A basis of cranking operators for the pairing-plus-quadrupole model

Abstract

Abstract. We investigate the RPA normal-mode coordinates in the pairingplus-quadrupole model, with an eye on simplifying the application of large amplitude collective motion techniques. At the Hartree-Bogoliubov minimum, the RPA modes are exactly the cranking operators of the collective coordinate approach. We examine the possibility of representing the self-consistent cranking operator by linear combinations of a limited number of one-body operators. We study the Sm nuclei as an example, and find that such representations exist in terms of operators that are state-dependent in a characteristic manner. The selection of proper collective variables is an important problem in the study of large amplitude collective motion. In the usual Constrained Hartree-Fock (CHF) or Hartree-Fock-Bogoliubov (CHFB) calculations, the collective subspaces are generated by a small number of one-body constraint (also called cranking) operators which are most commonly taken to be of the multipole form (r L Y LK ). In realistic calculations of processes such as fission, the number of coordinates to describe the full nuclear dynamics can easily become larger than can be dealt with in satisfactory manner, and a method to determine the optimal combination needs to be devised. Even assuming that such a method exists, there is no a priori reason to limit oneself to multipole operators, and the cranking operators should be determined by the nuclear collective dynamics itself, from the set of all one-body operators. This is clearly a difficult task, and one would like to be able to select a small group of operators, and find the optimal combination of these operators at each point of the collective surface.In our past work, we have investigated a theory of adiabatic large amplitude collective motion as a method to generate self-consistent collective subspaces (see reference [1] and references therein). The key ingredient of the method is the selfconsistent determination of the constraint operator, and as such it may provide an answer to the selection question discussed above. Using the local harmonic version (LHA) of the theory [1], we have recently embarked on a study of the properties of large amplitude collective motion in systems with pairing. We have dealt with two simple models: a semi-microscopic model of nucleons interacting through a pairing force, coupled to a single harmonic variable [2] and a fully microscopic O(4) model which may § Electronic address: T.Nakatsukasa@umist.ac.ukElectronic address: Niels.Walet@umist.ac.uk ¶ Electronic address: Giu.Dodang@th.u-psud.fr

Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

“…The more interesting study of the Baranger-Kumar pairing-plus-quadrupole model is described in Sec. IV E 2 [40,41]. The account of a model study that follows in Sec.…”

“…Though we hope that the reader will be impressed by the theoretical structure and by some of the new applications, it is also true, unfortunately, that so far we have not gotten back to the applications that stimulated this research in the first place. However, a new effort in this direction has been undertaken [40,41].…”

Mass dispersions from giant dipole resonances using the Balian–Vénéroni variational approach

“…The large number of computations required to evaluate (1) and the complexity of these calculations means that only a handful of calculations have been performed using this method and those calculations which have been performed have used simplified interactions and made use of symmetries (either spherical [5], or axial [6,7]) to render the problems tractable. However, modern advances in computing power mean that this approach can now be implemented using fully 3-dimensional TDHF codes with full Skyrme interactions [8,9,10,11].…”

Local harmonic approaches with approximate cranking operators

“…In Ref. [5], we have demonstrated for a few Sm isotopes that this can be remedied by introducing a cut-off energy for the 2qp matrix elements. For 208 Pb, which is spherical and has ∆ n = ∆ p = 0, the projection accidentally works well because the set of available 1p1h modes is very limited, ν(i 11/2 (i 13/2 ) −1 ), ν(g 9/2 (i 13/2 ) −1 ), π(h 9/2 (h 11/2 ) −1 ), π(f 7/2 (h 11/2 ) −1 ), all of which have almost the same particle-hole excitation energies.…”

“…The pairing-plusquadrupole (P+Q) Hamiltonian is adopted for the analysis because the RPA calculation is relatively easy due to the separability of the force (some of the results for the Sm isotopes have already been reported in Ref. [5]). Since the P+Q model is known to be able to describe collective phenomena involving both pairing and quadrupole degrees of freedom [6][7][8][9][10][11][12][13][14], we expect that a similar choice for the set of one-body operators should work for other realistic Hamiltonians as well.…”