Quasiboson Approximation Research Articles

Investigation of ground state correlations in an extended Lipkin-Meshkov-Glick model

The strength of ground state correlations calculated from the RPA and quasiboson approximation is known to differ by a factor of two from that, calculated by various other methods. This contradiction is investigated in detail in an extended Lipkin-Meshkov-Glick model. It is proved that the RPA + quasiboson approximation formula gives the correct ground state correlations if only collective states are included, as suggested by Rowe. The notion of a collective state is defined quantitatively.

Higher Random-Phase Approximation as an Approximation to the Equations of Motion

Starting from the equations of motion expressed as ground-state expectation values, we have derived a higher-order random-phase approximation (RPA) for excitation frequencies of low-lying states. The matrix elements in the expectation value are obtained up to terms linear in the ground-state correlation coefficients. We represent the ground state as eU|HF〉, where U is a linear combination of two particle-hole operators, and |HF〉 is the Hartree-Fock ground state. We then retain terms only up to those linear in the correlation coefficients in the equation determining the ground state. This equation and that for the excitation energy are then solved self-consistently. We do not make the quasiboson approximation in this procedure, and explicitly discuss the overcounting characteristics of this approximation. The resulting equations have the same form as those of the RPA, but this higher RPA removes many deficiencies of the RPA.

Application of a Higher RPA to a Model π-Electron System

We have applied a proposed higher-order random phase approximation (RPA) to the simple model system of two π electrons in the double bond of ethylene. The higher-order RPA removes some difficulties involved in the usual RPA, but retains the form of the RPA equations. To derive the higher RPA we retain all terms linear in the ground state correlation coefficients in the equations of motion for the excitation energy and in the equation determining the ground state. These equations are solved self-consistently and are simpler to handle than a configuration interaction solution in a realistic example. We do not use the quasiboson approximation.

RPA ground state correlations from perturbation theory

A correlated ground state wave function is derived for a non-degenerate system by explicitly summing up the perturbation theory diagrams of the RPA to all orders. Exact expressions are also given for the correlation energy, wave function normalisation and single-particle occupation probabilities. The major difference from previous results obtained with the quasi-boson approximation lies in the treatment of the lowest order diagrams. These are overcounted by a factor of two by the quasi-boson approximation if antisymmetrised matrix elements are used. As an example the case of 16O is discussed.

The pairing force in the quasi-boson approximation

A modified form of the BCS + RPA method for 0 + states is presented which changes the zero-frequency boson and the effective Hamiltonian. The modified theory is closely related to the Nogami method. Complete BCS + RPA state vectors are calculated and displayed graphically for a model system of two j = 9 2 levels, illustrating the improvements produced by the modified theory.

Comparison of various approximations for the ground state of 16O

The amount and nature of 2 particle-2 hole correlations in the ground state of 16O was investigated using various approximations. First 2ħω excitations were considered with the particles in the 2s-1d shell and the holes in the 1 p shell. Matrix diagonalization, with spurious states removed, resulted in 22% 2p-2h ground state correlations and a depression of the ground state energy of 7.9 MeV; if spurious states were incorrectly included, then both numbers were nearly doubled. The corresponding numbers for the RPA were 43% and 8.2 MeV using the quasi-boson approximation. This approximation is suggested to be inaccurate. Ordinary perturbation theory gave results remarkably close to matrix diagonalization. The corresponding wave functions had an overlap of .993. The correlations were spread over very many j-j coupling states but could be localized to relatively few states in the L-S representation. Perturbation theory was used in a larger space where the 2 particle-2 hole energy could be greater than 2ħω. This led to 32% 2p-2h correlations and the ground state depression was 24 MeV. The correlations with 2ħω and 4ħω energy were comparable. A dispersion method gave less correlation but qualitatively similar results. In the smaller configuration space the first excited 0 + state was at 24 MeV excitation energy relative to the ground state, but a more realistic estimate is probably about 14 MeV. This state looked slightly simpler in j-j rather than L-S coupling.

Intrinsic deformations of collective 0+ states

It is shown that 0+ pairing vibrations are associated with rather large mean square values of the quadrupole moment, indicating a pronounced intrinsic deformation. This deformation is demonstrated using a two level model, where the pairing vibration can be treated analytically within the quasi boson approximation. The results are compared with the data for the 6.06 MeV state in O16.

Grundzustandskorrelationen bei kollektiven Kernanregungen

Applying the Fermimodel ground state correlations are calculated in the quasi-particle formalism. Their dependence on the interaction potential strength, distance of energy levels and pairing is investigated. By means of the normalization constant of the ground state we criticize the quasi-boson approximation.

A remark on the generator-coordinate method and the quasi-boson approximation

By use of Bargmann's integral transform and the coherent state formalism, it is proved that the generator-coordinate method with quadratic approximation to the kernels (=Gaussian kernels) is completely equivalent to the quasi-boson approximation to the original Hamiltonian.

Isospin Invariance and the Pairing-Force Problem

The isoscalar pairing-force Hamiltonian is studied first in the BCS-like approximation, including neutron-neutron, proton-proton, and neutron-proton interactions. Constraining $〈\stackrel{^}{N}〉$ and $〈{\stackrel{^}{T}}^{2}〉$, we find a class of BCS-like wave functions which give the same minimum energy; they differ in the expectation value of ${\stackrel{^}{T}}_{z}$, $\ensuremath{-}T<~〈{\stackrel{^}{T}}_{z}〉<~T$. For the limit $〈{\stackrel{^}{T}}_{z}〉=T$, the minimum occurs at zero neutron-proton interaction. The residual interactions, those neglected by the BCS approximation, are treated by the quasiboson approximation. The spurious effects of number and isospin dispersion are identified. A procedure for explicitly displaying the number and isospin dependence of the energy is given, together with one for obtaining excited states with all possible isospins. Finally, as an example, the degenerate case is worked out, and agreement with the exact solution, including ground-state neutron-proton correlations, to the order considered, is demonstrated.

Even-parity states of 16O in a pairing-plus-quadrupole model

In this paper we give an outline of the generalized quasiboson approximation and use this method to investigate the interplay of pairing and Q-Q force in the lowest even-parity states of 16O. In particular we discuss the equilibrium deformation of the self-consistent field in the ground and first excited state and the dependence of the energies and some transition rates on the interaction parameters.

The ground state and the odd-parity spectrum of 40Ca

The energies and eigenvectors of the odd-parity states of 40Ca are given in the framework of the particle-hole model, using recently determined single-particle energies. The wave function of the ground state of 40Ca is evaluated within the framework of the quasi-boson approximation. It appears to be only 30% a pure shell-model state. Comparison is made between the theoretical occupation numbers and the experimental spectroscopic factors obtained in one-nucleon transfer reactions.

Beta-vibrations in even nuclei

The properties of β-vibrations in even nuclei are calculated within the quasi-boson approximation. A modification of the usual method is necessary in order to eliminate effects due to the existence of spurious states. The agreement of the theoretical results with the experimental values is satisfactory.