A numerical method has been developed to solve the RPA equation, exchange term included, in nuclear matter. The dynamic form factor S(q, ω) is extracted for several v4 and v6 phenomenological potentials, including the d1-potential of Gogny et al. The limits of validity of the long-wavelength (Landau) approximation and the often adopted local-kernel approximation are discussed. Substantial disagreements with the exact results are found for the latter. The method is then applied to solve a Jastrow-correlated extension of the RPA equation, using the hardcore OMY potential. Results of calculations performed in two-body cluster approximation and Fermi-Hypernetted-Chain (FHNC) approximation are compared. The two-body results predict an instability against density fluctuations, which disappears at the FHNC level. The validity and consequences of employing the FHNC effective potential within the self-consistent HF/RPA framework are discussed. Future developments include applying the method to other Fermi systems such as liquid 3He and the microscopic calculation of Landau parameters.
A numerical method has been developed to solve the RPA equation, exchange term included, in nuclear matter. The dynamic form factor S(q, ω) is extracted for several v4 and v6 phenomenological potentials, including the d1-potential of Gogny et al. The limits of validity of the long-wavelength (Landau) approximation and the often adopted local-kernel approximation are discussed. Substantial disagreements with the exact results are found for the latter. The method is then applied to solve a Jastrow-correlated extension of the RPA equation, using the hardcore OMY potential. Results of calculations performed in two-body cluster approximation and Fermi-Hypernetted-Chain (FHNC) approximation are compared. The two-body results predict an instability against density fluctuations, which disappears at the FHNC level. The validity and consequences of employing the FHNC effective potential within the self-consistent HF/RPA framework are discussed. Future developments include applying the method to other Fermi systems such as liquid 3He and the microscopic calculation of Landau parameters.
The hypotheticai system of nucleons known as nuclear matter provides a very interesting field of study with relevance to both nuclear physics and astrophysics. The task of finding nucleon-nucleon potentials that explain the empirical properties of nuclear matter, as well as the two-body data, has been hampered by the lack of an adequate many-body theory for treating the complicated forces involved. Realistic nucleon-nucleon potentials have strong spin, isospin, tensor and spin-orbit components which induce correspondingly strong correlations in nuclear matter. In this thesis we present a variational theory designed to cope with these complicated correlations. To place the theory in perspective we give an overview of several different many-body theories and compare their predictions in the somewhat simpler systems of the 4He and 3He liquids, and for a simple central potential model of nuclear matter. The variational theory is then developed, starting with the choice of wave function. A generalized Jastrow product of two-body correlation functions containing central, spin, isospin, tensor, and spin-orbit operators is used. The correlation function is generated by a series of two-body Schrodinger equations with boundary conditions which require the wave function to heal at a distance d. The wave function is parameterized by d and by the magnitudes of the non-central correlations. Expectation values with variational wave functions of the Jastrow type may be studied conveniently with generalized Mayer diagrams. Diagram rules are given and a general diagrammatic cluster expansion is derived. It is an expansion in powers of commutators of the non-central operators, and in their absence reduces to the usual Mayer cluster expansion. Evaluation of diagrams in the cluster expansion is complicated by the presence of the non-commuting operators. Some simple rules and useful methods for calculating the contribution due to the central, spin, isospin, and tensor operators are given. Spin-orbit operators are neglected beyond this point, but the approach used is extendible to them. Large classes of diagrams contributing to an expectation value can be summed by means of integral equations. The development of single chain, hypernetted chain, and Fermi hypernetted chain (FHNC) equations for central correlations is reviewed. The chain summation methods are extended to sum single operator chains (SOC) for the non-central operators. Methods are also given for treating the leading commutator corrections in the expansi0n by simple vertex factors. The energy expectation value for potentials with six operators are evaluated using the FHNC/SOC functions, with an exact treatment of the commutators involved. The energy is found to have a minimum with respect to variations in all parameters. Results of calculations with model potentials based on the Reid, Bethe-Johnson, Hamada-Johnston, and Gammel-Thaler potentials are reported. A crude estimate of the effect of the neglected spin-orbit potentials on the nuclear matter binding energy and saturation density indicates that it could be significant. A correct treatment of the spin-orbit potentials is the next major step that should be taken.
The generalized momentum distribution n(p,Q), related to the half-diagonal two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2\mathrm{h}}$(r1,r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$) by Fourier transformation in the variables r1-${\mathbf{r}}_{1}^{\ensuremath{'}}$ and r1-r2, plays a key role in the description of final-state interactions in the nuclear medium and other strongly interacting many-body systems. The function n(p,Q) is explored for two Jastrow-correlated models of infinite nuclear matter within a Fermi hypernetted-chain procedure. Significant departures from ideal Fermi gas behavior in certain kinematic domains provide signatures of the strong short-range correlations contained in these models. However, such deviations are less prominent than in earlier calculations based on low-order cluster truncations; correspondingly, violations of the sequential relation are greatly reduced. Simple prescriptions for improved low-cluster-order approximations to n(p,Q) are suggested by analysis of the results of the Fermi hypernetted-chain evaluation. These results are also used to assess the quality of Silver's approximation n(p,Q)\ensuremath{\approx}n(p)[S(Q)-1] for the generalized momentum distribution in terms of the ordinary momentum distribution n(p) and the static structure function S(Q), with findings that have potential implications for the interpretation of data from inclusive electron scattering by nuclei at high momentum transfers.
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.
We review self-consistent spectral methods for nuclear matter calculations. The in-medium T −matrix approach is conserving and thermodynamically consistent. It gives both the global and the single-particle properties the system. The T −matrix approximation allows to address the pairing phenomenon in cold nuclear matter. A generalization of nuclear matter calculations to the superfluid phase is discussed and numerical results are presented for this case. The linear response of a correlated system going beyond the Hartree-Fock+ Random-Phase-Approximation (RPA) scheme is studied. The polarization is obtained by solving a consistent Bethe-Salpeter (BS) equation for the coupling of dressed nucleons to an external field. We find that multipair contributions are important for the spin(isospin) response when the interaction is spin(isospin) dependent. 1
We review self-consistent spectral methods for nuclear matter calculations. The in-medium T -matrix approach is conserving and thermodynamically consistent. It gives both the global and the single-particle properties the system. The T -matrix approximation allows to address the pairing phenomenon in cold nuclear matter. A generalization of nuclear matter calculations to the super.uid phase is discussed and numerical results are presented for this case. The linear response of a correlated system going beyond the Hartree-Fock+ Random-Phase-Approximation (RPA) scheme is studied. The polarization is obtained by solving a consistent Bethe-Salpeter (BS) equation for the coupling of dressed nucleons to an external field. We find that multipair contributions are important for the spin(isospin) response when the interaction is spin(isospin) dependent.
A broad variety of many-body methods exists for the investigation of ground-state properties, ranging from sophisticated ab initio approaches to traditional, phenomenological models. The description of low-lying excited states of medium-mass nuclei with ab initio methods has also become possible through recent progress in many-body theory. For collective modes at higher energies, however, these methods usually cannot be applied. Therefore, when describing collective excitations either completely phenomenological, macroscopic models are employed or microscopic models using phenomenological interactions. One of the microscopic models well suited for the calculation of collective properties is the random-phase approximation (RPA). In the past, the use of phenomenological interactions for RPA has shown promising results. However, the application of chiral NN interactions yielded transitions at significantly too high energies, far from agreement with experimental data. This thesis focuses on the description of collective modes using both RPA and its second-order extension, SRPA. In contrast to previous research endeavors, we employ chiral NN+3N interactions. The use of chiral interactions is an important first step for describing ground-state, excitation and collective properties on an equal foundation. We find that the inclusion of 3N terms is crucial for RPA calculations and the prediction for collective modes is drastically improved through the 3N terms. For SRPA we show first-ever results with chiral interactions, again leading to an improvement in the predictions. For a successful ab initio description of ground-state properties the inclusion of correlations is of paramount importance. Past RPA calculations have been performed using the quasi-boson approximation, effectively neglecting ground-state correlations. Using RPA, the next step along the path towards an ab initio description of collective properties will, therefore, be the inclusion of correlations. To that end, we extend the RPA formalism to include ground-state correlations from two different many-body methods, the in-medium similarity renormalization group (IM-SRG) and coupled-cluster theory with singles and doubles excitations (CCSD). Both methods have been applied with great success for the calculation of ground-state energies. We develop a formalism based on density matrices for CC-RPA that enables RPA based on an CCSD ground state. The use of IM-SRG transformed matrix elements gives us the possibility to include ground-state correlations even at the level of SRPA. For both methods we observe a strong upward shift in the strength distributions, and, unexpectedly, we find a good agreement between IM-RPA and CC-RPA results. The structure of the transitions remains largely unchanged. We conclude that correlations have significant impact on the energetic positions, but not on the structure of the strength distributions. Employing IM-SRPA we find a strong downward shift in energy similar to the case of SRPA. The agreement of both methods with experiment is comparable.