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Physical Review

Application of the Concept of Ideal Collective Coordinate to a Simple Microscopic Model

Abraham Klein, Robert E. Johnson

Published Jul 20, 1968• 5 citations
Publisher

Abstract

In a previous paper the concept of an ideal collective coordinate was introduced as the basis for a phenomenological theory of nuclear collective motion. The application of this idea to the microscopic domain is illustrated by the study of a simple two-level model with monopole-monopole interactions, previously considered by Lipkin and collaborators. The complete scheme of calculation which is presented is seen as an alternative to the boson expansion methods currently in fashion for the study of anharmonic vibrations. The new scheme should, however, be applicable to a more extended range of problems such as the adiabatic limit for vibrations and the general problem of rotation-vibration interactions.

DOI
10.1103/physrev.171.1224

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In a previous paper the concept of an ideal collective coordinate was introduced as the basis for a phenomenological theory of nuclear collective motion. The application of this idea to the microscopic domain is illustrated by the study of a simple two-level model with monopole-monopole interactions, previously considered by Lipkin and collaborators. The complete scheme of calculation which is presented is seen as an alternative to the boson expansion methods currently in fashion for the study of anharmonic vibrations. The new scheme should, however, be applicable to a more extended range of problems such as the adiabatic limit for vibrations and the general problem of rotation-vibration interactions.
The concept of an ideal collective coordinate is introduced by means of the following example: Consider a one-dimensional vibration of a many-body system in the sense that a large subset of states ∣n〉|n〉∣n〉 of the system exhibits an energy spectrum and relative transition probabilities following the laws of the (in general anharmonic) oscillator described by H\mathcal{H}H (${p}_{\ensuremath{\alpha}}, \ensuremath{\alpha}$) $(\ensuremath{\alpha}|n)={\ensuremath{\omega}}_{n}(\ensuremath{\alpha}|n)$. We suppose the set of many-body states ∣n〉|n〉∣n〉 to extend indefinitely, and we take the transform $|\ensuremath{\alpha}〉=\ensuremath{\Sigma}|n〉$ ($n|\ensuremath{\alpha}$) to define a many-body generating state of the band which is precisely localized in $\ensuremath{\alpha}$ space. The basic assumption of collectivity, that changing the state of at most a few particles cannot much alter the value of $\ensuremath{\alpha}$, is shown to be sufficient to derive a phenomenological theory from the many-body starting point. The phenomenological aspects of a recent theory of rotations due to Villars is seen to be contained in the above formulation as a special case. A brief review is given of the generator coordinate and similar projection methods in order to exhibit their relationship with the present method.
In phenomenological descriptions of nuclear vibrations a close connection is implied or expressed between the most-collective transition operator and the harmonic-oscillator (boson) variables which are introduced to describe the collective motion. With the help of a widely used example - the monopole vibration model of Meshkov, Lipkin, and Glick - it is shown how this relationship suggests, for the microscopic case, a new and flexible class of boson representations. As shown, these are capable of yielding accurate descriptions of the system not only in the extremes of weak and strong coupling, but in the usually more inaccessible intermediate regions as well.
It is shown how a simple reinterpretation of the "transition-operator boson method" eliminates the difficulty noted by Okubo. The revised method is applied to the problem of anharmonic motion in the Lipkin model. The occurrence of a non-positive-definite mass parameter, as obtained by Holzwarth and by Giraud and Grammaticos, is then exposed as the easily avoidable consequence of an inauspicious choice of collective coordinate.NUCLEAR STRUCTURE Approximate many-body method applied to a solvable model.
The boson expansion method was developed with the purpose of continuously following even nuclear systems near their ground states, from the region of slightly anharmonic vibrations through the transitional region formed by a nonsharp phase transition to a region of permanent distortion characterized by the occurrence of rotational spectra. However, as the boson quantum was defined in the extreme vibrational limit, it was not possible to obtain quantitatively correct spectra in the transitional or deformed region. Believing that the major cause for the lack of success was the disregard of the self- consistent change of the basic degree of freedom, a simple variational method to be used with the boson expansion technique is presented. Its ability to cover the vibration-to-rotation transition is demonstrated by comparison with an exact solution for a four-level model with pairing force. Further the coverage of a quadrupole phase transition is shown for a system interacting via pairing-Plus- quadrupole forces. A successful calculation of static quadrupole moments of Cd and Te isotopes is presented. Finally the approach in general situations with more complicated forces is discussed. (auth)
A new method which enables us to transcribe the dynamics of the system of even number of fermions into that of the boson system is developed, with the purpose of analysing the “anharmonic effects” on the collective oscillations in spherical even nuclei from the standpoint of the microscopic theory of the collective excitations. In the “harmonic approximation”, a pair of bound quasi-particles is replaced by the “phonon” as an ideal boson. This replacement inevitably leads to neglecting two main effects; the dynamical effect due to the residual interactions omitted under the “harmonic approximation” and the effect of the Pauli-principle among the quasi-particles composing the different pairs. According to our theory, the two effects which cause “anharmonic effects” can be evaluated correctly, in principle. It is not our purpose to go into detailed quantitative calculations, but rather to develop the basic idea.