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Single-particle resonances in the continuum are crucial for studies of exotic nuclei. In this study, the Greens function approach is employed to search for single-particle resonances based on the relativistic-mean-field model. Taking $^{120}$Sn as an example, we identify single-particle resonances and determine the energies and widths directly by probing the extrema of the Greens functions. In contrast to the results found by exploring for the extremum of the density of states proposed in our recent study [Chin. Phys. C, 44:084105 (2020)], which has proven to be very successful, the same resonances as well as very close energies and widths are obtained. By comparing the Greens functions plotted in different coordinate space sizes, we also found that the results very slightly depend on the space size. These findings demonstrate that the approach by exploring for the extremum of the Greens function is also very reliable and effective for identifying resonant states, regardless of whether they are wide or narrow.
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The relativistic mean field theory with the Greens function method is taken to study the single-particle resonant states. Different from our previous work [Phys.Rev.C 90,054321(2014)], the resonant states are identified by searching for the poles of Greens function or the extremes of the density of states. This new approach is very effective for all kinds of resonant states, no matter it is broad or narrow. The dependence on the space size for the resonant energies, widths, and the density distributions in the coordinate space has been checked and it is found very stable. Taking $^{120}$Sn as an example, four new broad resonant states $2g_{7/2}$, $2g_{9/2}$, $2h_{11/2}$ and $1j_{13/2}$ are observed, and also the accuracy for the width of the very narrow resonant state $1h_{9/2}$ is highly improved to be $1times 10^{-8}$ MeV. Besides, our results are very close to those by the complex momentum representation method and the complex scaling method.
To study the exotic odd nuclear systems, the self-consistent continuum Skyrme-Hartree-Fock-Bogoliubov theory formulated with Greens function technique is extended to include blocking effects with the equal filling approximation. Detailed formula are presented.To perform the integrals of the Greens function properly, the contour paths $C_{rm b}^{-}$ and $C_{rm b}^{+}$ introduced for the blocking effects should include the blocked quasi-particle state but can not intrude into the continuum area. By comparing with the box-discretized calculations, the great advantages of the Greens function method in describing the extended density distributions, resonant states, and the couplings with the continuum in exotic nuclei are shown. Finally, taking the neutron-rich odd nucleus $^{159}$Sn as an example, the halo structure is investigated by blocking the quasi-particle state $1p_{1/2}$. It is found that it is mainly the weakly bound states near the Fermi surface that contribute a lot for the extended density distributions at large coordinate space.
Shell corrections of the finite deformed Woods-Saxon potential are calculated using the Greens function method and the generalized Strutinsky smoothing procedure. They are compared with the results of the standard prescription which are affected by the spurious contribution from the unphysical particle gas. In the new method, the shell correction approaches the exact limit provided that the dimension of the single-particle (harmonic oscillator) basis is sufficiently large. For spherical potentials, the present method is faster than the exact one in which the contribution from the particle continuum states is explicitly calculated. For deformed potentials, the Greens function method offers a practical and reliable way of calculating shell corrections for weakly bound nuclei.
We present the fundamental techniques and working equations of many-body Greens function theory for calculating ground state properties and the spectral strength. Greens function methods closely relate to other polynomial scaling approaches discussed in chapters 8 and 10. However, here we aim directly at a global view of the many-fermion structure. We derive the working equations for calculating many-body propagators, using both the Algebraic Diagrammatic Construction technique and the self-consistent formalism at finite temperature. Their implementation is discussed, as well as the inclusion of three-nucleon interactions. The self-consistency feature is essential to guarantee thermodynamic consistency. The pairing and neutron matter models introduced in previous chapters are solved and compared with the other methods in this book.
Microscopic calculations of the electromagnetic response of medium-mass nuclei are now feasible thanks to the availability of realistic nuclear interactions with accurate saturation and spectroscopic properties, and the development of large-scale computing methods for many-body physics. The purpose is to compute isovector dipole electromagnetic (E1) response and related quantities, i.e. integrated dipole cross section and polarizability, and compare with data from photoabsorption and Coulomb excitation experiments. The single-particle propagator is obtained by solving the Dyson equation, where the self-energy includes correlations non-perturbatively through the Algebraic Diagrammatic Construction (ADC) method. The particle-hole ($ph$) polarization propagator is treated in the Dressed Random Phase Approximation (DRPA), based on an effective correlated propagator that includes some $2p2h$ effects but keeps the same computation scaling as the standard Hartree-Fock propagator. The E1 responses for $^{14,16,22,24}$O, $^{36,40,48,52,54,70}$Ca and $^{68}$Ni have been computed: the presence of a soft dipole mode of excitation for neutron-rich nuclei is found, and there is a fair reproduction of the low-energy part of the experimental excitation spectrum. This is reflected in a good agreement with the empirical dipole polarizability values. For a realistic interaction with an accurate reproduction of masses and radii up to medium-mass nuclei, the Self-Consistent Greens Function method provides a good description of the E1 response, especially in the part of the excitation spectrum below the Giant Dipole Resonance. The dipole polarizability is largely independent from the strategy of mapping the dressed propagator to a simplified one that is computationally manageable
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