New developments on the recoil distance doppler-shift method

2010

Absolute transition probabilities are fundamental observables for nuclear structure. The recoil-distance-Doppler-shift (RDDS) technique, also called plunger technique, is a well established tool for the determination of these important experimental quantities via the measurement of lifetimes of excited nuclear states. Nowadays nuclear structure investigations are concentrated on exotic nuclei which are often produced with extremely small cross sections or with very low beam intensities. In order to use the RDDS technique also for the investigation of very exotic nuclei this method has to be adapted to the specific needs of these special reactions. This article gives an overview on recent RD…

rp-process nucleosynthesis at extreme temperature and density conditions

1998

We present nuclear reaction network calculations to investigate the influence of nuclear structure on the rp-process between Ge and Sn in various scenarios. Due to the lack of experimental data for neutron-deficient nuclei in this region, we discuss currently available model predictions for nuclear masses and deformations as well as methods of calculating reaction rates (Hauser-Feshbach) and beta-decay rates (QRPA and shell model). In addition, we apply a valence nucleon (NpNn) correlation scheme for the prediction of masses and deformations. We also describe the calculations of 2p-capture reactions, which had not been considered before in this mass region. We find that in X-ray bursts 2p-c…

Discovery of an Exceptionally Strong β -Decay Transition of F20 and Implications for the Fate of Intermediate-Mass Stars

2019

A significant fraction of stars between 7 and 11 solar masses are thought to become supernovae, but the explosion mechanism is unclear. The answer depends critically on the rate of electron capture on ^{20}Ne in the degenerate oxygen-neon stellar core. However, because of the unknown strength of the transition between the ground states of ^{20}Ne and ^{20}F, it has not previously been possible to fully constrain the rate. By measuring the transition, we establish that its strength is exceptionally large and that it enhances the capture rate by several orders of magnitude. This has a decisive impact on the evolution of the core, increasing the likelihood that the star is (partially) disrupte…

Nuclear moments of neon isotopes in the range fromNe17at the proton drip line to neutron-richNe25

2005

Nuclear moments of odd-A neon isotopes in the range 17 ≤ A ≤ 25 have been determined from optical hyperfine structures measured by collinear fast-beam laser spectroscopy. The magnetic dipole moments of 17 Ne, 23 Ne and 25 Ne, as well as the electric quadrupole moment of 23 Ne are either reported for the first time or improved considerably. The measurements also decide for a 1/2 + ground state of 25 Ne. The behavior of the magnetic moments of the proton drip-line nucleus 17 Ne and its mirror partner 17 N suggests isospin symmetry. Thus, no clear indication of an anomalous nuclear structure is found for 17 Ne. The magnetic moments of the investigated nuclei are discussed in a shell-model appr…

Magnetic dipole moments near 132Sn: new measurement on 135I by NMR/ON

1998

Abstract On-line low temperature nuclear orientation (OLNO) experiments have been performed on the isotope 135 I using the technique of nuclear magnetic resonance on oriented nuclei (NMR/ON). The magnetic moment of the 7 2 + ground state has been measured to be μ( 7 2 + 135 I ) = 2.940(2) μ N , thereby extending the known data on these states in odd- A I isotopes up to the neutron shell closure at N = 82. Shell-model calculations have been performed for the magnetic moments of 7 2 + states in the N = 82 isotones using free-nucleon and effective g -factors. The effective g -factors are obtained from a perturbation calculation that includes corrections for core polarisation and meson-exchange…

Low-lying level structure of Cu56 and its implications for the rp process

2017

The low-lying energy levels of proton-rich Cu56 have been extracted using in-beam γ-ray spectroscopy with the state-of-the-art γ-ray tracking array GRETINA in conjunction with the S800 spectrograph at the National Superconducting Cyclotron Laboratory at Michigan State University. Excited states in Cu56 serve as resonances in the Ni55(p,γ)Cu56 reaction, which is a part of the rp process in type-I x-ray bursts. To resolve existing ambiguities in the reaction Q value, a more localized isobaric multiplet mass equation (IMME) fit is used, resulting in Q=639±82 keV. We derive the first experimentally constrained thermonuclear reaction rate for Ni55(p,γ)Cu56. We find that, with this new rate, the …

Spins and Magnetic Moments ofK49andK51: Establishing the1/2+and3/2+Level Ordering BeyondN=28

2013

The ground-state spins and magnetic moments of $^{49,51}\mathrm{K}$ have been measured using bunched-beam high-resolution collinear laser spectroscopy at ISOLDE CERN. For $^{49}\mathrm{K}$ a ground-state spin $I=1/2$ was firmly established. The observed hyperfine structure of $^{51}\mathrm{K}$ requires a spin $Ig1/2$ and strongly suggests $I=3/2$. From its magnetic moment $\ensuremath{\mu}(^{51}\mathrm{K})=+0.5129(22){\ensuremath{\mu}}_{N}$ a spin-parity ${I}^{\ensuremath{\pi}}=3/{2}^{+}$ with a dominant $\ensuremath{\pi}1{d}_{3/2}^{\ensuremath{-}1}$ hole configuration was deduced. This establishes for the first time the reinversion of the single-particle levels and illustrates the prominen…

Nuclear structure theory for the astrophysical rp-process and r-process

2003

Abstract The astrophysical processes of rapid-proton capture and rapid-neutron capture require the knowledge of many nuclear properties which are not known from experiment. I will describe two examples of how theoretical models are used to provide this input. The first of these uses the Hartree-Fock method for displacement energies to obtain the masses of proton-rich nuclei needed for the rp-process. The second uses a model for configuration mixing near 132 Sn to provide Q values and beta-decay lifetimes for the r-process.

β-decay of22O

2005

A mass-separated 12 C 22 O molecular ion beam from the ISOLDE facility was used to study the decay of neutron-rich 22 O. The experimental results were compared with the results from an earlier experiment and predictions by shellmodel calculations using various effective interactions. The mechanism leading to the vanishing decay strength to the first 1 + level of the 22 F nucleus, predicted with the USD effective interaction but not supported by the experimental data, is analysed.

Octupole states in Tl207 studied through β decay

2020

The β decay of 207Hg into the single-proton-hole nucleus 207Tl has been studied through γ-ray spectroscopy at the ISOLDE Decay Station (IDS) with the aim of identifying states resulting from coupling of the πs−11/2, πd−13/2, and πh−111/2 shell model orbitals to the collective octupole vibration. Twenty-two states were observed lying between 2.6 and 4.0 MeV, eleven of which were observed for the first time, and 78 new transitions were placed. Two octupole states (s1/2-coupled) are identified and three more states (d3/2-coupled) are tentatively assigned using spin-parity inferences, while further h11/2-coupled states may also have been observed for the first time. Comparisons are made with st…

Structure And Decay Of Neutron-Rich Nuclides In The 115 ≤ A ≤ 138 Mass Range And r-Process Nucleosynthesis

2005

The structure and decay of neutron‐rich r‐process nuclides has been studied by a variety of means that take advantage of enhanced selectivity to permit identification of exotic nuclides. New level structures are presented for 134,135Sb along with data for Ag isomers and Cd yrast structures. Some of the properties measured play an important role in calculations of the yields of elements and isotopes produced in r‐process nucleosynthesis that takes place at high temperature in the presence of large densities of neutrons.

N=82Shell Quenching of the Classicalr-Process “Waiting-Point” NucleusCd130

2003

First $\ensuremath{\beta}$- and $\ensuremath{\gamma}$-spectroscopic decay studies of the $N=82$ $r$-process ``waiting-point'' nuclide $^{130}\mathrm{C}\mathrm{d}$ have been performed at CERN/ISOLDE using the highest achievable isotopic selectivity. Several nuclear-physics surprises have been discovered. The first one is the unanticipatedly high energy of 2.12 MeV for the [$\ensuremath{\pi}{g}_{9/2}\ensuremath{\bigotimes}\ensuremath{\nu}{g}_{7/2}]$ ${1}^{+}$ level in $^{130}\mathrm{I}\mathrm{n}$, which is fed by the main Gamow-Teller transition. The second surprise is the rather high ${Q}_{\ensuremath{\beta}}$ value of 8.34 MeV, which is in agreement only with recent mass models that include…

Two-Proton Radioactivity ofKr67

2016

In an experiment with the BigRIPS separator at the RIKEN Nishina Center, we observed two-proton (2p) emission from 67Kr. At the same time, no evidence for 2p emission of 59Ge and 63Se, two other potential candidates for this exotic radioactivity, could be observed. This observation is in line with Q value predictions which pointed to 67Kr as being the best new candidate among the three for two-proton radioactivity. 67Kr is only the fourth 2p ground-state emitter to be observed with a half-life of the order of a few milliseconds. The decay energy was determined to be 1690(17) keV, the 2p emission branching ratio is 37(14)%, and the half-life of 67Kr is 7.4(30) ms.

New experimental efforts along the rp-process path

2007

The level structure just above the proton threshold of the nucleus 30S has been studied using the neutron removal process on fast radioactive beams at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. In this work we provide a description of the experimental setup. The present status of the analysis is also discussed.

TheNd150(He3,t) andSm150(t,He3) reactions with applications toββdecay ofNd150

2011

The {sup 150}Nd({sup 3}He,t) reaction at 140 MeV/u and {sup 150}Sm(t,{sup 3}He) reaction at 115 MeV/u were measured, populating excited states in {sup 150}Pm. The transitions studied populate intermediate states of importance for the (neutrinoless) {beta}{beta} decay of {sup 150}Nd to {sup 150}Sm. Monopole and dipole contributions to the measured excitation-energy spectra were extracted by using multipole decomposition analyses. The experimental results were compared with theoretical calculations obtained within the framework of the quasiparticle random-phase approximation, which is one of the main methods employed for estimating the half-life of the neutrinoless {beta}{beta} decay (0{nu}{b…

Proton Inelastic Scattering onNi56in Inverse Kinematics

1994

Inelastic proton scattering to the first excited ${2}^{+}$ state of the doubly magic $^{56}\mathrm{Ni}$ nucleus was investigated in inverse kinematics, using a secondary beam of radioactive $^{56}\mathrm{Ni}$ nuclei. At an incident energy of 101 MeV/nucleon, a value $B(E2,{0}^{+}\ensuremath{\rightarrow}{2}^{+})=600\ifmmode\pm\else\textpm\fi{}120 {e}^{2}$ ${\mathrm{fm}}^{4}$ was measured. This result completes the set of experimental data for the first excited ${2}^{+}$ states in the $1f\ensuremath{-}2p$ shell with a closed shell of neutrons or protons. These data are compared to recent shell-model calculations.

Parity nonconservation in the γ decay of polarized17/2−isomers inTc93

2005

The determination of the ${0}^{\ifmmode^\circ\else\textdegree\fi{}}\ensuremath{-}{180}^{\ifmmode^\circ\else\textdegree\fi{}}$ asymmetry (${A}_{\ensuremath{\gamma}}$), which arises because of the parity nonconserving matrix element, in the 751-keV \ensuremath{\gamma} decay of polarized $17/{2}^{\ensuremath{-}}$ isomers in $^{93}\mathrm{Tc}$ with respect to the direction of polarization is reported. A combined analysis of the present results together with those from our earlier work yields an effect of two standard deviations.

Experimental determination of anIπ=2−ground state inCu72,74

2010

This article reports on the ground-state spin and moments measured in $^{72,74}\mathrm{Cu}$ using collinear laser spectroscopy at the CERN On-Line Isotope Mass Separator (ISOLDE) facility. From the measured hyperfine coefficients, the nuclear observables $\ensuremath{\mu}$(${}^{72}\mathrm{Cu})=\ensuremath{-}1.3472(10){\ensuremath{\mu}}_{N}$, $\ensuremath{\mu}({}^{74}\mathrm{Cu})=\ensuremath{-}1.068(3){\ensuremath{\mu}}_{N}$, $Q({}^{72}\mathrm{Cu})=+8(2) {\mathrm{efm}}^{2}$, $Q({}^{74}\mathrm{Cu})=+26(3) {\mathrm{efm}}^{2}$, $I({}^{72}\mathrm{Cu})=2$, and $I({}^{74}\mathrm{Cu})=2$ have been determined. Through a comparison of the measured magnetic moments with different models, the negative …

Gamow-Teller transitions in theA=40isoquintet of relevance for neutrino captures inAr40

2014

Background: The Gamow-Teller response of $^{40}\mathrm{Ar}$ is important for the use of liquid argon as a medium for neutrino detection. An ambiguity about the Gamow-Teller strength for the excitation of ${1}^{+}$ states at 2290 and 2730 keV in $^{40}\mathrm{K}$ results in a significant uncertainty for neutrino capture rates. This ambiguity is caused by the large discrepancy observed between strengths extracted from $^{40}\mathrm{Ar}$($p$, $n$) charge-exchange data and the transition strengths for the analog transitions studied in the $\ensuremath{\beta}$ decay of $^{40}\mathrm{Ti}$.Purpose: This study was aimed at resolving the ambiguity between the results from the $^{40}\mathrm{Ar}$($p$,…

Improving the nuclear physics input along the rp-process path

2007

The level structure of 30 S was studied at the NSCL by using neutron removal reactions with a radioactive 31 S beam. The γ -decay from excited states in 30 S was measured in a Ge-detector array. The results discussed for this work will reduce the uncertainties in the determination of the astrophysical 29 P(p, γ ) 30 S reaction rate under rp -process conditions.

Nucleosynthesis of proton-rich nuclei. Experimental results on the rp-process

2009

Experience NSCL; International audience; We report in this study the nuclear properties of proton-rich isotopes located along the rp-process path. The experiments have recently been performed at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. The level properties above the proton separation energy of the nuclei 30S, 36K and 37Ca were measured with precision of < 10 keV. This will allow a reduction in the determination of the astrophysical (p, ) reaction rate under rp-process conditions.

Spectroscopy ofB13via theC13(t,He3)reaction at115AMeV

2009

Gamow-Teller and dipole transitions to final states in $^{13}\mathrm{B}$ were studied via the $^{13}\mathrm{C}(t,^{3}\mathrm{He})$ reaction at ${E}_{t}=115A$ MeV. In addition to the strong Gamow-Teller transition to the $^{13}\mathrm{B}$ ground state, a weaker Gamow-Teller transition to a state at 3.6 MeV was found. This state was assigned a spin-parity of $3/{2}^{\ensuremath{-}}$ by comparison with shell-model calculations using the WBP and WBT interactions which were modified to allow for mixing between $n\ensuremath{\hbar}\ensuremath{\omega}$ and $(n+2)\ensuremath{\hbar}\ensuremath{\omega}$ configurations. This assignment agrees with a recent result from a lifetime measurement of excited…

Beta-decay of 56Cu

1998

Beta-decay studies of proton-rich isotopes near the doubly closed-shell nucleus 56Ni are of interest as (i) nuclei with a few nucleons outside a doubly-magic core are expected to represent comparatively simple configurations and thus be useful for testing nuclear shell-model predictions, and (ii) the large decay-energy window guarantees that a sizeable fraction of the strength of the allowed β-decay can be reached by the experiment. Moreover, nuclear structure properties of proton-rich N ~ Z isotopes are of astrophysical interest, e.g., concerning the EC cooling of supernovae and the astrophysical rp-process.

Nuclear spins, magnetic moments, and quadrupole moments of Cu isotopes fromN=28toN=46: Probes for core polarization effects

2010

Measurements of the ground-state nuclear spins and magnetic and quadrupole moments of the copper isotopes from $^{61}\mathrm{Cu}$ up to $^{75}\mathrm{Cu}$ are reported. The experiments were performed at the CERN online isotope mass separator (ISOLDE) facility, using the technique of collinear laser spectroscopy. The trend in the magnetic moments between the $N=28$ and $N=50$ shell closures is reasonably reproduced by large-scale shell-model calculations starting from a $^{56}\mathrm{Ni}$ core. The quadrupole moments reveal a strong polarization of the underlying Ni core when the neutron shell is opened, which is, however, strongly reduced at $N=40$ due to the parity change between the $\mat…

The endpoint of the rp-process

1997

Abstract The endpoint of rp-process nucleosynthesis in X-ray bursts determines the fuel consumption, the energy generation, and the abundance pattern of the produced nuclei. To investigate the time structure of rp-process nucleosynthesis, we used a nuclear reaction network including nuclei from H to Sn. We found that if 2p-capture reactions are included, the synthesis of nuclei heavier than Kr proceeds faster than previously thought. Therefore, in most X-ray bursts large amounts of nuclei in the A=80–100 region are expected to be produced. With an escape factor of about 1%, X-ray bursts could account for the large observed solar system abundances of the light p-nuclei like 92 Mo and 96 Ru t…

Single-particle shell strengths near the doubly magic nucleus 56Ni and the 56Ni(p,γ)57Cu reaction rate in explosive astrophysical burning

2019

Angle-integrated cross-section measurements of the $^{56}$Ni(d,n) and (d,p) stripping reactions have been performed to determine the single-particle strengths of low-lying excited states in the mirror nuclei pair $^{57}$Cu-$^{57}$Ni situated adjacent to the doubly magic nucleus $^{56}$Ni. The reactions were studied in inverse kinematics utilizing a beam of radioactive $^{56}$Ni ions in conjunction with the GRETINA $\gamma$-array. Spectroscopic factors are compared with new shell-model calculations using a full $pf$ model space with the GPFX1A Hamiltonian for the isospin-conserving strong interaction plus Coulomb and charge-dependent Hamiltonians. These results were used to set new constrain…

Measurement of the 2+→0+ ground-state transition in the β decay of F20

2019

We report the first detection of the second-forbidden, nonunique, 2+→0+, ground-state transition in the β decay of F20. A low-energy, mass-separated F+20 beam produced at the IGISOL facility in Jyvaskyla, Finland, was implanted in a thin carbon foil and the β spectrum measured using a magnetic transporter and a plastic-scintillator detector. The β-decay branching ratio inferred from the measurement is bβ=[0.41±0.08(stat)±0.07(sys)]×10-5 corresponding to logft=10.89(11), making this one of the strongest second-forbidden, nonunique β transitions ever measured. The experimental result is supported by shell-model calculations and has significant implications for the final evolution of stars tha…

β-decay studies of135–137Snusing selective resonance laser ionization techniques

2002

The decays of the very neutron rich Sn isotopes Sn135-137 were studied at CERN/ISOLDE using isotopic and isobaric selectivity achieved by the use of a resonance ionization laser ion source and mass spectroscopy, respectively. Neutron decay rates, gamma-ray singles, and gamma-gamma coincidence data were collected as a function of time. The half-life (T-1/2) and delayed neutron emission probability (P-n) values of 135 Sn were measured to be 530(20) ms and 21(3)%, respectively. For Sn-136, a T-1/2 of 250(30) ms was determined along with a P-n value of 30(5)%. For Sn-137, a T-1/2 of 190(60) ms and a P-n value of 58(15)% were deduced. Identification of low-energy transitions in Sb-135 was made p…

Determining therp-Process Flow throughNi56: Resonances inCu57(p,γ)Zn58Identified with GRETINA

2014

An approach is presented to experimentally constrain previously unreachable (p, γ) reaction rates on nuclei far from stability in the astrophysical rp process. Energies of all critical resonances in the (57)Cu(p,γ)(58)Zn reaction are deduced by populating states in (58)Zn with a (d, n) reaction in inverse kinematics at 75 MeV/u, and detecting γ-ray-recoil coincidences with the state-of-the-art γ-ray tracking array GRETINA and the S800 spectrograph at the National Superconducting Cyclotron Laboratory. The results reduce the uncertainty in the (57)Cu(p,γ) reaction rate by several orders of magnitude. The effective lifetime of (56)Ni, an important waiting point in the rp process in x-ray burst…

Measurement of the 2+→0+ ground-state transition in the β decay of 20F

2019

We report the first detection of the second-forbidden, nonunique, 2+→0+, ground-state transition in the β decay of 20F. A low-energy, mass-separated 20F+ beam produced at the IGISOL facility in Jyväskylä, Finland, was implanted in a thin carbon foil and the β spectrum measured using a magnetic transporter and a plastic-scintillator detector. The β-decay branching ratio inferred from the measurement is bβ=[0.41±0.08(stat)±0.07(sys)]×10−5 corresponding to logft=10.89(11), making this one of the strongest second-forbidden, nonunique β transitions ever measured. The experimental result is supported by shell-model calculations and has significant implications for the final evolution of stars tha…

Competition between allowed and first-forbidden β decays of At208 and expansion of the Po208 level scheme

2021

The structure of 208Po populated through the EC/β+ decay of 208At is investigated using γ-ray spectroscopy at the ISOLDE Decay Station. The presented level scheme contains 27 new excited states and 43 new transitions, as well as a further 50 previously observed γ rays which have been (re)assigned a position. The level scheme is compared to shell model calculations. Through this analysis approximately half of the β-decay strength of 208At is found to proceed via allowed decay and half via first-forbidden decay. The first-forbidden transitions predominantly populate core excited states at high excitation energies, which is qualitatively understood using shell model considerations. This mass r…

Spins and Magnetic Moments of $^{49}$K and $^{51}$K: establishing the 1/2$^+$ and 3/2$^+$ level ordering beyond $N$ = 28

2013

The ground-state spins and magnetic moments of $^{49,51}$K have been measured using bunched-beam high-resolution collinear laser spectroscopy at ISOLDE-CERN. For $^{49}$K a ground-state spin $I = 1/2$ was firmly established. The observed hyperfine structure of $^{51}$K requires a spin $I > 1/2$ and from its magnetic moment $\mu(^{51}\text{K})= +0.5129(22)\, \mu_N$ a spin/parity $I^\pi=3/2^+$ with a dominant $\pi 1d_{3/2}^{-1}$ hole configuration was deduced. This establishes for the first time the re-inversion of the single-particle levels and illustrates the prominent role of the residual monopole interaction for single-particle levels and shell evolution.

Discovery of an Exceptionally Strong β -Decay Transition of $^{20}$F and Implications for the Fate of Intermediate-Mass Stars

2019

A significant fraction of stars between 7-11 solar masses are thought to become supernovae, but the explosion mechanism is unclear. The answer depends critically on the rate of electron capture on $^{20}$Ne in the degenerate oxygen-neon stellar core. However, due to the unknown strength of the transition between the ground states of $^{20}$Ne and $^{20}$F, it has not previously been possible to fully constrain the rate. By measuring the transition, we have established that its strength is exceptionally large and enhances the capture rate by several orders of magnitude. This has a decisive impact on the evolution of the core, increasing the likelihood that the star is (partially) disrupted b…

Measurement of the 2+→0+ ground-state transition in the β decay of F 20

2019

| openaire: EC/H2020/654002/EU//ENSAR2 We report the first detection of the second-forbidden, nonunique, 2(+) -> 0(+), ground-state transition in the beta decay of F-20. A low-energy, mass-separated F-20(+) beam produced at the IGISOL facility in Jyvaskyla, Finland, was implanted in a thin carbon foil and the beta spectrum measured using a magnetic transporter and a plastic-scintillator detector. The beta-decay branching ratio inferred from the measurement is b(beta) = [0.41 +/- 0.08(stat) +/- 0.07(sys)] x 10(-5) corresponding to log ft = 10.89(11), making this one of the strongest second-forbidden, nonunique beta transitions ever measured. The experimental result is supported by shell-mode…

Measurement of the 2 + → 0 + ground-state transition in the β decay of F 20

Spectroscopy of 13B via the 13C(t,3He) reaction at 115 AMeV

2009

Gamow-Teller and dipole transitions to final states in 13B were studied via the 13C(t,3He) reaction at Et = 115 AMeV. Besides the strong Gamow-Teller transition to the 13B ground state, a weaker Gamow-Teller transition to a state at 3.6 MeV was found. This state was assigned a spin-parity of 3/2- by comparison with shell-model calculations using the WBP and WBT interactions which were modified to allow for mixing between nhw and (n+2)hw configurations. This assignment agrees with a recent result from a lifetime measurement of excited states in 13B. The shell-model calculations also explained the relatively large spectroscopic strength measured for a low-lying 1/2+ state at 4.83 MeV in 13B. …

The $^{150}$Nd($^3$He,$t$) and $^{150}$Sm($t$,$^3$He) reactions with applications to $\beta\beta$ decay of $^{150}$Nd

2011

The $^{150}$Nd($^3$He,$t$) reaction at 140 MeV/u and $^{150}$Sm($t$,$^3$He) reaction at 115 MeV/u were measured, populating excited states in $^{150}$Pm. The transitions studied populate intermediate states of importance for the (neutrinoless) $\beta\beta$ decay of $^{150}$Nd to $^{150}$Sm. Monopole and dipole contributions to the measured excitation-energy spectra were extracted by using multipole decomposition analyses. The experimental results were compared with theoretical calculations obtained within the framework of Quasiparticle Random-Phase Approximation (QRPA), which is one of the main methods employed for estimating the half-life of the neutrinoless $\beta\beta$ decay ($0\nu\beta\…