= Solution
<nuclear radius scaling> follows approximately $R=r_0A^{1/3}$ with $r_0\simeq1.2\,\mathrm{fm}$ for a sharp matter-radius convention. Consequently the interior <nucleon> density is nearly independent of <mass number>: <nuclear forces> are short range and exhibit <nuclear matter saturation>. A diffuse <nuclear charge distribution> has a charge-weighted <root mean square> radius, not a hard surface. For a uniform sphere, $r_{\rm ch}=\sqrt{3/5}R$; charge radii and matter radii need not coincide, especially with <neutron> skins.
Elastic <electron scattering> measures the <nuclear charge form factor>. In a simple one-photon, spherical approximation, with $q$ a wave number,
$$
F(q)=\frac{1}{Ze}\int\rho_{\rm ch}(\mathbf r)e^{i\mathbf q\cdot\mathbf r}\,d^3r,
\qquad F(q)=1-\frac{q^2\langle r^2\rangle}{6}+\cdots.
$$
The scattering <cross section> relative to a point-charge prediction is proportional to $|F|^2$, so its low-$q$ slope constrains the radius and <diffraction> structure constrains the radial distribution. Finite <momentum> coverage and <Coulomb distortion in electron scattering> limit reconstruction. In <Robert Hofstadter>'s experiment a <magnetic spectrometer> separated elastic and inelastic <electron> peaks; beam-charge normalization and closed-slit background checks prevented raw count structure from being mistaken for a <nuclear charge form factor>.<Electron Scattering and Nuclear Structure by Robert Hofstadter>[§III(b)(1), pp. 222–223; §IV(a), pp. 227–231]{ref}
A useful intrinsic-shape description is $R(\theta,\phi)=R_0[1+\sum_{\mu=-2}^{2}\alpha_{2\mu}Y_{2\mu}(\theta,\phi)+\cdots]$. An <intrinsic nuclear quadrupole moment> $Q_0=\int\rho_{\rm ch}(3z^2-r^2)\,d^3r$ distinguishes prolate and oblate positive-charge distributions by positive and negative signs. The laboratory <spectroscopic nuclear quadrupole moment> is state dependent: an $I=0$ state has zero laboratory quadrupole moment even when its intrinsic shape is deformed. <nuclear rotational bands> approximately follow $E_I\propto I(I+1)$; harmonic <nuclear quadrupole vibrations> instead have equally spaced <phonon> energies and multiphonon <spin> multiplets. Measured <electric quadrupole transitions> test the collective interpretation in addition to level energies. \b[Radius probes spatial extent; quadrupole observables and collective spectra probe shape.]
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