RMS Radius Difference in Nuclear Physics

• RMS radius difference is defined as the difference between the square roots of the mean-square radii of nuclear distributions, clarifying proton and neutron spatial disparities.
• Experimental methods such as electron scattering, muonic spectroscopy, and coherent elastic neutrino scattering enable precise measurement of these subtle nuclear size differences.
• The metric informs nuclear models and symmetry energy constraints, impacting our understanding of the neutron skin and astrophysical properties like neutron star radii.

The root-mean-square (RMS) radius difference quantifies the separation between the mean-square charge or matter radii of different nuclear or subnuclear distributions, most often between protons and neutrons in a given nucleus or between analogous nuclear states (e.g., isotope or mirror-pair differences). This metric is central to understanding nuclear structure, the distribution of neutrons relative to protons (the “neutron skin”), and to precision tests at the interface of atomic, nuclear, and particle physics. The RMS radius difference is rigorously defined as the difference between the square roots of the mean-square radii of the relevant distributions: for nucleons $i,j$, $\Delta R_{ij} = R_i - R_j$, where $R_{i,p} = \sqrt{\langle r^2 \rangle_{i,p}}$ is the RMS radius associated with species $i$ or $j$ (e.g., neutron, proton, or charge).

1. Mathematical Definition and Formalism

For a general distribution $\rho_x(r)$ ($x$ = charge, proton, neutron), the RMS radius is defined as

$$R_x = \sqrt{\langle r^2 \rangle_x} = \sqrt{ \frac{1}{N_x} \int r^2 \rho_x(r) \, d^3 r }$$

where $N_x$ normalizes the density. The difference of RMS radii, or “skin thickness” for protons and neutrons, is given by

$\Delta R_{np} = R_n - R_p$

This can be generalized to differences between isotopes, isomers, or mirror pairs:

$$\Delta R_{\rm ch} = R_{\rm ch}^{(A)} - R_{\rm ch}^{(A')}$$

for the charge radii of isotopes $A$ and $A'$. The formalism is foundational for extracting nuclear structure information from experimental observables such as charge-exchange reactions, electron scattering, atomic spectroscopy, and parity-violating processes.

2. Experimental Determination: Key Techniques

The experimental extraction of RMS radius differences spans multiple, mutually cross-checked methodologies:

• Elastic Electron Scattering: Directly probes the Fourier transform of the charge density. The RMS charge radius is extracted from the slope of the Sachs form factor $G_E(q^2)$ at $q^2 \to 0$ via $r_{\rm rms}^2 = -6\, dG_E/dq^2 |_{q=0}$. For neutron distributions, parity-violating electron scattering exploits the weak charge sensitivity to map $R_n$ independent of hadronic uncertainties, as realized in the PREX experiments (Mammei, 2012).
• Muonic Atom Spectroscopy: The muonic hydrogen Lamb shift is highly sensitive to the proton's charge radius due to the increased overlap between the muon and the nuclear charge density. The finite-size effect appears in the transition energies and, for $2S \to 2P$ in muonic hydrogen, produces a term proportional to $r_p^2$ with a calculable coefficient (Carroll et al., 2011).
• Coherent Elastic Neutrino-Nucleus Scattering (CEnNS): The weak interaction cross-section amplitude is dominated by the neutron distribution at low $Q^2$, as neutrinos couple primarily to neutrons. The COHERENT experiment constrains the neutron RMS radius $R_n$, and, by combination with atomic parity violation (APV), enables precise extraction of $\Delta R_{np}$ (Cadeddu et al., 2018, Corona et al., 2023).
• Atomic Parity Violation (APV): The weak charge measured via APV in atoms with closed neutron and proton shells is sensitive to the neutron skin, as the overlap of the electron wave function with the neutron density directly affects parity-violating observables (Cadeddu et al., 2018, Corona et al., 2023).
• Collinear Laser Spectroscopy and Isotope Shifts: The difference in mean-square charge radii between isotopes or mirror nuclei is determined from isotope shifts in atomic transitions, with high-precision calculations accounting for field shifts and QED recoil (Skripnikov et al., 2024, Pineda et al., 2021).

3. Theoretical Approaches and Model Dependencies

Extraction and interpretation of RMS radius differences crucially depend on modeling nuclear densities and accounting for all relevant theoretical corrections:

• Nuclear Density Parameterizations: Fermi (2pF) distributions and expansion in basis functions (e.g., Laguerre or Fourier–Bessel) are employed to represent nuclear densities. The extracted radius is sensitive to the choice of parameterization, especially the behavior at large $r$, which influences the low-$q$ curvature of form factors and, thus, extrapolations required in electron scattering (Sick et al., 2014, Sick et al., 2017).
• Finite-Size and QED Corrections: In atomic and muonic systems, the leading finite-size shift is proportional to the mean-square radius, but higher-order QED, two-photon exchange, and proton-polarizability corrections also play roles at the sub-percent level. The resolution of the “proton radius puzzle” required carefully including previously neglected dynamic vacuum polarization contributions for muonic atoms (Walcher, 2023).
• Statistical and Systematic Uncertainties: Physical constraints (e.g., monotonic nuclear densities), experimental resolution, and theoretical uncertainties (atomic theory, nuclear correlations) are tightly controlled in recent work. Uncontrolled parameterizations or neglect of higher moments in the charge distribution induce spurious shifts in extracted radii at the $\sim0.04$ fm level in protons (Sick et al., 2017, Sick et al., 2014).

4. Key Results and Applications

A selection of recent RMS radius difference measurements and their physical implications includes:

System/Process	$\Delta R$ (fm)	Experimental Technique	Reference
$^{133}$Cs neutron skin $\Delta R_{np}$	$0.62 \pm 0.31$	COHERENT CEnNS + APV	(Cadeddu et al., 2018)
$^{54}$Ni–$^{54}$Fe $\Delta R_{\rm ch}$	$0.049(4)$	Collinear laser spectroscopy	(Pineda et al., 2021)
$^{208}$Pb neutron skin	$0.33^{+0.16}_{-0.18}$	Parity-violating electron scattering	(Mammei, 2012)
$^{133}$Cs–$^{127}$I $\Delta R_n$	$0.02^{+0.42}_{-0.67}$	COHERENT CEnNS + atomic parity violation	(Corona et al., 2023)
Proton radius difference (method-dependent)	$(0.03496 \pm 0.00096)$	$e$–$p$ vs. $\mu$H Lamb shift	(Carroll et al., 2011)

The neutron skin thickness connects directly to the symmetry energy slope $L$ in the nuclear equation of state, impacting nuclear structure and astrophysical observables such as neutron star radii and tidal deformabilities. For example, the difference $\Delta R_{\rm ch}=0.049(4)$ fm in the $^{54}$Ni–$^{54}$Fe mirror pair constrains $20 \leq L \leq 70$ MeV, favoring a soft neutron matter equation of state (Pineda et al., 2021).

5. Conceptual Issues and Controversies

The Proton Radius Puzzle

Discrepancies of up to 4% between proton RMS charge radii determined from electron scattering and atomic hydrogen spectroscopy versus the muonic hydrogen Lamb shift once raised the possibility of physics beyond the Standard Model. Careful re-analyses, including nonresonant quantum interference corrections and dynamic vacuum polarization effects, have resolved these to high accuracy, aligning both approaches at $r_p \approx 0.8745$ fm (Solovyev, 2018, Walcher, 2023).

Methodological Pitfalls

• Parameterization Extrapolation: Low-$q$ power-series truncations in electron scattering analyses ignore substantial higher moments, systematically biasing extracted radii downward (Sick et al., 2017).
• Unphysical Tails: Fit functions that allow for long-range Gaussian tails or poles outside the data region can dominate the extracted RMS radius, introducing systematic uncertainties much greater than experimental errors (Sick et al., 2014).

6. Implications and Future Directions

High-precision RMS radius difference measurements are at the forefront of nuclear structure and weak-interaction studies, providing model-independent access to neutron distributions and enabling robust constraints on nuclear matter properties. Future experiments—such as upgraded CEvNS detectors (COHERENT, SNS), new-generation precision electron scattering (PRad-II), and advances in atomic theory for isotope shift analysis—promise sub-percent-level determination of radius differences and related observables (Corona et al., 2023, Gasparian et al., 2020, Skripnikov et al., 2024). Continued refinement of experimental and theoretical approaches will further solidify the role of RMS radius differences as fundamental parameters in nuclear and atomic physics.