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Robert Hofstadter's Stanford installation selected electrons up to $190\mathrm{MeV}$. Magnetic spectrometer scans counted shielded Lucite Cherenkov radiation pulses per monitor charge. Carbon-12 at $187\mathrm{MeV}$ and $80^{\circ }$ resolved $185.1\mathrm{MeV}$ elastic scattering and $180.7\mathrm{MeV}$ inelastic scattering ($4.43\mathrm{MeV}$ excitation). Calibration used forward-beam energy, hydrogen nuclear recoil and carbon-12 energy levels. Closed entrance slits checked background.^[ref]

The key distinction is between an electron counter and an energy measurement. Here the counter registers particles admitted by the magnetic spectrometer; changing its magnetic field selects another outgoing momentum. A peak is therefore reconstructed from a scan, rather than read as one pulse whose height equals the scattered electron energy. Dividing counts by monitored incident electric charge compensates for changes in beam intensity. A peak's height alone is insufficient: its area and the instrument's accepted momentum interval determine the selected yield.

An inelastic scattering peak corresponds to a different final nuclear quantum state, so including it in the elastic scattering yield would answer a different structural question. Conversely, finite energy resolution can hide sufficiently nearby excitations beneath an apparent elastic scattering peak. Background subtraction, target-loss corrections, angular acceptance and counter efficiency all enter before counts become a differential scattering cross-section. A dip in uncorrected counts is therefore not automatically a zero of a nuclear charge form factor.

nuclear radius scaling follows approximately $R=r_0{A}^{1/3}$ with $r_0\simeq 1.2\mathrm{fm}$ 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, $r_{\mathrm{ch}}=\sqrt{3/5}R$; 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,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.^[ref]

A useful intrinsic-shape description is $R(\theta ,\varphi )=R_0[1+{\sum }_{\mu =-2}^2{\alpha }_{2\mu }{Y}_{2\mu }(\theta ,\varphi )+\cdots ]$. An intrinsic nuclear quadrupole moment $Q_0=\int {\rho }_{\mathrm{ch}}(3z^2-r^2)d^{3}r$ 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. Radius probes spatial extent; quadrupole observables and collective spectra probe shape.

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,Here $q$ is physical momentum transfer divided by $\hslash$, not the momentum itself. The low-$q$ slope, $F(q)=1-q^2⟨r^2⟩/6+\cdots$, determines the charge r.m.s. radius. A sharp uniform sphere hasIts first diffraction minimum is at $qR\simeq 4.493$, 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.^[ref]

Shape information comes from quadrupole moments, Coulomb-excitation transition strengths and rotational spectra. An intrinsic axial moment $Q_0=\int \rho (3z^2-r^2)d^{3}r$ 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 $E_J={\hslash }^{2}J(J+1)/(2\mathcal{I})$, 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.

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: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.

From scattering to size and shape. For a spherical nuclear charge distribution,Thus the nuclear charge form factor is a normalized Fourier transform. Its low-$q$ expansion,measures the nuclear charge radius $r_{\mathrm{ch}}=\sqrt{⟨r^2{⟩}_{\mathrm{ch}}}$. 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 $qR\simeq 4.493$, while $r_{\mathrm{ch}}=\sqrt{3/5}R$.

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.

Observed pattern. Apart from the lightest nuclei, the central density is approximately constant; the size grows approximately as $R\simeq 1.2A^{1/3}\mathrm{fm}$ 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 $4a\ln 3\simeq 2.2\mathrm{fm}$. Shell effects, deformation and the lightest systems give departures from this smooth pattern. These qualitative features are established by the electron scattering density measurements.^[ref]

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 $⟨r^2⟩$. 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.

Robert Hofstadter investigated nuclear charge distributions through energetic electron scattering. His 1956 review compares $400\mathrm{MeV}$ electrons scattered from helium gas with predictions for point charges and several extended charge profiles. Beyond about $70^{\circ }$, the elastic scattering was over a hundred times smaller than the point-charge prediction. The extracted nuclear charge form factor distinguishes candidate spatial profiles, connecting measured angular counts with a model of nuclear size and shape.^[ref]

A physicist.^[ref]