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And, by the way, what is, or are, the measured values?

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  • $\begingroup$ Define "measured". What counts as measuring a length scale? The smallest meaningful length scale is the Planck length. But we can't really stick a ruler up to it and measure it that way. So what do you mean? $\endgroup$
    – Jim
    Nov 14 '14 at 16:19
  • $\begingroup$ Nuclear radii are of a length of a few hundreds of femtometers ( 10^{-13} )to a few picometers ( 10^{-12} ) but I don't know if they were measured directly or estimated. $\endgroup$
    – Sofia
    Nov 14 '14 at 16:35
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The smallest length that has been directly measured is about $10^{-18}$m, and the measurement was done at the Large Hadron Collider. In a collider the length scale of the phenomena you are studying is related to the energy of your collision by:

$$ \Delta x \approx \frac{h c}{E} $$

where $h$ is Planck's constant, $c$ is the speed of light and $E$ is the collision energy. At the LHC the collision energy is around $10^{12}$eV - the total energy of the colliding protons is $7 \times 10^{12}$eV, but only a fraction of this energy is used in any particular quark-quark collision. If you feed these values into the equation above you get a value for $\Delta x$ of around $10^{-18}$m.

This isn't just playing with numbers. This is a real length measurement. For example you might ask if a quark is a fundamental particle or composite like protons. Well the measurements at the LHC tell us that if it's composite its size must be less than $10^{-18}$m or we would have seen evidence for its size at the LHC.

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    $\begingroup$ Hey John, I considered posting a similar answer using the LHC energy however it occurred to me that the IceCube 1 PeV neutrino events could in some sense be measurements of size. Do you think this is too much of a stretch? $\endgroup$ Nov 14 '14 at 17:11
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    $\begingroup$ @BrandonEnright: that's why I included the weasel words directly measured. I think that the IceCube measurements are really an indirect measurement. There are all sorts of indirect measurements because energies higer than 1TeV may have to be included when calculating scattering cross sections i.e. very high energy physics will affect probabilities at lower (observable) energies. As a result arguably we have probed energies far higher than 1TeV. $\endgroup$ Nov 14 '14 at 17:22
  • $\begingroup$ I considered a similar answer as well; however I'd have said $10^{-19}$ m, i.e. 1 TeV. I wondered why you used 100 GeV instead... $\endgroup$
    – xi45
    May 10 '16 at 12:36
  • $\begingroup$ @JohnRennie do you mean $10^{12}$ eV? $\endgroup$
    – jim
    May 10 '16 at 12:53
  • $\begingroup$ @jim: oops, yes, thanks. A one terajoule collider would be nice though :-) $\endgroup$ May 10 '16 at 15:19
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Observation of a single electron in a Penning trap shows the upper limit of the particle's radius is 10^−22 meters

Reference.

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