Ikeda sum rule,self-consistency and double-beta decay in the renormalized quasiparticle random phase approximation

The effect of the inclusion of ground-state correlations into the QRPA equation of motion for the two-neutrino double-beta (ββ_2ν) decay is carefully analyzed. The resulting model, called renormalized QRPA (RQRPA), does not collapse near the physical value of the nuclear force strength in the particle-particle channel, as happens with the ordinary QRPA. Still, the ββ_2ν transition amplitude is only slightly less sensitive on this parameter in the RQRPA than that in the plain QRPA. It is argued that this fact reveals once more than the characteristic behavior of the ββ_2ν transition amplitude within the QRPA is not an artifact of the model, but a consequence of the partial restoration of the spin-isospin SU(4) symmetry. It is shown that the price paid for bypassing the collapse in the RQRPA is the violation of the Ikeda sum rule.     

The quasiparticle rpa as a model for gamow-teller ß-decay

The validity of the quasiparticle random-phase approximation is studied by comparing it with full shell-model calculations in the sd-shell. Phenomenological interactions relevant for the sd-shell have been applied, and Gamow-Teller ß⁺ strengths are calculated. It is found that in the QRPA, the ß⁺ strength is a factor of two higher than the shell-model result. The shell-model results are less sensitive to the particle-particle interaction strength than the QRPA. With u, v factors from the shell-model wavefunctions, the ß⁺ strength in the QRPA reproduces the shell-model result. This suggests that the QRPA does not contain enough ground-state correlations.     

Renormalized proton- neutron QRPA and double beta decay of 82Se to excited states in 82Kr

We apply the self-consistent renormalized proton-neutron QRPA (RQRPA) method to calculate the two-neutrino double beta (2νββ) decay matrix elements associated with the ground-state and excited-state transitions of the ⁸²Se → ⁸²Kr decay. The RQRPA method is an extension of the pnQRPA method and promotes the Pauli exclusion principle violated by the pnQRPA ground state and yields more stable nuclear matrix elements with increasing strength of the proton-neutron interaction. In the present work the RQRPA wave functions are also used to evaluate 2νββ-decay rates to excited final states. The resulting theoretical half lives are compared with the new stringent experimental limits obtained by using a HPGe detector and external sources of enriched selenium.     

Effects of ground-state correlations on 2νββ decay rates and limitations of the QRPA approach

Half-lives of the 2νββ decay are calculated in the proton-neutron QRPA for⁷⁶Ge,⁸²Se,¹⁰⁰Mo,^128,130Te,¹³⁶Xe and¹⁵⁰Nd. The strength of the particle-particle interaction, which plays a decisive role for a reliable evaluation of the half-lives, is determined from a QRPA calculation of singleβ ⁺ decays. The 2ν decay rates calculated with the interaction strength fitted in this way are strongly suppressed and found to be consistent with the existing experimental data. Effects of the ground-state correlations on the suppression are investigated. On the other hand, the present calculation indicates limitations of the QRPA approach.     

A comparative study of double beta decay by shell model and quasiparticle RPA

Nuclear matrix elements of the two-neutrino and neutrinolessββ decays of⁴⁸Ca(0 _g.s. ⁺ )→⁴⁸Ti(0 _g.s. ⁺ ) are calculated by shell model and QRPA. The two-neutrino matrix elementM _GT ^2v is rather reliably evaluated in the QRPA approach by a careful fit of the particle-particle interaction strength in the 1⁺ channel, which governs the spinisospin ground-state correlations. The shell-model value ofM _GT ^2v depends not only on the 1⁺ interaction but largely on the pairing and quadrupole interactions. Concerning the neutrinoless-mode nuclear matrix elements, the shell model gives generally smaller values than the QRPA. A detailed analysis indicates that the discrepancies originate mainly from the truncation of shell-model configurations (fp-space). The QRPA calculation in a larger model space well takes into account transitions from/to single-particle orbits far from the Fermi surface, and those transitions give rise to sizable contributions because of large momentum transfers due to the exchange of a virtual neutrino.     

On microscopic theory of radiative nuclear reaction characteristics

A survey of some results in the modern microscopic theory of properties of nuclear reactions with gamma rays is given. First of all, we discuss the impact of Phonon Coupling (PC) on the Photon Strength Function (PSF) because it represents the most natural physical source of additional strength found for Sn isotopes in recent experiments that could not be explained within the standard HFB + QRPA approach. The self-consistent version of the Extended Theory of Finite Fermi Systems in the Quasiparticle Time Blocking Approximation is applied. It uses the HFB mean field and includes both the QRPA and PC effects on the basis of the SLy4 Skyrme force. With our microscopic E1 PSFs, the following properties have been calculated for many stable and unstable even–even semi-magic Sn and Ni isotopes as well as for double-magic ¹³²Sn and ²⁰⁸Pb using the reaction codes EMPIRE and TALYS with several Nuclear Level Density (NLD) models: (1) the neutron capture cross sections; (2) the corresponding neutron capture gamma spectra; (3) the average radiative widths of neutron resonances. In all the properties considered, the PC contribution turned out to be significant, as compared with the standard QRPA one, and necessary to explain the available experimental data. The results with the phenomenological so-called generalized superfluid NLD model turned out to be worse, on the whole, than those obtained with the microscopic HFB + combinatorial NLD model. The very topical question about the M1 resonance contribution to PSFs is also discussed.     

Self-consistent QRPA in the theory of nuclear structure

The conventional QRPA is extended to take into account the effect of the Pauli principle and the ground state correlations. The coupling between quasiparticle and phonons is found by minimizing the ground state expectation value of the auxiliary Hamiltonian. The model with pairing plus2-2 interaction is used.     

Quasiparticle shell-model calculation within multi-phonon subspace and SU(6) boson model

The hamiltonian with quadrupole interaction is diagonalized within the multi-phonon subspace for the cases of ⁷⁴Se, ¹¹⁴Cd and ¹²⁶Xe. The results are compared with those of the SU(6) boson model based on the Tamm-Dancoff phonon and the applicability of the boson model is discussed. As a by-product, the applicability of the quasiparticle random-phase approximation (QRPA) is investigated. It is shown that the SU(6) boson model is much better than the QRPA. The contribution from non-collective phonon degrees of freedom to the many-phonon high-spin states is also discussed.     

Sum rules for two-particle operators and double beta decay

Sum rules for the double Gamow-Teller and Fermi operators are derived. They are exact when additional symmetries, isospin and SU(4) invariance, hold. Moreover, they represent a useful approximation independent of further assumptions for the cases of practical importance, nuclei with N − Z⪢ 1. The 2ν mode of double beta decay exhausts only ≈ 10⁻⁴ of the GT sum rule. Examples of the double strength distribution calculated within QRPA are shown and the importance of experimental determination of the double strength is stressed. Several processes which can be used for that purpose are identified and briefly discussed.     

Non-collapsing QRPA for neutrinoless double beta decay

The standard quasiparticle random phase approximation(QRPA) is widely used to describe the neutrinoless double beta decay process. Although it has been quite successful in many cases of interest, it has some shortcomings. The most important one is that its solutions collapse for physical values of the particle-particle strength. We shall show that modifications can be done which can extend the validity of this standard QRPA beyond the point of collapse. Such modifications are: The introduction of proton-neutron pairing, the inclusion of the Pauli principle and the extension of the Hilbert space. If all these modifications are introduced into the standard QRPA then the collapse does not occur for physical values of the particle-particle strengths. Thus, one might be able to extract more accurate values on the effective neutrino mass by using the best available experimental limits on the half life of neutrinoless double beta decay. Presented by G. Pantis at the Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997.     

Test of physics beyond the standard model in nuclei

The modern theories of Grand Unification (GUT) and SuperSymmetric (SUSY) extensions of Standard Model (SM) suppose that the conservation laws of the SM may be violated to some small degree. The nuclei are well-suited as a laboratory to test fundamental symmetries and fundamental interactions like lepton flavor (LF) and lepton number (LN) conservation. A prominent role between experiments looking for LF and total LN violation play not yet observed processes of neutrinoless double-beta decay (0νββ decay). The GUT and SUSY models offer a variety of mechanisms that allow 0νββ decay to occur. They are based on mixing of Majorana neutrinos and/or R-parity-violation hypothesis. Although the 0νββ-decay has not been seen, it is possible to extract from the lower limits of the lifetime upper limits for the effective electron Majorana neutrino mass, effective right-handed weak-interaction parameters, the effective Majoron coupling constant, R-parity-violating SUSY parameters, etc. A condition for obtaining reliable limits for these fundamental quantities is that the nuclear matrix elements governing this process can be calculated correctly. The nuclear structure wave functions can be tested by calculating the two-neutrino double-beta decay (2νββ decay) for which we have experimental data and not only lower limits as for the 0νββ decay. For open-shell nuclei, the method of choice has been the quasiparticle random-phase approximation (QRPA), which treats Fermion pairs as bosons. It has been found that, by extending the QRPA including fermion commutation relations, better agreement with 2νββ-decay experiments is achieved. This increases also the reliability of conclusions from the upper limits on the 0νββ-decay transition probability. In this work, the limits on the LN-violating parameters extracted from current 0νββ-decay experiments are listed. Studies in respect to future 0νββ-decay experimental projects are also presented.     

Extension of the renormalized quasiparticle random phase approximation

The quasiparticle random phase approximation is extended in order to restore of the Pauli principle beyond the renormalized approach by treating the so-called scattering terms in the QRPA phonon operators. It has been shown that this new framework can be described in a case of a single nuclear shell occupied by both protons and neutrons in terms of the QRPA(14,3) algebra. An application of the formalism to the double beta decay of calcium⁴⁸Ca is discussed to some extent. Presented by W.A. Kamiński at the Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997. This work has been supported in part by the State Committee for Scientific Research (Poland), Contract No. 2 P03B 189 09.     

Looking for an Optimal Ground State Within Quasiparticle Random Phase Approximation

A new ansatz for the correlated ground state of the many-nucleon system is proposed which results in obtaining a modified Quasiparticle Random Phase Approximation (QRPA). An additional degree of freedom is introduced which allows to determine variationally the ground state simultaneously with solving the QRPA equations. This new approach, QRPA with an optimal ground state, is studied within the proton-neutron Lipkin model. New solutions have been found, in the range of the interaction strength where the standard QRPA formalism does not work. A relation between one of them and the solution obtained within a semi-classical approach is established. A detailed study of the expectation value of the quasiparticle number operator in the ground state and the transition amplitude for the two-neutrino double beta Fermi decay, is also presented.     

Spin and orbital contributions to collective M1 transitions in 46,48Ti

Low- and high-lying K^π = 1⁺ states and M1 transitions in ^46,48Ti are studied. The model hamiltonian is treated in the quasi-particle particle random phase approximation (QRPA) with an exact restoration of its rotational invariance. A considerable spin contribution to the transition matrix elements is found for the low-energy (about 4 MeV) strong M1 transition (the orbital contribution being 30–70% of the spin one), although the microscopic structure of this state in ⁴⁶Ti is typical for an orbital isovector excitation. The calculated energies and B(M1) values are in good agreement with the experimental data. The results are compared to the estimates of the isovector scissor model.     

Calculations of the β-decay half-lives of neutron-deficient nuclei

In this work,β~+/EC decays of some medium-mass nuclei are investigated within the extended quasiparticle random-phase approximation(QRPA),where neutron-neutron,proton-proton and neutron-proton(np) pairing correlations are taken into consideration in the specialized Hartree-Fock-Bogoliubov(HFB) transformation.In addition to the pairing interaction,the Br¨uckner G-matrix obtained with the charge-dependent Bonn nucleon-nucleon force is used for the residual particle-particle and particle-hole interactions.Calculations are performed for even-even proton-rich isotopes ranging from Z =24 to Z =34.It is found that the np pairing interaction plays a significant role inβ-decay for some nuclei far from stability.Compared with other theoretical calculations,our calculations show good agreement with the available experimental data.Predictions of β-decay half-lives for some very neutron-deficient nuclei are made for reference.     

Self-consistent approach to β-decay and delayed multi-neutron emission

The beta-decay half-lives and delayed multi- neutron emission branchings for the nuclei near the new neutron shell N = 34 are treated within self-consistent Density Functional + Continuum QRPA model (DF + CQRPA). A comparison with the recent self-consistent calculations from relativistic QRPA and standard (semi-microscopic) FRDM is performed.     

Calculation of the beta decay of Pd. Quasiparticle random phase approximation results

Single beta decay transitions in ^114–120Pd are calculated. Theoreticalβ⁺ and β⁻ strength distributions, for transitions to 1⁺ states in the nuclei ^114–120Ag and ^114–120Rh, are obtained in the framework of the quasiparticle random phase approximation (QRPA). The effective two-body interaction which is used in the calculations is constructed from the Bonn one-boson-exchange potential (OBEP). Particle-hole and particle-particle like channels of the two-body force are included in the definition of the QRPA matrix equations. Effects associated with the particle number violation of the quasiparticle mean field are accounted for by using a particle-number-projected version of the QRPA formalism. Theoretical strength distributions for the β⁻ and β⁺ braches are shown and compared with data.