If you’ve done any physics work before, you might have noticed that the formulas tend to include a lot of constants: the speed of light, Planck’s constant, the Bohr magneton, electron/proton/neutron masses, and so on. It makes sense that we would need to use constants, since it would be pretty odd/coincidental if the relationships defined between real, physical quantities would be some pleasant numbers in our decimal system. Unlike in mathematics, physics constants are generally real things you have to measure and apply models to in order to calculate.
So, they’re not usually defined abstractly in the same way that pi or e are. Though there are still a few useful constants in physics that have abstract definitions like in pure mathematics. One of those constants is the Madelung constant—what I have fittingly dubbed “the realest abstraction”—and it’s pretty damn cool.
The Madelung constant, named after Erwin Madelung (not to be confused with the other, more famous Erwin in physics)*, is used to determine the total amount of electrical potential for an ion in a given crystal lattice. If that sounds bloated, don’t worry—the exact physical interpretation won’t be important in our discussion, but you can basically think of it as the answer to this question:
Assuming an infinite structure (so whatever pattern the atoms take on just continues on forever) and approximating atoms as point charges (so any weird charge distribution is ignored), what’s the total effect from electrical forces on a single atom by all the others in the structure?
One important thing to note is that this value converges. In other words, if I start summing the effects of each atom individually and go outwards from the center by distance, the sum will tend towards a specific value (the Madelung constant). Since the effect of any single atom falls off (though not exponentially) as you increase the distance, this should make some intuitive sense.
Another interesting property of the constant is that it’s unitless—a pure maths number. In practice, it’s intended to be multiplied by some combination of the electric charge and atomic distance, but you can think of the constant itself as a fundamental property of a crystal’s structure, or even a fundamental maths constant. You’ll see why this is a good description soon.
For the crystal of NaCl (also known as salt), there are two Madelung constants—you get different values if you use sodium (Na) or chlorine (Cl) as the reference atom (otherwise, the constant will always be the same). Since the two types of atoms occupy positions in a pattern that maintains some level of symmetry if you start switching between the two, the effects of each are the same magnitude and differ only by a sign.
The NaCl crystal has a very simple pattern, which makes it an ideal example for this. It occupies a cube structure where sodium and chlorine atoms switch off as you move across adjacently. You can think of it like a checkerboard that extends infinitely, with Na placed where the white squares are and Cl on the black ones. Add another layer by placing another checkerboard on top of the one you already have, except shifted one space over. Keep adding layers, and pretty soon you’ll have the lattice we’re looking for.
To simplify things, before I show you the calculation, let’s set the charges of Na and Cl to be one fold and opposite to one another, so that the charge of Na is just 1 and the charge of Cl is -1. Let’s also set the distance between the reference atom and its nearest neighbors—the ones just adjacent to it on our checkerboard pattern (there are 6 in total)—as a distance of 1 away.
With all those assumptions, the formula for finding the Madelung constant of NaCl looks something like this:
It’s stuffed, but I’ll try to explain each part: M is the Madelung constant, and the subscripts represent that constant from either the reference atom Na or Cl (remember how the charges were reversed). The summation goes from negative infinity to infinity for (j,k,l), hitting every combination of the three along the way. You can think of (j,k,l) as the three coordinates describing a particular atom’s position in the lattice (this means (j,k,l) from negative to positive infinity will describe every possible atom). The origin (0,0,0) is our reference atom, (0,0,1) would be one of the 6 nearest neighbors, and so on (if you’ve done anything with 3d coordinates, it’s literally the exact same thing).
You might have also noticed that there’s a way to tell if the atom we’re looking at is sodium or chlorine just by looking at its coordinates: add them all together—if it’s an even number, it’ll be the same atom as the reference/origin, and odds are the other type. If you consider how the nature of this checkerboard pattern works in your head, it should start to be clear exactly why that works.
With that in mind, we can understand the numerator, which represents the electrical “effect”—It’ll work out to be positive if the atoms are the same and negative if they’re different. Lastly, the denominator is just the distance, with larger distances giving smaller effects.
So what happens when you actually do the sum? It depends on how you sum it, and this is where things get really interesting. There are two ways to do it that make the most intuitive sense, and I’ll describe them both here.
One way is to add them up like spheres of increasing radii. Add the 6 nearest neighbors, then the next closest set ((0,1,1) and the other atoms at distance 
There are some really interesting properties to this summing pattern (OEIS #A005875):
- The number of atoms at each distance going outwards follows a peculiar sequence: 6, 12, 8, 6 (the first four already described), then 24, 24, 12, 30, 24, 24, 8, 24, 48, 6, 48, 36, 24, 24, 48, 24, 24, 30, 72, and so on…?
- It’s especially weird when you consider that the number of atoms at each distance is the same as the number of equidistant points from a cubic center, which seems like something pretty abstract/purely mathematical.
- This pattern is equivalent to the number of ways of writing a nonnegative integer n as a sum of 3 squares when 0 is allowed. For example, n=1 can be written as 1 or -1 squared in any of the three square places with the other two as zero, giving 6 unique ways (With some effort, you can figure out why that works to give you the right pattern).
And that’s already a fairly interesting result from seemingly unassuming beginnings.
But here’s the real kicker: doing the sum this way doesn’t actually get you the right constant. In fact, it won’t even get you a constant—it doesn’t converge. And I don’t mean that in the sense that it will tend towards infinity or negative infinity, which would be boring but somewhat understandable. It doesn’t converge in the sense that as you increase the distance, it just sums to random values around the actual Madelung constant that never seem to get any closer (though taking the average of those fluctuations over a long period can work, albeit slowly).
You might have already realized why that’s really weird: As you get further away, the distance increases, and the effect of any individual atom is lessened. This should really be lending itself to converging. You might have noticed something else, though: While distance increases as you get further away, meaning each individual atom has a lower effect, so does the amount of atoms.
There are just generally more atoms at further distances, a fact you can pick up on just by picturing the cubic lattice. Still, the value doesn’t even want to go towards either infinity, so this means that the distance and atom increases somehow “balance out” in the sum, creating a sort of self-regulating parity. This is even more surprising when you consider that every other atom impacts the origin in an opposite manner, which should add to the difficulty of a potential balancing act.
It also makes the simplicity of the next summing method surprising: Sum using expanding “cubes” instead of spheres, taking all the atoms in the 3x3x3 cube, then all the additional atoms you add in the 5x5x5 “shell” surrounding, then the 7x7x7 and so on, and it converges almost instantly. For NaCl, the value comes out to be about ±1.748 (depending on if you used cholrine or sodium as the reference).
As a side note, it converges even faster if you only take the “fraction” of each atom that’s in the current shell. In other words, “face” atoms are 1/2 inside or outside (and so you add only half the value until the next shell), atoms on the edge are either 1/4 or 3/4, and corners count for 1/8 or 7/8. I’ll probably post some code for this soon (edit: it’s posted).
I really do think this is amazing, and I may just be scratching the surface. If I could, I’d do my thesis on this (though apparently, someone else already did a fairly exhaustive analysis).
So what other weird and interesting properties of the Madelung constant can you come up with? Have at it, and tell me how it goes.