Towards the Limits of Existence of Nuclear Structure: Observation and First spectroscopy of the Isotope $^{31}$K by measuring its three-proton Decay

TL;DR

This paper reports the first observation and spectroscopy of the extremely proton-rich isotope $^{31}$K, measuring its three-proton decay and determining its energy states and separation energy, thus exploring the limits of nuclear existence.

Contribution

It provides the first experimental data on $^{31}$K, including its decay modes, energy states, and separation energy, extending knowledge of nuclear stability near the proton drip line.

Findings

Observed $^{31}$K beyond the proton drip line.

Measured three-proton decay and determined energy states.

Established an upper half-life limit of 10 ps.

Abstract

The most-remote from stability isotope $^{31}$K, which is located four atomic mass units beyond the proton drip line, has been observed. It is unbound in respect to three-proton (${3p}$) emission, and its decays have been detected in flight by measuring trajectories of all decay products using micro-strip detectors. The $3p$-emission processes have been studied by means of angular correlations $^{28}$S+3$p$ and the respective decay vertexes. The energies of the previously-unknown ground and excited states of $^{31}$K have been determined. This provides its $3p$ separation-energy value $S_{3p}$ of $-4.6(2)$ MeV. Upper half-life limits of 10 ps of the observed $^{31}$K states have been derived from distributions of the measured decay vertexes.

Equations

$${\rho }_{\theta }=\sqrt{{\theta }_{p1{-}^{28}S}^2+{\theta }_{p2{-}^{28}S}^2}.$$

$${\rho }_3=\sqrt{{\theta }_{p1{-}^{28}S}^2+{\theta }_{p2{-}^{28}S}^2+{\theta }_{p3{-}^{28}S}^2}$$

Full text

three-proton decays of 31K [from 36Ar fragmentation and subsequent charge-exchange reaction of 31Ar at 620 MeV/u; measured angular proton–proton–proton–28S correlations, derived decay energies and half-life values for three states in 31K.

In recent experimental Mukha et al. (2018) and theoretical Grigorenko et al. (2018) studies of the lightest isotopes in the argon and chlorine isotope chains, limits of existence of the corresponding nuclear structure were addressed. For the issue of existence of nuclear structure, we adopt the approach used in Ref. Grigorenko et al. (2018). Namely, a nuclear configuration has an individual structure with at least one distinctive state, if the orbiting valence protons of the system are reflected from the corresponding nuclear barrier at least one time. Thus the nuclear half-life may be used as a criterion here, and two extreme cases can be mentioned. The very long-lived particle-emitting states may be considered as quasistationary. For example, the half-lives of all known heavy two-proton ($2p$) radioactivity precursors (e.g., 45Fe, 48Ni, 54Zn, 67Kr) are a few milliseconds Pfützner et al. (2002); Pomorski et al. (2011); Blank et al. (2005); Goigoux et al. (2016). For such long-lived states, modifications of nuclear structure by coupling with continuum are negligible. In the opposite case of very short-lived unbound ground states (g.s.), the continuum coupling becomes increasingly important, which can be regarded as a transition to continuum dynamics. For example, the discussion of the tetra-neutron (4n) system has shown that its spectrum is strongly affected both by the reaction mechanism and by the initial nuclear structure of the participants Grigorenko et al. (2004). In Ref. Grigorenko et al. (2018), the isotopes 26Ar and 25Cl were predicted as the most remote nuclear configurations with identified g.s.. Similar predictions allow to expect a number of previously-unknown unbound isotopes located within a relatively broad (by 2–5 atomic mass units) area along the proton drip line. For more exotic nuclear systems beyond such a domain, no g.s. of isotopes (and thus no new isotope identification) are expected. Therefore a new borderline indicating the limits of existence of isotopes in the nuclear chart and the transition to chaotic-nucleon matter may be inspected.

In this work we continue the “excursion beyond the proton dripline” of Ref. Mukha et al. (2018) by presenting the results of additional analysis of the data obtained with a 31Ar secondary beam Xu et al. (2018). In the experiment, described in detail in Refs. Xu et al. (2018); Mukha et al. (2018), the 31Ar beam was produced by the fragmentation of a primary 885 MeV/u 36Ar beam at the SIS-FRS facility at GSI (Germany). The previous objectives of the experiment were studies of $2p$ decays of 29,30,31Ar isotopes. We briefly repeat the general description of the experiment and the detector performance. The FRS was operated with an ion-optical settings in a separator-spectrometer mode, where the first half of the FRS was set for separation and focusing of the radioactive beams on a secondary target in the middle of the FRS, and the second half of FRS was set for the detection of heavy-ion decay products. The secondary 620 MeV/u 31Ar beam with an intensity of 50 ions $\text{s}^{-1}$ bombarded a 27-mm thick 9Be secondary target located at the FRS middle focal plane. In the cases addressed in Refs. Xu et al. (2018); Mukha et al. (2018), the 29,30Ar nuclei were produced via neutron knockout reactions from the 31Ar ions. The decay products of unbound 29,30Ar nuclei were tracked by a double-sided silicon micro-strip detector (DSSD) array placed just downstream of the secondary target. Four large-area DSSDs Stanoiu et al. (2008) were employed to measure hit coordinates of the protons and the recoil heavy ions (HI), resulting from the in-flight decays of the studied $2p$ precursors. The high-precision position measurement by DSSDs allowed for reconstruction of all fragment trajectories, which let us to derive the decay vertex together with angular HI-$p$ and HI-$p_{1}$-$p_{2}$ correlations. For example, the trajectories of measured 28S+$p$+$p$ coincidences served for the analysis, and the spectroscopic information on 30Ar was concluded Xu et al. (2018). The spectra of 30Ar were observed by using 2p angular correlations as function of their root-mean-square angle relative to 28S,

[TABLE]

A number of by-product results was obtained in a similar way from the data recorded in the same experiment. In particular, a 3p-unbound nuclear system of 31K was populated in a charge-exchange reaction. This mechanism has lower cross section than knockout reactions, and the obtained data have smaller statistics than in the previously-mentioned case Xu et al. (2018). In spite of poor statistics with few events registered, we have obtained several nuclear-structure conclusions from the data. The 31K spectrum was derived from trajectories of all decay products 28S+$p_{1}$+$p_{2}$+$p_{3}$ measured in four-fold coincidence. All detector calibrations were taken from the analyses reported in Refs. Xu et al. (2018); Mukha et al. (2018).

In Fig. 1, we present 3p correlations observed in decays of 31K as function of their root-mean-square angle

[TABLE]

derived from the measured trajectories of 28S+$3p$ coincident events. The kinematical variable $\rho_{3}$ is introduced because the decay protons share the 3p-decay energy, in analogy with 2p decays [see eq. (1)]. One can see three peaks (i), (ii) and (iii) reflecting the population of states in 31K isotope, and their respective $3p$-decay energies $Q_{3p}$ of about 4.5, 9 and 16 MeV may be estimated from the upper axis.

In order to establish decay schemes of the states in 31K, we have produced angular $\theta_{p-^{28}S}$ correlations projected from the 28S+$3p$ events which are selected by the gates around the $\rho_{3}$-peaks (i), (ii), (iii) in Fig. 1. These projections are shown in the respective panels in Fig. 2. In particular, the lowest-energy 4.5 MeV peak (i) may correspond to the 31K g.s., which decays by emission of a proton first into an intermediate 30Ar g.s., whose 2p-decay energy of 2.45(15) MeV is known Xu et al. (2018). Then the corresponding $\theta_{p-^{28}S}$ correlations in Fig. 2(i) should consist of two contributions. The firstly-emitted proton should cause a peak in the observed $\theta_{p-^{28}S}$ correlations. The second component should be the known broad $\theta_{p-^{28}S}$ distribution from the 30Ar g.s. 2p-decay Xu et al. (2018), which is centered at $E_{p-^{28}S}\simeq$1.2 MeV (because the 2 protons, which can not be distinguished, share the 2p-decay energy). We have fitted the data by a sum of two respective components: 1) the Monte-Carlo simulation including the response of the experimental setup to 1p-emission by 31K (the simulation procedure is described in details in Refs. Mukha et al. (2007); Agostinelli et al. (2003); 2) the known detector response to the 2p-decay of 30Ar g.s. (see Ref. Xu et al. (2018)). One may see, that the small-angle region of the $\theta_{p-^{28}S}$ distribution agrees with the 2p-decay of 30Ar g.s. (the dotted-line taken into account the literature value of the 2p-decay energy of 2.45${}^{+0.05}\!\!\!\!\!\!\!\!\!\!\!\!_{-0.10}$ MeV), while the large-angle correlations can be described by the 1p-emission of 31K into 30Ar g.s. (solid line) with the best-fit decay energy of 2.15(15) MeV.

The illustration of procedure of the data fit is given in Fig. 3, where the probability that the simulated response of the setup to the 1p decay of 31K into the 30Ar g.s. matches the measured angular $\theta_{p-^{28}S}$ correlations is shown in dependence on the 1p-decay energy. The best-fit energy and its uncertainty have been derived from the distribution centroid and width, respectively. Thus we may assign the 3p-decay energy of the 31K g.s. as 2.15(15)+2.45${}^{+0.05}\!\!\!\!\!\!\!\!\!\!\!\!_{-0.10}\simeq$4.6(2) MeV.

Similar angular $\theta_{p-^{28}S}$ projections made with the 9 and 16 MeV gates (see Fig. 2) are less conclusive. One may see that both distributions in Fig. 2 (ii) and (iii) contain no contribution from the 2p-decay of 30Ar g.s., and therefore the 9 and 16 MeV excited states in 31K should proceed via excited states in 30Ar.

The result of the data analysis, the assigned levels of 31K and their decay scheme, is shown in Fig. 4. The derived information may be improved in following experiments with higher statistics and increased resolution. The mass of the 31K g.s. may be derived by using the masses of 28S+3p and the $Q_{3p}$ value, which then may be compared with available theoretical predictions. The energy of the 31K g.s. has been predicted by the systematics proposed for the mass differences of mirror nuclei (the improved Kelson-Garvey mass relations Tian et al. (2013)), which is shown in Fig. 4 on the right-hand side. One sees quite a large disagreement with the experimental value. Such a difference may be explained by the effect of Thomas-Ehrmann shift Thomas (1952); Ehrman (1951) which is often observed in 1p-unbound nuclei. Alternatively, as the 31K g.s.  decays via the long-lived 30Ar g.s., we may use the empirical $S_{p}$ systematics of 1p-emitting states in light nuclei based on parametrization of experimental mirror energy differences (MED) Fortune (2018). The definition is MED=$S_{n}$(neutron-rich nucleus)$-S_{p}$(its proton-rich mirror), and MED=(Z/A**1/3)MED’, where the MED’ value does not depend on the nuclear charge * Z* and mass A Fortune (2018). This parametrization can be scaled to the presumably $d_{3/2}$ g.s. of 31K by using the known thresholds of the A-2 mirror pair 29Cl∗(3/2*+)–29Mgg.s.(3/2+), which results in not much better agreement with the data (see Fig. 4 on right-hand side). The similar estimate can be done by using the A-2 mirror states 29Clg.s.(1/2+)–29Mg∗(1/2+) where excitation energy of the experimentally identified 29Mg∗(1/2+*) is 55 keV according to Ref. Baumann et al. (1989). Then the evaluated $S_{3p}$ value is of -6.0 MeV which still disagrees with the data. Such a difference in the observed and predicted energies of the 31K g.s. requires further investigation, for example the influence of three-nucleon forces may be studied like in Ref. Holt et al. (2013).

The width of the 31K g.s. derived by the fit in Fig. 2 (i) provides only the upper-limit value $\Gamma_{g.s.}<$ 400 keV, as it reflects the experimental resolution. For comparison, the upper-limit Wigner estimate for a single-particle 1$d_{3/2}$-shell width of 31K g.s. is about 30  keV only. The widths of the excited 9 and 16 MeV states were estimated from the $\rho_{3}$ distribution in Fig. 1 giving the values of 1 and 2 MeV, respectively. One should also note, that the spectrum of 31Mg, which is the mirror nucleus of 31K, displays a number of low-energy levels assigned to two rotational bands and to a spherical configuration Nishibata et al. (2017), which provides an evidence of shape coexistence in this nuclear system. Most of these states de-excite by $\gamma$-ray emission with half-lives in the nanosecond range. As all 31K states are unbound, the isospin-symmetrical rotational bands are unlikely to be excited.

Nevertheless, we have evaluated the half-life values of the observed 31K states in the picosecond range by measuring distributions of their decay vertexes. Figure 5 (a) shows the profile of 2p-decay vertexes of the 30Ar∗ short-lived excited states first published in Mukha et al. (2015) by using the measured 28S+$p$+$p$ trajectories. This profile serves as a reference in the evaluation of the 3p-decay vertexes from the 31K “ground” and “first excited” states which are shown in Fig. 5 (b) and (c), respectively. The two latter profiles are derived from the measured 28S+$p$+$p$+$p$ events by applying the selection gates around the peaks (i) and (ii) in the $\rho_{3}$-spectrum of 31K shown in Fig. 1. The Monte Carlo simulations Agostinelli et al. (2003); Mukha et al. (2007) of the reference case of 30Ar short-lived excited states are shown in Fig. 5(a). They assume $T_{1/2}\simeq$0 ps for the 30Ar states and take into account the experimental angular uncertainties in tracking the fragments and reconstructing the vertex coordinates. The simulations reproduce the data quantitatively. The half-life uncertainty is illustrated by the $T_{1/2}$=5 ps simulation which fails fitting the data. The asymmetry of the rising and falling slopes of the vertices is due to multiple scattering of the fragments in the thick target. Similar Monte Carlo simulations with $T_{1/2}$ of 0, 5 and 10 ps of the 31K states are compared with the corresponding data in Fig.5(b). One may see that the $T_{1/2}$=5 ps simulation is the best fit for the 31K g.s. data. However, production of few events of 31K inside the 27-mm thick secondary target result in the $T_{1/2}$ uncertainties of 10 ps. Thus we conclude that the half-life value of the 31K g.s.  is shorter than 10 ps, which is our upper-limit estimate. For the 31K excited-state, the best fit of its vertex profile is shown in Fig.5(c) giving $T_{1/2}$=0 ps. Though we found no indication on long-lived states in 31K, a dedicated experiment with improved resolution and larger statistics inspecting possible shape coexistence in 31K may provide a strict test of isospin-symmetry conservation/violation of such exotic nuclear systems.

In conclusion, the first spectroscopy of the previously-unknown isotope 31K, located four atomic mass units beyond the proton drip line, has revealed states whose widths are much smaller than the values of 3–5 MeV which are mandatory for the formation of a nuclear state Grigorenko et al. (2018). Therefore the half-lives of the observed 31K states are much longer than those predicted at the limits of existence of nuclear structure, and one can conclude that a transition region to chaotic nuclear systems is not reached yet. Looking to the future, similar charge-exchange reactions with more exotic beams like 48Ni or 67Kr prospect investigations of nuclear systems located by seven mass units beyond the proton drip line, where the basic mean-field concept and Pauli principle may be ultimately tested. Last but not least, the mass of the 31K g.s. can be derived from the measured $S_{3p}$ value, which is the most challenging test of predictions of nuclear mass models.

This work was supported in part by the Helmholtz International Center for FAIR (HIC for FAIR); the Helmholtz Association (Grant No. IK-RU-002); the Russian Science Foundation (Grant No. 17-12-01367); the Polish National Science Center (Contract No. UMO-2015/17/B/ST2/00581); the Polish Ministry of Science and Higher Education (Grant No. 0079/DIA/2014/43, Grant Diamentowy); the Helmholtz-CAS Joint Research Group (Grant No. HCJRG-108); the Ministry of Education & Science, Spain (Contract No. FPA2016-77689-C2-1-R); the Hessian Ministry for Science and Art (HMWK)through the LOEWE funding scheme; the Ministry of Education, Youth and Sports, Czech Republic (Projects LTT17003 and LM2015049); the Justus-Liebig-Universität Giessen (JLU) and the GSI under the JLU-GSI strategic Helmholtz partnership agreement; DGAPA-PAPIIT UNAM IA103218. This work was carried out in the framework of the Super-FRS Experiment Collaboration. This article is a part of the Ph.D. thesis of D. Kostyleva.