Validity domain of the generalized Bernstein formula for extracting the neutron-to-proton transition-matrix-element ratio $M_n/M_p$ beyond few-tens-of-MeV proton scattering

The ratio $M_n/M_p$ of neutron to proton transition matrix elements for low-lying quadrupole excitations is a standard diagnostic of neutron-proton density decoupling in exotic nuclei. A generalized Bernstein formula (Khan, Phys. Rev. C 105, 014306 (2022)) extends the classic phenomenological extraction prescription by allowing different neutron and proton radii and diffusenesses, but it has been numerically validated only for proton scattering at a few tens of MeV per nucleon. The open problem is to establish the validity domain of this generalized formula for other hadronic probes (neutrons, deuterons, alpha particles, pions) and other energies, to derive any probe-dependent modifications it requires, and to delimit the nucleus classes for which dedicated input systematics are needed, notably halo nuclei, for which the adopted radius parameterization is explicitly not valid.

Background

Excited nuclear states are characterized by their energy and their excitation probability. For a collective excitation such as the first $2^+$ state of an even-even nucleus, the excitation strength splits into a neutron and a proton contribution, encoded in the neutron and proton transition matrix elements $M_n$ and $M_p$ . The proton part is fixed by electromagnetic measurements — typically Coulomb excitation, in which the target nucleus is excited by the electric field of a passing heavy projectile, yielding the reduced transition probability $B(E2)$ — while the neutron part is only accessible through hadronic probes such as inelastic proton scattering, which couples to both neutron and proton densities. Combining one electromagnetic and one hadronic measurement therefore determines the ratio $M_n/M_p$ , whose deviation from the neutron-to-proton number ratio $N/Z$ diagnoses how differently the neutron and proton fluids participate in the excitation (for an isoscalar vibration of a homogeneous fluid, $M_n/M_p=N/Z$ ). In the phenomenological analysis, the hadronic measurement is reduced to a deformation length $\delta_F=R_0\beta$ , where $\beta$ parametrizes the quadrupole deformation of the optical potential — the effective complex potential, fitted to elastic-scattering data, that describes the average interaction between the probe and the nucleus — and $R_0$ is the nuclear radius. The excitation is induced by the transition potential, i.e. the term linear in the deformation parameters obtained by expanding the deformed optical potential. In 1981 Bernstein, Brown and Madsen derived a formula relating $M_n/M_p$ to the ratio of the hadronic deformation length $\delta_F$ to the electromagnetic (Coulomb) deformation length $\delta_p$ , using the ratio $b_n/b_p$ of the probe's interaction strengths with neutrons and protons (tabulated values: about 3 for 10–50 MeV protons, 0.95 at 1 GeV protons, 1/3 for 10–50 MeV neutrons, 1 for alpha particles, 1/3 for $\pi^+$ and 3 for $\pi^-$ at 160–200 MeV). This original Bernstein formula implicitly assumes identical neutron and proton radii and diffusenesses, an assumption that fails for exotic neutron-rich nuclei. Khan (Phys. Rev. C 105, 014306 (2022)) generalized the derivation, allowing separate neutron and proton radii $R_{n,p}$ and surface diffusenesses $a_{n,p}$ (the thickness of the density falloff at the nuclear surface), and supplied nuclear-chart-wide phenomenological parameterizations for these quantities, with $a_{n,p}$ tied to neutron and proton separation energies. The generalized formula was validated numerically only for proton scattering at about 50 MeV per nucleon against microscopic analyses of the oxygen, sulfur and argon isotopic chains, where it typically differs from the original Bernstein formula by less than about 10% except for very neutron-rich systems ( $N/Z\gtrsim1.6$ ), proton-rich ones ( $N/Z\lesssim0.8$ ), or cases in which the electromagnetic deformation length exceeds the hadronic one. Two stated limitations border the validated domain: the adopted light-nucleus radius parameterization is explicitly not valid for halo nuclei (weakly bound systems with an extended, dilute neutron density tail), and the derivation still assumes that the deformation parameter of the transition potential equals that of the densities and that the macroscopic liquid-drop relation $M_n/M_p=(N/Z)(\beta_n/\beta_p)$ holds. Meanwhile, modern rare-isotope experiments already operate outside the validated window: inverse-kinematics proton scattering on $^{42}$ Si and $^{44}$ S at 82–91 MeV per nucleon has forced experimentalists to interpolate the untabulated $b_n/b_p$ with an ad hoc value carrying a 50% uncertainty.

Problem Statement

Establish the validity domain of the generalized Bernstein formula of Khan (Phys. Rev. C 105, 014306 (2022)) for phenomenologically extracting the ratio $M_n/M_p$ of neutron to proton transition matrix elements from combined electromagnetic and hadronic measurements, beyond the domain in which it has been validated — proton scattering at a few tens of MeV per nucleon. Concretely: (1) determine for which hadronic probes (neutrons, deuterons, alpha particles, pions) and which incident energies the generalized formula remains accurate — 'accurate' meaning here that the $M_n/M_p$ it extracts agrees, to within a relative tolerance that the answer states explicitly, justifies against the precision required for the intended rare-isotope application, and applies uniformly across probes, energies and nucleus classes, with the benchmark $M_n/M_p$ obtained for the same nucleus and state from a microscopic reaction analysis that is specified independently rather than tuned per case — and derive any probe-dependent modifications it requires where the single-ratio $b_n/b_p$ description of the probe's neutron-versus-proton coupling breaks down; and (2) delimit the nucleus classes, notably halo and other weakly bound or strongly deformed systems where the adopted radius/diffuseness systematics fail or the assumed equality of potential and density deformation parameters breaks down, for which dedicated input systematics must be developed. For each probe the validity boundary is characterized by the incident-energy range over which the formula remains accurate, and for each nucleus class by the conditions under which the adopted radius/diffuseness systematics fail or the assumed equality of potential and density deformation parameters breaks down; proton scattering at incident energies above the validated few-tens-of-MeV window is part of component (1), not merely its baseline. The two components and the cases in (1) are independent sub-questions: an established boundary for a single probe family or a single nucleus class settles that case but does not by itself resolve the probe-general objective, which covers every listed probe, higher-energy proton scattering, and every listed nucleus class. These components jointly serve one objective: a validated, probe-general prescription that experimental programs at rare-isotope facilities can use to extract trustworthy $M_n/M_p$ values outside the few-tens-of-MeV proton-scattering window.

References

E. Khan, 'Derivation of the M_n/M_p ratio in exotic nuclei', Physical Review C 105, 014306 (2022). DOI: 10.1103/PhysRevC.105.014306, https://doi.org/10.1103/PhysRevC.105.014306; arXiv:2110.01299, https://arxiv.org/abs/2110.01299
A. M. Bernstein, V. R. Brown, V. A. Madsen, 'Neutron and proton transition matrix elements and inelastic hadron scattering', Physics Letters B 103, 255 (1981). DOI: 10.1016/0370-2693(81)90219-7, https://doi.org/10.1016/0370-2693(81)90219-7
L. A. Riley, I. Conroy, A. M. Himmelreich, et al., 'Determination of proton and neutron contributions to the 0_gs^+ → 2_1^+ excitations in 42Si and 44S using inelastic proton scattering in inverse kinematics and intermediate energy Coulomb excitation', Physical Review C 112, 014331 (2025). DOI: 10.1103/b8xj-ycqk, https://doi.org/10.1103/b8xj-ycqk; arXiv:2508.00703, https://arxiv.org/abs/2508.00703
P. D. Cottle, L. A. Riley, A. Gade, K. W. Kemper, M. Spieker, 'Observing the effects of numbers of valence nucleons on 0_gs^+ → 2_1^+ transitions in deformed nuclei by comparing proton and neutron transition matrix elements', Physical Review C (2026). DOI: 10.1103/q4ws-71zc, https://doi.org/10.1103/q4ws-71zc; arXiv:2608.13893, https://arxiv.org/abs/2608.13893