Charge Radius Operator

• Charge Radius Operator is a key observable defining the spatial distribution of electric charge in nuclei and atomic systems.
• It is implemented via one-body and two-body operators in ab initio and χEFT frameworks to analyze isotope shifts and transition strengths.
• Rigorous studies address basis convergence, truncation effects, and error budgets to reliably extract mean-square radii and related properties.

The charge radius operator is a fundamental observable in nuclear and atomic physics, encapsulating how electric charge is distributed within a nucleus or nucleon. Its matrix elements are crucial for extracting physical root-mean-square (rms) radii from both theory and experiment, tightly linking few- and many-body structure calculations with high-precision spectroscopy. The charge radius operator underlies the quantitative determination of properties such as isotope shifts, Lamb shifts, and transition strengths, and is implemented differently depending on the physical system and theoretical formalism.

1. Operator Definition and Formal Structure

The charge radius operator quantifies the spatial distribution of electric charge in a quantum system. In a multi-nucleon nucleus, the intrinsic point-proton mean-square radius operator, removing center-of-mass (CM) contributions, is given by

$$\hat r_p^2 = \frac{1}{Z} \sum_{i=1}^A \frac{1+\tau_{i,z}}{2} \bigl(\mathbf{r}_i - \mathbf{R}_{\rm CM}\bigr)^2,$$

where $\tau_{i,z}$ projects onto protons, $\mathbf{r}_i$ is the coordinate of nucleon $i$, and $\mathbf{R}_{\rm CM}$ is the many-body CM coordinate. Expanding the squared term yields a sum of one-body and two-body operators: $$\sum_i(\mathbf r_i - \mathbf R_{\rm CM})^2 = \sum_i \mathbf r_i^2 - \frac{1}{A}\sum_{i \ne j} \mathbf r_i \cdot \mathbf r_j,$$ ensuring translational invariance (Caprio et al., 16 Jan 2025).

Experimentally, the nuclear charge radius $r_c$ differs from the point-proton radius $r_p$ by several corrections: $$r_c^2 = r_p^2 + R_p^2 + \frac{N}{Z} R_n^2 + \frac{3\hbar^2}{4 m_p^2 c^2} + \cdots,$$ where $R_p^2, R_n^2$ are the intrinsic mean-square charge radii of the proton and neutron, and the Darwin–Foldy term incorporates relativistic effects. Additional corrections (e.g., spin–orbit, two-body meson-exchange currents) may be included depending on the required precision (Caprio et al., 16 Jan 2025).

In the context of effective field theory (EFT) for atomic systems, the charge-radius effects enter as a tower of local operators: $$\Delta H_{\rm pp} = \int d^3r \left\{ C_{r_p} \delta^{(3)}(\mathbf{r}) \psi^\dagger \psi + C_h \nabla^2 \delta^{(3)}(\mathbf{r}) \psi^\dagger \psi + \cdots \right\},$$ encoding both the mean-square radius $r_p^2$ and higher moments (e.g., the "second moment" parameter $h$) in atomic Hamiltonians (Burgess et al., 2016).

2. Implementation in Ab Initio and Chiral Effective Field Theory

In ab initio no-core configuration interaction (NCCI) calculations, Slater determinants are constructed from single-particle harmonic oscillator (HO) orbitals. The charge radius operator matrix elements are evaluated in this basis, requiring both one-body radial integrals (analytic in terms of Laguerre polynomials) and two-body terms from the CM subtraction: $$\langle n'l'jm | r^2 | nl jm \rangle = \delta_{l',l} \delta_{j',j} \delta_{m',m} \int_0^\infty R_{n'l}(r) r^2 R_{nl}(r) dr.$$ The expectation value in a truncated ($N_{\rm max}$, $\hbar \omega$) basis is given by

$$\langle r_p^2 \rangle = \langle \Psi(N_{\max}, \hbar \omega) | \hat{r}_p^2 | \Psi(N_{\max}, \hbar \omega) \rangle,$$

where $|\Psi\rangle$ is the many-body ground state (Caprio et al., 16 Jan 2025).

In chiral effective field theory (χEFT) for light nuclei, the two-nucleon charge density operator through N$^4$LO in Weinberg power counting includes:

• A one-body piece $\rho_{1N}$ (at leading and subleading orders)
• A two-body one-pion-exchange piece $\rho_{2N}^{1\pi}$ (at N$^3$LO)
• A short-range two-body "contact" term $\rho_{2N}^{\rm cont}$ (at N$^4$LO): $$\rho_{2N}^{\rm cont}(\mathbf{q}) = 2e G_E^S(q^2) [A q^2 + B q^2 (\sigma_1 \cdot \sigma_2) + C (q \cdot \sigma_1)(q \cdot \sigma_2)]$$ All low-energy constants combine into a single parameter (often labeled $C_{\rm short}$) and enter calculations of the deuteron form factor, structure radius, and ultimately the mean-square neutron charge radius (Filin et al., 2019).

3. Basis Convergence, Truncation Effects, and Extrapolation

Operators like $r^2$, which probe long-range correlations, converge slowly in localized oscillator bases. Observables show pronounced dependence on both basis truncation $N_{\max}$ and oscillator frequency $\hbar \omega$. Standard practice is to scan $\hbar \omega$ over a suitable range (e.g., 12–30 MeV) for several $N_{\max}$ values (e.g., 4, 6, 8, 10, 12), seeking regions of minimal sensitivity as an indicator of convergence.

Empirically, convergence can often be modeled by exponential fits of the form

$X(N_{\max}) \approx X_\infty + a e^{-c N_{\max}},$

where $X$ may denote the mean-square radius, quadrupole moment, or $B(E2)$. By fitting three consecutive $N_{\max}$ points, the asymptotic value $X_\infty$ can be extracted, with the spread across fits providing an estimate of the residual theoretical uncertainty (Caprio et al., 16 Jan 2025).

In χEFT calculations for the deuteron, truncation errors are primarily estimated using Bayesian models for the expansion parameter, alongside propagated statistical errors from LECs and uncertainties from experimental input and fit regions (Filin et al., 2019).

4. Correlation with Transition Strengths and Dimensionless Ratios

There exists a systematic correlation between observables derived from long-range charge operators—most notably, between electric quadrupole ($E2$) transition strengths and the mean-square nuclear charge radius. The dimensionless ratio

$\frac{B(E2; J_i \to J_f)}{e^2 r_p^4}$

exhibits much greater stability under variations in the underlying basis than the individual quantities in numerator and denominator. In axially symmetric rotor models, this ratio is proportional to the squared quadrupole deformation parameter $\beta^2$ and known Clebsch–Gordan coefficients.

When experimental data provides a precise value for the ground-state charge radius $r_c$, calibrated theoretical predictions of $B(E2)$ strengths can be constructed as

$$B_{\rm pred}(E2) = \left[\frac{B(E2)}{e^2 r_p^4}\right]_{\rm calc} \times (e r_c^2)^2,$$

thus anchoring the slowly convergent theoretical observable to an experimentally robust quantity. This procedure yields absolute $B(E2)$ predictions that agree well with measured and high-precision Green's-Function Monte Carlo values across the $p$-shell (Caprio et al., 16 Jan 2025).

5. Charge Radius Operators in Few-Body and Atomic Systems

In atomic physics, particularly in the analysis of hydrogenic atoms, charge radius effects are systematically encoded via local operator expansions in an EFT framework. Charge radius contributions and higher moments are incorporated as delta-function and derivative operators: $$O_{r_p}(0) = C_{r_p} \delta^{(3)}(\mathbf{r}) + C_h \nabla^2\delta^{(3)}(\mathbf{r}) + \cdots,$$ with $C_{r_p}$ and $C_h$ matching coefficients for the mean-square and fourth moment of the charge distribution, respectively. The finite-size effects modify the small-radius boundary condition of the Schrödinger wavefunction, leading to energy shifts in S-states: $$\Delta E_{n\ell} = \delta_{\ell 0} \frac{2 (Z\alpha)^4 m_r^3}{3 n^3} \biggl[ r_p^2 - \frac{Z\alpha m_r}{2} h \biggr] + \mathcal{O}((Z\alpha)^6),$$ where $h$ encodes the fourth central moment of the charge distribution and can become significant in muonic hydrogen due to the large reduced mass. Neglect of $h$ leads to bias in extracted charge radii if only $r_p^2$ is fit, and inclusion of both parameters reconciles electronic and muonic extractions (Burgess et al., 2016).

6. Error Budgets and Parameter Extraction

Precision extraction of the mean-square charge radius and its moments depends critically on error analysis, including:

• Truncation of expansion series (e.g., $\chi$EFT order)
• Statistical uncertainties in fitted low-energy constants (LECs)
• Propagation of experimental uncertainties (e.g., isotope shift measurements)
• Modeling choices such as the fitting window in momentum transfer

For the neutron mean-square charge radius, extraction proceeds via inversion of the structure radius definition, using high-precision values for hydrogen–deuteron isotope shifts and incorporating propagating all error sources. The result: $r_n^2 = -0.106^{+0.007}_{-0.005}~\mathrm{fm}^2$ represents a comprehensive uncertainty assessment from both theoretical and experimental origins (Filin et al., 2019).

The charge radius operator thus plays a central, quantitatively controlled role in modern nuclear structure, precision atomic physics, and the extraction of fundamental hadronic properties. Its continued refinement and rigorous treatment remain vital for the close comparison of theory and experiment in the pursuit of Standard Model tests and nucleon/nuclear structure determination.