Basic Concepts Underlying Singular Perturbation Techniques

In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate x^* = ε^(-1)x. In a secular-type problem x and x^* are used simultaneously. This paper discusses layer-type problems in which x^* is used in a thin layer and x outside this layer. Assume one seeks approximations to a function f(x,ε), uniformly valid to some order in ε for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define "domain of validity" one needs to consider intervals whose endpoints depend on epsilon. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions "matching." The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions

Generalization of the NpNnN_pN_n Scheme and the Structure of the Valence Space

The $N_pN_n$ scheme, which has been extensively applied to even-even nuclei, is found to be a very good benchmark for odd-even, even-odd, and doubly-odd nuclei as well. There are no apparent shifts in the correlations for these four classes of nuclei. The compact correlations highlight the deviant behavior of the Z=78 nuclei, are used to deduce effective valence proton numbers near Z=64, and to study the evolution of the Z=64 subshell gap.Comment: 10 pages, 4 figure

Quadrupole collectivity in random two-body ensembles

We conduct a systematic investigation of the nuclear collective dynamics that emerges in systems with random two-body interactions. We explore the development of the mean field and study its geometry. We investigate multipole collectivities in the many-body spectra and their dependence on the underlying two-body interaction Hamiltonian. The quadrupole-quadrupole interaction component appears to be dynamically dominating in two-body random ensembles. This quadrupole coherence leads to rotational spectral features and thus suggests the formation of the deformed mean-field of a specific geometry

Evolution of the N=50 gap from Z=30 to Z=38 and extrapolation towards 78Ni

The evolution of the N=50 gap is analyzed as a function of the occupation of the proton f5/2 and p3/2 orbits. It is based on experimental atomic masses, using three different methods of one or two-neutron separation energies of ground or isomeric states. We show that the effect of correlations, which is maximized at Z=32 could be misleading with respect to the determination of the size of the shell gap, especially when using the method with two-neutron separation energies. From the methods that are the least perturbed by correlations, we estimate the N=50 spherical shell gap in 78Ni. Whether 78Ni would be a rigid spherical or deformed nucleus is discussed in comparison with other nuclei in which similar nucleon-nucleon forces are at play.Comment: 7 pages, 8 figures, accepted for publication PRC (22 december 2011

SU(3) quasidynamical symmetry underlying the Alhassid--Whelan arc of regularity

The first example of an empirically manifested quasi dynamical symmetry trajectory in the interior of the symmetry triangle of the Interacting Boson Approximation model is identified for large boson numbers. Along this curve, extending from SU(3) to near the critical line of the first order phase transition, spectra exhibit nearly the same degeneracies that characterize the low energy levels of SU(3). This trajectory also lies close to the Alhassid-Whelan arc of regularity, the unique interior region of regular behavior connecting the SU(3) and U(5) vertices, thus offering a possible symmetry-based interpretation of that narrow zone of regularity amidst regions of more chaotic spectra.Comment: 4 pages, LaTeX, 5 eps figure

New magic number for neutron rich Sn isotopes

The variation of E(2+_1) of (134-140)Sn calculated with empirical SMPN interaction has striking similarity with that of experimental E(2+_1) of even-even (18-22)O and (42-48)Ca, showing clearly that N=84-88 spectra exhibit the effect of gradual filling up of \nu(2f_{7/2}) orbital which finally culminates in a new shell closure at N=90. Realistic two-body interaction CWG does not show this feature. Spin-tensor decomposition of SMPN and CWG interactions and variation of their components with valence neutron number reveals that the origin of the shell closure at 140Sn lies in the three body effects. Calculations with CWG3, which is obtained by including a simple three-body monopole term in the CWG interaction, predict decreasing E(2+_1) for (134-138)Sn and a shell closure at 140Sn.Comment: 4 pages, 5 figure