On the “Anharmonic Effects” on the Collective Oscillations in Spherical Even Nuclei. I

Abstract

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-pa:I;ticles 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. § 1. IntroductionRecently the microscopic theory of the collectiye oscillations of spherical even nuclei 1 ) has clarified that the "phonon" used in the Bohr-Mottelson model is a strongly correlated (bound) pair of quasi-particles with a definite total angular momentum. It was also confirmed by using the so-called "pairing plus quadrupole force" that this picture for the "phonon" reproduces the approximately correct energy of the first excited state (with spin and parity 2+) of spherical even nuclei and the E2 transition probability from this leveP)The essential approximation used in this theory is the "harmonic approximation" which is known under such various names as Sawada's approximation, random phase approximation, method of approximate second quantization, method of linearized equation of motion and Dyson's new Tamm-Dancoff method. Under this "harmonic approximation ", the second excited state of spherical even nuclei corresponds to the two-" phonon" state, which consists of a degenerate triplet (0+, 2+, 4+), and the E2 cross-over transition (2 2 + ---c)O) and the Ml transition (2 2 + ---c)2 1 +) are forbidden.As is well kno-vvn, actual properties of the second excited states of real nuclei show appreciable deviations from the simple regularities. Thus it has been desired to investigate the "anharmonic effects " giving rise to such deviations in the framework of the microscopic theory. 3 ) at colostateuniv on July 17, 2015 http://ptp.oxfordjournals.org/ Downloaded from T. Marumori, 1"'\d. Yamamura and A. TokunagaThe main purpose of this paper is to develop a new method which enables us to transcribe the dynamics of the system of even number of fermions into that of the boson system'correctly, with the purpose of analysing the "anharmonic effects". In .the "harmonic approximation", a pair of quasi-particles is replaced by an ideal boson. This replacement inevitably leads to neglecting two main effects; the dynamical effects due to the. residual interactions (in the original Hamiltonian) omitted under the " harmonic approximation " and the effect of the Pauli-principle among the quasi-particles comp...

Basic Notions

“…These studies have confronted by their own problems of not having hermicity and/or not conserving the boson number, although possible solutions have been considered. The concept of boson mapping idea has been applied to the many-body problems like in the treatments of Belyaev-Zelevinski type [51] and of Marumori type [52]. The former study proposed to map the operators so that the commutation relations of all the physically relevant operators are preserved.…”

“…The wavelet function is localized both in time and frequency domains. The wavelet function (denoted by Ψ ) must have the properties that it has zero mean and that it is square integrable (so-called admissibility condition) [60]: 52) which mean that Ψ must oscillate in a finite duration. These conditions allow one to analyze efficiently the localized signal, some part of which is particularly important like the relevant low-energy region around the absolute minimum of the microscopic constrained energy surface.…”

Microscopic studies of two-phonon giant resonances

“…(234) can be traced back to the last term of multipole operator (223) which cannot be projected onto the space of the phonon operators. On the other hand, applying Marumori expansion technique [89], one may expand the operator B τ (jj ′ ; λ − µ) ∼ α + α in an infinite sum of even-number phonon operators. Keeping only the first term of this expansion, the non-diagonal term of the model Hamiltonian, H int.…”

Boson dominance in nuclei

“…Of special importance are the works of Beliaev and Zelevinsky [3], who constructed a composite boson operator requiring its commutation with the nuclear Hamiltonian, and of Marumori, Yamamura, and Togunaga [4], who developed a method based on a map of fermion into boson matrix elements.…”