= Solution
High-energy elastic <electron scattering> probes nuclear charge through the <electromagnetic interaction>. Resolve the elastic peak with a spectrometer, measure its angular dependence and normalize by beam charge, target thickness and <detector acceptance>. In a plane-wave, approximately static treatment,
$$
\frac{d\sigma}{d\Omega}\simeq\left(\frac{d\sigma}{d\Omega}\right)_{\rm point}|F(q)|^2,
\qquad F(q)=\frac1{Ze}\int\rho_{\rm ch}(\mathbf r)e^{i\mathbf q\cdot\mathbf r}d^3r.
$$
Here $q$ is <physical momentum> transfer divided by $\hbar$, not the <momentum> itself. The low-$q$ slope, $F(q)=1-q^2\langle r^2\rangle/6+\cdots$, determines the charge r.m.s. radius. A sharp uniform sphere has
$$
F(q)=3\frac{\sin(qR)-qR\cos(qR)}{(qR)^3},\qquad
\langle r^2\rangle=3R^2/5.
$$
Its first <diffraction> minimum is at $qR\simeq4.493$, illustrating how angular structure determines size. Radii approximately follow $R\propto A^{1/3}$, but a sharp radius, a surface half-density radius and an r.m.s. radius are different definitions.
A motivating experimental implementation is Hofstadter's magnetic-spectrometer measurement: selecting outgoing <electron> <momentum> separates <elastic scattering> from nuclear excitation, rather than treating every scattered <electron> as an elastic event. Beam-energy and angle calibration, empty-target or blocked-acceptance backgrounds, and target-thickness checks matter. Finite resolution mixes nearby excitations; multiple scattering and Coulomb distortion require correction before a precision density is inferred.<Electron Scattering and Nuclear Structure by Robert Hofstadter>[§III(b), pp.222–223; §IV(a), pp.227–231, spectrometer scans and elastic/inelastic separation]{ref}
Shape information comes from quadrupole moments, Coulomb-excitation transition strengths and rotational spectra. An intrinsic axial moment $Q_0=\int\rho(3z^2-r^2)d^3r$ is positive for a prolate charge <distribution> and negative for an oblate one. A laboratory quadrupole moment is not identical to $Q_0$: a rotationally averaged spin-zero state can have zero laboratory moment despite intrinsic deformation. Collective rotational levels approximately follow $E_J=\hbar^2J(J+1)/(2\mathcal I)$, while measured $E2$ strengths test the associated charge collectivity. Muonic-atom X-rays and <isotope shifts> supply complementary charge-radius constraints; hadronic probes provide additional, more interaction-dependent matter and <neutron> information. Neither a charge radius alone nor a finite-range form-factor measurement uniquely fixes the entire nuclear shape.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2014/paper-1-charge-form-factor.png]
{title=Original model charge form factors at equal r.m.s. radius; the diffraction structure carries information beyond the radius}
Back to article page
