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Christine Nattrass | USLHC | USA

View Blog | Read Bio

The size of the proton

There is a new measurement of the size of the proton and it turns out that protons are smaller than we thought they were.

At some point in your education you probably got introduced to the Bohr model of the atom.  The nucleus is made up of protons and neutrons, and electrons orbit around the nucleus.  In the Bohr model, electrons orbit the nucleus in circular orbits like the Earth orbits the Sun, but these orbits are only allowed to have some radii (which correspond to an integer number of de Broglie wave lengths).  Electrons can transition between these levels and when they do, they either absorb a photon (in the case of an electron being excited from, say, the ground state to an excited state) or emit a photon (in the case of an electron going from an excited state to a lower state.)  This is shown below:

The Bohr model isn’t exactly right – but it’s close enough to get some feel for what’s going on.  In a more precise quantum mechanical picture, the electron isn’t actually orbiting the nucleus – it’s smeared out in what we call a wave function.  The square of the wave function tells us how likely we are to find the electron in a given place.  The ground state orbital (the shape of the wave function of the electron in the atom) is spherical.  The lowest excited state has four different possible orbitals, one spherical (S) and three which are shaped like a dumbbell (P), a sort of 3D figure-8.

What you probably learned in school was that these S and P orbitals have exactly the same energy – and they almost do.  In a simple model, the nucleus is just a point particle – meaning it exists just at a single point, with no size in any dimension.  But protons aren’t point particles – they’re just very small.  In the S orbitals, the electron spends most of its time near the nucleus, but in the P orbitals, the electron spends less of its time near the nucleus.  This difference in how much time the electron spends near the nucleus leads to a very small shift in the energy of the orbitals, called the Lamb shift.  The Lamb shift is measured by measuring the photon emitted when an electron goes from the P to the S orbital in the second shell.  It depends on the mass of the electron and the size of the proton.  (Here’s the explanation of the Lamb shift on the experiment’s web site.)

In this new measurement, they looked at hydrogen with a muon (the heavier cousin of the electron) instead of an electron.  Because the muon is about two hundred times heavier than the electron, it spends more time near the nucleus than the electron, meaning it’s more sensitive to the Lamb shift than the electron.  Previously, the best measurement of the diameter of the proton was 0.877±0.007 femtometers (m) and this measurement measured it to be 0.8418±0.0007 fm.  A femtometer is 10-15 meters.  If you were a proton (you’re somewhere between 1-2m tall), this would mean traveling one millimeter would be like traveling from the Earth to the Sun (1011 m).  This measurement would be like finding out that you’re 5’5″ instead of 5’8″ by looking at how long it takes for you to walk between Milwaukee, WI and Chicago, IL (150 km) and Milwaukee, WI and Madison, WI (141 km)*.

The fact that this measurement is so far off from our expectations indicates one of the following:

  • The precise calculation we’re comparing to is flawed.The proton is actually a really complicated object – perhaps we forgot an important component.
  • The measurement has some flaw we haven’t figured out yet.  Maybe there was some systematic shift that wasn’t taken into account.
  • Our theory is flawed.  This could indicate some physics beyond the Standard Model – exactly what we’re looking for at the LHC.

We have to seriously consider the first two options, but the third would obviously be very exciting.

So why I am writing about this here?  First, it illustrates that there are other ways of studying fundamental particle physics than by slamming things together.  Second, it’s an interesting result that may hint at exciting new physics we’re hoping to see at the LHC.  Third, it’s a great segue into my next post…

*Yes, this analogy breaks down at some point.  Don’t take it too far.

  • Demian Cho

    Excellent explanation in elementary level! I hope you don’t mind me linking the post to my facebook page.


  • Jonathan Clift

    Hello Christine.

    Are physicists sure that the proton is spherical? Isn’t it possible that it is a bit lumpy, because of its internal structure, and that both measurements are right? (In my very naive way, I could imagine the shape of the (P) wave functions of the electron/muon aligning somehow with the internal structure of the proton and hence giving different measures depending on the particular experiment being done and which (P) dominates.)

  • Christine Nattrass

    Demian – thanks! Glad you like it. And any publicity would be appreciated! You can also like the US LHC on Facebook and get updates on the LHC posted to your wall.

    Jonathan – this is a really good question.
    You’re right that in principle the proton is lumpy. We can study the proton’s structure using deep inelastic scattering, which means we hit the proton with a very high energy electron. By doing this we can see the lumps – which are actually quarks and gluons. But the quarks and gluons are “moving” very quickly. (Moving is in quotes because they actually have wave functions so this isn’t quite techinically correct, but it’s close. We’re really talking about the period of the time-dependent wave function.) The electron or muon “moves” much slower than the quarks and gluons so it can only see the average. This is analogous to taking pictures. If you take pictures at night you have to use a slow shutter speed so that the film is exposed to enough light – but if you take a picture of someone running using a slow shutter speed, all you’ll see is a blur because the shutter was open long enough for the person to move across the frame. So looking at the proton using electrons or muons in an atomic orbital is like taking a picture of the proton with a slow shutter speed.
    There all sorts of fluctuations in a proton, but there’s no particular reason, for instance, for an up quark to be on the top rather than the bottom in this experiment. If we extend the picture analogy, our proton is like a bunch of kids (quarks and gluons) on a playground. If you look at any given point in time, each kid is on a particular piece of playground equipment. But if you took a picture of the playground where you exposed the film for, say, five minutes, each kid would show up as a blur in the photograph. The average distribution would be fairly smooth. So at any given point in time, the proton is lumpy – but we average over long times, so the proton looks smooth in this experiment.
    But I can design an experiment where the average distribution of quarks and gluons isn’t uniform. Protons have a magnetic field (called spin) and electrons have a spin. If I put a hydrogen atom in a magnetic field, the proton’s spin will line up with the magnetic field and then I can see a difference in energy for electrons with their spin lined up with the proton spin and those with their spin in the opposite direction. This would be equivalent to going to the playground with a chocolate cake and passing out chocolate cake – the distribution of kids on the playground would no longer be uniform. This effect is called hyperfine splitting. Unfortunately, while we know that the different orientations of the proton’s spin correspond to different configurations of quarks and gluons in the proton, we don’t know what those configurations are. We know the proton has spin, but we haven’t figured out where it comes from. (This is another field of study in physics.)
    So yes, protons are lumpy, but whether we see those lumps depends on how we look at them. This experiment is analogous to measuring how far away from the playground equipment the kids are on average. There are other ways to define the size of the proton and in what will likely be my next post I’ll talk about another way from ALICE data.

  • tom


    I was wondering. You mention that the current best estimate for the diameter of a proton is 0.8418±0.0007 fm. That is about a 0.1% deviation. Can we expect atoms from one edge of the universe to match atoms plucked from the opposite reaches ? Is it possible that some atoms would be scaled up or down by multiple factors with respect to the lot we are familiar with ?


  • tom – as far as we know, the laws of physics are the same everywhere in the observable Universe. Therefore we expect protons to be the same, identical, and indistinguishable, everywhere. There are symmetry arguments that corroborate this as well: https://secure.wikimedia.org/wikipedia/en/wiki/Noether%27s_theorem#Applications

  • How about protons being wavefunctions, too, whose location is somewhat uncertain? Or maybe the quarks in the protons are perturbed randomly by charged virtual particles from vacuum fluctuations. That might prevent a very precise estimate of proton diameter.

  • Christine Nattrass

    Hi Dave – so what you’re referring to is the Heisenberg Uncertainty Principle, which says (roughly) that the more precisely you know the momentum, the less precisely you know the position. However, if you know the wave function you know everything about the electron or muon. (Assuming you are not working at such a small scale that quantum mechanics is no longer sufficient to describe the data and you need to use quantum field theory… but this experiment is still at scale where quantum mechanics is sufficient.) You can get quarks (and anti-quarks and gluons) inside protons from vacuum fluctuations – but these are not the quarks we mean when we say that a proton is made up of three quarks. Vacuum fluctuations are already taken into account in what we call the Standard Model – so that’s not sufficient to explain why we measured the proton to be smaller than we expected.

  • Jha

    Have physicists sorted out wave without medium dilemma ?

  • Christine Nattrass

    Jha, I’m not exactly sure what you’re asking. If you’re referring to the question of whether or not an ether is necessary, it’s not – in fact, the evidence is strongly against an ether. The Michelson-Morley experiment looked for evidence of an ether and found none:

  • I could imagine the shape of the (P) wave functions of the electron/muon aligning somehow with the internal structure of the proton and hence giving different measures depending on the particular experiment being done and which (P) dominates.)Thank you.
    Regard : http://www.anekataskulit.com/

  • Victor

    In case the size of the proton is smaller wouldn’t it be noticeable in the LHC experiments? I mean they had been smashing protons together and the discrepancy in size should have affected the statistical data of number of collisions.

  • Jonathan,

    Your idea that proton may not be spherical is a great idea; I was also thinking similarly;


    ———> DR Hota

  • Fitting question!!

  • Nice explanation!

    Particularly, I am impressed by the following statements:

    “Our theory is flawed. This could indicate some physics beyond the Standard Model – exactly what we’re looking for at the LHC”.

    ———–> DR Hota

  • GeorgeM. Webb

    You say the proton diameter is 0.8418fm. Other references say the proton radius is 0.8418fm. Which is it?

  • ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    When the strings of a particle are balled up or collapsed… they are a proton or neutron.

    The individual radii (1 of 20 strings) are the connectors used to connect neutrons to protons (balled up) and protons to electrons (full length but twisted together).

    Everything is made out of the same particle and every particle has 20 strings unless it is smashed up deformed matter.
    A proton has one string balled (tightly wound together) with a neutron, 18 balled by themselves and one full length twist connected to an electron.

    A free proton would look like this   ~~~●~~~     (that’s one free string, 18 balled, one free string)

    A free neutron would look like this   ●~~~     (19 balled, and one free string)

    A free electron would look like this   ---∗---     (one free string, 18 free strings in a disc shape, one free string)

    A proton can grab a neutron and an electron.

    ●~~~ ~~~●~~~ ---∗---     (NPE on the loose)

               ●●~~~∗---     (NPE combined)

    (that’s a neutron with its previously free string balled up together with one of the proton’s previously free strings (now also balled up) and the other proton string is twisted with an electron string (that free proton string and electron string twists are still full length))

    Two free protons   ~~~●~~~ ~~~●~~~  
    can combine and still be 2 protons   ~~~●●~~~   (that might look like 2 free neutrons but it is not because there are also balled up strings in the middle of the package holding them together.

    To clarify: two free neutrons
      ●~~~ ~~~●   that are now combined would look like this   ●●  

    If you throw another free proton into that 2 proton package   ~~~●●~~~ ~~~●~~~
    you will get one changing into a neutron when they combine   ~~~●●●~~~   that’s Helium-3

    If 4 free protons  
    ~~~●~~~ ~~~●~~~ ~~~●~~~ ~~~●~~~

    …grab each other 2 will change into neutrons   ~~~●●●●~~~

    And then the outer two that still have a free string can grab electrons…

    ---∗~~~●●●●~~~∗---   that’s regular Helium, it can also be called Helium-4

    If you understand the way this works… with a little thinking anyone can figure out isotopes.
    For instance why 3 protons would not make lithium-3…

    i.e. why there can be extra neutrons but not just a bunch of protons (or extra protons)… we’ve just seen that above the way Helium-3 was created.
    Nucleus 3 can only be helium-3   ~~~●●●~~~   or Hydrogen-3 (tritium)   ●●●~~~  
    (Lithium-3 would be a nucleus with 3 protons and zero neutrons… and that can’t be a nucleus)

    “Lithium-4 contains three protons and one neutron. This is the shortest-lived known isotope of lithium. It decays by proton emission to helium-3 with half-life of about 10^−23 seconds.”

    ~~~●~~~ ~~~●~~~ ~~~●~~~ ~~~●~~~     (start with 4)

    ~~~●●●~~~ ~~~●~~~     (combine in only way possible to make 3 protons, 1 neutron… notice one proton is on the loose… not attached to nucleus, say goodbye, it’s unstable, eject it)


    Something like an atom with Protons, Neutrons and Electrons has to be the correct model.
    Things are different weights, different colors, different properties, etc. but everything has to be made out of the same thing.
    An atom is the way to do it.
    They almost have the model correct… but everything is actually just strings and tension

    ELECTRON    ---∗---

    An electron is shaped like the metal spines of an umbrella (without the hinges or fabric of course).
    One string extents from where your hand would hold it up to the center of axis. There, eighteen strings (or radii) extent out in the same curved disc type shape as the umbrella. The last string goes straight up (the same length as all the rest) and connects with the field in space (space is made of the same stuff by the way).

    Notice the way some elements in vertical columns in the Periodic table chart have an atomic number with difference of 18 between them. Most of the chart is like that (notice how many columns there are).

    It’s because 18 is the determinant number in electron shell configuration.

    Every electron particle has 20 strings.
    One string is attached to the proton.
    One string connects with space (or an electron in the next outer shell).
    The other 18 strings form the electron disc.
    When electrons connect with each other they have 18 strings to play with.

    Check the larger noble gases: Argon 18, Krypton 36, Xenon 54, Radon 86, the amount of electrons in outermost shells will always sum to 18, the first three even have atomic numbers that are multiples of eighteen. Three groups of six radii from one electron can form (along with seven other electrons) the corners of a cube or the "Octet Rule" and seal off the package.

    Important note: Electrons are actually particles but they (the strings they are made from) form a mesh-like cage around the nucleus. They are also held in place by string connections to the protons.
    An electron is actually not moving… only the vibrations that are traveling around the strings are moving… and that’s what everyone mistakenly thinks an electron is.

    Electrons (particles) cannot orbit around a nucleus.

    The protons are stationary and the (multiple) electrons that supposedly are orbiting would require a massive amount of bearings and axles. And they would also interfere with each others orbits.
    You can’t use “force” as the holder (or carrier) because any force is also made from particles or their connection.

    To make matters worse… an equatorial orbit (supposedly happening) would need something like a circular track around the proton (actually the nucleus as a whole) with a sliding connection. That’s ridiculous.

    PROTON    ~~~●~~~

    The proton is 20 strings (like everything else) one string radii is attached to a neutron, one is attached to a electron and the other 18 remaining string radii are balled up or collapsed.

    If the strings collapse in groups of three each that would make 6 groups (3 * 6 = 18) or six types of (what they call) Quarks.
    And if they collapse in groups of six each that would make 3 groups (6 * 3 = 18) or three (what they call) Quarks in three flavors.
    Maybe the grouping during collapse happens in different numbers like… 3, 6 and 9 …that still sums to 18 strings.
    The jury is still out on all of this Quark business. When they smash up protons they assume they have found different subatomic particles because of the different weights. That is just a different number of strings being smashed apart.

    If you magnified a proton until it was the size of the dot above the letter “i” then the strings could be compared to something a lot finer than the web of a spider extending out a few hundred meters. Fine enough where eighteen strings can curl into a space the size of the proton and have a spaghetti ball type configuration with a very loose string (or filament) pack.

    It is the way to make the most universe with the least amount of material. And only one type of material.

    NEUTRON    ●~~~

    A neutron is the same as proton but with 19 string radii balled up or collapsed. And when it is in the nucleus all 20 are collapsed (although one of the 20 is collapsed in unison with a proton string)

    One Proton string and one Neutron string balled up or collapsed together is called a Meson.


    A Neutrino is a completely balled up or collapsed particle (all 20 strings) or a group of completely balled up particles ●● NOT connected to the field or anything else.
    The speed of light is completely irrelevant to a Neutrino. The speed of light is field stuff, the neutrino is on its own.
    You could say the Neutrino is in the “ultimate time” zone.

  • Yes, you can call stuff like that “counter-intuitive” — it explains that and makes everything else completely allowable!

  • ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    If you were going to test if there is a medium for the conveyance of light, would you…

    A) Test if the Earth is rushing through the medium.
    2) Test if the Earth is NOT rushing through the medium.
    C3) Both of the above (same as: just test for medium, no constraints)

    Here is your chance to agree with those great men and pick “A”, everything you think you know is based on that.

    NOTE: The correct answer is of course “C3” but modern physics is based on Michelson-Morley experiment and they picked “A”
    Michelson-Morley picked “A” and everything you think you know is based on that. Michelson-Morley “confirmed” there is no medium with their experiment (it’s actually a pillar of modern science)
    The only problem is if “2” is happening they are completely in the dark about it.

  • Anthony Dean

    Why can’t we think of mass as acting at right angles to light (i.e. light travels along spacetime plane, mass bends spacetime around it). E.g. if we used a Tau instead of a Muon we likely would find the size is even smaller.