The N = 28 neutron magic number is one that originates from the large spin-orbit splitting of the 0f orbital, with the f 7/2 getting separated and forming its own shell above the N = 20 shell gap and the f 5/2 joining the 1p orbitals to form the f p shell. Focusing on the N = 28 nuclei isotones: for 48 Ca the N = 28 gap is large, demonstrated by the near spherical nature of the 48 Ca ground state and its high-lying excited states. However, as protons are removed from the sd shell, the aforementioned shell gap gets diluted. Just two protons fewer, for 46 Ar the energy of the first excited 2 + state falls sharply [1], while another two protons lower, 44 S is a deformed nucleus [2,3] with co-existing spherical and prolate ground shapes [4,5]. Even 43 S with N = 27 shows a signature of the disappearing N = 28 gap with the inversion of the 0f 7/2 and 1p 3/2 orbitals [6]. Below 44 S in the isotonic chain, both 42 Si and 40 Mg are considered deformed with an oblate shape for Si and prolate shape for Mg [7].
Along with the degeneracy of the ν(f p) orbitals with the filling of the 0f 7/2 orbital is the reduction of the spacing of the proton 1s 1/2 and 0d 3/2 orbitals [8]. For 37 P (Z = 15; N = 24), the first excited 3/2 + state is at 865 keV with a ground state of 1/2 + , however as the neutron number increases the 3/2 + falls down to 184 keV for 43 P at N = 28. These two effects together are responsible for the deformation in this region. For the Mg isotopes the deformation trend continues down to 32 Mg (N = 20). Thus, the roughly triangular region formed by 32,40 Mg and 46 Ar displays sudden changes in shape [9]. These structural changes affect the β-decay properties of these nuclei. The Gamow Teller (GT) strength distribution depends on the deformation and affects the half life (T 1/2 ) and β-delayed neutron emission probability (P n ).
The β decay of neutron-rich nuclei approaching the N = 28 magic number and beyond is not very well studied. These isotopes are also important for astrophysical and reaction network calculations and thus to understand the effect of evolving nuclear structure in this region on the β-decay properties is vital. One open question is to better understand the β-decay mechanism with neutrons and protons occupying orbitals in different shells leading to parent and daughter nuclei having opposite parities. The allowed GT transitions can populate high lying states as has been observed in some heavier Zn and Ga isotopes near 78 Ni [10], however in our recent study of the β decay of 38,40 Si [11], clear indication of low-lying core excited states was seen which were well reproduced by large-scale shell-model calculations [12,13]. More information is needed to fully understand this issue. Another question relates to the large feeding to low-lying opposite-parity states found experimentally in many cases, which cannot be attributed to allowed GT transitions. Two possibilities exist, one is that First Forbidden (FF) transitions are feeding these opposite-parity levels, while the other relates to γ decay (mainly E1 transitions) from GT-fed neutron-unbound resonance states to the low-lying states which can go largely undetected [14]. If the FF decays are important then their effect should also be pronounced on the half lives (T 1/2 ) and β-delayed neutron emission probabilities (P n ).
Recent large-scale shell-model calculations by Yoshida et al., [13,15] using the SDPFSDG-MU , have highlighted the significant influence of FF transitions on the half-life values and β-delayed neutron emission probabilities especially for isotopes with N ∼ 28. In FF β decay, the electron-neutrino pair carries one unit of angular momentum and hence the decay involves a parity change with ∆J ≤ 2. As neutrons start filling up the 1p 3/2 orbital, its transformation to a proton in the 0d 3/2 orbital becomes feasible and thus FF transitions can compete with the allowed GT decay. Yoshida has documented the states that would be populated via FF transitions along with their logf t values for isotopes with 13 < Z < 20 and 38 < N < 48 based on shell-model (SM) calculations [15]. Prior to that, Warburton et al.
[16] calculated FF transitions for less exotic isotopes with A ∼ 40, and as an example they predict a 1% branch to the 2 + 1 state at ∼ 3.2 MeV for the β decay of 36 P to 36 S. Another recent theoretical work, Marketin et al., [17] also highlighted that in certain regions of the nuclear chart (one being neutron-rich sd-shell nuclei), first-forbidden transitions constitute a large fraction of the total decay rate and must be taken into account consistently in modern evaluation of half-lives.
In the current work we studied the β -decay of P and S isotopes around and beyond N = 28 namely, 42,43,44 P and 44,46 S, to explore the aspects of β decay relevant for large neutron excess. For all these isotopes β-decay half-lives have been reported earlier [18], however, some with limited precision. Similarly P n values have been reported in an earlier work, both experiments performed at GANIL in its early implementation [19,20]. The current work, to our knowledge, will be the second reported measurement of half-lives and P n values in this very exotic region of the chart of nuclides [21] and is the first to measure β-delayed γ rays to study excited states in the daughter nuclei (both β and βn). Negative-parity intruder states were observed here for the first time in 42 S following β -decay of 42 P.
The observation of γ transitions allowed us, in some cases, to confirm half lives by following the characteristic γ rays in the daughter nucleus. We also used the intensity of γ transitions in the grand-daughter nuclei in the β and βn chains to infer the β-delayed neutron emission probabilities. We confirm previous measurements of a 100% P n for 43 P [20] following the γ transitions in 43 S and 43 Cl. Additionally, high P n values have been derived for the N > 28 44 P and 46 S. The experimental results are interpreted within configuration interaction shell model calculations using the SDPFSDG-MU interaction [12,13].
The experiment to investigate the β -decay of very exotic 42-44 P and 44,46 Si was carried out at the National Superconducting Cyclotron Laboratory (NSCL) [22] at Michigan State University. A 48 Ca primary beam, which is the best candidate to produce these fragments was accelerated to 140 MeV/u and then fragmented on a thick Be target at the target position of the fragment separator, A1900 [23], to produce the nuclei of interest. A wedge-shaped Al degrader was positioned at the intermediate dispersive image of A1900 to increase the energy dispersion for different isotopes and hence provide a cleaner particle identification of the cocktail beam. Two magnetic rigidity settings of the A1900, with 1% and 2% momentum acceptance, were employed, with the larger momentum acceptance essential for the most exotic isotopes 44 P and 46 S. The selected isotopes in the two settings were transported to the Beta Counting System (BCS) [24]. The heart of the BCS is a pixelated (40 strips x 40 strips) Double-Sided Silicon Strip Detector (DSSD). An Al degrader upstream ensured that the implants stopped at the middle of the ≈ 1mm thick DSSD. The DSSD was followed by a Single-Sided Silicon Strip Detector (SSSD) which served as a veto detector. For the particular A1900 settings used for this experiment a large flux of light particles was transmitted which needed to be eliminated to ensure good implant-β correlation. The DSSD recorded the time and position of implants (GeV energy depositions), as well as subsequent decays (keV to MeV energy depositions) by using dual gain pre-amplifiers. The implant rate was kept low, about 150/s on an average, to maximize the efficiency of correlating the implanted ion with the decay products. Two Si PIN detectors placed upstream of the DSSD provided the energy loss and time of flight information (along with the scintillator at the intermediate dispersive image of the A1900) for particle identification of the incoming implants. The DSSD and SSSD stack was surrounded by 16 Clover detectors to record the β-delayed γ rays with an efficiency of about 5% at 1 MeV. Efficiency was measured using a standard mixed gamma-ray source, SRM-4275 from NIST, and 56 Co placed outside of the DSSD and then GEANT4 simulations were performed to correct for the dimensions of the DSSD. The SRM source provides photons from 125 Sb (with 125m Te in equilibrium) and 154,155 Eu with energies ranging from 27 to 1596 keV while 56 Co allowed us to extend the energy range to 3.55 MeV. The data were collected event by event using the NSCL digital data acquisition system [25]. Each channel provided its own time-stamp signal, which allowed coincidences and correlations to be built in the analysis. Along with the isotopes mentioned above, 43,45 S and 45,47 Cl were also transported to the BCS and they will be discussed in an upcoming publication.
As pointed out earlier, the half lives and P n values have been reported before for 42-44 P [18][19][20] but no delayed γ emission was measured as the experimental setup did not include γ detection. For P with Z = 15, in the simple shell model ordering of levels the odd proton would occupy the 1s 1/2 orbital, however with neutron number inching toward N = 28 the π1s 1/2 and π0d 3/2 orbitals are almost degenerate [8], leading to increased occupancy of the π0d 3/2 orbital. For the odd-odd 42 P, theoretical predictions favor a 0 -ground state [26] with a proposed configuration of π0d 1 3/2 ν1p 1 3/2 and the same could be expected for 44 P though no experimental confirmation exists. Other spin states from the same configuration also lie close in energy in the calculations. In the nuclear physics picture where the decays of the nucleus are caused by transformation of the nucleon constituents, for the present case of β decay of P isotopes, one of the neutrons from either the f p or sd shells has to transform to a proton. As the 1s 1/2 and 0d 3/2 proton orbitals are not fully occupied, transformation of a 1s 1/2 neutron to 1s 1/2 proton or 0d 5/2,3/2 neutron to 0d 3/2 proton is possible (GT transitions), creating core neutron excited 1p1h states in the daughter. The change of 0f 7/2 neutron is less likely as it will populate a state at higher excitation energy. Moreover, the 1p 3/2 neutron can transform to a proton in 1p orbitals (GT decay) but can also switch to a 0d 3/2 proton and that will create a 0p0h state of opposite parity in the daughter populated by a FF transition.
To extract the half-life of 42 P a decay curve was generated by histograming the time difference between the position-correlated implanted 42 P ions and its decay products within a 9-pixel grid. The time correlation window was kept at 500 ms for this case. The fit to the decay curve as shown in Fig. 1 (a) includes the contributions from the decay of daughter, 42 S, and β-delayed neutron daughter, 41 S, as well a background component to account for the long-lived activity not accounted for explicitly. Previously known half-lives of 41,42 S [27,28] and β-delayed neutron emission probability for 42 P (50(20)% [29]), were used for the fit shown. Changing the mean value of P n by 20% did not affect the extracted half-life. As the statistics allowed, the obtained half-life of 47.1(20) ms was confirmed by fitting a decay curve which was gated on the β-delayed 903 keV transition from 42 S as shown in Fig. 1
This fit only included one exponential for the parent decay and a background due to the selectivity.
The β-delayed γ spectrum as recorded by the Clover array (addback between the four crystals was employed) is shown in Fig. 2 and the transitions associated with both 42 S and 41 S are marked. All the transitions except for the 3003(1)-keV are found to be in coincidence with the 903(1)-keV line. The 3003-keV transition proceeds to the ground state directly and shows clear coincidence with the 2010(1)-keV γ ray. Some of the coincidence spectra are shown in Fig 3 (a-c). The 903(1)-, 1820(1)-, 2100(1)-, 2679(2)-and 3003(1)-keV transitions were known before from in-beam spectroscopy of 42 S [30] and the states generated are candidates for positive parity states. The placement of the 2805-keV transition is tentative and differs from Ref. [30]. The 2010-keV and the high energy γ transitions above 4 MeV [4110(2)-, 4243(2)-, 4591(1)-and 4936(2)-keV] are associated with the β -decay of 42 P and observed for the first time. The observed γ-ray are listed in Table I along with the absolute intensities per 100 decays.
The partial level scheme of 42 S based on the coincidences and energy and intensity balance is shown in Fig. 4. The absolute β branches were extracted using the measured efficiency of the Clover array and the total implanted 42 P nuclei (from the fit to the decay curve).
Further, the logf t values were extracted using the measured half life (T 1/2 ) and the known [31]. The levels above 5 MeV have logf t values in the range for allowed GT transitions while the lower energy states could be populated via FF transitions as will be discussed later. The level schemes for the β1n daughter 41 S and the grand-daughter 41 Cl were also confirmed and improved from literature information [30,32] and are shown in Fig. 5.
The β-delayed one neutron emission probability was calculated by the counting of γ-rays emitted by descendants in both the β0n-decay and the β1n-decay chains [33]. The absolute intensity of the 118-keV transition in 42 Cl (0.87% branching [34]) and the 130-keV transition in 41 Cl were used to evaluate a P 1n value of 39(6)%. The γ counts were corrected for the growth and decay of activities and the error quoted includes the uncertainty in the absolute branching of the 130-keV transition. This represents the lower limit of the P n value and is consistent with the value of 50(20)% obtained in Ref. [29].
43 P is proposed to have a 1 2 + ground state [35] because of the unpaired proton in the sd shell, and its β -decay should feed 1 2 + and 3 2 + states in the daughter nucleus via GT transitions. 43 S has recently been studied several times [6,30,36,37] and confirmed to have a 320-keV 7 2 -isomeric excited state with 3 2 -as the ground state. This inversion is suggestive of the melting of the N = 28 shell gap as mentioned before. More recently, in a one-neutron knockout reaction from 44 S [37], several states with proposed negative parity have been identified along with a 2.6-MeV state with a likely J π = 3 2 + assignment decaying directly to the ground state. It is purported to be the neutron knockout from the 0d 3/2 orbital in 44 S. In β decay, this state can be formed by the transformation of the 0d 3/2 neutron to 0d 3/2 proton creating a core excited 1p1h state in an allowed GT transition. The low-lying 1 no 43 S was formed in its β decay. In the only earlier work to measure P n value for 43 P [19], a value of 100% was measured by observing neutrons, and here we confirm it by a completely different approach. One of the reasons for the 100% βn decay is the low S n value of 2629 (9) keV in 43 S [31]. Still it is surprising as a 2600 keV 3/2 + was observed in 43 S [37] and there are several negative parity states below S n which could have been populated in FF transitions.
Figure 7 shows the decay curve for 43 P for a correlation time window of 1000 ms. An excellent fit to the decay curve was obtained by excluding the decay of the β0n daughter 43 S consistent with our prior discussion on P n being 100%. The T 1/2 thus obtained is 37.5 (22) ms consistent with the evaluated value of 36.5(15) ms [18,31].
We cross the N = 28 shell gap with 44 P and as already discussed for the odd-odd N = 27 isotope 42 P, the low-lying states are built on the πd 3/2 -νp 3/2 configuration. Shell model calculations [13,26] predict 0 -as the ground state from that configuration with the 1 -and 2 -lying close by. A half-life of 18.5(25) ms is known from the prior GANIL work [18],
showing a drop of a factor of 2 from 43 P. Beyond that no other experimental information exists for 44 P, though an extrapolation from 43 P would imply a P n close to 100%.
The decay curve for 44 P is shown in Fig. 8 where a fit assuming 100% P n gives an excellent reproduction of the data similar to 43 P with a half-life of the parent of 18(1) ms agreeing very well with the previous measurement as well as a SM calculation by Yoshida et al. [13,15]. Inclusion of a small β0n branch did not change the half-life beyond the error quoted. As was seen for 43 P, the T 1/2 extracted from the decay curve is not very sensitive to P n value, likely because the half-lives of both β0n and β1n daughters are long compared to their parent.
Going beyond the decay curve, we look at the β-delayed γ spectrum for 44 P in Fig. 9.
In the short correlation window, a weak 1330-keV transition is seen which represents the 2 + → 0 + of the β0n daughter, 44 S, and perhaps the 955-keV transition. The 955-keV transition from the proposed 2 + state at 2283 keV as seen in the inverse kinematics proton scattering [35]. No other transition is observed feeding these 2 + states. There are no candidates for any strong decay from the negative-parity states at higher energies fed by allowed GT decay as were observed in 42 S.
On the other hand, the prominent 185-and 1155-keV transitions are from 43 S, the β1n daughter. The 1155-keV depopulates a state at the same energy with a likely J π of 3/2 - as was observed in a neutron-knockout reaction [37] and in the same study the 185-keV transition was proposed to decay from the 1162-keV state to the 977-keV second 7/2 -state which decays to the ground state. The 977-keV state was verified in a recent Coulombexcitation experiment [38]. As can be seen from Fig. 9 (top panel) we have very tentative indications of 983 keV transition instead of 977 keV if at all. However the crucial point is that the 185-keV transition is stronger in contradiction to the neutron knockout experiment. In an earlier measurement to measure lifetimes [6], the order of the 977-and 185-keV transitions was proposed to be reversed; 977 keV depopulating the 1162-keV (5/2 -) state to the 185-keV state with J π = 1/2 -and the 185-keV transition depopulating that. The current β-decay data seems to corroborate this ordering which is also consistent with the SM predictions of a low-lying 1 2 -state. 44 P is proposed to have a 0 -ground state, feeding a 1 -neutron resonance which can emit an ℓ = 0 neutron populating 1 2 -and 3 2 -states. If the J π of the 1162-keV state is indeed 5 2 -as proposed then it is less likely to be populated in the β1n decay if 44 P has a 0 -ground state but possible if it has higher spin for its ground state. The γ spectrum when followed for 1000 ms after the initial implantation shows the transitions from β0n and β1n granddaughters, namely 2789 keV in 44 Cl (see next section) and 328 and 880 keV in 43 Cl. The lowest, 118-keV transition from 42 Cl is also observed with a rather small intensity implying that β2n is much weaker than β1n. From the yields of the γ rays in the granddaughter nuclei, the P n value is estimated to be 75(18)%.
The β -decay of an even-even nucleus with a 0 + ground state is special in that it is characterized by strong low-energy GT transitions to the odd-odd daughter nucleus. By that it is implied that the B(GT ) to a discrete state below roughly 5 MeV is > 0.1 or a logf t value < 4.78 as discussed in Ref. [13]. As the allowed GT decay from the 0 + parent will feed only 1 + states, there exists a 1 + state in the odd-odd daughter with a strong β transition strength. The neutron-proton pairing force in the J π = 1 + channel is responsible for lowering the energy of this 1 + state. Yoshida et al. [13] demonstrate in their calculations that switching off the proton-neutron 1 + pairing matrix greatly reduces the low energy GT strength.
In 40 S and 42 S, β -decay strong population of the 1 + state around 2 MeV in the daughters (2306 keV in 40 Cl and 2123 keV in 42 Cl) is seen. It would be interesting to see how the lowenergy GT strength evolves as a we move to more exotic neutron-rich nuclei and in turn give an idea of the proton-neutron paring strength when the proton and neutron occupy different shells. The β decay of 44 S and 46 S with neutron numbers 28 and 30, respectively, studied here for the first time with delayed γ rays, allowed us to explore this effect.
The delayed γ spectrum for the 44 S implants, 100 ms after the implantation, is shown in Fig. 10. The strongest transition observed is at 2789(1) keV with a half-life consistent with 44 S decay as will be discussed next. The single escape peak of this transition was also seen and is indicated in the spectrum. The 2789-keV transition is proposed to depopulate a level at the same energy as no coincidences are observed and based on prior discussion would correspond to the 1 + state in 44 Cl populated strongly by allowed GT transitions. This follows the trend of the lighter isotopes of a strong low-lying 1 + state in the daughter nucleus, though the energy is higher than in 42, 40 Cl. Also seen in the spectrum are weak transitions at 891 and 1000 keV which have previously been tentatively assigned to 44 Cl [39]. Here they would be populated through FF decay as the 1 + intruder 1p1h states are predicted around 3 MeV. These states at 891 keV and 1000 keV are good candidates for the calculated 0 - 1 (844 keV), 1 - 2 (1047 keV) and 0 - 2 (1378 keV) or 2 - 2 (971 keV) states which would decay directly to the 2 -ground state of 44 Cl and their intensity (1-2%) is consistent with the logf t values calculated for the FF decays [15]. The other lower lying states seen in [39] are not observed here. Also seen are the lowest transitions in the granddaughter 44 Ar (1158 keV and 853 keV). Out of scale in the figure is the 328-keV transition from 43 Cl, the β1n daughter. The P n value is estimated to be 40(5)%, based on the γ intensity observed in the descendants and is consistent with intensity found for the bound states in 44 Cl.
The decay curves for 44 S implants and with the additional gate of the 2789 keV γ transition are shown in Fig. 11. The extracted half-lives are 125.5(25) ms and 119(6) ms respectively. The two previous measurements were 100(1) ms [18] and 123 (10) ms [19] and our measurement supports the longer half-life. The SM calculation by Yoshida et al. [13] gives 121.3 ms as the half life in excellent agreement with the measurement.
46 S with N = 30 is the most exotic isotope produced and studied in this experiment.
Grevy et al. [18] earlier measured its half life to be 50( 8) ms with no further information.
The first and only study of the excited levels of the β0n daughter 46 Cl was via fragmentation of 48 K where two low-lying states at 118 keV and 151 keV were clearly identified [39] though no spin/parity assignment could be made.
The β-delayed γ spectrum for 46 S is shown in Fig. 12. The strongest and clearest transitions marked by blue squares belong to 45 Cl which is the β1n daughter as illustrated by the cartoon in the figure. The spectrum is shown for a correlation window of 250 ms and in that time one can also see the transitions from the β1n granddaughter, 45 Ar, namely 542 keV and 1158 keV. Unlike the decay of 44 S, no strong transition is observed which can be associated with the decay of the 1 + state likely to be populated in the GT decay of 0 + parent. This illustrates that for N = 30, the states populated by GT transitions have moved above the 3.5-MeV neutron separation energy (S n ) limit and lie amongst the neutron unbound states leading to strong delayed neutron emission. A weak transition at 1554 keV corresponding to the 2 + → 0 + decay in 46 Ar, the β0n granddaughter is observed. Though the β decay of 46 Cl is not reported, if we assume that the 1554-keV transition is indeed from this path, then there must be a bound state populated in 46 Cl, albeit weakly, which cannot be isolated in the current data. It is thus clear that the GT strength has moved above the neutron threshold and 46 S, though an even-even isotope, does not display the strong low-energy GT strength as was expected in Ref. [13].
The decay curve for 46 S, shown in Fig. 13, was used to extract a half-life of 41(4) ms.
Though smaller than the literature value of 50(8) ms [18], it is quite consistent with the shell-model prediction discussed earlier. The contribution of the β0n and β1n daughter were allowed to vary for this fit and the best reproduction of the decay curve is consistent with a large P n value. A β2n contribution was not included as no γ transitions from the β2n daughter or granddaughter were seen. The P n value is estimated to be 80(16)% for this case.
As we know, β decay is a well established probe of nuclear structure; with the halflives providing the first insight. The N = 28 isotones below 48 Ca are characterized by large deformation and shape co-existence, in spite of the shell closure which should be reflected in the measured half-lives. The shell-model calculations using the SDPFSDG-MU interaction in the large sd + f p + sdg valence space reproduce the properties of these exotic nuclei quite well with the oblate deformation of 42 Si and shape co-existence in 44 S.
In Refs. [13,15], Yoshida et al. carried out calculations of β -decay half-lives and βdelayed neutron emission probabilities using this interaction for neutron-rich nuclei with 13 ≤ Z ≤ 18 and 22 ≤ N ≤ 34. The half-lives were calculated in two conditions, one, when only GT transitions were allowed and then also including the FF transitions (GT+FF) both using experimental Q - β -values. The measured half-lives of P and S isotopes are compared to the calculations in the two scenarios in Fig. 14 and Table II. The numbers within the red box are those from the current work, while the rest are the evaluated values from NNDC [31]. The half-lives of the odd N nuclei, 43,45 S, were not discussed here, but will be part of a forthcoming publication. As can be seen, there is not much difference between the GT only and GT+FF predictions both for P and S isotopes and the measured and calculated numbers agree pretty well except for the most exotic 44 P. For both calculations, experimental Q(β -) values were used from the AME2016 database when measured; for the most exotic 44 P and 46 S the numbers are from systematic trends. Not using the experimental Q(β -) for the calculation systematically underestimates the half lives by about 66% on average. For the N = 27 nucleus 42 P, using different ground-state spins (0 -to 4 -) gives divergent answers with the favored 0 - possibility giving the lowest value of ∼37 ms which is not consistent with the measurement.
The decrease is primarily due to the small logf t value of the ground-state to ground state FF decay. In Fig. 14 we show the value corresponding to the 2 -as ground state spin which gives the overall best reproduction of the data. For the N = 28 nucleus 43 P for which we determined a 100% P n value, the measured half-life is longer by about 30% from the calculated value including FF but there is better agreement for 44 P when considering the FF transitions. It is also noticeable that for 42 P (N = 27) and 43 P (N = 28), the measured half-lives are very close and then fall by a factor of 2 for 44 P whereas for the equivalent S isotopes there is a smooth trend. This could signal a somewhat larger N = 28 shell closure for P than S. There is enough evidence already to suggest that the N = 28 gap is completely quenched for the S isotopes.
The measured β-delayed neutron emission probabilities, P n , for the P and S isotopes are also compared to the above-mentioned SM calculations with both GT only and GT+FF estimates in Fig. 15. The measured values track the calculation well especially for the odd Z P isotopes which display a higher P n for the same neutron number. For the N = 28 isotope 43 P, 100% of the β decay proceeds via the delayed neutron path; for 44 S (N = 28) the P n value is of the order of 40% only. This difference is due to the strong low-energy GT transition for the decay of the even-even S isotope. The 100% P n for 43 P is predicted for GT only while it is somewhat less for GT+FF calculations. For 44 P the measured value is explained with the inclusion of FF which is consistent with the observation of the lowlying positive-parity states in 44 S (consistent with the half-life discussion). For the N ≥ 28 nuclei in the shell-model calculations, some low-lying FF matrix elements are enlarged, thus contributing substantially to the total half-lives and reducing the delayed neutron emission. This is mainly attributed to the neutron Fermi surface moving to the 1p orbitals.
The disappearance of the N = 28 shell gap and the appearance of exotic shape coexistence can be correlated with the changed Fermi level for protons and neutrons as a function of deformation. In antisymmetrized molecular dynamics (AMD) calculations by Y.
Suzuki and M. Kimura [40], it is shown that for prolate deformation, as found in exotic S isotopes with N = 26 to 30, the neutron Fermi level is the 1p 3/2 orbital while for proton it is the 1s 1/2 orbital. This will greatly enable the FF β transitions to occur. The half-lives and P n values which are bulk β-decay properties already showcase the influence of FF decay but, additionally, we can specifically look for the states populated in these decays. We review the 42 P decay as was shown in Fig. 4. The detailed level scheme is based on energy and intensity balance and coincidences where possible. The logf t values were deduced using the measured branching, T 1/2 , and the Q β -from NNDC. As for the S isotopes, it is expected the 27 th neutron is in the 1p 3/2 orbital while the last proton is in the 0d 3/2 orbital leading to a favored 0 -ground state in the SM calculations using the SDPFSDG-MU interaction [13,15] and also using the SDPF-U interaction [26]. However other J π combinations from the same configuration (0 -to 3 -) lie very close in energy and as we saw the measured half-life aligns better with higher spin assignment. The states observed in 42 S above 5 MeV have logf t values which fall in the range for allowed GT transitions and hence should have negative parity as the odd-odd parent has a negative parity ground state. All of them decay to the 2 + 1 and 2 + 2 states likely via E1 transitions. The absence of direct decay to the 0 + ground state of 42 S makes it less likely that the populated states are of 1 -spin. Hence the assignment of 0 -to the ground state of 42 P is also unlikely.
In a recent intermediate-energy Coulomb-excitation study [38], a 4150(110)-keV transition was observed in coincidence with the 903-keV γ transition and a 2 + state at 5050 keV was proposed with the same uncertainty. Though close in energy to the 5012-keV state in this work, we think they are different as we also observe the decay to the 2 + 2 state at 3003 keV and more so the large β branching to it makes it a candidate for a negative parity state via GT decay. However because of the large error bars quoted for the 5050-keV level it could correspond to the 5146-keV that we observe with a weak branching and a decay to the 2 + 1 state only. If so, it could be fed directly through FF for a 2 + assignment to that state. The states just above 3 MeV, i .e. 3483, 3582, and 3708 keV could also be candidates for FF decays as they are not likely to be the negative-parity 1p1h states. The 2724-keV level has already been assigned a 4 + while the 3003-keV state has been earlier proposed to be a 2 + state and is consistent with the SM predictions.
Figure 16 compares the experimental logf t values with the SM predictions for the β - decay of 42 P with different ground state possibilities (0 -, 2 -, 3 -). In the calculations the yrast 0 -to 4 -in 42 P are very close and one cannot predict definitely the actual ground state. The predicted decay from the 0 -ground-state gives several 1 -states as excellent candidates for the states observed above 5 MeV with logf t values consistent with the experimental values; however as noted before the decay of the states merely to the 2 + states makes it a less likely possibility. Both the decay from 2 -and 3 -states also gives many 2 -and 3 -candidates in the energy range of 5 to 6 MeV with logf t values consistent with the measurement and which can decay to the lower 2 + states and likely not to the 0 + ground state of 42 S. There are also candidates for the states below 5 MeV via FF decays, especially a 3 -ground state could feed the 4 + state also via a FF decay. The apparent feeding observed to the states around and below 3 MeV of a few percent is consistent with the predicted logf t [15] values.
Thus based on the current work we propose either 2 -or 3 -as the spin and parity of the ground state of 42 P. The population of higher spin states in the β1n channel (Fig. 5) also favors a higher than 0 spin for the 42 P ground state.
Next we look at the systematics of the decay of the even-even S isotopes and the predictions for the SDPFSDG-MU interaction in that regard. Shown in Fig. 17 are the partial level schemes as populated in the GT allowed decay of 40,42,44,46 S. The even-even parent ( 0 + ) nucleus in all cases is expected to strongly populate the 1 + states in odd-odd Cl isotopes that have large B(GT ) values via GT transitions [13]. The spin-isospin component of the N N interaction may favor proton-neutron pairs in the daughter nucleus, created by the action of the GT operator on a BCS-type neutron-neutron pair contained in the ground state of the parent nucleus. The trend of the 1 + states to move to higher energies as the neutron number increases is reproduced by the SM calculation, where for 46 Cl the strong GT branches with small logf t values lie above the neutron separation energy. Couple this to the lowering of the neutron separation energy S n with increasing neutron excess and the P n value makes a rapid change from 44 S to 46 S reaching almost 100%. As was noted earlier, in 44 S decay, we did observe weak population of low lying negative parity states in 44 Cl.
Significant larger contribution of FF will further reduce the P n and it would be interesting to see what will be the case for the next exotic even-even S isotope, 48 S, which is yet to be observed and studied. Also, we should note that the SM predictions for 44 S decay are two states, 1 + 3 and 1 + 4 (i.e. the 3rd and 4th in energy), which have small logf t values. The energies of the two are very close and have nearly equal logf t with either being a good candidate of the experimentally observed state. Examining the occupancies of the two states from the calculations in comparison to those of the ground state of parent 44 S reveals that the 1 + 3 state is formed predominantly by the transformation of the sd neutrons to sd protons, namely ν1s 1/2 to π1s 1/2 and ν0d 3/2 to π0d 3/2 . For the 1 + 3 it is the transformation of the f p neutrons which is the dominant component. It is difficult to determine which of these we are not observing. A second state with a similar logf t decaying to the ground state (like the 2789 keV) should have been measurable. It is likely then that one of them is higher in energy and above S n which would be consistent with a higher P n value observed than what the SM predicts for 44 S (Fig. 15).
β -decay of very exotic P and S isotopes reaching and beyond the N = 28 shell gap were studied at the NSCL using a 40x40 DSSD as the implant and β detector, surrounded by a γ array of 16 Clover detectors in close geometry. The half-lives were extracted from the fitting of decay curves for 42-44 P and 44,46 S and are mostly consistent with the earlier work from GANIL, except for 44 S which at 125.5(25) ms is somewhat longer than the literature value of 100(1) ms and 46 S which is smaller than previously measured. The P n values were also calculated and for 43 P decay we have a case of 100% P n and for both 44 P and 46 S, with N = 29 and 30 respectively, it is reaching 100% too. The impact of FF transitions which can be important for neutron-rich nuclei in this region with neutrons filling the 1p 3/2 orbital was explored by comparing the measured T 1/2 and P n values with SM calculations with and without FF transitions included. The P isotopes with N = 27, 28, 29 displayed a larger contribution from FF transitions in their β decay. For the case of 42 P decay there are clear indicators of states which could be populated in FF decay with logf t values consistent with the predictions of the SM calculations. Both the half-life and P n -value of 44 P also indicate influence of FF transitions. Comparison with SM calculations of states populated in 42 S in the decay of 42 P both via allowed GT and FF decays suggests a J π of 2 -or 3 -for the ground state of 42 P contrary to earlier suggestion of 0 -. Further, for the even-even S isotopes studied, 44,46 S, the predicted strong low-energy GT strength is observed for 44 S with a 1 + state strongly populated at 2789 keV. On the other hand, for 46 S the 1 + states move higher in energy and lie above S n leading to a large P n value. These interesting systematics for P and S isotopes with neutron number close to 28 are crucial for refining the calculations in this exotic region and set the stage for more detailed studies of the even more exotic nuclei to be explored at the Facility for Rare Isotope Beams (FRIB) in the coming months and [1] S. Calinescu, L. Cáceres, S. Grévy, O. Sorlin, Z. Dombrádi, M. Stanoiu, R. Astabatyan, C. Borcea, R. Borcea, M. Bowry, W. Catford, E. Clément, S. Franchoo, R. Garcia, R. Gillibert, I. H. Guerin, I. Kuti, S. Lukyanov, A. Lepailleur, V. Maslov, P. Morfouace, J. Mrazek, F. Negoita, M. Niikura, L. Perrot, Z. Podolyák, C. Petrone, Y. Penionzhkevich, T. Roger, 5/2 -(1/2 -,3/2 -) (1/2 + ) 7/2 -41 Cl 451 1.99 s 42 P J π = (2 -,3 -) 0 (1/2 -,3/2 -) (9/2 -) Q βn = 12.03 (10) MeV T 1/2 = 47.1 (20) ms (3/2 + ) 130 891 1445 1475 (2712) 1614 1551 1308 (5/2 + ) (2850) βn β -1405 1346 761 41 S 584 554 130 (7/2 + ) FIG. 5. Partial level scheme of β-delayed neutron daughter 41 S and granddaughter 41 Cl. The transitions in red have been confirmed in this work though they had been tentatively suggested before [31]. The arrow widths are only approximately proportional to intensity. A. Ratkiewicz, K. P. Rykaczewski, D. W. Stracener, E. H. Wang, M. Woli ńska Cichocka, and E. F. Zganjar, Phys. Rev. Lett. 117, 092502 (2016). L. Olivier, C. Portail, L. Qi, B. Roussire, L. Stan, D. Testov, J. Wilson, and D. Yordanov, Physics Letters B 772, 359 (2017).
[15] S. Yoshida, "Database (beta decay), https://github.com/sotayoshida/betadecaydataprc97.054321,"
[16] E.
Warburton, J. Becker, B. Brown, and D. Millener, Annals of Physics 187, 471 (1988). [17] T. Marketin, L. Huther, and G. Martínez-Pinedo, Phys. Rev. C 93, 025805 (2016). 0 100 200 300 400 500 Time (ms) 1 10 100 1 k Decays / 5ms Exp. 44 P 43 S Background Total fit 44 P T 1/2 = 18(1) ms [24] J. Prisciandaro, A.
Morton, and P. Mantica, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 505, 140 (2003), proceedings of the tenth Symposium on Radiation Measurements and Applications. [25] C. Prokop, S. Liddick, B. Abromeit, A. Chemey, N. Larson, S. Suchyta, and J. Tompkins, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 741, 163 (2014). [26] L. Gaudefroy, Phys. Rev. C 81, 064329 (2010). 0 500 1000 1500 2000 Time (ms) 1 10 100 1 k Decays / ms 44 S 44 Cl 43 Cl Background Exp. Total fit 44 S T 1/2 = 125.5(25) ms (a) 0 200 400 600 800 Time (ms) 10 100 1 k Decays/ 10ms Exp. (2789 keV gate) 44 S Background Total fit (b) T 1/2 = 119(6) ms FIG. 11. (a): Decay curve derived for 44 S from β-correlated implants within a grid of 9 pixels for 2s along with the fit used to extract half-life and the initial activity. The components of the fit are (i) exponential decay of parent, 44 S, (ii) exponential growth and decay of daughter nuclei, 44 Cl (β 0n) and 43 Cl (β 1n), and (iii) exponential background. Known half-lives were used for the daughter nuclides [31]. (b): Decay curve gated by the 2789-keV transition which represents the 1 + → 2 -in 22 24 26 28 30 32 34 Neutron # (N) 0 50 100 P n (%) GT+FF GT only GT +FF GT only P (Exp.) S (Exp.) S (Z=16) P (Z=15) FIG. 15. Comparison of experimental P n values for P and S isotopes from N = 22 to N = 30 with shell model calculations using the SDPFSDG-MU interaction [13]. The numbers extracted in the current analysis are augmented with values from NNDC.
0 2 -0 2 -0 2 -0
) 889 1 + 1293 (1 + ) 1998 (1 + ) 2306 1 + 1 + 1 + (1 + ) 1 + 1 + (1 + ) 5830 (30) 5600 (90) 4390 (110) 3540 (170) 1162 1740 2099 2237 1268 2123 2241 3030 1350 2025 2238 2792 2789 2225 2700 2925 3059 3511 3520 3354 3609 3992 4185 4310 40 S 42 S 44 S 46 S Q β = 4720 (30) Q β = 7190 (60) Q β = 11270 (90) Q β = 14400 (400) 40 Cl 42 Cl 44 Cl 46 Cl
3.7 4.7 5.42 5.93 3.8 5.14 4.33 4.13 4.99 6.22 4.81 4.17 4.78 4.41 5.76 5.59 4.43 4.73 4.47 5.6 5.25 5.27 4.33 4.51 Exp Exp Exp Exp SM SM SM SM
FIG. 17. Beta decay of even-even S isotopes from A = 40 to 46. Data for 40,42 S are from NNDC [31].
Only the 1 + states expected to be populated via allowed GT transitions along with SM prediction are shown. The focus is to illustrate that with increasing neutron number the GT strength imbibed by the 1 + states moves to higher energies eventually residing in the neutron unbound states for 46 S.