= Solution
\b[What is measured.] A monoenergetic <Electron> beam is directed at a thin target. A movable <magnetic spectrometer> selects the scattering angle and outgoing momentum, separating <elastic scattering> from nuclear excitation and other backgrounds. Incident beam current and target thickness supply the normalization of the <differential scattering cross-section>. The <Hofstadter nuclear electron scattering experiment> is a classic implementation of this electromagnetic probe.
For a heavy, approximately spin-zero <atomic nucleus>, neglecting recoil and treating the electron in the <Born approximation>, the point-charge result is multiplied by the <nuclear charge form factor>:
$$
\frac{d\sigma}{d\Omega}\simeq\frac{Z^2\alpha^2\cos^2(\theta/2)}{4E^2\sin^4(\theta/2)}|F(q)|^2,\qquad q\simeq2E\sin\frac\theta2.
$$
Here $q$ is the magnitude of the spatial <momentum transfer> in <natural units>. The cosine factor includes the <Electron> <spin>. Dividing the elastic data by the point-charge factor isolates $|F(q)|^2$ within these approximations. Recoil, magnetic terms for targets with spin, <radiative corrections> and particularly <Coulomb distortion in electron scattering> must be included when significant.
\b[From scattering to size and shape.] For a spherical <nuclear charge distribution>,
$$
F(q)=\frac{4\pi}{Ze}\int_0^\infty r^2\rho_{\rm ch}(r)\frac{\sin(qr)}{qr}\,dr,\qquad F(0)=1.
$$
Thus the <nuclear charge form factor> is a normalized <Fourier transform>. Its low-$q$ expansion,
$$
F(q)=1-\frac{q^2\langle r^2\rangle_{\rm ch}}6+\cdots,\qquad |F(q)|^2=1-\frac{q^2\langle r^2\rangle_{\rm ch}}3+\cdots,
$$
measures the <nuclear charge radius> $r_{\rm ch}=\sqrt{\langle r^2\rangle_{\rm ch}}$. Larger <momentum transfers> resolve the density profile through <diffraction> maxima and minima. For a uniform sphere of sharp radius $R$, the <uniform-sphere nuclear charge form factor> is $3[\sin(qR)-qR\cos(qR)]/(qR)^3$; its first zero is at $qR\simeq4.493$, while $r_{\rm ch}=\sqrt{3/5}R$.
A practical extraction fits a family of densities to all measured cross sections, rather than treating a few measured values of $|F|^2$ as a complete density reconstruction. Finite angular coverage, resolution and the missing sign information make this an <inverse problem>.
\b[Observed pattern.] Apart from the lightest <nuclei>, the central density is approximately constant; the size grows approximately as $R\simeq1.2A^{1/3}\,\mathrm{fm}$ and there is a diffuse surface. The coefficient describes an equivalent sharp nuclear radius, rather than the root-mean-square radius itself. Heavy <nuclear charge distributions> are often described by the <two-parameter Fermi nuclear charge distribution>, $\rho(r)=\rho_0/[1+\exp((r-c)/a)]$, with half-density radius $c$ scaling roughly as $A^{1/3}$ and diffuseness $a$ around $0.5\,\mathrm{fm}$. The surface thickness is then about $4a\ln3\simeq2.2\,\mathrm{fm}$. Shell effects, deformation and the lightest systems give departures from this smooth pattern. These qualitative features are established by the <electron scattering> density measurements.<Electron Scattering and Nuclear Structure by Robert Hofstadter>{ref}
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2017-pnp-charge-density-models.png]
{title=Original sharp-edge and diffuse nuclear charge models, showing how a surface profile changes the <nuclear charge form factor>}
\b[Complementary probes.] Transition energies in a <muonic atom> are especially sensitive to nuclear size because the heavy negative <muon> orbits close to the charge. Electronic <isotope shifts>, after separating nuclear-mass and finite-size contributions, measure changes in $\langle r^2\rangle$. Both require atomic-structure calculations. Electromagnetic excitation and transition strengths constrain deformation and charge multipoles. Hadronic scattering can also constrain nuclear dimensions, but carries uncertainties from the <strong interaction> and probes matter rather than purely electric charge.
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