First-principles results for nuclear – decay properties of shell nuclei
We evaluate the allowed - decay properties of nuclei with systematically under the framework of the nuclear shell model with the use of the valence space Hamiltonians derived from modern methods, such as in-medium similarity renormalization group and coupled-cluster theory. For comparison we also show results obtained with the newly developed shell-model interactions from chiral effective field theory and phenomenological USDB interaction. In this work, we have tested predictive power of effective interactions by comparing calculated results with the experimental data for – decay properties of shell nuclei. We have performed calculations for O F, F Ne, Ne Na, Na Mg, Mg Al, Al Si, Si P and P S transitions. Theoretical results of , log values and half-lives, are discussed and compared with the available experimental data. calculations of -decay properties are very limited for the shell nuclei, thus present comprehensive study will add more information to it.
pacs:21.60.Cs - shell model, 23.40.-s --decay
The study of -decay properties of unstable-nuclei have been extensively investigated in the past brownwild (). There are several experimental data available for half-lives, log values, Gamow-Teller () strengths, values and branching fractions nndc (); ame2012 (); ame2016 (). On the other hand with the recent development in approaches it is possible to predict these properties. The -decay half-lives are very important for understanding -process nucleosynthesis. The strength is one of the important tools to study the structure of atomic nuclei. The experimental strengths can be obtained from -decay studies and charge-exchange (CE) reactions. The reliable estimates of strength distributions in neutron-rich nuclei can be of great interest for description of -decay properties.
Study of -decays based on methods is very limited, and used to be available only for few-body systems and several light nuclei. Study of transitions with the inclusion of the effects of three-nucleon forces and two-body currents from chiral effective field theory are reported for N and O in Ref. andres (). The origin of anomalous long life time for the -decay of C to N was investigated using no-core shell model with the Hamiltonian from the chiral effective field theory including three-nucleon force terms moris (). The no-core shell model results for strengths for -shell nuclei with , using two-nucleon () and three-nucleon () interactions derived from chiral effective field theory, were reported in Ref. PRL990425012007 (). The chiral low-energy constants and are constrained by means of accurate calculations of the binding energies of the system and half-life of triton gazit (). The study of the tritium -decay with the chiral effective field theory was reported in Ref. baroni (). The uncertainties in constraining low-energy constants from H decay were reported in Ref. klos ().
In very early years, features of the beta-decay of neutron-rich shell nuclei with five or more excess neutrons were predicted by Wildenthal et al wcb (). The comprehensive study of -decay properties of -shell nuclei for = 17–39 were reported by Brown and Wildenthal in Ref. brownwild (). The Gamow-Teller beta-decay rates for A18 nuclei were reported in Ref. chou (). For the -shell nuclei, the calculated Gamow-Teller matrix elements of 64 decays of nuclei in the mass range A=41–50 were reported in Ref. pinedo (). In these studies, a substantial quenching of the axial-vector coupling in the strength was found in -shell and -shell nuclei about by 20 and also in -shell nuclei by 25-26. Theoretical studies on the quenching of the strengths were carried out for nuclei with closed-core 1 nucleon Arima (); Towner (), and the effects of coupling to 2p-2h configurations and roles of two-body meson-exchange currents were found to be important.
In later years, the shell model calculations for -decay properties of neutron-rich nuclei with were reported by Li and Ren in Ref. li (). The nuclear -decay half-lives for and shell nuclei were reported in Ref. kumar_jpg (). The shell model description of strengths in -shell nuclei were available in Ref. vik_epja (). The systematic shell-model study of -decay properties and strength distributions in neutron-rich nuclei were reported in Ref. otsuka (). Theoretical calculations for half-lives of medium-mass and heavy mass neutron-rich nuclei from QRPA based on the Hartree-Fock Bogoliubov theory or other global models were available in the literature suhonen (); q1 (); q2 (); q3 (); ser ().
More recently, results of the study on and double-beta decays of heavy nuclei within a framework of an effective theory were presented in Ref. perez (). The calculations of strengths in shell nuclei for 13 different nuclear transitions including electron-capture reaction rates for NaNe and MgNa were reported in Ref. archana_gt (). Here, a need for the quenching of the strength was reported also for the shell-model interactions.
In the present work we have reported – decay properties of nuclei using nuclear shell model based on interactions and newly developed shell-model interactions from chiral effective field theory. The purpose of the present work is to study how well the recent and newly developed shell-model interactions based on chiral interactions can describe the -decay properties in -shell, and also to find how much quenching is necessary for these interactions by comparing with many more experimental data than in Ref. archana_gt (). This work will add more information to earlier works wcb (); brownwild (); li (), where shell model results with phenomenological effective interactions were reported.
This paper is organized as follows. In Sec. II, we present details of interactions. The formalism of the calculations for -decay properties are presented in Sec. III. In Sec. IV, we present theoretical results along with the experimental data. Finally, a summary and conclusions are drawn in Sec. V.
Ii Ab initio Hamiltonians
To calculate , log values and half-lives for the shell nuclei, we have performed shell-model calculations using two interactions : IM-SRG stroberg () and CCEI jansen (); jan1 (). Also we have performed calculations with newly developed shell-model interaction derived from the chiral effective field theory Huth (). For comparison, we have also performed calculations with the phenomenological USDB effective interaction usdb () in addition to the above three interactions. For the diagonalization of matrices we used J-scheme shell-model code NuShellXnushellx ().
Stroberg et al. stroberg () derived a mass-dependent valence-space Hamiltonians using the IM-SRG Tsukiyama (); bogner () including the chiral and nucleon interactions (see Refs. hergertpr (); hergertps (); lecture () for more details about IM-SRG). In the IM-SRG, a continuous unitary transformation, parametrized by the flow parameter s, is applied to the initial normal-ordered A-body Hamiltonian:
where the and are the diagonal and off-diagonal part of the Hamiltonian, respectively. The off-diagonal Hamiltonian become zero as
The resulting Hamiltonian is used in the shell-model calculations for shell nuclei. In the present work we use effective interactions with = 24 MeV (see details in Ref. stroberg () for other parameters).
We can expand the Hamiltonian for the suitable model-space using the valence-cluster expansion lisetskiy () given as
Here A is the mass of the nucleus for which we are doing calculations, is the core Hamiltonian, is the valence one-body Hamiltonian, and is the two-body Hamiltonian. The two-body term is derived from Eq. (3) by using the Okubo-Lee-Suzuki (OLS) similarity transformation okubo (); suzuki (). After using this unitary transformation the effective Hamiltonian becomes non-Hermitian. For changing the non-Hermitian to Hermitian effective Hamiltonian the matrix operator  = is used, where S is a matrix that diagonalize the Hamiltonians (see Ref. nbarrett () for further details ). The Hermitian shell-model Hamiltonian is obtained as .
The and parts are taken from a next-to-next-to-next-to leading order (N3LO) chiral nucleon-nucleon interaction, and a next-to-next-to leading order (N2LO) chiral three-body interaction, respectively. For both IM-SRG and CCEI, we use = 500 MeV for chiral N3LO interaction entem (); machleidt (), and = 400 MeV for chiral N2LO interaction navratil (), respectively. The low-energy constants (LECs) of 3NF are given as = 0.098 and navratil ().
In the CCEI to achieve faster model-space convergence, the similarity renormalization group transformation has been used to evolve two-body and three-body forces to the lower momentum scale (see Ref. Jurgenson () for further details). Also, for the coupled-cluster calculations, a Hartree-Fock basis built from thirteen major harmonic-oscillator orbitals with frequency = 20 MeV have been used. Using IM-SRG targeted for a particular nucleus imsrg_ragnar () and CCEI interactions, the shell model results for spectroscopic factors and electromagnetic properties are reported in Refs. pcs_prc (); archana_prc ().
Recently, L. Huth et al. Huth () derived a shell-model interaction from chiral effective field theory. The valence-space Hamiltonian for shell is constructed as a general operators having two low energy constants (LECs) at leading order (LO) and seven new LECs at next-to-leading order (NLO) and fitted the LECs of CEFT operators directly to 441 ground- and excited-state energies. For the chiral EFT interaction they have taken the expansion in terms of power of based on Weinberg power counting wibnernpb (), where Q is a low-momentum scale or pion mass and 500 MeV is the chiral-symmetry-breaking scale.
In the beta decay, the value corresponding to transition from the initial state of the parent nucleus to the final state in the daughter nucleus is expressed as Piechaczek ()
where is the Gamow-Teller transition strength, and is the axial vector phase space factor that contains the lepton kinematics. In this work, we have calculated the phase space factor with parameters given by Wilkinson and Macefield wilkinson () together with the correction factors given in Refs. Sirlin (); wga (). The values are very large, so they are defined in term of “log” values. The log is expressed as log= log.
The total half-life is related to the partial half-life as
where runs over all the possible daughter states that are populated through transitions.
The partial half-life is related to the total half-life of the allowed -decay as
where, is called the branching ratio for the transition with partial half-life .
The Gamow-Teller strength is calculated using the following expression:
where (=-1.260) is the axial-vector coupling constant of the weak interaction, and and are the initial and final state shell-model wave functions, respectively, here the refers to isospin operator for the decay, for the -decay we use the convention = , is the initial-state angular momentum, and is the quenching factor.
which is independent of the direction of the transitions.
values are defined as
where the total strength is defined by
In the -decay the endpoint energy of the electron (in units of MeV) is an essential quantity to calculate the phase space factor . is given by the expression brownwild ()
where the Q is the -decay Q value, and and are excitation energies of the initial and final states. Here, we have taken Q values from the experimental data ame2016 ().
Iv Results and Discussions
In Table 1 we compare calculated and experimental values of the matrix elements . Calculated values of presented here are those with =1. The -decay energies (E (decay)), branching ratios () and log values as well as the values of W are given in Table 1. The quenching factors are obtained by chi-squared fitting of the theoretical values to the corresponding experimental values. The quenching factors as well as the root-mean-square (RMS) deviations for the effective interactions considered here are given in Table 2. The quenching factors are slightly different for different effective interactions. Their values are in the range of =0.62-0.77. The value for USDB, =0.770.02, is consistent with the one, =0.764, in Ref. Richter (). The RMS deviations for the and CEFT interactions are enhanced compared with the USDB by 25-32 and 52, respectively.
In Fig. 2 we show the distribution of calculated log values with the experimental data for some –decays nuclei for which experimental log values are available for excited states also. In case of F, although results of the interactions for excitation energy for the excited and states slightly differ from the experimental data, all the interactions give calculated log values close to the experimental data. For Ne, the calculated log values with the CCEI are better in comparison to other interactions. The calculated value for excitation energy for state is in good agreement for all the interactions for Na. For this nucleus all the four interactions give reasonable results for log values.
In Table 3, we compare the theoretical and the experimental log and -decay half-lives of several selected shell nuclei. Calculated and experimental half-lives are in fairly good agreement in the cases shown in Fig. 2. For F(5/2) Ne, Ne(0) Na and Na(1) Mg transitions, the difference in the half-lives is almost determined by the difference of the B(GT) value of the transition to the state with the largest branching ratio, and also difference in phase space factors. For F(5/2) Ne, the transition to the g.s. (3/2) gives some contributions. For Mg(0) Al transition, the agreement between calculations and experiment is not so good for log and half-lives. This is due to large values of the calculated B(GT) strength compared with the experimental one. For Mg(0) Al() transition, the phase space factor, which is estimated to be roughly proportional to (decay energy), depends very much on the interactions as the value for this transition is as small as 1.832 MeV. The excitation energies for the 1 state of Al obtained for the interactions are smaller than the experimental one, =1.373 MeV, by 0.175, 0.571, 0.251 and 0.018 MeV for USDB, IM-SRG, CCEI and CEFT, respectively, which leads to an enhancement of the phase space factor by nearly 10 times for IM-SRG. Though the difference of the B(GT) values is within a range of a factor of about 3, large difference in the phase space factors leads to larger difference in the half-lives.
Here, we make some general comments on the half-lives. (1) For O and F isotopes, calculated half-lives are in fair agreement with the experimental values within a factor of 2.1-2.2, except for O obtained with IM-SRG. (2) The discrepancy between calculated and experimental half-life becomes large (a) when the discrepancy between the calculated and experimental is large, or (b) when the transition with the dominant branching ratio is different between the calculation and the experiment, or (c) when the value for the transition is small and the difference between the calculated and experimental excitation energies is large enough to lead to a substantial change of the phase space factor for the transition. In case of O with IM-SRG, a large discrepancy comes from combined effects of (a) and (b). Nuclei in the island of inversion such as Mg can not be well described for both the and phenomenological interactions due to the reason (a). Mg discussed above corresponds to the case (c). (3) For isotopes with =10-13 (=14-15), there are one or two cases (or more cases) for each in which the calculated half-lives differ from the experimental ones by a factor more than 3 due to the reasons (a), (b) or (c) in case of IM-SRG and CCEI. More extensive study of the half-lives as well as branching ratios and values will be presented elsewhere.
V Summary and conclusions
In the present work we have performed shell model calculations using approaches along with interaction based on chiral effective field theory and phenomenological USDB interaction, and evaluated , log values and half-lives for the shell nuclei. We also obtained quenching factors corresponding to different interactions for the calculation of the strengths.
All the interactions as well as the interaction based on chiral effective theory considered here need certain quenching of the strengths, as large as by 44-62, as for the phenomenological USDB interaction.
It would be an interesting problem to study how much of them can be explained by the contributions from two-body currents.
The effects of the two-body currents have been studied for transitions in tritium gazit (); baroni () as well as in C and O andres ().
The effects have been studied also for electromagnetic moments and transitions in few-body and light nuclei with 9 Pia (); Pastore ().
The two-body currents have been taken into account in the study of electromagnetic moments and transitions in -, - and -shell nuclei Parzu ()
though they are limited to the induced part.
It is desirable to extend the study of the two-body currents systematically to nuclei with medium and heavy masses.
We would like to thank P. Navrátil, S. R. Stroberg, Jason Holt and G. R. Jansen for useful discussions on effective interactions. We would also like to thank Vikas Kumar and Archana Saxena for useful discussions during this work. AK acknowledges financial support from MHRD for his Ph.D. thesis work. TS would like to thank a support from JSPS KAKENHI under Grant No. 15K05090.
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