Spatial dependence of pairing in deformed nuclei
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NUCLEI Theory
Spatial Dependence of Pairing in Deformed Nuclei* E. B. Balbutsev1), L. A. Malov1)** , and P. Schuck2) Received December 21, 2010
Abstract—The solution of time-dependent Hartree–Fock–Bogoliubov equations by the Wignerfunction-moments method leads to the appearance of refined low-lying modes whose description requires the accurate knowledge of the anomalous density matrix. It is shown that calculations with Woods–Saxon potential satisfy this requirement, producing an anomalous density matrix of the same quality as more complicated calculations with realistic forces. DOI: 10.1134/S1063778811110020
1. INTRODUCTION
calculations with various realistic forces (Argonne, Gogny, Skyrme).
The problem of the spatial dependence of the pairing field is at the moment the object of strong interest of nuclear theorists [1–6] because of the necessity to explain the properties of nuclei disposed far from the beta-stability line. We met this problem when studying the nuclear scissors mode. It is known [7, 8] that one must take into account pair correlations to describe correctly nuclear scissors motion, therefore Time-Dependent Hartree–Fock– Bogoliubov (TDHFB) equations should be the natural instrument to work with. Being the isovector mode the nuclear scissors appear in the frame of our approach (Wigner-function moments) together with isoscalar low-lying excitations (ISLLE), which are generated by quantum corrections to the semiclassical limit of TDHFB equations, implying their subtle structure. In this sense ISLLE modes turn out to be even more subtle and refined modes than the scissors modes. Naturally, one cannot use semiclassical expressions for the anomalous density and pairing field to describe such excitations—one needs quantum-mechanical expressions, which can be found by solving static HFB equations. In the previous paper [9] we performed methodical calculations of the anomalous density, pairing field and coherence length for the spherical nucleus 134 Ba using Woods–Saxon singleparticle wave functions as a test case. Our results turned out to be in very good agreement with the respective results of [1–5] obtained in self-consistent ∗
The text was submitted by the authors in English. Joint Institute for Nuclear Research, Dubna, Russia. 2) ´ Institut de Physique Nucleaire, CNRS and Universite´ ParisSud, Orsay, France. ** E-mail: [email protected] 1)
The aim of this paper is to repeat the analogous calculations for deformed nuclei (again with the Woods–Saxon potential). It is necessary to note that all known HFB calculations of deformed nuclei [6, 10–12] use the oscillator basis, which allows one to transform the anomalous density to the relative and center-of-mass coordinates with the help of Moshinski coefficients. In this case one is forced to use very large basis sets to get the correct asymptotic behavior of the solution. Our approach is free from this problem because the Woods–Saxon potential ensures the correct asymptotics automatically.
2. ANOMALOUS DENSITY MATRIX The anomalous density matrix is defined [
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