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How big is an electron?

April 11, 2015

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 10^{10} 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:

a_B \sim 4 \pi \epsilon_0 \hbar^2/me^2.

[Here, \hbar is Planck’s constant, \epsilon_0 is the vacuum permittivity, and m is the electron mass, and e 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 U \sim -e^2/4 \pi \epsilon_0 r that pulls the electron toward the nucleus, and that gets stronger as the electron size r (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 KE \sim p^2/2m.  The typical momentum of the electron gets larger as r gets smaller, as dictated by the Heisenberg uncertainty principle, p \sim \hbar/r, which means that the electron kinetic energy gets larger as the atom size r shrinks: KE \sim \hbar^2/mr^2.

In the balanced state that is an atom, PE and KE are about the same, which means that e^2/4 \pi \epsilon_0 r \sim \hbar^2/mr^2.  Solve for r, and you get that the electron size is about equal to the Bohr radius, which numerically works out to about 1 Angstrom, or  10^{-10} m.

atom

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 10^{-10} 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.

sphere_puzzle

In fact, the energy required to “build” the electron is E \sim e^2/4 \pi \epsilon_0 r, where r 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, E = mc^2, suggested that even a very small piece of matter was really an intensely concentrated form of energy.  One electron represents about 10^{-13} 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, E = e^2/4\pi \epsilon_0 r = mc^2 gives an estimate for r that is called the “classical radius of the electron”:

r_c = e^2/(4 \pi \epsilon_0 m c^2) \sim 10^{-15} m

So that’s our first estimate for the intrinsic electron size: about  10^{-15} 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 10^5 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 10^{-15} 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 1/(\text{distance})^3.  Another way of saying this same thing is that the electron has a “magnetic moment” whose value is equal to the “Bohr magneton”, \mu_B = e \hbar/2m.  (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 e is something like \mu \sim e \omega r^2, where \omega is the sphere’s rotation frequency.  If the sphere is spinning very quickly, then \omega is large, and the equator of the sphere is moving at a very fast speed v \sim \omega r.

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 \mu = e \omega r^2 = e v r = \mu_B, and require that v < c, then you get that the size of the electron r must be bigger than

\lambda_c \sim \hbar/(m c) \approx 10^{-12} 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 10^{-12} m.  This is about a thousand times larger than the classical electron radius.

Coincidentally, this value \lambda_c 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 mc^2.  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:

asymmetric_electron

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 d, which is defined as something like d \sim (charge in the bottom half of the electron - charge in the top half of the electron ) \times ( 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 d < 10^{-30} \text{ electron charges} \times \text{meters}.  This means that either the electron is an extremely symmetric object (for example, a very perfect sphere) or its size is smaller than about 10^{-30} 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 10^{-18} 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 10^{-18} and 10^{-30}, 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”, \ell_p.  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 \ell have very large energy, E \sim \hbar c/\ell.  It should therefore be possible to convert that energy, if only for a brief instant, into a large mass M = E/c^2. (After a short time t \sim \hbar/E, 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, R_s \sim G M/c^2.  (Here, G is Newton’s gravitational constant).  This starts to be a real problem when R_s 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 R_s becomes equal to the wavelength \ell of the photon that created it.  If you work through the chain of algebra above, this will bring you to a length scale of

\ell_p = \sqrt{\hbar G/c^3} \approx 10^{-35} m.

At any length scale smaller than \ell_p, 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 10^{-10} meters down to 10^{-35} 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.

15 Comments leave one →
  1. April 12, 2015 1:46 am

    Excellent post! (Excellent blog!) This is the first time I’ve seen another reference to the experiments testing the roundness of the electron. It’s nice to see how that fits in with the picture.

    Intuitively (ha!), it seems an electron must have some spacial extent. How can a truly dimensionless point have properties? (Unless reality is math, maybe? Or if this were a simulated reality?) I was initially very taken with string theory’s ability to give size and shape to fundamental particles. There is something lyrically beautiful about all such being vibration modes of one thing.

    Personally, I vote for the little e being no bigger than 10^-30. I’m old-fashioned, so I’d be okay with it being a wee round boojum of some kind.

    You mentioned Planck length space… just for the record, I’m a crank who wants Einstein to have been right. In the conflict between GR and QM, I want GR to have nailed it. Space is smooth (not quantized), gravity is not a “force,” there’s no such thing as a graviton (there’s no graviton field). QM turns out to be another case of Epicycles; there’s something else going on. Someday we’ll look back and smile fondly at how laughably wrong QM was.

    I jest, but I don’t think we’ve ruled out the possibility. Matter and energy pretty clearly seem quantized. Perhaps time and space are not? Pity the Planck level is so far down.

    • Brian permalink*
      April 12, 2015 11:22 am

      Secretly, I have some similar feelings. The great Landau thought that quantum mechanics was an incomplete theory that would one day be replaced with something more correct. But in recent decades there has been a push to just accept QM as literally, exactly true. I personally hope proponents of that perspective turn out to be wrong.

      • April 12, 2015 2:02 pm

        There was that PBR paper not long ago that seemed to put QM squarely into ontological territory. And we seem stuck with quantum weirdness like entanglement as fact. But QM has *known* weaknesses whereas GR seems pretty bullet-proof in that regard.

  2. Haydon Knight permalink
    April 13, 2015 11:48 pm

    Hi Brian, thankyou for your thoughtful and informative post.

    I personally don’t know much about physics (in both an absolute and relative sense), but have heard of Kaluza-Klein theory. In this, as I (poorly) understand it, the metric of general relativity is extended to 5 dimensions and Maxwell’s equations (in regards to the effect of electromagnetism on the momentum of matter) “pop out”. Electromagnetism is then interpreted as motion along a compactified 4th spatial dimension.

    The electron has heavier analogues (the muon and the tauon); it seems reasonable to me then that an (over) simple model could be constructed of the electron as a bare charge – the simplest oscillation along this 4th spatial dimension, with its rest mass somehow(?) deriving from the energy of this oscillation. In this model then, the “size” of the electron is the size of the compactified dimension. Wikipedia:

    http://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory#The_Quantum_Interpretation_of_Klein

    suggests that the size of this dimension is ~10^-32 m.

    Now, of course, the electron is more complicated than this and has other properties – for example it is involved in the weak interaction (e.g. neutron decay). So this simple model cannot be particularly accurate. And of course, M/string theory has superceded (enormously) the Kaluza-Klein model and has its own predictions (which I don’t know) for the scale of electrons as “strings”.

    What are your thoughts on estimating the size of electrons from extra-dimensional theories? Valid or invalid? Presumably there are actually proper derivations by actual physicists with actual math involved?

    • Brian permalink*
      April 14, 2015 12:11 pm

      This is interesting, but I can’t say that I understand where the relation between the electron mass and the dimension size came from. If I figure it out (or have someone explain it to me), I’ll add another comment.

      As for general comments about extra-dimensional theories, I am probably not qualified to say much. All I can say is that (as a condensed matter theorist) I have never felt compelled to take any of them particularly seriously.

  3. Matthias permalink
    June 18, 2015 6:15 am

    Maybe you are interested in this:
    Simply atoms – atoms simply
    http://www.lajpe.org/icpe2011/8_Friedrich_Herrman.pdf

    • Brian permalink*
      June 18, 2015 12:40 pm

      I like the visual images in this paper — I have long enjoyed thinking about the electron wavefunction as something like a flowing fluid. But I disagree with the first conclusion that “Bohr’s first postulate is not needed.” The existence of stationary, non-radiating states is already a result that contradicts classical physics. It is implicit in the Schrodinger equation, which is what this article starts from!

  4. September 5, 2015 6:15 pm

    Great article.

    WRT your comment “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.”

    I would like to expand on your picture of a spinning electrical field trapping an orthogonal magnetic field in the center of your article.

    Dirac imagined the electron as coupled 2-spinors and derived the mass of the electron from this coupling constant. Imagine if these coupled spinors spin, at the speed of light, orthogonal to themselves exactly as your picture depicts. Much of the argument regarding the “speed” of the spinning material goes away and all of a sudden, the amount of energy required to “flip” an electron makes perfect sense and the whole thing fits within the classical Lorentz radius of 3 femtometers (10^-15 meters).

    If interested, I have done some video animations of this type of particle at

    http://www.animatedphysics.com/videos/electrons.htm

    Thanks for the thought provoking article.

  5. Simon Simple permalink
    September 28, 2015 5:51 am

    Dear Brian,

    This was the first article I read on Gravity and Levity, and it is delightful. Thanks! Especially, and finally, a good discussion for the normal person on spin.

    I am a biologist, so excuse some physically stupid questions, but, I thought wave-functions were conceived to extend (very thinly) out to “infinity” wherever that might be. So doesn’t that imply that electrons are simultaneously very small, as you so well describe, but also quite large?

    Second is: is there any rational answer to the question: how fast does an electron travel “around” the nucleus? Apart from the old stand-by: somewhat slower than the speed of light. Or is the wave function everywhere and nowhere (“that’s where it’s at”) at the same time, which was my current understanding. Does it progress, around the nucleus at all, or is it just a stable entity, transforming into something resembling a particle when it is needed/queried by external prodding?

    • Brian permalink*
      September 28, 2015 8:27 am

      Hi Simon. Thanks for the kind words. Your questions are good ones!

      1: You are right that the electron actually lives in a “probability cloud” that extends, mathematically, all the way to infinity. But the intensity of that cloud decays very quickly with distance, exponentially in fact: e^{-r/a_B}. So by the time you’re talking about distances bigger than a nanometer, you can say that the electron spends only about a billionth of its time out that far from the nucleus. But you are right that the electron inside an atom does not have one orbit radius, but rather a whole distribution of different radii.

      2: Sure, the electron has a typical speed of travel around the nucleus (in the same way that it has a typical radius), and its value is e^2/(4\pi \epsilon_0 \hbar) \sim 2.2 \times 10^{6} m/s, or about 5 million mph. As with the radius, it is best to think that the electron lives in a superposition of a whole bunch of different orbits, with an (exponential) distribution of different velocities.

  6. Omega permalink
    January 3, 2016 3:58 am

    If we were to take the electron-as-assembled-charge model seriously, wouldn’t we also have to take into account the negative contribution to the mass-energy from the attractive magnetic force due to the relative motion of the charge elements, and the positive contribution from the momentum and hypothetical rest mass of the moving charge stream? My favorite model pictures the electron as a massless ring of charge rotating at the speed of light. The repulsive coulomb force is precisely cancelled by the attractive magnetic force, meaning that no net energy is required to assemble the ring, allowing us to set the radius to any value we choose. The mass-energy of the electron in this case comes entirely from the momentum of the rotating charge stream (in much the same way that a box containing light, whose mass energy is pure momentum, will still appear to have inertial mass from the outside). I suppose there’s still the problem of a nonzero repulsive centrifugal effect due to the change in direction of a momentum stream, as well as the ample experimental evidence that the electron is point-like down to at least 10^-20 m. It’s still a fun model to think about!

    • Brian permalink*
      January 3, 2016 11:33 am

      I’m a little confused by your comment. parallel currents attract each other, but antiparallel currents (like on opposite sides of a current loop) repel each other. So wouldn’t the magnetic force be repulsive?

      Put another way, a current loop has a finite (positive) self-inductance, which means that it takes energy to get a current flowing. So adding a rotation to the ring of charge would only add to the electrostatic part of the self-energy rather than mitigating it.

      Am I missing something?

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