directly to its ground state — releasing all the energy it absorbed as one photon. If the energy of the photon is the same, so is its wavelength (you can thank E=ℏv for that). As a result, the reflected light will have the same colour as the ray that came in:

In elastic scattering, no energy loss occurs between when the electron is excited and when it emits a photon. As a result, the photon that is released has all the energy that was in the original photon.

Additionally, as dictated by the law of reflection, the light that reflects back will do so at exactly the same angle above the surface as the incident beam. This process, as a whole, is called elastic scattering.

But like we hinted at earlier, not all photons release their energy like this. Here’s where fluorescence comes in.

Our second option is that our incoming photon energizes an electron to an entirely new energy shell and subshell. In this situation, the electron is no longer obligated to fall directly down to the ground state and release one photon with all the energy it received.

Rather, it can make a sort of electronic pit-stop at a lower energy level within the same shell and release a small portion of the energy it originally carried. Not with a photon, but by other modes of energy loss that are non-radiative, like making its atom move faster, or leaking away that energy as heat.

But this doesn’t end the affair. After this vibrational relaxation, the electron has to complete the rest of its descent to ground state by releasing a photon.

Because of non-radiative energy loss before an electron re-emits a photon, the excited electron will always emit light that is less energetic than the light it was illuminated with.

Except, things have changed. Because the electron already vented out some of its energy to reach a lower subshell, even directly going back to ground from here won’t release a photon with as much energy as our original beam of light. This is fluorescence.

That is, the light that comes out of our sample will have a consistently different wavelength and colour than the light we put in.

To be more specific, less energy will give it a longer wavelength, closer to the end of the electromagnetic spectrum where you can find micro and radio waves. This difference is more technically termed a Stokes Shift, and by using E=ℏv, we can calculate exactly how much energy the electron must have lost non-radiatively.

So maybe…we might even be able to infer the specific energy gaps and arrangement of orbitals in its atom. Hold on to this idea.

Applying Fluorescence Experimentally

The most useful part of fluorescence is this Stokes shifted light, which contains a surprising amount of information.

Because the shells and subshells of essentially every atom and molecule have a unique spacing pattern, their electrons only have a handful of possible relaxation pathways.

Given a certain wavelength of photon that initially crashes into an electron, we can first use the principle of quantization to check if will even get excited to a nearby shell.

If it does, we can map out every relaxation pathways it could take to arrive back at ground, and predict the exact energy and wavelength of every photon it could possibly emit in the process.

This is the closest thing we have to an atomic barcode: the next time we observe a reflected photon with an identical energy and wavelength, we can trace it right back to the particular molecule that released it.

Imagine we’re biologists that want to measure the quantity of a molecule like nicotinamide adenine dinucleotide (NAD+) in some liver cells that we’ve cultured on a petri dish.

Side note: For those of you wondering about this oddly specific example, NAD+ happens to be a key electron carrier in cell respiration. Quantifying it can help us gain a substantial amount of information on cells, including their biological age and how efficiently they can extract energy from food.

To do this, we can create a basic fluorescence spectroscopy setup with a 350nm deep red laser.

Note that all the wavelength values in this NAD+ experiment are very loosely based on these fluorescent spectra from the Blizard Institute, and are only meant to outline how the general procedure would work. Also note that this setup is extremely, extremely simplistic. If you tried this experiment out in the real world, the results would be disappointing.

With filters, we can selectively hone in on light fluoresced by NAD+ at 341nm and 343nm, as shown by γ(fluorescent). All other wavelengths, which must have come from other molecules, can be discarded.

With filters, we can selectively hone in on light fluoresced by NAD+ at 341nm and 343nm, as shown by γ(fluorescent). All other wavelengths, which must have come from other molecules, can be discarded.

Regardless of whether wavelength is 350nm or 370nm, a good choice for your would essentially need to meet two requirements.

  1. It would have to provide the electrons in NAD+ the precise quanta of energy they’d need to reach their next highest energy shell.
  2. It should ensure that the Stokes Shift — or the difference in wavelength between the absorbed light and emitted light — was large enough so we could tell the two apart.

Knowing that NAD’s specific relaxation paths at this wavelength could only ever produce light at 343nm and 341nm, we can then add a selective lens that excludes light from any other wavelength, and then place a wavelength-specific sensor directly behind it.

When we fire our first pulse of light with our laser, the photons will penetrate the cells — some colliding with the membrane, others bombarding DNA in the nucleus and proteins in the cytoplasm. Some will even collide with NAD+.

But that doesn’t matter, since we know a key fact. Any light that comes out at 341 or 343nm must have come from NAD+. Theoretically, it would be almost impossible for it to have come from another molecule, since that would suggest it had the exact energy spacing between its orbitals as NAD+ to let it emit photons of the same colour.

Thus, the concentration of NAD+ in this otherwise mysterious sample of cells becomes quite simple to determine. After all, it would be directly proportional to the ratio between the amount of light we released and the amount of light that exited with the right wavelengths to traverse our filters:

A relatively low concentration of NAD+ in our sample (illustrated on the left) would naturally make it less likely for our incident photons to collide with them. This would reduce the quantum yield, ϕ, which would be defined as the ratio of 341 and 343nm photons picked up by the sensor. With a higher [NAD+], more collisions would occur and more of the light that’s released would be able to cross the filter. This would eventually make ϕ approach 1. Note that wavelength is denoted by the lowercase Greek letter lambda (λ)

Formally, this ratio would be referred to in percent form as the quantum yield of our fluorescent system, meaning that the sum of the quantum yields at 341 and 343nm would express the fraction of light that ended up fluorescing the NAD in the cells and help us predict how much of it these cells contain.

Can I have my Ph.D. Now?

Not yet! Of course, this is just the beginning of what fluorescence can do. The spectroscopy process we just described happens to be one of many technologies that rely on it.

Building on this basic principle, we can make individual components of cells glow under a microscope. We can conjugate fluorophores to molecules of glucose so they can make tumours visible to the naked eye. Now, we can even analyze the spectrum of an object’s fluorescently-reflected light to infer how molecules interact, using a technique called FRET.

Stated differently, not only can we detect the individual quantities of molecules A and B in a sample, but also when and how much of A and B have decided to bind, collide, or displace each others’ atoms in a chemical reaction.

But fundamentally, all this tech is the answer to one of our most wild and ancient pursuits as humans: How do you see without actually seeing?

Fluorescence unlocks that possibility. And now, hopefully you can better appreciate all the pieces that go into making it work.

TLDR (for real this time)

  • Fluorescence is a form of luminescence, where the electrons of an atom molecule can accept and release very specific quanta of energy.
  • Electrons can absorb photons (along with their energy) to temporarily suspend themselves in a higher energy orbital that’s further from the nucleus. But when they descend back to their ground state, they can do so in more than one way.
  • Fluorescence specifically occurs when an excited electron first releases some of its energy non-radiatively (without light) to reach a lower subshell, then releases its remaining energy in the form of a photon that’s less energetic than the photon that excited it.
  • Because of their different attractive forces (due to a different number of protons and electrons), essentially all atoms have a unique spacing and configuration of their electron orbitals. Thus, a particular fluorophore will only release a handful of very specific wavelengths of light based on all of its potential relaxation pathways.
  • By comparing the light that we used to illuminate our sample to the light it fluorescently reflected, we can make educated guesses about its material composition.

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