May 2025 - Fitting Charges Into Spheres

Today we're going to approach a problem, rather than a paper or a larger topic. In a sphere (or a blob, if we want to give away why I need to know), there is a number of electrons. They of course all have a charge of e-, which means that they should repel one another. The force applied from the charge is obviously stronger than that of their gravity, so every particle should repel every particle with the force given by Coulomb's law. Given that, there should be a stable position within the sphere at which all of the particles have found a stable position. Naively, one might make the assumption that the distances between all electrons would be equal, but let's investigate this first.

First, let's consider the 2D case. For 4 electrons, one would place an electron in the center of a perfect triangle for the maximal distance. However, this should already give us some insight into our problem, as from one of the points of the triangle, the distance to the middle electron will always be (significantly) shorter than to the other two points in the configuration. This already discounts the assumption that all points were equidistant. In the case for 5 electrons, it'll retain the electron in the center, and form a square along the edge of the sphere. This continues for a while, until a geometric constraint is violated. When making sketches of these configurations, one might notice that they are Euclidean shapes, with a point at the center, meaning all these are, are a series of Isosceles triangles, touching their long sides. The angles around the center point sum to 360 degrees, obviously, so the narrow angle of these triangles are 360/n. Let's call this angle a, and the other two angles b. The question becomes, when the distance between the neighbouring points on the circle becomes smaller than half that between the outside points and the center. This is a fairly simple trig problem.

At this point, at the very latest, the single point in the center should fan out to another perfect triangle. At least one of the three points of the smaller shape will have to create another isosceles triangle. There doesn't seem to me to be an easy mathematical formula to determine the size of the inside triangles. My own approximation would place of the circle that the point of the Euclidean shape would lie on at about a third of the outer radius, perhaps slightly bigger, seeing as that would place the distance between the outer radius equal to that to the inner circle's diameter. Perhaps charge screening will get rid of the electrical force originating from the other rows beyond the first that acts onto any given particle. Looking at how other people have solved this problem in 2D, this seems more or less how they've approached it. At a certain particle number, the offset of each of the concentric Euclidean shapes will create a spiral structure, though in truth, it doesn't seem to matter that it does, and it's not even commonly understood to be an optimal solution to the problem, but rather a good enough approximation with enough of a pattern to be memorable.

In 3D, I this same approach seems to not have been embraced as much as for 2D, though I'm sure it works just as well. In essence, we should still expect to work with Euclidean geodesics, and the constraint at which a new geodesic shape would be required is going to be effectively the same, though it will result in a 4-particle pyramid. The resulting structure, as particle number increases, will result in less of a pretty pattern, though the "spiral arm" might be kept intact from one chosen perspective. My worry is actually about those numbers of particles that can't span a Euclidean geodesic. Six particles, for example, wouldn't span such a shape, and neither would five, meaning that when the sum of particles of the inside shape reaches 6, including the center particle, there is no shape this can take. This might result in particles moving back and forth between shells. In the meanwhile, we might expect irregular shapes, if we only really have one shell, meaning a total of 6 particles.

Now, having not found a mathematically rigorous proofing for what I'm trying, there are papers by people with more wits and more time.

"Central Configurations in Three Dimensions" by Battye, Gibbons and Sutcliffe approach presents an approximation of the problem I was looking into by assuming a linear central potential pulling the particles back toward the center, while the particle-particle interaction is constructed for gravitation. Coulomb potentials function basically the same, so that doesn't bother us too much (yet).

at which two particles feel no mutual force. In the context of our problem, this is the interparticle screening, so there will be a particle within R that The idea still eventually leads to the sphere packing problem, that I touched on once, but more importantly, the solutions conform to the known configurations that solve the Thomson problem at low particle numbers N, which is a good indicator that the approach is sound. The approach consists of several considerations, taking into account a number of things. One is the geometric consideration that for equal spheres, one sphere can only be in contact with 12 others at the most, which gives rise to the shell structure that was already evident in my own attempt to broach the subject. Assuming all charges exist in some positively charged spatial domain Ω, then given the specific effects of the potential functions

are only dependent on the geometry of Ω. The upper bounds of energy can be found by the Williamson averages of the distances and thus the potentials, and the lower bounds derive from the ion radius. By the separation probability distribution, the computation for large numbers can be streamlined by not populating the domain with singular points, but rather with triples of points, creating triangles. For lower particle numbers, FCC can be assumed due to a conjecture by Harriot and Kepler (proven 2002 by Hales), though it's possible for there to be a configuration with the same density, i.e. same energy distribution between particles. Since FCC is a crystalline configuration, its application will result in structures that are like hexagonal one way, and square the other, which is not so helpful for large numbers of particles in a sphere, where FCC packing will necessarily leave out layers of empty space. While there are an uncountable number of packings with the same density, conveniently, local variations of density smooth out with large enough particle number. This is perhaps somewhat obvious in terms of laws of large numbers, but good to keep in mind.

Apparently my assumption that regular polyhedra (with one additional at the mass point) are solutions was correct, but it seems that they're not necessarily stable solutions, but most solutions can be successfully approximated by structures with regular triangles as faces. They are classed as regular deltahedra with exactly as many vertices as included in the outer shell. Up to N = 12, this will be all of them, and those numbers that don't have a corresponding deltahedron will necessarily include some square faces. Recursively, the 12-adjacent structure will repeat itself wherever possible to define the outermost shell. Notably, the most symmetrical solutions are significant drop-off points in the energy spectrum. Charting the distances from the particles to the center in the order they're added (holistically, there's no good way to phrase this), the shell structures becomes very clear, and is immediately reminiscent of the different energy shells of electrons in molecules, even though each new shell becomes significantly larger (longer) than the last, since the Pauli-Exclusion principle doesn't apply and what's left is basically the spatial restriction.