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.     

pn-QRPA and higher-order approximations

The proton-neutron Quasi-particle Random Phase Approximation (pn-QRPA) is reviewed and higher-order approximations discussed with reference to the beta decay physics. The approach is fully developed in a boson formalism. Working within a schematic model, we first illustrate a fermion-boson mapping procedure and apply it to construct boson images of the fermion Hamiltonian at different levels of approximation. The quality of these images is tested through a comparison between approximate and exact spectra. Standard QRPA equations are derived in correspondence with the quasi-boson limit of the boson Hamiltonian. The use of higher-order Hamiltonians is seen to improve considerably the stability of the approximate solutions. The mapping procedure is also applied to Fermi beta operators and transition amplitudes are discussed. The range of applicability of the QRPA formalism is examined. Presented at Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997.     

Self-consistent random phase approximation within the O(5) model and Fermi transitions

Self-consistent quasiparticle random phase approximation (SCQRPA) is considered in application to the Fermi transitions within the O(5) model. It is demonstrated that SCQRPA improves on renormalized QRPA (RQRPA), a method that has recently become rather popular in this context. The analytical form of the SCQRPA vacuum is used to evaluate all the matrix elements. The SCQRPA results show a general trend similar to the exact solutions. The necessity to change the single particle basis beyond the transition point, and to include the proton-proton and neutron-neutron channels in the QRPA operator, in addition to the proton-neutron one, is pointed out.     

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.     

Effect of pairing correlation on low-lying quadrupole states in Sn isotopes

We examined the low-lying quadrupole states in Sn isotopes in the framework of fully self-consistent Hartree-Fock+BCS plus QRPA.We focus on the effect of the density-dependence of pairing interaction on the properties of the low-lying quadrupole state.The SLy5 Skyrme interaction with surface,mixed,and volume pairings is employed in the calculations,respectively.We find that the excitation energies and the corresponding reduced electric transition probabilities of the first 2~+ state are different,given by the three pairing interactions.The properties of the quasiparticle state,two-quasiparticle excitation energy,reduced transition amplitude,and transition densities in~(112)Sn are analyzed in detail.Two different mechanisms,the static and dynamical effects,of the pairing correlation are also discussed.The results show that the surface,mixed,and volume pairings indeed affect the properties of the first 2~+ state in the Sn isotopes.     

Neutron-proton pairing

The two topics discussed here are the treatment of neutron-proton pairing and the problem of phase transitions in systems of neutrons and protons. The conclusions are based on exact calculations, possible in simplified situations, and the exact results are compared with the BCS treatment. QRPA, and many of its modifications, which are the most popular approaches to double beta decay, involve the quasiparticle transformation as a decisive first step. It is therefore imperative to understand its strengths and limitations. Presented at Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997. The work reported here is based on Refs. 3, 4, and 5. The credit for it belongs to my collaborators, in particular to Osvaldo Civitarese, Jonathan Engel, and Stuart Pittel. This work was supported by the U.S. Department of Energy under Grant. No. DE-FG03-88ER-40397.     

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.     

An equations-of-motion method for anharmonic vibrations: Application to a degenerate shell model

An equations-of-motion method is proposed for calculating anharmonic effects in doubly even spherical nuclei. The method can be viewed as an extension of the QRPA theory for spherical nuclei, and its applicability is demonstrated for the case of a degenerate shell model of configuration ($\text{21}\text{2}$)⁶ by comparison with the results of a previous exact diagonalization.     

Study of the muon number violating (μ, e conversion in a nucleus by using quasi-particle RPA

The exotic (μ⁻, e⁻) conversion reaction for the ⁴⁸Ti nucleus is studied in the framework of the quasi-particle random phase approximation (QRPA). For the non-coherent processes the relevant total rate is calculated by summing over partial rates for all the possible intermediate states constructed in the above approximation. For the coherent process the contribution is obtained by using an uncorrelated BCS vacuum. In order to check the validity of closure approximation, which is almost unavoidable in shell-model calculations, we also evaluate the total (μ⁻, e⁻) conversion rates by QRPA sum-rules by first explicitly calculating a suitable mean excitation energy of the nucleus. The influence of the ground state correlations to the (μ⁻, e⁻) conversion matrix elements is estimated by using a correlated RPA vacuum. The fraction of the transition rate of the coherent process for each of these methods is calculated and the results are compared to those found previously by using shell-model closure approximation.     

Pygmy and Giant Dipole Resonances in Proton-Rich Nuclei ~(17,18)Ne

The pygmy and giant dipole resonances in proton-rich nuclei~(17,18)Ne are investigated with a fully self-consistent approach. The properties of ground states are calculated in the Skyrme Hartree-Fock with the Bardeen-CooperSchrieffer approximation to take into account the pairing correlation. The quasiparticle random phase approximation(QRPA) method is used to explore the properties of excited dipole states. In the calculations the SLy5 Skyrme interaction is employed. In addition to the giant dipole resonances, pygmy dipole resonances(PDR) are found to be located in the energy region below 10 MeV in both 17,18 Ne. The strength and transition density show that the low-lying states are typical PDR states. However, analyzing the QRPA amplitudes of proton and neutron 2 quasiparticle(2 qp) configurations for a given low-lying state in ~(17,18)Ne, we find that the PDR state is less collective, more like a single 2 qp excitation.     

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.     

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.     

Validity of the two-determinantal deformed Hartree-Fock approximation

The two-fold degeneracy of the deformed Hartree-Fock (HF) solutions in thes-d shell suggests the use of a two-determinantal intrinsic state. The validity of this two-determinantal variational method is established in the case of the exactly solvable Lipkin Hamiltonian, for which the usual HF solution is known to be 2 fold degenerate. The approximation of using a two-determinantal intrinsic state turns out to be an exceedingly good one and in general as the particle number and the strength of the interaction increase, the exact solution is rapidly approached. The states of positive (negative) parity which are obtained by projection from the intrinsic two-determinantal state are found to reproduce the exact ground (first excited) state with a high accuracy.     

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 use of coherent states in the variational treatment of proton-neutron interactions

Coherent states are used as trial states to determine, variationally, the structure of the eigenvectors belonging to a schematic Hamiltonian consisting of single-particle, pairing and residual proton-neutron interaction terms. It is shown that the standard proton-neutron quasiparticle random-phase approximation (pn-QRPA) is recovered, as a variational theory, by replacing quasiparticle pair creation and annihilation operators by bosons. It is also shown that an exact, algebra preserving, mapping of the Hamiltonian is needed to describe the spectrum beyond the QRPA phase transition. The role of the spurious components of the trial wave functions is discussed. Received: 19 February 2002 / Accepted: 3 May 2002     

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.     

Nonequilibrium kinetics: Exact and approximate solutions

The relaxation of an internal state distribution in the presence of an excess of an inert gas is considered. The explicit time dependence of the nonequilibrium contributions to the transition rate coefficients is approximated using the Kapral-Hudson-Ross method. The resulting solution contains cross-correlation terms which do not appear when a single reaction is considered. It is shown that the first term of a perturbation expansion of an exact formal solution gives the Kapral-Hudson-Ross solution for short times, and the Chapman-Enskog solution at long times if there is a wide separation in time scales. The Kapral-Hudson-Ross, Chapman-Enskog, and exact solutions are compared for a two-state, hard-sphere model system.Based on a dissertation by S. Hudson, Massachusetts Institute of Technology, 1972.National Science Foundation predoctoral fellow, 1967–1971.     

Proton-neutron quasiparticle RPA and charge-changing transitions

General formulae for a proton-neutron QRPA calculation are presented. All terms in the RPA order are retained, and no specific assumptions on the residual interaction and one-body charge-changing transitions are made. QRPA phonon correlations are introduced in first-order perturbation for quasiparticle transitions from odd-mass and odd-odd parent nuclei.