Particle-hole Phonons Research Articles

Particle-hole excitations within a self-consistent random-phase approximation

We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the $X$ and $Y$ amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.

Extension of the random phase approximation including the self-consistent coupling to two-phonon contributions

A microscopic formalism is developed that includes the coupling to two particle-hole phonons in the particle-hole propagator by extending the dressed random phase approximation (DRPA) equation for a finite system. The resulting formalism is applied to study the low-lying excitation spectrum of 16O. It is observed that the coupling to two-phonon states at low energy generates excited states with quantum numbers that cannot be obtained in the DRPA approach. Nevertheless, the two-phonon states mix weakly with particle-hole configurations and participate only partially in the formation of the lowest-lying positive-parity excited states. The stability of the present calculation is tested vs. the truncation of model space. It is demonstrated that when single-particle strength fragmentation is properly considered, the present formalism exhibits convergence with respect to the chosen model space within the confines of the chosen approximation scheme.

Faddeev treatment of long-range correlations and the one-hole spectral function of16O

The Faddeev technique is employed to study the influence of both particle-particle and particle-hole phonons on the one-hole spectral function of O16. Collective excitations are accounted for at a random phase approximation level and subsequently summed to all orders by the Faddeev equations to obtain the nucleon self-energy. An iterative procedure is applied to investigate the effects of the self-consistent inclusion of the fragmentation in the determination of the phonons and the corresponding self-energy. The present results indicate that the characteristics of hole fragmentation are related to the low-lying states of O16.

Faddeev description of two-hole–one-particle motion and the single-particle spectral function

The Faddeev technique is employed to address the problem of describing the influence of both particle-particle and particle-hole phonons on the single-particle self-energy. The scope of the few-body Faddeev equations is extended to describe the motion of two-hole one-particle (two-particle one-hole) excitations. This formalism allows to sum both particle-particle and particle-hole phonons, obtained separately in the Random Phase Approximation. The appearance of spurious solutions for the present application of the Faddeev method is related to the inclusion of a consistent set of diagrams. The formalism presented here appears practical for finite nuclei and achieves a simultaneous inclusion of particle-particle and particle-hole phonons to all orders while the spurious solutions are properly eliminated.

Self-consistent determination of the one-body density matrix and particle-hole excitations

We develop a procedure recently proposed for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The one-body density matrix in the correlated ground state is now no more assumed to be diagonal in the Hartree-Fock basis. An expression is rather provided where this quantity is expressed in terms of the X and Y amplitudes of the particle-hole phonon operators. A set of RPA-like equations depending only on the one-body density matrix is then constructed by means of the method of linearized equations of motion and solved via an iterative procedure. The procedure is employed to calculate spectra and strength distributions of several neutral and charged Na and Li metal clusters within the jellium approximation. Improvements are observed with respect to standard RPA calculations.

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

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.

Two-body propagators with dynamic effective interactions in a many-fermion medium: II. s-Channel

In open-shell nuclear structure calculations collective excitations of the closed-shell core play an important role. When the energies of these excitations are comparable to the spread of energies in the model space a dynamic treatment of the core states is essential. In a Green function formulation dynamic core states, represented by particle-hole phonons, are included in the effective interaction of the s-channel integral equation for the fourfermion vertex function Γ. The solution of the equation is obtained analytically and from it the s-channel two-body propagator K is calculated. The structure of the propagator K reveals that, in addition to poles corresponding to the normal shell model states, many new poles of K are obtained. These correspond to two-particle- n-phonon configurations. Furthermore, it is shown that, due to the phonons, correlations of the ground state are induced in the calculation of the propagator K. The difference between these correlations and the well-known particle-particle-RPA correlations also included in the formalism is pointed out. The expressions derived can easily be handled numerically. In the calculations the physical states of single-particle and single-hole nuclei can be incorporated by an appropriate dressing of the single-fermion Green function.