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The central system of protons and neutrons in an atom. Its mass number $A$ is its baryon number in ordinary nuclear states; $Z$ is the proton number and $N$ the neutron number.

The nucleus contains almost all of an atom's mass and carries positive electric charge $Ze$, where $e$ is the elementary charge. The atomic number $Z$ identifies the chemical element; changing $N$ at fixed $Z$ gives another isotope. The mass number $A=Z+N$ counts nucleons, rather than specifying the exact nuclear mass: nuclear binding energy lowers the mass of a bound system relative to its separated constituents. The simplest nucleus is a single proton; other atomic nuclei can contain many protons and neutrons bound by the nuclear force.

A nuclear quantum state has properties such as spin and a spatial nuclear charge distribution. These are different questions from merely counting nucleons. A small-angle elastic scattering measurement can reveal a nuclear charge radius, while a wider range of momentum transfer tests the distribution's shape. The Hofstadter nuclear electron scattering experiment made this connection between scattering patterns and nuclear structure experimentally concrete.

An isotope shift is a difference between corresponding atomic transition frequencies in different isotopes. One contribution comes from nuclear mass, through recoil and electron correlations; another comes from the changing nuclear charge distribution. For a specified transition, the leading finite-size contribution is often written $\delta {\nu }_{\mathrm{field}}=F\delta ⟨r^2⟩$, with an electronic factor $F$. Extracting changes of nuclear charge radius therefore requires separating mass and finite-size contributions rather than identifying the entire shift with radius.

A muonic atom has a negative muon bound in the Coulomb potential of an atomic nucleus. Its much larger reduced mass than in an electronic chemical atom greatly reduces the orbital scale, increasing sensitivity of transition energies to the nuclear charge radius and nuclear charge distribution. Measuring muonic X-ray transitions therefore complements electron scattering. Strong relativistic and nuclear-size effects in heavy systems require a treatment beyond a point-nucleus nonrelativistic Coulomb spectrum.

A spatial charge density describing the electric charge inside an atomic nucleus. In a static description centered on the nucleus, normalize it bywhere $Z$ is the atomic number and $e>0$ the elementary charge. A nuclear charge distribution is an average over a nuclear quantum state, rather than a photograph of immobile protons. Even when the average has spherical symmetry, individual nucleons need not sit on spherical shells. The proton charge radius and the internal charge density of the neutron also contribute: a distribution of pointlike proton centers is an approximation to a nuclear charge distribution, not its definition.

The experimental attraction is that an electron probes electric charge through the electromagnetic interaction, without itself feeling the nuclear force. Varying its momentum transfer changes the spatial scale tested. Long wavelengths mainly constrain a nuclear charge radius; shorter wavelengths can distinguish a diffuse edge from a sharper one through diffraction. Recovering the entire nuclear charge distribution from a finite range of noisy scattering data remains an inverse problem.

Put $k_Z=Z{e}^2/(4\pi {ϵ}_0)$. Gauss's law for a uniform spherical nuclear charge distribution, with the electrostatic potential matched at $R$, gives the electronic potential energySubtracting the point-nucleus Coulomb potential leaves the nonnegative perturbationThe normalized nonrelativistic hydrogen-like ion $1s$ wavefunction has $|\psi (r){|}^2=Z^3{e}^{-2Zr/a_0}/(\pi a_0^3)$. Since $ZR/a_0\ll 1$, approximate it inside the atomic nucleus by $|\psi (0){|}^2$. First-order time-independent perturbation theory then givesDefine $E_H=-Z^2\mathrm{Ry}$, as the paper specifies for the hydrogen-like ion, rather than the neutral-hydrogen value. Using $\mathrm{Ry}=e^2/(8\pi {ϵ}_0{a}_0)$ givesto leading order in $ZR/a_0$. For $A=208$, $R\simeq 7.11\mathrm{fm}$, $ZR/a_0\simeq 0.0110$, andThe finite nuclear charge radius makes the energy level less bound. The quoted frequency is a level shift divided by $h$: a line frequency shifts by $(\Delta E_u-\Delta E_l)/h$, so a decay into this shifted $1s$ level alone has the opposite sign.

The formula is a nonrelativistic, small-nuclear charge radius estimate. Keeping the exponential in the first-order integral gives $\Delta E_{1s}=\Delta E_{\mathrm{leading}}[1-5ZR/(6a_0)+O((ZR/a_0{)}^2)]$, about a one-percent correction here. More seriously, $Z{\alpha }_{\mathrm{fs}}\simeq 0.598$ for lead, so relativistic contraction is substantial. A precise physical prediction needs a finite-nucleus Dirac equation calculation; the boxed estimate is the model requested, not precision heavy-ion spectroscopy.

The mirror nuclei have interchanged neutron and proton numbers. Neglect their strong-interaction binding difference and describe them with the same uniformly charged sharp sphere of radius $R$. The nuclear Coulomb energy isChanging one neutron into a proton decreases the nucleon rest-mass sum by $m_n-m_p=1.294\mathrm{MeV}/c^2$, but raises the Coulomb energy. The diagram gives $(M_F-M_O)c^2=2.25\mathrm{MeV}$ for nuclear masses, soWith the standard electromagnetic conversion $\alpha \hslash c\simeq 1.44\mathrm{MeV}\mathrm{fm}$, the uniform-sphere Coulomb displacement energy estimate isThis $R$ is the uniform sphere's outer radius. If “charge radius” means the rms nuclear charge radius, the same distribution givesThe numerical result is an estimate within the charge-density model, not a model-independent radius measurement. A discrete-proton correction replaces $Z^2$ by $Z(Z-1)$, giving $\Delta [Z(Z-1)]=16$ and $R\simeq 3.90\mathrm{fm}$. The main answer above follows the literal continuum charged-sphere model. Strong isospin-breaking, nonuniform density and changes in the valence-nucleon distribution also limit the extraction.

A muonic atom binds a negative muon in the nuclear Coulomb potential. The muon is a spin one-half lepton with the electron's charge but about 207 times its mass. For a point nucleus of mass $M_N$ and charge $Ze$, the reduced mass and nonrelativistic hydrogenic scales, with fine-structure constant $\alpha$, areFor a heavy nucleus ${\mu }_r\simeq m_{\mu }$, giving a radius roughly $1/207$ of the corresponding electronic radius and transition energies roughly 207 times larger. For hydrogen, nuclear recoil matters: $m_{\mu }\simeq 106\mathrm{MeV}/c^2$ gives ${\mu }_r\simeq 95\mathrm{MeV}/c^2$ and $a_{\mu }\simeq 284\mathrm{fm}$ rather than the infinite-nuclear-mass value about $255\mathrm{fm}$.

A slowed negative muon is captured into an excited orbit and undergoes a radiative cascade, ejecting atomic electrons and emitting muonic X-rays. Inner orbits penetrate the nuclear charge distribution, particularly at large $Z$. Replacing a point charge by an extended charge makes the central attraction less singular and shifts states with substantial nuclear overlap. Measured transition energies consequently constrain the nuclear charge radius. Sensitivity is particularly strong for $S$ states, whose contact density scales as $(Z\alpha {\mu }_r{)}^3$.

An instructive experiment is Randolf Pohl's team's muonic-hydrogen spectroscopy. A stopped-muon target prepared a metastable $2S$ population; a tunable pulsed laser drove a $2S\to 2P$ transition, followed by a detectable $2P\to 1S$ X-ray. Scanning laser frequency located a resonance whose interpretation included the proton finite-size contribution to the Lamb shift. Delayed detection selected the long-lived population against prompt cascade backgrounds. Laser-off or detuned measurements provide a background control; energy and time selection alone would not establish a laser-induced resonance.^[ref]

Extracting a radius requires quantum electrodynamics, including vacuum polarization, recoil and nuclear-structure corrections. A free muon decays on a microsecond timescale; a bound negative muon can instead undergo muon capture, for example ${\mu }^{-}p\to n{\nu }_{\mu }$. These processes limit observation time. In heavy atoms, relativistic motion and nuclear overlap make the simple Bohr formulas only qualitative. Muonic spectroscopy converts a smaller orbital scale into enhanced sensitivity to nuclear charge structure, provided atomic and nuclear corrections are controlled.

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.

The electrostatic self-energy of a uniformly charged sphere of sharp radius $R$ is $3Z^2\alpha \hslash c/(5R)$. For mirror nuclei, subtract the changed neutron/proton rest-mass contribution from their nuclear mass difference before identifying the residual with a nuclear Coulomb energy difference. The inferred $R$ is a sharp-sphere radius; the corresponding nuclear charge radius defined as an rms radius is $\sqrt{3/5}R$. Replacing $Z^2$ by $Z(Z-1)$ removes a proton self-interaction approximation and shifts the estimate.

Consider an ideal nuclear charge distribution that is constant inside a ball of radius $R>0$ and zero outside:This is a deliberately sharp-edged model. Substitution into the spherical nuclear charge form factor gives, for $q\ne 0$,At $q=0$ take the continuous limit $F(0)=1$. Direct radial integration gives $⟨r^2{⟩}_{\mathrm{ch}}=3R^2/5$, and the Taylor series provides an independent check:The first positive zero solves $\sin x-x\cos x=0$, equivalently $\tan x=x$, with $x=qR\simeq 4.49340946$. Consequently the first diffraction minimum moves toward smaller momentum transfer as $R$ increases.

For comparison, a smooth Gaussian function charge density with the same nuclear charge radius isThe Fourier transform of a Gaussian stays positive and has no zero. Both models have the same leading low-$q$ curvature, but they disagree at higher momentum transfer: at $qR=2$, the sphere gives $F\simeq 0.653097$ and the Gaussian function gives $F_G\simeq 0.670320$. Measuring only a nuclear charge radius cannot distinguish these two nuclear charge distributions.

As a chosen scale illustration, take $R=3.00\mathrm{fm}$. The predicted first zero is $q=1.49780{\mathrm{fm}}^{-1}$, or $\hslash cq\simeq 295.557\mathrm{MeV}$ using $\hslash c\simeq 197.327\mathrm{MeV}\mathrm{fm}$. With a $200\mathrm{MeV}$ ultrarelativistic electron beam and negligible nuclear recoil, it occurs at $\theta \simeq 95.27^{\circ }$. These are consequences of chosen model parameters, not measured properties of carbon or a claim that all atomic nuclei are uniform balls.