Collective motion from various aspects
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ollective Motion from Various Aspects¶ E. B. Balbutsev Joint Institute for Nuclear Research, 141980 Dubna, Moscow oblast, Russia Abstract—Three methods to describe collective motion, the Random Phase Approximation (RPA), Wigner Function Moments (WFM), and Green’s Function (GF) methods, are compared in detail and their physical content analyzed in the example of a simple model, a harmonic oscillator with quadrupole–quadrupole residual interaction. It is shown that they give identical formulae for eigenfrequencies and transition probabilities of all collective excitations of the model. The exact relation between the RPA and WFM variables and the respective dynamical equations is established. The transformation of the RPA spectrum into one of WFM is explained. The very close connection of the WFM method with the GF method is demonstrated. A differential equation describing the current lines of RPA modes is established and the current lines of the scissors mode are analyzed as a superposition of rotational and irrotational components. The orthogonality of the spurious state to all physical states is proved rigorously. PACS numbers: 21.60.Ev, 21.60.Jz, 24.30.Cz DOI: 10.1134/S1063779608060038
1. INTRODUCTION The aim of the present paper is the systematic comparison of three methods to describe the collective motion. As an example, their competition in the description of the nuclear scissors mode will be considered. This very curious excitation was predicted thirty years ago [1, 2]. Its experimental discovery [3] has initiated a cascade of theoretical studies. An excellent review of their development over twenty years was given by Zawischa [4]. Very briefly, the situation can be described in the following way. All microscopic calculations with effective forces reproduce experimental data with respect to the position and the strength of the scissors mode, some of them [5] also giving reasonable fragmentation of its strength. However, the situation is more obscure in regard to simple phenomenological models whose aim is to explain the physics of the phenomenon and to interpret it in the most simple and transparent terms. Noticeable discord of the opinions of various authors must be observed here [4]. It will be interesting to compare the possibilities, advantages, and disadvantages of various methods in the description of all subtleties of this mode. Full analysis of the scissors mode in the framework of a solvable model (harmonic oscillator with quadrupole–quadrupole residual interaction (HO + QQ)) was given in [6]. Several points in understanding the nature of this mode were clarified, for example, its coexistence with the isovector giant quadrupole resonance (IVGQR), the decisive role of the Fermi surface deformation, and several more. The Wigner Function Moments (WFM) method was applied to derive analytical expressions for currents of both coexisting modes, their excitation energies, and magnetic and electric ¶ The
text was submitted by the author in English.
transition probabilities. Our formulae for energies turned o
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