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When you fire an alpha particle at a nucleus, the electrostatic forces repel each other, causing it to slow down to a stop before accelerating back in the opposite direction. So the initial kinetic energy is equal to the electrostatic potential energy it has at the point of instantaneous rest, and then you can rearrange the equation to look like this: $$r=\frac{1}{4 \pi \epsilon_0}\frac{Q_{\alpha}Q_n}{E_k}$$ The concept makes intuitive sense to me, you should be able to deduce something about a nucleus' radius by firing positively charged particles at it. But the equation has me confused, can you not just fire the alpha particle with a greater velocity, thus greater kinetic energy, and now your estimation for the radius will be smaller?

So if you choose the velocity at which the alpha particle is released carefully, you could make the estimation equal whatever you want it to equal?

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    $\begingroup$ Well, one keeps increasing the $\alpha$ energy until the scattering process stops being Rutherford-like. Then you know something changed. (Of course, one keeps increasing it to fully map out the scattering process to get detail on excited nuclear states, induced nuclear reactions, and whatnot - classic nuclear physics stuff). $\endgroup$ – Jon Custer Jul 11 '18 at 13:15
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That is not the way the size of a nucleus is measured. It is for all practical purposes impossible to fire a particle directly at the center of a nucleus, partly because it is effectively impossible to position a nucleus at a precise location, and partly because it is effectively impossible to aim the particle with that much precision. Instead, a beam of particles is fired at a collection of nuclei whose locations are unknown. Most of the particles miss the nuclei, but the percentage of particles that pass a nucleus at a given distance from the center of the nucleus can be calculated. When a particle passes a nucleus at a given distance and at a given energy, it will be deflected by a predictable angle. The closer it approaches to the nucleus, the greater the deflection angle; but if it actually hits the nucleus the deflection will abruptly become different.

So, the number of particles deflected into each angle is measured and plotted as a curve. The curve should suddenly change at the deflection angle corresponding to the radius of the nucleus. When that experiment was first performed, the big surprise was that the curve was smooth until very large deflection angles were reached, which showed that the nucleus was much smaller than was expected. A good explanation of all this is given by this Youtube video.

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