Week of November 7, 2021

Let's begin with basics. When science students encounter the subject of atomic physics they typically will encounter the phrase "electron shell-filing" in some context or varying form. What this means is that electrons in atoms tend to form configurations that grow with number in such a way that eventually, with a certain number of electrons, a new shell begins to form much like a layer of an onion. But to get at the basic physics of electron shell-filling, we first often introduce the principle quantum number, n, that labels each shell. Sometimes n is introduced in the context of quantum mechanics within complicated mathematical expressions. The integer value of n appears in these equations almost as if by magic and without as much explanation as most students need. In fact, students interested in the subject want to feel comfortable with the appearance of n in these expressions if only to know that they are integer values (n = 1, 2, 3 . . . ), that they determine the "energy level" (almost synonymous with "electron shell"), and eventually that n is one of four quantum numbers that they're told describe all that's needed to identify all the physical properties of an electron in a given atom.

There are two things we need to know about quantum numbers associated with electrons in an atom:

1. They determine geometric properties associated with each electron, and
2. From these geometric properties we can determine all the physical properties of the system.

The subject of the first is left for a later blog entry since we'll want to have a more advanced framework in which to convince ourselves of the statement but also the fact is more of a historical note than a commonly presented fact. The subject of the second is very much at the heart of this entire body of work -- I do know yet know all the mathematical interconnections between these geometric properties and all physical properties of each electron in a given atom except that their associated quantum numbers usefully form a kind of accounting system in the quantum mechanical framework to arrive at the physical properties we want to evaluate.

Thus, for the moment, we are left with the want of an interested science student. That is, a want to understand the principle quantum number well enough to know what it's all about. Of course, I will also take this want a step deeper, because I'm more interested in a new generation of students who actually build an intuition about the underlying physics of the concept. So let's get started.

Nature Loves to Be Lazy

After years of thinking about my thinking about the body of work I've formed over the years, I believe I've settled on one of the simplest ways to explain the phenomenon of electron shell-filling. The first scientific principle that one should be comfortable with is that things in the natural physical world prefer to move toward places where they experience the least about energy or have the smallest amount of force acting upon them as possible. Our everyday experience is filled with great examples.

• Water falls over a cliff that we call a waterfall.
• If you tip over a glass of milk, the milk spills onto the table and eventually onto the floor.
• A vehicle in neutral rolls down a hill.

Electrons behave the same way, but not through the force of gravity since we're interested in the interactions involving their charge. Electrons prefer to move to locations where they experience the least amount of electric energy. This happens naturally and means that the total electric force acting on them is as small as possible. Once they arrive in these locations of least energy, we call the resulting configuration of electrons their electrostatic configuration. In science courses, we don't give much explanation of how or why electrons form multiple shells, but that's what I'd like to do here in the simplest way I can. There may be a simpler way, but this way also helps us build up to other ideas I'd like to develop in this blog.

Electrons on a Circle

Now that we understand -- given many examples -- that physical objects naturally move toward positions of least energy, let's consider a simple example using electrons that will demonstrate the formation of electron shells.

Consider electrons restricted to move on the circumference of a circle. To minimize the total energy of the system, each electron in a two-electron system will be found on opposite sides of the origin. Three electrons will be found at the vertices of an equilateral triangle, four electrons will be found at the vertices of a square, five electrons will be found at the vertices of a regular pentagon, six electrons at the vertices of a regular hexagon, and so on. I've shown these cases in the figure below so you can visualize these a bit more easily than relying upon your imagination.

Individual configurations of N = 1, 2, 3, 4, 5 and 6 electrons on the perimeter of a circle form familiar geometric shapes.

You may have merely believed that these are the lowest energy configurations of electrons in part because the configurations look very symmetric, very appealing to your eyes, your senses and sensibilities, but of course, we can do the math to back up these statements. The only equation you need to use here is called Coulomb's law,

Here, q_i and q_j are the the charge of individual electrons. We can then relate this to the elementary charge, e. So, q_i = q_j = e =1.602 × 10^-19 coulombs . As well, r_ij is the distance between the two electrons, and ɛ₀ = 8.854×10⁻¹² F⋅m⁻¹ is the permittivity of free space,

The task is to apply this equation to every interacting electron pair in each configuration, then add up the total energy. For example, in the 4-electron case, you need to calculate the Coulomb energy among pairs, q₁ with q₂, q₃, and q₄, AND q₂ with q₃ and q₄, AND q₃ with q₄. As a simple check to help ensure you've calculated all the pair-wise terms, for each N-electron system there are N(N-1)/2 terms. So, for example, for N=4, there are 4(4-1)/2 = 12/2 = 6 terms. We may continue adding an electron to the circle, calculate the total electrostatic potential energy until we find a global minimum, and find that the result is a regular polygon having N vertices, or an N-gon. This means that the distance between every neighboring pair of electrons is the same. The details of global energy minimization are a bit more involved than I'd like to address here, but the result is a very aesthetically-pleasing, and frankly intuitive, one of highly symmetric regular polygons. This fact is beneficial to developing the subject as it emphasizes the geometric symmetry properties of the system as they pertain to a physically realistic collection of electrons constrained to move on a two-dimensional circle. The system may be a thin circular metallic disk or the potential well of a two-dimensional quantum dot -- another fun subject to discuss at a later date in this blog.

Electrons on the Full Circle

So far, we've been interested in calculating only the total potential energy of electrons constrained to the perimeter of the circle, but what if we allow them to move around within the circle? So, for example, suppose you wanted to add an electron to the 4-electron solution that forms a square in the figure above. Since we know that the new electron's charge interacts with the charge of the original four electrons through the Coulomb force, and the resulting energy stored within these interactions is inversely related to the distance between each electron pair, we would expect that less energy would be stored when the electron pairs are separate by larger distances. In this example, the location within the circle that is most distance from the original four electrons is the center. This is a geometric symmetry based guess that helps us develop a procedure to compare the energy of different electron configurations. So if we place the fifth electron at the center and calculate the total energy of the system, we can compare it with the total energy of the system when all five are located on the perimeter of the circle (like we presumably did to draw the N=5 system in the figure above). We find that this charge-centered configuration is not lower energy, so the configuration with all five electrons on the perimeter is considered the electrostatic configuration. Now, we could also try other 5-electron configurations, but let us not overthink the problem too much so we can examine some important results.

Next, try this process for every other N-electron system. Place one charge at the most-distance location in the circle from all the other N-1 electrons -- the center-- and calculate the total energy of the resulting charge-centered configuration. How does the energy of these charge-centered configurations compare to their respective N-electron perimeter-constrained configurations? The energy differences between the charge-centered and perimeter-constrained configurations for each N-electron system are plotted in the figure below.

For up to 5 electrons, the energy difference increases. This means that the charge-centered configurations of these N-electron systems are progressively greater in energy than the perimeter-constrained configurations, and the perimeter-constrained configurations are the electrostatic configurations. Continuing to add electrons to the system, however, notice that the energy difference decreases, until N=12 when the energy of the perimeter-constrained system is higher than for its charge-centered configuration. Thus, the global minimum electrostatic potential energy configuration of the 12 electron system is charge-centered with 11 electrons remaining on the perimeter -- notably, in the same geometric configuration for the perimeter-constrained 11-electron case as you can see in the figure below.

What we just described is what I believe to be the simplest example of electron shell-filling. A new ring, or shell, of electrons has naturally begun to form at the center of the 12-electron system!

Electron Shell-Filling 101

From the plot of energy differences above, we know that adding a 13th electron to the system yields a lower energy in a charge-centered configuration, so we know the center shell will remain. What we won't know without further calculations, though, is whether or not the 13th electron pushes the centered electron in the 12-electron configuration aside to form a 2-electron center shell, or if all 12 electrons remain on the perimeter. Spoiler-alert: A 2-electron inner shell forms for N=13 because it's the lowest energy configuration for a 13-electron system. In fact, as more electrons are added to the system, not only does the inner shell continue to grow both in radius and electron number, but a third shell eventually begins to form at the center. The shell-filling process in this 2-dimensional circular system visually appears to be the formation of new inner electrons as new electrons repel other electrons away from the center. In other words, the center is the location where a new electron can be most-distant from all the other electrons already in the system. Something to think about: Is the center-most electron actually the new electron added to the system? If all the electrons have the same elementary charge, e, does it matter?

This entire process is a natural result of global energy minimization. Electron shell-filling occurs because Nature prefers to move electrons into configurations of the least electrostatic potential energy within a constraining system. Electron shell-filling is a natural process that requires nothing more energy minimization to understand. We were guided by geometric symmetry using Coulomb's force law.

Practical Implications & Beyond?

As noted earlier, the practical implications of this fundamental exercise includes any system in which charged particles are constrained to a 2-dimensional circle. Devices can be manufactured such that this kind of constraint is enforced.

What happens in the 3-dimensional case of electrons constrained to the surface or volume of a sphere? The surface case is the Thomson Problem I've mentioned in earlier blog entries. The connection I discovered between the Thomson Problem and the entire periodic table of elements, however, is precisely the same procedure outlined here, but with the evaluation of global energy due to just a single charge-centered configuration for every N-electron Thomson problem solution. Key to an understanding of how and why these geometric solutions are so intimately related to natural atomic structures is to notice, for example, how the 11-electron perimeter configuration in the 12-electron electrostatic configuration is identical in geometric symmetry to the 11-electron configuration. Additionally, notice how the distance between the center-electron in the 12-electron case in the figure above is exactly the same distance from all the other 11 electrons on the perimeter. How does the centered charge contribute to the geometric symmetry of the system?

So, About Quantum Numbers Again

We have examined how new electron shells form. Notably, each electron shell represents a different principal quantum number, n. The outer shell may be n=1, and the inner shell may be n=2. As more electrons are added, more shells form and are represented by n=3, 4, 5, and so on. If we examine the actual potential energy landscape of the system, we would begin to see that electrons on the inner shell experience a different potential energy than the potential energy experienced by electrons on the outer shell. The difference between these shell energies is what we mean by energy levels.

The other quantum numbers that describe electrons in an atom are similar in geometric origin to the spatial relationship among electrons within a particular electron shell. The N=3 case, for example, have electrons separated by 60° while in the N=4 case, the electrons are separated by 90°. This is part of the geometric relationship associated with yet another type of quantum number.

A third type of quantum number helps identify pairs of electrons that have shared geometric relationships. For example, In the N=5 electron case above, two pairs of electrons share similar geometric relationships, but the remaining fifth electron is unpaired. In the N=6 electron case there are three pairs of electrons that are geometrically similar. The geometric orientation of these pairs in space, represent this third type of quantum number.

A fourth type of quantum number simply differentiates the spatial location of each electron in a particular pair. is the electron to the right or left of the center in the pair? or is the electron above or below the center in the pair? This geometric location within a pair determines the value of a fourth type of quantum number.

And so it is with the four different types of quantum numbers for electrons in real atoms. We arrived here by only thinking about geometry.

Next Week

In next week's blog entry I will discuss the units of energy shown in the plot above and how Coulomb's law can be reduces to mere geometry. This is how scientists like to simplify their lives when doing mathematical calculations without losing sight of the physical quantities.