Matt Strassler [January 14, 2013]
What keeps the electrons in an atom trapped around the nucleus of the atom?
Well, at first glance (especially when we look at the cartoon version of an atom, whose limitations I described) the electrons orbiting the nucleus of an atom look naively like the planets orbiting the sun. And naively, there’s a similar effect at work. Only it will turn out there’s a twist.
What keeps the planets orbiting the sun? In Newton’s view of gravity (Einstein’s is a bit more complicated, but we don’t need it here) any pair of objects pull on each other via gravitational forces proportional to the products of the two objects’ masses. In particular, the sun’s gravity pulls the planets toward the sun (with a force that is inversely proportional to the square of the distance between them… in other words, if you make the distance half as big, the force gets stronger by a factor of four.) The planets each pull on the sun, too, but the sun is so heavy that the pull has almost no effect on how the sun moves.
The tendency (called “inertia”) of all objects to travel in straight lines, when unaffected by other things, works against this gravitational pull, in such a way that the planets travel in orbits around the sun. This is shown in Figure 1, for a circular orbit. In general these orbits are ellipses — though these orbits are nearly circular for the planets, as a result of how they formed. (Not so for various smaller rocks [asteroids] and iceballs [comets] that orbit the sun.)
In a similar way, all pairs of electrically charged objects pull or push on each other, again with a force that varies with the inverse of the square of the distance between the objects. Unlike gravity, however, which (for Newton) always pulls objects together, electrical forces can push or pull. Objects which both have positive electric charge push each other away, as do those which both have negative electric charge. Meanwhile a negatively charged object will pull a positively charged object toward it, and vice versa. Hence the romantic phrase: “opposites attract”.
So the positively-charged atomic nucleus at the center of an atom pulls the lightweight electrons at the outskirts of an atom toward it, much as the sun pulls on the planets. (And similarly the electrons pull on the nucleus, but the mass of the nucleus is so much larger than that of the electrons that the pull has almost no effect on the nucleus. The electrons also push on each other, which is part [but only part] of the reason they don’t tend to spend too much time very close to each other.) Naively, then, the electrons in an atom could travel on orbits around the nucleus in much the same way as the planets travel around the sun. And naively, at first glance, that’s what they seem to do, especially in the cartoon atom.
But here’s the twist — actually it is a double-twist, each twist creating an opposite effect, so that they cancel each other out!
The Double Twist: How Atoms Differ From Planet-Sun Systems
First twist: Unlike planets around the sun, electrons orbiting the nucleus should — naively — be able to radiate light (or more generally, electromagnetic waves, of which visible light is just an example). And this radiation— naively — should cause the electrons to slow down and spiral into the nucleus! (There is actually a similar effect in Einstein’s theory of gravity; the planets can radiate gravitational waves. But the effect is incredibly tiny. Not so for electrons.) Naively an atom’s electrons should spiral into the nucleus in a tiny fraction of a second!
And if it weren’t for quantum mechanics, they would! This potential disaster is illustrated in Figure 2.
Second twist: But our world operates according to principles of quantum mechanics! And quantum mechanics comes armed with an amazing counter-intuitive principle, the uncertainty principle. This principle, which reflects the fact that electrons are as much waves as particles, is something that deserves a long article of its own; but for today, here’s what we need to know about it. The general consequence of the uncertainty principle is that not all aspects of an object can be simultaneously known; there are certain aspects for which measuring one precisely makes the other uncertain, and vice versa. A specific case involves the position and velocity of “particles” like electrons; if you know exactly where an electron is, you don’t know where it’s going, and vice versa. You can compromise, and know to some extent where it is and to some extent where it is going. In fact, that’s the situation in an atom. Let’s see why.
Suppose an electron spiraled down toward the nucleus as in Figure 2. Well as it did so, we would know better and better where it was located. The uncertainty principle would then tell us that its velocity would have to become more and more uncertain. But if the electron came to a stop at the location of the nucleus, its velocity would not be uncertain! So it can’t stop. Instead, as it tried to spiral in, it would have to move with a faster and faster random motion. And this increased motion would in turn fling the electron away from the nucleus!
The tendency to spiral inward is thus counteracted by the tendency to move more quickly due to the uncertainty principle. A balance is found when the electron reaches a preferred distance from the nucleus, and that distance that sets the size of atoms!
If the electron is initially very far from the atomic nucleus, it will initially spiral, as shown in Figure 2, by radiating electromagnetic waves. But eventually its distance from the nucleus is small enough that, as described in the preceding paragraph, the uncertainty principle prevents any further approach. At that stage, where the balance between radiation and uncertainty is found, the electron establishes a stable “orbit” around the nucleus (or better, an “orbital”, a name which is chosen to reflect the fact that, unlike a planet, an electron spreads out in a very non-particle-like way, thanks to quantum mechanics, and doesn’t really have an orbit like a planet does). The radius of that orbital sets the radius of an atom. See Figure 3.
There’s another feature — the fact that electrons are fermions — that causes the electrons not all to go down to the same radius, instead they stack up into “orbitals” of different radius. I won’t go into this here, but it will come up again in another article.
How Big are Atoms? An Estimate from the Uncertainty Principle
In fact (for readers who are so inclined — there’s a little bit of algebra involved) we can estimate the size of an atom by using only the force law for electrical forces, the mass of the electron, and the uncertainty principle! You can skip this part if you want, but it’s so much fun: watch how easy it is to deduce the basic size of atoms! To keep this simple, let’s do this for hydrogen, where the nucleus consists of just one proton, and there’s just one electron around it.
- Let’s call the electron’s mass me.
- Let’s call the uncertainty in an electron’s position Δx
- Let’s call the uncertainty in an electrons velocity Δv
The uncertainty principle says
- me (Δ v) (Δ x) ≥ ℏ
where ℏ is Planck’s constant h divided by 2 π. Notice this says that (Δ v) (Δ x) can’t be too small, which means the two uncertainties can’t both be really small — that’s the point! — though you could have one uncertainty be really small as long as the other uncertainty was really big.
When an atom settles down to its preferred “ground state”, we can expect that the “≥” sign turns into a “~” sign, where A ~ B means “A and B aren’t quite equal, but they aren’t very different either.” This is a very useful symbol when making estimates!
In short, for a hydrogen atom in its ground state, for which the uncertainty in position Δx will be of order the radius of the atom R, and the uncertainty in velocity Δv will be of order the typical speed V of the electron as it orbits the atom, we have
- me V R ~ ℏ
Now how do we figure out V and R? There’s a relationship between them and the force that holds an atom together. In non-quantum physics, an object of mass m in circular orbit of radius r and moving with speed v around a central object which pulls on it with mass F satisfies
- F = mv2/r
This won’t apply exactly to an electron in an atom, but it will do so approximately. The force acting on the atom is an electric force exerted by the proton (of charge +1) on the electron (of charge -1), and that takes the form
- F = ke2/r2 = α ℏ c / r2
where k is Columb’s constant, e is the fundamental unit of electric charge, c is the speed of light, again ℏ is Planck’s constant h divided by 2π, and we have defined the “fine-structure constant” α = ke2/ℏc ~ 1/137.04… or so. Putting the previous two equations for F together as an approximate relationship gives
- α ℏ c / r2 ~ mv2/r
Now let’s apply that for our atom, letting v → V, r → R, and m → me. And let’s also multiply the above equation by me R³. This gives
- α ℏ c me R ~ me2 V2 R2 = (me V R)2 ~ ℏ2
where in the last step we used the uncertainty relationship for our atom, me V R ~ ℏ. Now we can solve for the radius of the atom R, finding
R ~ ℏ / (α c me) ~ 137 (10-34 kg m²/s) / (3·108 m/s · 9·10-31 kg) ~ 0.5 · 10-10 m
which is just about right! Simple-minded estimates like this won’t get you precise answers, but they will give you accurate answers!