Direct mass test of the predicted quenching of the $N=82$ neutron shell gap along the isotonic chain from $^{132}$Sn toward Zr and Sr, and its $r$-process consequences

The identification of the $\pi 1p_{3/2}$ proton-hole state in $^{131}$In (J. Taprogge et al., Phys. Rev. Lett. 112, 132501 (2014)) led, through shell-model calculations, to the prediction that proton subshell closures are eroded along the $N=82$ isotonic chain below doubly magic $^{132}$Sn and that the $N=82$ neutron shell gap is consequently quenched between Sn ($Z=50$) and Zr ($Z=40$), with direct consequences for $r$-process flow near the $N=82$ waiting points. The problem remains open: twelve years on, no direct mass measurement exists for any $N=82$ isotone below $^{130}$Cd. Partial progress — ISOLTRAP cadmium masses confirming gap reduction at $Z=48$, a tentative experimental location of the $\pi 0f_{5/2}$ proton-hole orbital, and the single-ion Rare-RI Ring storage-ring technique — leaves the quantitative test of the predicted gap quenching for $Z=38$–$47$, and its $r$-process quantification, as the surviving open core.

Background

The nucleus $^{132}$Sn with 50 protons ($Z=50$) and 82 neutrons ($N=82$) is 'doubly magic': both proton and neutron shell closures make it an inert core whose neighboring nuclei can be described as single protons, neutrons, or holes coupled to that core. In this region the proton orbitals between the magic numbers $Z=28$ and $Z=50$ are, in order of filling, $0f_{5/2}$, $1p_{3/2}$, $1p_{1/2}$ and $0g_{9/2}$ (the notation $nlj$ gives the radial quantum number $n$, orbital angular momentum $l$, and total angular momentum $j$; $\pi$ marks a proton orbital). Gaps between these orbitals create sub-shell closures at $Z=38$ (gap above $1p_{3/2}$) and $Z=40$ (gap above $1p_{1/2}$) along the chain of $N=82$ isotones — nuclei sharing $N=82$ with $Z$ decreasing from 50 toward Sr and Zr. In $^{131}$In, one proton hole below $^{132}$Sn, the $\pi 1p_{1/2}$ and $\pi 0g_{9/2}$ hole states were known, but the positions of the $\pi 1p_{3/2}$ and $\pi 0f_{5/2}$ hole states were not. Taprogge et al. (Phys. Rev. Lett. 112, 132501 (2014)) observed a state at $E_{x}=1353$ keV — populated both in the $\beta$ decay of $^{131}$Cd and after $\beta$-delayed neutron emission from $^{132}$Cd — and identified it, with a tentative $3/2^{-}$ assignment, as the previously unknown $\pi 1p_{3/2}$ proton-hole state. Inserting this energy into large-scale shell-model calculations (diagonalizing a Hamiltonian in the $Z=28$–$50$, $N=50$–$82$ valence space), they found a 'surprising absence of proton subshell closures' along the $N=82$ chain below $^{132}$Sn. Because the energies of the neutron orbitals bounding the $N=82$ gap shift as protons are removed along the chain — a monopole-driven shell-evolution effect (the energy of a single-particle orbital depends on the occupation of the other orbitals through the angle-averaged, or monopole, part of the residual nucleon–nucleon interaction): the attractive monopole coupling between the filled $\pi 0g_{9/2}$ proton orbital and the $\nu 0h_{11/2}$ neutron orbital below $N=82$ is what makes the $Z=50$ and $N=82$ closures at $^{132}$Sn mutually enhanced, so emptying $\pi 0g_{9/2}$ while moving from Sn ($Z=50$) toward Zr ($Z=40$) removes support from $\nu 0h_{11/2}$ and narrows its separation from the $\nu 2f_{7/2}$ orbital above $N=82$ — the calculations predicted substantial quenching of the $N=82$ neutron shell closure between Sn ($Z=50$) and Zr ($Z=40$). The standard empirical measure is the two-neutron shell gap $\delta_{2n}(Z,N=82)=S_{2n}(Z,82)-S_{2n}(Z,84)$, where the two-neutron separation energy is $S_{2n}(Z,N)=M(Z,N-2)-M(Z,N)+2m_{n}$ for nuclear masses $M$ and neutron mass $m_{n}$; equivalently $\delta_{2n}(Z,82)=M(Z,80)+M(Z,84)-2M(Z,82)$, so a gap determination at each element $Z$ requires masses at $N=80$, $82$ and $84$. This matters for astrophysics: in the rapid neutron-capture ($r$-)process in neutron-star mergers and supernova ejecta, nucleosynthesis at the $N=82$ waiting points (nuclei whose neutron-capture flow stalls because the $N=82$ shell closure suppresses neutron capture and the $\beta$-decay half-lives are comparatively long, of order tens to hundreds of milliseconds for these very neutron-rich species) shapes the solar $r$-process abundance pattern near $A\approx 130$. The relevant masses feed capture $Q$-values, photodissociation rates and separation energies in reaction-network simulations. At the time of the 2014 paper, among the $N=82$ isotones below $^{132}$Sn only $^{131}$In had a directly measured mass (from the Penning-trap measurement of its $1/2^{-}$ $\beta$-decaying isomer, Kankainen et al., Phys. Rev. C 87, 024307 (2013)); the $^{130}$Cd mass was still derived from evaluated $\beta$-decay data (it was first measured directly only in 2015), and everything below $Z=48$ rested on divergent extrapolations from global mass models. Precision mass spectrometry of such exotic nuclei uses Penning traps (highest precision but needing lifetimes of order $\gtrsim 100$ ms and usable beam intensities), multi-reflection time-of-flight spectrometers, and isochronous storage-ring mass spectrometry such as the Rare-RI Ring at RIKEN's RIBF, which can measure a single pre-identified ion. The shell-model interaction used in 2014 was a realistic effective interaction derived from the CD-Bonn nucleon–nucleon potential renormalized by way of the $V_{low-k}$ approach, but it still required empirical adjustments (a proton-proton pairing reduction and multipole modifications) to reproduce known spectroscopy in the $^{132}$Sn region, so the gap prediction is only as trustworthy as the extrapolation. Taprogge et al. also showed that the predicted disappearance of the proton subshell closures depends neither on the tentative $3/2^{-}$ assignment of the 1353-keV state nor on the assumed position of the $\pi 0f_{5/2}$ proton-hole state, but on the mere proximity (less than 1 MeV) of the next proton single-particle orbital to $\pi 1p_{1/2}$, independent of its character.

Problem Statement

Determine quantitatively how the $N=82$ neutron shell closure evolves along the isotonic chain below doubly magic $^{132}$Sn — specifically, whether the two-neutron shell gap $\delta_{2n}(Z,N=82)=M(Z,80)+M(Z,84)-2M(Z,82)$ is quenched between Sn ($Z=50$) and Zr ($Z=40$), as predicted by the shell-model conclusion of Taprogge et al. (Phys. Rev. Lett. 112, 132501 (2014)) that proton subshell closures at $Z=38$ and $Z=40$ are eroded along $N=82$, and identify the direct mass measurements that would provide the decisive test. The central, still-unmet deliverables are: (i) the set of $N=80$, $N=82$ and $N=84$ masses whose direct measurement would extract $\delta_{2n}$ at each element — for the $N=82$ isotones $^{129}$Ag, $^{128}$Pd, $^{127}$Rh, $^{126}$Ru, $^{125}$Tc, $^{124}$Mo, $^{123}$Nb, $^{122}$Zr, $^{121}$Y and $^{120}$Sr none of these masses has yet been measured, so the determination must presently be made from existing masses and mass models rather than from new measurements; (ii) the quantitative $\delta_{2n}(Z,N=82)$ evolution along the chain that these masses and models imply, compared against the predicted quenching and against the spread among global mass models; and (iii) quantification, via reaction-network calculations with those masses, of what the implied gap evolution does to $r$-process nucleosynthesis near the $N=82$ waiting points and to the solar $r$-process abundance pattern in the $A\approx 120$–$132$ region. A complete answer settles, as far as the best available masses and mass models allow, whether the $N=82$ magic number persists, weakens, or collapses between $Z=50$ and $Z=38$, identifies the measurements that would confirm or refute the prediction, and quantifies what the resulting gap evolution implies for $r$-process abundances.

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