Radii of proton emitters

TL;DR

This work addresses how to define and interpret the radius of proton-emitting resonances, where standard rms radii are ill-defined due to decay. It combines a complex-energy (Rigged Hilbert Space) approach with exterior complex scaling to define a complex radius $\tilde{r}_{\rm rms}$ and uses time-dependent propagation to connect this quantity to measurable rms radii, identifying an early-time plateau where $r_{\rm rms}$ tracks $\mathrm{Re}(\tilde{r}_{\rm rms})$. The study, focusing on the $^{15}$F ($^{14}$O+$p$) $d_{5/2}$ resonance, reveals a non-monotonic dependence of $\mathrm{Re}(\tilde{r}_{\rm rms})$ on decay energy and an imaginary part that grows with the decay width, with a proton-halo signature near threshold. It provides a practical route to access charge radii in proton-unbound nuclei and presents a robust, extensible framework for complex observables in open quantum systems, applicable beyond nuclear physics.

Abstract

Nuclear radius is a fundamental structural observable that informs many properties of atomic nuclei and nuclear matter. Experimental studies of radii in dripline nuclei are in the forefront of research with radioactive ion beams. Of particular interest are charge radii of proton-unbound nuclei that will soon be approached in laser spectroscopy. In this paper, using the complex-energy approach and direct time propagation, we investigate the radius of the proton resonance whose size is ill-defined in the standard stationary quantum-mechanical description. An early-time plateau is identified during which the radius of the Gamow resonance coincides with the real-energy radius accessible experimentally. We demonstrate a non-monotonic dependence of the complex radius on decay energy and a local increase of the charge radius across the threshold (a proton halo effect).

Figures (4)

• Figure 1: Real part of the $d_{5/2}$ radial wave function $u$ in $^{15}$F at $Q_p=2.06$ MeV evaluated along the real axis $r$ (solid line) and along the complex-rotated coordinate $\tilde{r}$ (dashed line), see Eq. (\ref{['coordinate transformation']}). The inner ($r_{\rm i}$) and outer ($r_{\rm o}$) turning points are indicated by dotted lines.
• Figure 2: (a) Complex-rms radius $\tilde{r}_{\rm rms}$ and (b) decay width $\Gamma$ as a function of decay energy $Q_p$ calculated for the proton resonant $d_{5/2}$ state in $^{15}\mathrm{F}$. The real and imaginary parts of $\tilde{r}_{\rm rms}$ are shown by the solid and dashed lines, respectively. The dash-dotted line shows the rms radius in the HO basis, i.e., without continuum coupling. The proton threshold is marked by the dotted line.
• Figure 3: Complex radius integrand $r^4 \psi^2$ (a, c) and the corresponding cumulative integral $\int_0^r r^{\prime 4} \psi^2 \, d r^\prime$ (b,d) for the proton resonant $d_{5/2}$ state in $^{15}$F at $Q_p=0.69$ MeV (top) and 1.40 MeV (bottom). All quantities are evaluated along the real axis without complex rotation. The dashed line indicates the potential barrier. The dash-dotted line and shaded bar mark the real and imaginary part of the complex radius $\tilde{r}_{\rm rms}^2$, respectively. The inner and outer turning points are indicated. The intersection point at which the cumulative integral becomes equal to the complex rms radius in shown by an arrow.
• Figure 4: Time evolution of the rms radius $r_{\rm rms}$ of the $d_{5/2}$ resonance in $^{15}\mathrm{F}$ (solid line) as a function of time $t$ (in units of the half-live $T_{1/2}$) for (a) $Q_p=0.69$ MeV and (b) $Q_p=2.06$ MeV. The real (dashed line) and imaginary (shaded band) parts of the complex radius $\tilde{r}_{\rm rms}$ are shown for comparison. The uncertainty in $r_{\rm rms}$ related to the variation of $r_{\rm TPA}$ by $\pm 1$ fm is marked by a gray band. Arrows mark the time at which $r_{\rm rms}$ start deperting from $\Re(\tilde{r}_{\rm rms})$ within the uncertainty $\Im(\tilde{r}_{\rm rms})$.