I have been taught that you can find out the size of a nucleus of an atom by firing electrons at high velocities at the atom. This causing scattering due the positive charge of the nucleus and diffraction due to the small de Broglie wavelength. Part of this relies on the first minimum of the diffraction pattern occurring at $\sin \theta = \frac{1.22 \lambda}{d}$, where $d$ is the diameter of the nucleus. This is a variant on the same equation that I came across in light diffraction where $d$ is the width of the slight, or spacing between elements of a grating. Why should $d$ here now become the diameter of a nucleus rather than the distance between nuclei? Thank you for all your help
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$\begingroup$ @rob: I worked on LHC for a while, so my notion of "small" is based on that... SLAC, for instance, did 20GeV with a 2 mile long linac, so 1GeV is probably something like a football field+. To me that's still "small" (coincidentally I also had a short stint with a collaboration that built one of those machines). Having said that, I agree... we aren't exactly talking about tabletop experiments, for sure! Thanks for suggesting to make this an answer. Since nobody else wrote something, I will oblige. $\endgroup$– CuriousOneCommented Jan 4, 2015 at 17:42
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If we evaluate the DeBroglie wavelength of electrons (e.g. with this online calculator), then we find that an electron energy of 1 eV leads us to $1.2\times10^{-9}\,\mathrm m$ or about 1 nm (that's the size of small organic molecules with 2-3 benzene rings), 15 eV for the size of a hydrogen atom. If we use 1 GeV electron energy we finally get down to $1.2\times10^{-15}\,\mathrm m$, which is roughly nuclear size. So if we want to make atomic measurements, a little tabletop experiment will do, to measure the size of nuclei with scattering directly takes a medium-sized accelerator.
answered Jan 4, 2015 at 17:44
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2$\begingroup$ Although you are very clear in the practicalities of the matter at hand would you please be able to explain how the equation has come about rather than its usage? $\endgroup$– SomniareCommented Jan 5, 2015 at 9:03
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$\begingroup$ @Somniare: The "derivation" is on the same page as the online calculator: hyperphysics.phy-astr.gsu.edu/hbase/debrog.html#c1 and there is a good Wiki page: en.wikipedia.org/wiki/Matter_wave. I don't think that I could add much to what's there. Maybe one word of caution: these are all handwaving arguments. The actual scattering theory of electrons on nuclei is hard core quantum field theory. $\endgroup$ Commented Jan 5, 2015 at 9:11
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$\begingroup$ Sorry, I'm wondering about the derivation of the angle of minima in the diffraction pattern rather than one of the de Broglie wavelength. $\endgroup$– SomniareCommented Jan 6, 2015 at 12:31