Sizes of Nuclei

Just as different atoms have different overall sizes depending on how many electron orbits they occupy, different nuclei have different sizes depending on how many nucleons they contain. Experimentally, the radius $R_X$ of a nucleus with mass number A is found to obey the simple relation $R_X=R_0A^{1/3}$, where $R_0=1.2\times10^{-15}$ m (1.2 fm) is a constant that is essentially the same for every nucleus.

This $A^{1/3}$ scaling has a striking consequence for nuclear DENSITY. Treating the nucleus as a sphere of volume $\frac{4}{3}\pi R_X^3$ containing mass $M\approx mA$ (where m is taken as roughly the common average mass of a proton and a neutron, since they are so close in mass), the density works out to $\rho=\frac{M}{\frac{4}{3}\pi R_X^3}=\frac{mA}{\frac{4}{3}\pi R_0^3A}=\frac{3m}{4\pi R_0^3}$. Notice that the mass number A cancels out completely -- so this predicts that EVERY nucleus, regardless of how many nucleons it contains, should have exactly the same density. Substituting the known values of m, $\pi$ and $R_0$ gives a numerical value of about $2.3\times10^{17}$ kg/m$^3$. …