T.T.S. Kuo

Convergence of effective hamiltonian expansion and partial summations of folded diagrams

Abstract Starting from a time-dependent formulation of the energy-independent effective hamiltonian, a sequence of partial summations for the folded-diagram series is defined and a connection between the energy-independent and energy-dependent effective hamiltonians is shown. The partial summations are shown to be convergent in the presence of intruder states which are weakly coupled to the model space.

The nuclear reaction matrix

Different definitions of the reaction matrix G appropriate to the calculation of nuclear structure are reviewed and discussed. Qualitative physical arguments are presented in support of a two-step calculation of the G-matrix for finite nuclei. In the first step the high-energy excitations are included using orthogonalized plane-wave intermediate states, and in the second step the low-energy excitations are added in, using harmonic oscillator intermediate states. Accurate calculations of G-matrix elements for nuclear structure calculations in the Aapprox. =18 region are performed following this procedure and treating the Pauli exclusion operator Q/sub 2//sub p/ by the method of Tsai and Kuo. The treatment of Q/sub 2//sub p/, the effect of the intermediate-state spectrum and the energy dependence of the reaction matrix are investigated in detail. The present matrix elements are compared with various matrix elements given in the literature. In particular, close agreement is obtained with the matrix elements calculated by Kuo and Brown using approximate methods. (AIP)

Shell-model calculations and realistic effective interactions

A review is presented of the development and current status of nuclear shell-model calculations in which the two-body effective interaction between the valence nucleons is derived from the free nucleon-nucleon potential. The significant progress made in this field within the last decade is emphasized, in particular as regards the so-called Vlow−k approach to the renormalization of the bare nucleon-nucleon interaction. In the last part of the review we first give a survey of realistic shell-model calculations from early to present days. Then, we report recent results for neutron-rich nuclei near doubly magic 132 Sn and for the whole even-mass N = 82 isotonic chain. These illustrate how shell-model effective interactions derived from modern nucleon-nucleon potentials are able to provide an accurate description of nuclear structure properties.

Folded diagrams and 1s-Od effective interactions derived from Reid and Paris nucleon-nucleon potentials

Abstract The sd-shell effective-interaction matrix elements are derived from the Paris and Reid potentials using a microscopic folded-diagram effective-interaction theory. A comparison of these matrix elements is carried out by calculating spectra and energy centroids for nuclei of mass 18 to 24. The folded diagrams were included by both solving for the energy-dependent effective interaction self-consistently and by including the folded diagrams explicitly. In the latter case the folded diagrams were grouped either according to the number of folds or as prescribed by the Lee and Suzuki iteration technique; the Lee-Suzuki method was found to converge better and yield the more reliable results. Special attention was given to the proper treatment of one-body connected diagrams in the calculation of the two-body effective interaction. We first calculate the (energy-dependent) G-matrix appropriate for the sd-shell for both potentials using a momentum-space matrix-inversion method which treats the Pauli exclusion operator essentially exactly. This G-matrix interaction is then used to calculate the irreducible and non- folded diagrams contained in the Q - box . The effective-interaction matrix elements are obtained by evaluating a Q - box folded diagram series. We considered four approximations for the basic Q - box . These were (C1) the inclusion of diagrams up to 2nd order in G, (C2) 2nd order plus hole-hole phonons, (C3) 2nd order plus (bare TDA) particle-hole phonons, and (C4) 2nd order plus both hole-hole and particle-hole phonons. The contribution of the folded diagrams was found to be quite large, typically about 30%, and to weaken the interaction. Also, due to the greater energy dependence of higher-order diagrams, the effect of folded diagrams was much greater in higher orders. That is, the contribution from higher-order diagrams for most cases was greatly reduced by the folded diagrams. The convergence of the folded-diagram series deteriorates with the inclusion of higher-order Q - box processes in the method which groups diagrams by the number of folds, but remains excellent in the Lee-Suzuki method. Whereas the inclusion of the particle-hole phonon was essential to obtain agreement with experiment in earlier work, when the folded diagrams are included the effect of the particle-hole phonon is to reduce the amount of binding. All four approximations to both potentials produce interactions which badly underbind nuclei. The excitation spectra given by these interactions are, however, all rather similar to each other. The Paris interaction produces more binding than does the Reid, but differences between results obtained with the two interactions were often less than differences obtained in the four approximations. Essentially no difference was found between the effective non-central interactions from the Reid and Paris potentials after including the folded diagrams, although these two potentials themselves are quite different, especially in the strength of the tensor force. Comparisons between.calculated spectra and experiment were done for 18O, 18F, 19F, 20O, 20Ne, 22Ne, 22Na and 24Mg.

Shell Model Description of the {sup 14}C Dating {beta} Decay with Brown-Rho-Scaled NN Interactions

We present shell model calculations for the beta decay of 14C to the 14N ground state, treating the states of the A=14 multiplet as two 0p holes in an 16O core. We employ low-momentum nucleon-nucleon (NN) interactions derived from the realistic Bonn-B potential and find that the Gamow-Teller (GT) matrix element is too large to describe the known lifetime. By using a modified version of this potential that incorporates the effects of Brown-Rho scaling medium modifications, we find that the GT matrix element vanishes for a nuclear density around 85% that of nuclear matter. We find that the splitting between the (J(pi),T)=(1(+),0) and (J(pi),T)=(0(+),1) states in 14N is improved using the medium-modified Bonn-B potential and that the transition strengths from excited states of 14C to the 14N ground state are compatible with recent experiments.

Low momentum nucleon–nucleon potentials with half-on-shell T-matrix equivalence

We study a method by which realistic nucleon-nucleon potentials V_NN can be reduced, in a physically equivalent way, to an effective low-momentum potential V-low-k confined within a cut-off momentum k-cut. Our effective potential is obtained using the folded-diagram method of Kuo, Lee and Ratcliff, and it is shown to preserve the half-on-shell T-matrix. Both the Andreozzi-Lee-Suzuki and the Andreozzi-Krenciglowa-Kuo iteration methods have been employed in carrying out the reduction. Calculations have been performed for the Bonn A and Paris NN potentials, using various choices for k-cut such as 2 fm-1. The deuteron binding energy, low-energy NN phase shifts, and the low-momentum half-on-shell T-matrix given by V_NN are all accurately reproduced by V-low-k. Possible applications of V-low-k directly to nuclear matter and nuclear structure calculations are discussed.

Infinite order summation of particle-particle ring diagrams in a model-space approach for nuclear matter

Abstract A ring diagram model-space nuclear matter theory is formulated and applied to the calculation of the binding energy per nucleon ( BE A ) , saturation Fermi momentum (kF) and incompressibility coefficient (K) of symmetric nuclear matter, using the Paris and Reid nucleon-nucleon potentials. A model space is introduced where all nucleons are restricted to have momentum k ⩽ kM, typical values of kM being ∼3.2 fm−. Using a model-space Hartree-Fock approach, self-consistent single particle spectra are derived for holes (k ⩽ kF) and particles with momentum kF kM, we use a free particle spectrum. Within the model space we sum up the particle-particle ring diagrams (both forward- and backward-going) to all orders. A rather simple expression for the energy shift ΔE0pp is obtained, namely ΔE0pp is expressed as integrals involving the trace of Y(λ)Y+(λ)GM where GM is the model-space reaction matrix, the Ys are transition amplitudes obtained from solving RPA-type secular equations and λ is a strength parameter to be integrated from 0 to 1. We have used angle-average approximations in our calculations, and in this way different partial wave channels are decoupled. For the 3S1-3D1 channel, the effect of the ring diagrams is found to be particularly important. The inclusion of the ring diagrams has largely increased the role of the tensor force in determining the nuclear matter saturation properties, and consequently we obtain saturation densities which are significantly lower than those given by most other calculations. For the Paris potential, our results for BE A , kF and K are respectively 17.38 MeV, 1.42 fm−1 and 96.3 MeV. For the Reid potential, the corresponding results are 15.15 MeV, 1.30 fm−1 and 110.7 MeV. Our calculated values for the binding energy per nucleon and saturation density are both in rather satisfactory agreement with the corresponding empirical values.

Shell model calculations of two to four identical-“particle” systems near 208Pb

Abstract The structure of the nuclei 204–206Pb, 210–212Pb, 210Po, 211At, and 212Rn is studied in terms of conventional nuclear shell models. An inert 208Pb core is assumed, and active particles (holes) are distuibuted in the low-lying single-particle (hole) orbits. Experimental single-particle energies are used for the one-body part of the effective residual interaction. Realistic interaction matrix elements developed for this mass region by Kuo and Herling are used for the matrix elements of the two-body part of the residual interactions. As much as possible, other effective one-body operators for electromagnetic observables are derived from experimental data on the single-particle (hole) nuclei 207Pb, 209Pb, and 209Bi. Observables treated are ground state binding energies, excitation energies, strengths for one- and two-particle transfers, and E2 and M1 observables. Generally, excellent agreement is found. The configuration mixing calculations do not remove anomalies in the magnetic moments of excited states in 206Pb and 212Rn. Many states in these nuclei are predicted by the models which have not been observed as yet. It is found that a truncation scheme for doubly even nuclei treated here in which only seniority-0 and seniority-2 states are allowed is potentially very useful.

A simple method for evaluating Goldstone diagrams in an angular momentum coupled representation

Abstract A simple and convenient method is derived for evaluating linked Goldstone diagrams in an angular momentum coupled representation. Our method is general, and can be used to evaluate any effective interaction and/or effective operator diagrams for both closed-shell nuclei (vacuum to vacuum linked diagrams) and open-shell nuclei (valence linked diagrams). The techniques of decomposing diagrams into ladder diagrams, cutting open internal lines and cutting of one-body insertions are introduced. These enable us to determine angular momentum factors associated with diagrams in the coupled representation directly, without the need for carrying out complicated angular momentum algebra. A summary of diagram rules is given.

Ground-state properties of closed-shell nuclei with low-momentum realistic interactions

Ground-state properties of