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Progress of Theoretical Physics

Unitary-Model-Operator Approach to Nuclear Effective Interaction. I: Description of Short-Range Correlations

K. Suzuki

Published Nov 1, 1982• 16 citations
Publisher

Abstract

A general theory is proposed for the description of short-range correlations of two nucleons in nuclei in the framework of the unitary-model-operator approach. A new effective interaction is derived through the unitary transformation of the original Hamiltonian. The transformed Hamiltonian is given in a cluster expansion form, and the structure of the one- and two-body cluster terms is studied. New aspects of the present approach, compared with the G-matrix theory, are; (i) the new effective interaction is E-independent and Hermitian, (ii) it satisfies the decoupling condition between low- and high-momentum two-particle states, and (iii) it contains both of the contributions of ladder and folded diagrams. It is suggested that the properties (i)∼(iii) of the new effective interaction may offer considerable advantages in accounting manybody correlations and deriving effective interaction among valence nucleons.

DOI
10.1143/ptp.68.1627

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•Jan 1, 2006•0 citations
A general theory is proposed for the description of short-range correlations of two nucleons in nuclei in the framework of the unitary-model-operator approach. A new effective interaction is derived through the unitary transformation of the original Hamiltonian. The transformed Hamiltonian is given in a cluster expansion form, and the structure of the one- and two-body cluster terms is studied. New aspects of the present approach, compared with the G-matrix theory, are; (i) the new effective interaction is E-independent and Hermitian, (ii) it satisfies the decoupling condition between low- and high-momentum two-particle states, and (iii) it contains both of the contributions of ladder and folded diagrams. It is suggested that the properties (i)∼(iii) of the new effective interaction may offer considerable advantages in accounting manybody correlations and deriving effective interaction among valence nucleons.
A theory is formulated to derive an effective interaction in nuclei in the framework of the unitarymodel-operator approach.A unitary operator is introduced to describe two-body correlations, and it is determined from the condition that the effective interaction should be decoupled between the lowmomentum model space and the high-momentum space.The unitary-transformed Hamiltonian is represented in a cluster expansion form.The one-and two-body cluster terms are mainly discussed.The theory is applied to the calculation of the energies of the ground-and one-body states with the Reid soft core potential.A large gain of the energy has been obtained for the ground state in comparison with the result of the usual G-matrix theory.As for the one-body energies obtained, agreement with the experimental values is fairly good.Although the discrepancies with the experiments still remain at the 1O~ 15 % level, the present approach shows to have some desirable features compared with the nuclear many-body theories proposed so far. § 1. IntroductionA fundamental problem in nuclear physics is to clarify the properties of nuclei in terms of the interaction among their constituent particles.Although there have been many theories or approaches, we have not yet reached the goal of reproducing successfully the experimental data.The investigation of nuclear properties, in the usual nonrelativistic approach, proceeds in two steps: The first one is to determine the interaction among nucleons, and the second is to solve the many-body SchrOdinger equation.Concerning the nucleon-nucleon force, it seems that the phenomenological potential, such as the Reid potential, would be incomplete and one should wait until quark dynamics, probably quantum chromodynamics (QCD), solves finally the problem o~ nuclear force.It is of no doubt that in order to fit the experiments we need more accurate knowledge on the nuclear force including the threebody force.However, if we want to reach the goal, we need also a good many-body theory.Regardless of the accuracy of the nuclear force, we shall not be able to solve the problem unless we have a way of constructing an accurate solution for a many-body system.We believe that many-body theory should share the responsibility for the discrepancy between theory and experiment.It will be sure that nuclear many-body theory is still to be improved.Among many theories for nuclear many-body problem, the most widely applied method is, needless to say, the linked-diagram perturbation theory.This approach may also be referred to as the folded-diagram theory based on the G-matrix interaction especially due 'to Brandow 1l and Kuo et al. 2l An outstanding problem in this approach is the convergence of the perturbation expansion.Since Schucan and Weidenmliller 3 ) pointed out the divergence of the perturbation expansion due to the presence of intruder states, this difficulty has long been a serious problem.Present status of this problem is
In this work a unitary correlation operator is presented that explicitly describes the short-ranged central and tensor correlations in the nuclear many-body system. These short-ranged correlations are induced by the repulsive core and the pronounced tensor force of realistic nucleon-nucleon interactions and cannot be described by the simple many-body states of a mean-field or shell model approach. The unitary correlation operator is discussed for the Argonne V18 and the Bonn-A interactions. Applying the correlation operator onto the Hamiltonian a common effective interaction for low momenta is obtained. Calculations for 4He using the one- and two-body part of the correlated Hamiltonian compare favorably with exact many-body methods. Calculations for 16O and 40Ca which are not possible with exact methods are performed using harmonic oscillator shell model states. The observed deviation from the experimental binding energies and radii is attributed to the missing three-body forces. The correlated interaction in a basis-free representation is used in the Fermionic Molecular Dynamics model to calculate the nuclei of the p- and the sd-shell. In this model the uncorrelated many-particle state is given as a Slaterdeterminant of Gaussian wave-packets with spin and isospin degrees of freedom which allows a consistent description of spherically symmetric, intrinsically deformed and clustered nuclei.
Correlations play a crucial role in the nuclear many-body problem. We give an overview of recent developments in nuclear structure theory aiming at the description of these interaction-induced correlations by unitary transformations. We focus on the Unitary Correlation Operator Method (UCOM), which offers a very intuitive, universal and robust approach for the treatment of short-range correlations. We discuss the UCOM formalism in detail and highlight the connections to other methods for the description of short-range correlations and the construction of effective interactions. In particular, we juxtapose UCOM with the Similarity Renormalization Group (SRG) approach, which implements the unitary transformation of the Hamiltonian through a very flexi-ble flow-equation formulation. The UCOM- and SRG-transformed interactions are compared on the level of matrix elements and in many-body calculations within the no-core shell model and with Hartree-Fock plus perturbation theory for a variety of nuclei and observables. These calcu-lations provide a detailed picture of the similarities and differences as well as the advantages and limitations of unitary transformation methods.
The general nuclear contact matrices are defined, taking into consideration all partial waves and finite-range interactions, extending Tan's work on the zero range model. The properties of these matrices are discussed and the relations between the contacts and the one-nucleon and two-nucleon momentum distributions are derived. Using these relations, a new asymptotic connection between the one-nucleon and two-nucleon momentum distributions, describing the two-body short-range correlations in nuclei, is obtained. Using available numerical data, we extract a few connections between the different contacts and verify their relations to the momentum distributions. The numerical data also allows us to identify the main nucleon momentum range affected by two-body short-range correlations. Utilizing these relations and the numerical data, we also verify a previous independent prediction-connecting between the Levinger constant and the contacts. This work provides an important indication for the relevance of the contact formalism to nuclear systems, and should open the path for revealing more useful relations between the contacts and interesting quantities of nuclei and nuclear matter.
The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
The equation of state of symmetric nuclear matter is addressed starting both from a realistic interaction derived from nucleon-nucleon scattering processes and from a low-momentum effective potential. The approach is based on finite temperature Green’s functions. The internal energy per particle is estimated from the summation of diagrams and through the Galitskii-Koltun’s sum rule. At present, there is not a unique and reliable theory of nuclear interactions. From the study of nucleon-nucleon scattering processes different models for the basic interaction between nucleons have been developed, but difficulties arise when they are used in calculations which aim to reproduce the properties of bound nuclei. These models all reproduce the experimental N-N phase shifts up to the threshold for pion production with high accuracy, but differ in the short-wavelength region, where less experimental constraints are available. In deriving the properties of dense nuclear matter, if we start from a bare N-N potential we have to take into account the complex correlations induced