= Nuclear charge radius
{title2=$r_{\mathrm{ch}}=\sqrt{\langle r^2\rangle_{\mathrm{ch}}}$}
The <root mean square> radius of a <nuclear charge distribution>, referred to its center. Define its charge-weighted second moment by
$$
\langle r^2\rangle_{\mathrm{ch}}=\frac1{Ze}\int r^2\rho_{\mathrm{ch}}(\mathbf r)\,d^3r.
$$
For a spherical <charge density> with finite fourth moment, insert the <Taylor series> $\sin u/u=1-u^2/6+O(u^4)$ into the <nuclear charge form factor>:
$$
F(q)=1-\frac{q^2}{6}\langle r^2\rangle_{\mathrm{ch}}+O(q^4),
\qquad
\langle r^2\rangle_{\mathrm{ch}}=-6\left.\frac{dF}{d(q^2)}\right|_{q=0}.
$$
Thus a radius is a low-<momentum transfer> slope, not necessarily the position of a sharp edge. In this real static model,
$$
|F(q)|^2=1-\frac{q^2}{3}\langle r^2\rangle_{\mathrm{ch}}+O(q^4).
$$
The factor of two between the slopes of $F$ and $|F|^2$ matters when interpreting a <differential scattering cross-section>. A <proton charge radius> is the corresponding size concept for the <proton>; the spatial arrangement of many <nucleons> introduces an additional scale in a heavier <atomic nucleus>.
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