How big is an electron?

In the year 1908, while the great minds of natural philosophy were puzzling over how to understand the structure of atoms in terms of the recently-discovered electron, Vladimir Lenin (yes, that Vladimir Lenin) declared that:

The electron is as inexhaustible as the atom

Lenin was making a philosophical point, but you can view his declaration as something like a scientific prediction.  Just as scientists of his era were discovering that atoms were made from smaller constituents like electrons, so too, he said, would we eventually find that the electron was made from even smaller components.  And on the cycle would go, with each new piece of matter being an “inexhaustible” source of new discovery.

It was a pretty reasonable expectation at the time, given the relentless progress of reductionist science.  But so far, despite more than a century of work by scientists (including many presumably extra-motivated Soviets), we still have no indication that the electron is made from anything else, or that it has any internal structure.  Students of physics in the 21st century are taught that the electron is a point in space with certain properties — mass, charge, and spin — but that it cannot be thought of as a spinning sphere or anything else that has a size and shape.  The electron looks pretty exhausted.

In this post, though, I want to take Lenin’s side, and ask the heretical question: “If the electron actually is a real, physical object with a finite size, then how big is it?”  Not surprisingly, there is no clear answer to this question, but some of the candidate answers turn out to be pretty interesting.

Option #1: The Bohr radius

If you’re a physicist, and someone asks you “how big is an electron?”, then the most canonically correct thing to say is “There is no concept of an electron size other than the spatial extent of the electron wavefunction.  The size of the electron wavefunction is the electron size.”

This opinion essentially amounts to telling someone that they need to stop trying to think about a quantum electron as a physical object that you could hold in your hand like a baseball if only it was enlarged times.  People who profess this opinion are presumably also the ones who get annoyed by the popular declaration that “matter is 99.999% empty space.”

So, if such a person were pressed to give a numerical value for the “size of the electron”, they might say something like “Well, most electrons in the universe are bound into atoms.  So the typical ‘size’ of an electron is about the same as the typical size of an atom.”

As has been discussed here before, the typical size of an atom is given by the Bohr radius:

.

[Here, is Planck’s constant, is the vacuum permittivity, and is the electron mass, and is the electron charge.]

Just to remind you how this length scale appears, you can figure out the rough size of an atom by remembering that an electron bound to an atom exists in a balance between two competing energies.  First, there is the attractive potential energy that pulls the electron toward the nucleus, and that gets stronger as the electron size (which is about the same as the average distance between the electron and the nucleus) gets smaller.  Second, there is the kinetic energy of the electron .  The typical momentum of the electron gets larger as gets smaller, as dictated by the Heisenberg uncertainty principle, , which means that the electron kinetic energy gets larger as the atom size shrinks: .

In the balanced state that is an atom, and are about the same, which means that .  Solve for , and you get that the electron size is about equal to the Bohr radius, which numerically works out to about 1 Angstrom, or   m.

So now we have our first candidate answer to the question “how big is an electron?”.  If someone asks you this question, then you can sort of roll your eyes and then say “usually, about meters”.

But let’s keep going, and see if there are any other concepts of an “electron size”.  Perhaps you don’t really like the answer that “the size of the electron wavefunction is the electron size”, because the electron wavefunction can be different in different situations, which means that this definition of “size” isn’t really an immutable property of all electrons.  Maybe you prefer to think about the electron as a tiny little ball, and the wavefunction as a sort of probability distribution that describes where the ball tends to be from time to time.  If you insist on this point of view, then what can you say about the size of the ball?

 

Option #2: The classical electron radius

If you’re going to think about the electron as a tiny charged ball, then there is one thing that should bother you: that ball will have a lot of energy.

To see why this is true, imagine the hypothetical process of assembling your tiny charged ball from a bunch of smaller pieces, each with a fraction of the total charge.  Since the pieces have an electric repulsion from each other, and since you are bringing them very close to each other, the ball will be very hard to put together.

In fact, the energy required to “build” the electron is , where is the electron size.  So the smaller the electron is, the harder it is to build, and the more energy gets stored in the form of electric repulsion between all the pieces.  The large self-energy of the  electron also means that if the electron were to get “broken”, then all the tiny pieces would fly apart from each other, and release a tremendous amount of energy.

But, in fact, by the end of the first decade of the 20th century, people already knew that a single electron stored a lot of energy.  Einstein’s famous equation, , suggested that even a very small piece of matter was really an intensely concentrated form of energy.  One electron represents about Joules (500,000 electron volts), which means that it only takes a few cups’ worth of electrons to have enough rest mass energy to equal the energy of an atomic bomb.  (Of course, if you ever actually removed all the electrons from a few cups’ worth of matter, you would create something much worse than an atomic bomb.)

So one of the early attempts to estimate a size for the electron was to equate these two ideas.  Maybe, the logic goes, the large “mass energy” of an electron is actually the same as the energy stored in the electric repulsion of its constituent pieces.  Equating those two expressions for the energy, gives an estimate for that is called the “classical radius of the electron”:

m

So that’s our first estimate for the intrinsic electron size: about   m.

This classical radius, by the way, works out to be about the same size as the typical size of an atomic nucleus.  And this is, perhaps, where we get the typical middle school picture of the atom: if you think about the electron using its classical radius, then the picture you get is of a tiny negatively-charged speck orbiting another equally tiny positively-charged speck, with about a factor difference between the speck sizes and the orbit sizes.

Option #3: The Compton Wavelength

This picture of the electron as a little charged ball with size meters seemed relatively okay up until the 1920s, when it was discovered that the electron had more properties than just its charge and its mass.  The electron also has what we now call a “spin”.

Basically, the concept of spin comes down to the fact that the electron is magnetic: it has a north pole and a south pole, and it creates a magnetic field around itself that is as large as about 1 Tesla at a distance of 1 Angstrom away, and that decays in strength as .  Another way of saying this same thing is that the electron has a “magnetic moment” whose value is equal to the “Bohr magneton”, .  (As it turns out, this magnetic moment is essentially the same as the magnetic moment created by the orbit of an electron around a nucleus.)

At first sight, this magnetic-ness of the electron doesn’t seem like a problem.  A charged sphere can create a magnetic field around itself, as long as the sphere is spinning.  Remember, for example, that electrical currents create magnetic fields:

and that current is just moving electric charge.  So a spinning charged sphere also creates magnetic fields by virtue of the movement of its charge-containing surface:

This sort of image is, in fact, why the electron’s magnetic-ness was first called “spin”.  People imagined that the little electron was actually spinning.

A problem arises with this picture pretty quickly, though.  Namely, if the electron is very small, then in order for it to create a noticeable magnetic field it has to be spinning really quickly.  In particular, the magnetic moment of a spinning sphere with charge is something like , where is the sphere’s rotation frequency.  If the sphere is spinning very quickly, then is large, and the equator of the sphere is moving at a very fast speed .

We know, however, that nothing can move faster than the speed of light (including, presumably, the waistline of an electron).  This puts a limit on how fast the electron can spin, which translates to a limit on how small the radius of the electron can be if we have any hope of explaining the observed electron magnetic field in terms of a physical rotation.

In particular, if you set , and require that , then you get that the size of the electron must be bigger than

m.

In other words, if you hope to explain the electron magnetic field as coming from an actual spinning motion, then the size of the electron needs to be at least m.  This is about a thousand times larger than the classical electron radius.

Coincidentally, this value is called the “Compton wavelength”, and it has another important meaning.  The Compton wavelength is more or less the smallest distance to which you can confine an electron.  If you try to squeeze the electron into an even smaller distance, then its momentum will become so large (via the uncertainty principle) that it its kinetic energy will be larger than .  In this case, there will be enough energy to create (from the vacuum) a new electron-positron pair, and the newly-created positron can just annihilate the trapped electron while the newly-created electron flies away.

Option #4: the empiricist’s view

At this point you may start to feel like none of the above estimates for the electron size seems meaningful.  (Although don’t be too harsh in discarding them: the concepts of the Bohr radius, the classical electron radius, and the Compton wavelength appear over and over again in physics.)

So let’s consider a purely empirical view: what do experiments tell us about how big an electron is?

I know of two types of experiments that qualify as saying something about the electron size.  The first is a measurement of the electric dipole moment of the electron.  The idea is that, perhaps, the electron does not always have its charge arranged in a perfectly spherical way.  Maybe its “shape” can be slightly asymmetric, like this:

An asymmetric electron

In this case the electron would would want to align its “head” with an external electric field.  The strength of the electron asymmetry is quantified by the electron dipole moment , which is defined as something like charge in the bottom half of the electron charge in the top half of the electron electron size).

So far, however, experiments looking for a finite value of the electron dipole moment have not found one, and their experiments place an upper limit of .  This means that either the electron is an extremely symmetric object (for example, a very perfect sphere) or its size is smaller than about meters.

 

Another set of experiments looks for corrections to the way that the electron interacts with the vacuum.  At a very conceptual level, an electron can absorb and re-emit photons from the vacuum, and this slightly alters its magnetic moment in a way that is mind-bogglingly well-described.  If the electron were to have a finite size, then this would alter its interaction with the vacuum a little bit, and the magnetic moment would very slightly change relative to our theories based on size-less electrons.

So far, however, experiments have seen no evidence of such an effect.  The accuracy of the experimental observations places an apparent upper limit on the electron size of about m.

 

Option #5: The Planck length

At this point, seeing the apparent failure of the classical description to produce a coherent picture, and seeing the experimental appearance of such spectacularly small numbers as and , you may be willing to abandon Lenin’s hope of an “inexhaustible” electron and simply declare that the electron really is size-less.  From now on, you may resolve, when you draw an electron, you’ll draw it as a single pixel.  But only because you can’t draw it as a half-pixel.

But this brings up the last, and strangest, question: what is the smallest conceivable length that anything can have?  In other words, if the universe has a fundamental “pixel size”, then what is it?

Of course, I don’t know the answer to this question.  Whether there really is a “smallest possible length” is interesting to consider, but at the moment it can only be addressed with speculation.  Nonetheless, we do know that there is a length scale below which our most basic theories of the universe stop making sense.  This is called the “Planck length”, .  At distances smaller than the Planck length, we are unable to describe even what empty space is like.

As I understand it, the Planck length problem can be viewed like this.  Our modern understanding of the vacuum (i.e., of empty space) is that it contains one photon mode for every possible photon wavelength.  This means that empty space is essentially full of photons of all conceivable energies and with all conceivable wavelengths.

However, photons with very small wavelength have very large energy, .  It should therefore be possible to convert that energy, if only for a brief instant, into a large mass . (After a short time , the mass will have to disappear again and give its energy back to the vacuum).  If that mass is large enough, though, it can create a small black hole.  A black hole gobbles up all other photons around it if they have a wavelength smaller than the black hole’s Schwarzschild radius, .  (Here, is Newton’s gravitational constant).  This starts to be a real problem when gets as large as the wavelength of the photon that first created the black hole, because then the black hole can consume the original photon and all other photons with smaller wavelength.

Do you see the problem with that (semi-contorted) logical sequence?  If very short length scales exist, then very high energy photons exist.  But if high energy photons exist, then they should be able to create, for a brief moment, very high masses.  Those high masses will create black holes.  And those black holes will eat up all the high energy photons.

It’s sort of a logical inconsistency.

As mentioned above, this problem first arises when the Schwarzschild radius becomes equal to the wavelength  of the photon that created it.  If you work through the chain of algebra above, this will bring you to a length scale of

m.

At any length scale smaller than , we don’t know what’s going on.  At such small length scales either quantum theory should be different from what we know, or photons should be different, or gravity should be different.

Or maybe at such small length scales there is no good notion of continuous space at all.  Such thinking, as I understand it, gives rise to lots of picturesque ideas about “quantum foam”, and is the playground of (as yet mostly non-existent) theories of quantum gravity.

This picture is supposed to illustrate the idea that space might smooth and continuous until you try to look at it at the scale of the Planck length.

 

What shall we say about Lenin?

So in the end, was Lenin right about the electron being “inexhaustible”?  For the moment, it looks like the answer is no, in the sense that there isn’t really any serious candidate for the intrinsic size of the electron.  In that sense, we could all have saved some time by just accepting the standard dogma that an electron is a sizeless point in space.

But I personally tend to resist dogmatism in all its forms, even the kind that is almost certainly correct.  Because sometimes those heretical questions lead you through all sorts of interesting ideas, ranging from from meters down to meters.

And if it becomes clear some day that Lenin’s statement really only works on the Planck scale, then we can probably say that his prediction came several centuries before its time.