Delayed neutron emission near the shell-closures

The self-consistent Density Functional + Continuum QRPA approach (DF+CQRPA) provides a good description of the recent experimental beta-decay half-lives and delayed neutron emission branchings for the nuclei approaching to (and beyond) the neutron closed shells N = 28, 50, 82. Predictions of beta-decay properties are more reliable than the ones of standard global approaches traditionally used for the r-process modelling. An impact of the quasi-particle phonon coupling on the delayed multi-neutron emission rates P2n, P3n, ... near the closed shells is also discussed.


Introduction
The β-decay half-lives and β-delayed neutron emission probabilities of very neutron-rich nuclei are of great value for nuclear structure theory.Reliable masses and betarates of extremely neutron-rich nuclei are indispensable for the r-process modeling.Accurate beta-decay data on the fission products are important for safety studies of advanced nuclear reactors.
A purely integral quantities, β-decay half-lives and βdelayed neutron emission probabilities, give an insight to isospin response of nuclei far from the stability.A combined analysis of total β-decay half-lives and multineutron emission rates (P xn -values) enables one to reconstruct the beta-strength functions carrying back information on the nuclear density functional at high isospinasymmetry regime.The beta-decay studies show that not only the neutron-halo nuclei near the drip-line reveal the features of weakly bound open quantum systems.Similar effects arise in neutron-rich nuclei because of "shellerosion" induced by proton-neutron tensor interaction.
A substantial amount of more precise β-decay data on the fission and fragmentation products are expected to come from the acting radioactive beam facilities: ISOLDE-CERN, ALTO, RIKEN, TRIUMF, NSCL, as well as constructed FAIR, Spiral-2, HIE-CERN facilities.An importance of this field has been stated in the IAEA Project of creating new data base of the beta-decay and delayed neutron emission rates [1].It will acquire accurate experimental data and theoretical predictions for nuclides beyond the reach of existing or planned facilities.In this respect, the fully microscopic models are of a special importance, as they ensure more reliable extrapolation of nuclear data to extreme N/Z ratios.
Based on the self-consistent description of the ground state properties within the local energy-density functional theory [2], the approach to the large-scale continuum a e-mail: ibor48@mail.ruquasi-particle random phase approximation (QRPA) calculations of the allowed Gamow-Teller (GT) and firstforbidden (FF) beta decays has been developed in [3].In the report, the ground state properties and half-lives as well as delayed neutron-emission rates of (near) spherical nuclides near the neutron closed shells N = 28, 50, 82 is presented and compared with the recent experimental data [4][5][6][7].Comparison with the standard global Finite Range Droplet Model (FRDM) [8] used for the astrophysical modeling is also be done.An emphasis is made on the constraints imposed by the half-lives as well as delayed neutron-emission rates on the beta-decay strength functions.A possible impact of the quasi-particle phonon coupling [9,10] on the delayed multi-neutron emission rates P 2n , P 3n, ... near the closed shells is discussed as well.

Theoretical background
The self-consistent model for the GT and first-forbidden decays has been developed in [3] in which a detailed description and comparison with existing semi-microscopic global models can be found.The DF+CQRPA model treats the ground state and β-decay properties of (quasi)spherical nuclei.
1.The ground state properties are derived self-consistently in the energy-density functional theory.The present framework also exploits a new functional DF3a [11] and has a provision to fix (before variation) the ground state spin-parity of the parent (daughter) isobaric companion [4].
2. For excited states we follow an approximate treatment [3] in which the scalar and spin-isospin components of the DF can be approximately decoupled.This allows for independent nucleon-nucleon (NN) interactions in the scalar and spin-isospin channels.The strength parameters are considered as universal (A-independent) constants.
3. The β-strength functions are calculated within the continuum QRPA of the finite Fermi system theory.For the spin-isospin effective NN-interaction in the p-h channel a finite-range δ + π + ρ form is assumed.The nuclear medium modified one-π and one-ρ exchange terms, are important for the spin-isospin responses [3]. 4. The correlations beyond the QRPA are included by re-scaling the spin-dependent multipole operators by the same energy-independent quenching factor Q 1/2 = (g A /G A ).The one-pion component of the residual interaction is quenched by the same factor Q.
5. For the calculation of the β-decay half-lives we have considered allowed and first-forbidden transitions treated in terms of the reduced multipole operators depending on the space and spin variables [3].
6.The DF+CQRPA framework was successfully used for calculations of the beta-decay half-lives [3] and magnetic moments [12].
7. For a simple estimate of an impact of the np − nh configurations on the β-strength functions, we use a spreading width of excitations Γ↓≈αω as in [13].Also we studied an influence of quasiparticle-phonon coupling included via finite rank separable approximation (FRSA) [9,10].

Results
Below the performance of the DF+CQRPA approach is exemplified.The main features of the β-decay and multineutron emission rates are described for the isotopes near the magic shell-closures in the neutron-rich sector.

48 Ca region
The region of Z∼0, N>28 is of great interest in the view of new experimental possibilities opened at FLNR, JINR for exploiting the Ca beams.In Fig. 1, the integral betadecay characteristics calculated from the DF3+CQRPA beta-strength functions are compared with the recent data for the potassium isotopes.The experimental β-decay half-lives and total P n values are reasonably well described by our model.The experimentally observed drop of the P n value at A = 50 is related to occurrence of the additional first-forbidden transitions mediated by the odd-particles in the course of filling of the N = 32 and 34 subshells.Further steady increase of the P n value is due to the impact of strong GT decays built on the ν1 f 7/2 -π1 f 7/2 transitions.

Nuclei below Z = 28 shell
The Co isotopes with N 50 are a clean case for which the beta-decay characteristics are defined by the Gamow-Teller strength function.In Fig. 2, we show the GT strength distribution for 74 Co (in terms of log(ft) values).Importantly, the resulting T 1/2 and P n values are very sensitive to the GT strength near the one-neutron separation energy.
As seen from Fig. 3, our calculations underestimate the half-lives and overestimate a bit the P n values.In general, a very regular A-dependence of the calculated beta-decay observables agrees with the data and confirms a predominantly Gamow-Teller character of the beta-decay in this isotopic chain.(Color online) Beta-decay half-lives and total P n values for the K (potassium) isotopic chain.The experimental data have been obtained from the NUBASE evaluation [14].The data of Ref. [15] are also shown.

78 Ni region
The most neutron-rich region of the nuclear chart in vicinity of 78 Ni may serve as a laboratory for exploring nuclear structure under extreme conditions.The Z∼28 isotopes beyond N=50 may be affected by weakened neutron binding due to the ground-state spin inversion and possible existence of new neutron shell closures N = 56 − 58.An existence of neutron-halo structures is not excluded revealing unusual features of weakly bound open quantum system.These structural changes naturally impact the betastrength function.
An evidence for new spherical subshell closure at N=58 was experimentally obtained recently for 83−86 Ga isotopes [4,5].This subshell formed of two nearly degenerated orbitals may originate from migration of the neutron 2d 5/2 and 3s 1/2 orbitals causing a reduction of N=50 shell gap and some weakening of the 78 Ni doubly-magic core.Another peculiarity of this region is that the neutron- proton tensor interaction triggers a crossing of the proton 2p 3/2 and 1 f 5/2 levels for nuclei beyond Z = 28.Resulting ground-state spin inversion has well been confirmed in magnetic moment measurements for Cu isotopes (see Ref. [12]).
Studying the β-decay of very neutron-rich nuclides with N > 50 in vicinity of the 78 Ni one meets with another difficulty.After crossing the closed neutron shells the neutrons fill positive parity orbitals while protons occupy negative parity orbitals and the first-forbidden transitions become possible.The share of the FF decays to the total halflife increases substantially for isotopes beyond the N = 50 shell.Clearly, in the scheme including both the GT and FF decays, the total half-lives for Ga chain at N > 50 depend on the ordering of the orbitals in the neutron (N = 58) as well as proton subshells.Additional measurements of the delayed multi-neutron probabilities would have provided us with important information on the beta-strength distributions within the sub-spaces corresponding to emission of n = 0, 1, 2 etc neutrons.
Below we discuss the case of 86 Ga: the most "powerful" 2n emitter in medium heavy nuclei region (P 2n = 20 ± 10% [16]).The "DF3a+CQRPA" model is extended to account for spin inversion in the 78 Ni region.The approach is consistent with the systematics of the spin-parity of the ground states of Cu isotopes (understood as driven by the neutron-proton interaction when filling the ν1g 9/2 shell [17]).The calculation fixing the ground state to π1 f 5/2 single-particle state reproduces both 81−84 Ga and new 86 Ga half-lives.No local adjustment was made.
Description of the P xn values in 86 Ga is more difficult.They are mostly determined by the GT strength function, since the FF strengths (Δ J = 0, 1, 2) being concentrated at the excitation energies below S 1n (transition energies higher the Q βn ).As seen from Fig. 4, our CQRPA calculation shows some deficiency of the GT strength within one-neutron emission sub-space.This is due to neglect of the GT strength fragmentation which may be caused by the effects beyond the simple 1p − 1h description and by deformation (both are not included in the DF3a+CQRPA model).
For the Ga isotope chain we estimated an impact of the np − nh configurations beyond the CQRPA via spreading width Γ↓ ≈ αω, as suggested in Ref. [13].Including the spreading (Figs.5-7) makes the half-lives shorter and closer to the data [5], total P n values increase, and the P xn values are now P 1n = 28% > P 2n = 22% [16].This is a strong argument in favour of accounting for a quasiparticle-phonon coupling.

132 Sn region
In 132 Sn region the GT and FF decays contribute differently to the beta-decay rates for the Z < 50 and Z > 50 isotopes.The intensive GT decays in the Z < 50 nuclei mostly correspond to the (ν1g 7/2 , π1g 9/2 ) configuration.The high-energy GT decays contribute strongly to the total half-lives of the nuclei with Z < 50, N < 82.Then at N > 82, the high-energy first-forbidden decays become opened which are mainly related to the (ν1h 11/2 , π1g 9/2 ) configuration.In Fig. 8, we exemplify the above said for the Pd isotopic chain.Importantly, the 1p − 1h DF3+CQRPA calculations [18] slightly overestimate the recent RIKEN data [7].At the same time our FRSA calculation [10] including both the tensor interaction and 2p−2h configuration effects shows a very good agreement for 128 Pd.      [7,14,19] and halflives based on the FRDM [8], the FRSA model including both the tensor interaction and 2p−2h configuration effects [10], the shell model (SM) [20].
As for nuclei beyond Z = 50 the π1g 9/2 orbital is fully occupied, the GT transitions are related to low-energy (ν1g 7/2 , π1g 7/2 ) configuration.Due to the phase space effect, the higher energy forbidden transitions which are opened at N > 82 and related to the (ν2 f 7/2 , π1g 7/2 ), (ν1 f 7/2 , π2d 5/2 ) configurations dominate the total halflives.Though, for nuclei with Z > 52 and N > 82, π1g 9/2 orbital is partially de-blocked due to pairing correlations, the GT transition strength is quenched by the occupancy factor of the proton level (1 − v 2 π1g 9/2 ).Probably, the most striking example of half-life reduction due to high-energy forbidden transitions is the kink at 134 Sb (N > 82) in Fig. 9. Notice, that our calculated half-lives are in rather good agreement with the data.For instance, in 136 Sb the T 1/2 = 0.80 s which is rather close to the experimental value T 1/2 = 0.923 ± 0.014 s, as the FRDM [8] gives a factor 5 longer half-life.
In Fig. 10, we display the Q β values for the Sb isotopes beta-decay and neutron emission thresholds in the daughter nuclei (S xn ) calculated with the DF3 functional.One may conclude that two-neutron emission for A > 135 and three neutron emission for A > 138 are possible.However, compared to Z < 50 isotopes, the delayed neutron emission branching should be rather suppressed in the Sb isotopes due to the fact that the dominating FF transitions undergo outside the Q βn window.Another complication is that the P xn values are extremely sensitive to the S xn values.Possible redistribution of the beta-strengths near the multi-neutron emission thresholds due to quasiparticlephonon coupling is that counts.As seen from Fig. 11, in the 136 Sb isotope increasing the Q βn value or decreasing the Q β2n thresholds by about 0.5 MeV would have changed the P 1n and P 2n values considerably.One has to mention that the accuracy of our calculation of the Q βxn values is about 0.5-1 MeV.For instance, in 136 Sb our calculations with different versions of the density functional give the total P n value from 5.4% to 15.6% and P 2n value

Summary
With the universal set of the NN-interaction parameters the self-consistent approach to nuclear beta-decay [3] reliably describes both ground-state properties and small amplitude nuclear spin dynamics of (quasi)spherical nuclei in a wide region of the nuclear chart.Microscopic treatment of the GT and FF beta-decays within the single DF+CQRPA framework makes it possible to explain a number of peculiar effects observed in the beta-decay and delayed neutron emission of very neutron-rich nuclei beyond the closed neutron shells.Further extension of the self-consistent DF+CQRPA is related to description of deformed nuclei [22].Due to a deficiency in extrapolating the model parameters (defined near the line of stability), predictions of semi-empirical and schematic global models should always be taken with some reservation for unstable isotopes near the shell-closures.
The search for delayed multi-neutron emission at JINR, GSI, ALTO, HRIBF and RIKEN would be promising in the region beyond 48 Ca (K, Ca, Sc isotopes with N > 30).It is expected to be more favorable (though more difficult) for Z < 50 (Ag, Cd, In isotopes) than for Z 50 (Sn, Sb, Te isotopes).The dedicated (β − n) and (β − n − γ) experiments at JINR, GSI, HRIBF, ALTO, RIKEN using fragmentation reactions and post-accelerated beams at FRIB facilities are indispensable for testing the nuclear structure models far off stability.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 201

Figure 1 .
Figure1.(Color online) Beta-decay half-lives and total P n values for the K (potassium) isotopic chain.The experimental data have been obtained from the NUBASE evaluation[14].The data of Ref.[15] are also shown.

Figure 2 .
Figure 2. (Color online) The Gamow-Teller strength function for the decay of the 74 Co parent nucleus.

Figure 3 .
Figure 3. (Color online) The β-decay half-lives and total P n values for the Co isotopic chain.Experimental data are taken from Refs.[6, 14].

Figure 4 .
Figure 4. (Color online) The Gamow-Teller β-decay strength functions for 86 Ga plotted as a function of transition energy.

Figure 5 .
Figure 5. (Color online) The β-decay half-lives for the Ga isotopic chain calculated without and with inclusion of the spreading width.The experimental data are taken from Refs.[5,14,16].

Figure 6 .
Figure 6.(Color online) The total delayed neutron emission probabilities for the Ga isotopic chain calculated without (P nMIN ) and with inclusion of the spreading width (P nMAX ).

Figure 7 .
Figure 7. (Color online) The P 1n,2n,3n and total P n emission probabilities for the Ga isotopic chain calculated with inclusion of the spreading width.

Figure 9 .
Figure 9. (Color online) Beta-decay half-lives for the Sb isotopic chain.The experimental half-lives were obtained from the NUBASE evaluation[14].

Figure 10 .
Figure 10.(Color online) Q β values and the emission thresholds in daughter nuclei for the Sb isotopes

Figure 11 .
Figure 11.(Color online) The strength functions for the GT and FF decays in 136 Sb (plotted in terms of the transition energy)