Folded diagrams and s-d shell effective interactions
Abstract
Folded diagrams with one to four folds are calculated for the two-body s-d shell effective interaction. Their main effect is to renormalize the non-folded terms by a wound-integral factor (1 − 2κ), κ being approximately 0.15. When used in a conventional A = 18 to 20 shell-model calculation, the folded diagram effective interaction yields reasonable excitation energy spectra, but not enough ground state binding energies.
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Shell-model calculations and realistic effective interactions
2009, Progress in Particle and Nuclear PhysicsA review is presented of the development and current status of nuclear shell-model calculations, in which the two-body effective interaction between 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 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 132Sn, and for the whole even-mass 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.
Finite nuclei calculations with realistic potential models
1991, Physics Letters BAn effective interaction is deduced from a microscopic point of view for different realistic N-N interactions. The effect of second order and folded diagrams is considered in the calculation of the low-energy spectra of some light s-d shell isotopes. We confirm that the Bonn-A potential gives the best agreement with data. The Paris and Argonne ν14 potentials display a very similar behaviour. The general trend of the results shows better agreement with data for potentials with weaker tensor force components, but other details of the construction of the potentials might contribute to this effect.
Folded diagrams and 1s-Od effective interactions derived from Reid and Paris nucleon-nucleon potentials
1983, Nuclear Physics, Section AThe 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 . The effective-interaction matrix elements are obtained by evaluating a folded diagram series. We considered four approximations for the basic . 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 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.
A folded diagram microscopic calculation of nuclear Coulomb displacement energies
1981, Nuclear Physics, Section AWe have performed a rather extensive microscopic calculation of the 17F-17O Coulomb energy differences ΔEC. Our main purpose has been to study the effects of (i) folded diagrams, (ii) core polarization and (iii) the effects due to different nucleon-nucleon potentials, different single-particle spectra and different radial wave functions. Using a proton-neutron representation we have included higher-order Coulomb corrections like e.g. the coupling of valence particles to collective vibrations of the core.
The inclusion of folded diagrams is very important; it is equivalent to a self-consistent treatment of the Q-box starting energies. It causes a significant suppression of the effect due to core polarization, its contribution to ΔEC becoming small, about 0.03 MeV. As a consequence of this “self-correcting” behavior our results for ΔEC are quite stable with respect to the choice of different single-particle spectra. Two rather different nucleon-nucleon potentials, the Reid soft-core potential and a meson exchange potential of the Bonn-Jülich group also lead to quite similar results for ΔEC. In the case of , for example, ΔC calculated with the Reid potential ranges, depending on other assumptions, from 3.36 to 3.43 MeV and from 3.36 to 3.40 MeV with the Bonn-Jülich potential: Both are significantly smaller than the experimental value of 3.54 MeV. Optimizing the radial wave functions in the spirit of a Brueckner-Hartree-Fock theory improves the calculated values of ΔEC for the state but not for .
We feel that other processes such as the explicit inclusion of core deformation or mesonic degrees of freedom or both are needed to explain the Coulomb displacement energies.
A simple method for evaluating Goldstone diagrams in an angular momentum coupled representation
1981, Annals of PhysicsA 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.
An equation of motion method for deriving microscopic effective interactions in a correlated model space
1980, Nuclear Physics, Section AStarting from a diagrammatic analysis of the equation of motion method, we derive an effective interaction theory for a correlated model space where the basis vectors correspond physically to the addition of valence particles and/or holes to the true ground state of the core nucleus. The resulting effective interaction is valence linked and connected, energy independent, and contains folded diagrams. In addition, it gives directly model eigenvectors with amplitudes that correspond to spectroscopic factors. With terms having the same number of folds grouped together, the general structure of is very simple. This is very useful in the application of the present theory to actual microscopic nuclear structure calculations. The treatment of core projection insertions is discussed in some detail. A proof of the cancellation of the disconnected diagrams is given. When folded diagrams are summed up using a partial summation method, the present effective interaction theory is shown to be consistent with the usual Green function theory for many-body problems.