Toward extremes of angular momentum: Application of the Pfaffian algorithm in realistic calculations

Abstract

In a calculation of rotated matrix elements with angular momentum projection, the generalized Wick's theorem may encounter a practical problem of combinatorial complexity when the configurations have more than four quasi-particles (qps). The problem can be solved by employing the Pfaffian algorithm generally applicable to calculations of matrix elements for Hartree-Fock-Bogoliubov states with any number of qps. This breakthrough in many-body techniques enables studies of high-spin states in a shell-model framework. As the first application of the Pfaffian algorithm, the configuration space of the Projected Shell Model is expanded to include 6-qp states for both positive and negative parities. Taking 166 Hf as an example, we show that 6-qp states become the main configuration of the yrast band beyond spin I ≈ 34 , which explains the observed third back-bending in moment of inertia. Structures of multi-qp high-K isomers in 176 Hf are analyzed as another example.PACS numbers: 21.10. Re, 21.60.Cs, 23.20.Lv, 27.70.+q All nucleons of even-even nuclei couple pairwise in their ground state. Nuclear rotation brings an effect into the system that tends to break the nucleon pairs. This effect due to the rotation of nuclei was suggested in 1960 [1] in analogy to the electron pair breaking in superconductivity due to the existence of external magnetic fields. However, it was soon after realized that a sharp phase transition with pair collapse does not occur in nuclear systems due to the fact that nuclei have a finite size [2]. Another reason is that nucleons have an orbital angular momentum in addition to spin. The nucleons in the vicinity of the Fermi surfaces belong to subshells with rather different j-values, and therefore, they feel the Coriolis force very differently. Pairs in those orbitals with the highest angular momentum j, as for instance the neutron i 13/2 shell in the rare earth region, feel a strong Coriolis force, and therefore, break first (the Stephens-Simon effect [3]). They contribute to formation of 2-quasiparticle (qp) state as the main configuration of the yrast state. The Stephens-Simon effect [3] successfully explained the experimental observations of backbending in moment of inertia for rotating nuclei [4]. As a nucleus rotates faster and faster, subsequent pair-breakings can occur for the pairs from the next highest j orbitals. In the rare earth region, proton pairs in the h 11/2 shell are expected to break next, which was observed experimentally [5]. Measurements for further pair breakings at higher angular momenta are difficult. However, there have been early [6] and recent evidences [7,8] of breaking of three nucleon pairs, which form 6-qp states as the main configuration of the yrast sequence.Pair-breaking in nuclei can lead to formation of another special group of excited states: nuclear isomers. In a deformed potential a j-shell splits up into 2 j+1 K-states. Two or more states with high K quantum numbers can couple to form a high-K multi-qp configuration. The selection rules of elec- * El...

Shape-coexisting rotation in neutron-deficient Hg and Pb nuclei

“…Also, shape fluctuations can strongly affect K mixing and hence the decay properties of states [28]. The projected shell model (PSM) [18,19], as a calculation truncated in an angularmomentum-projected basis with qp configurations [29], is suitable for studying the behaviors of different qp states and mixing among them. However, the existing PSM applies to deformation-fixed calculations, i.e., assuming a fixed deformation in calculations.…”

Variation after projection with a triaxially deformed nuclear mean field

“…For efficiency, we use the new techniques of Refs. [19][20][21] to evaluate the kernels for projections.…”

Recent developments of the projected shell model based on many-body techniques

“…The Nilsson-BCS calculation defines a deformed qp basis from which the PSM model space is constructed. The multi-qp configurations up to 6-qp states for even-even nuclei are given as [18] …”

“…Moreover, it has been shown that the Pfaffian algorithm is very efficient also for calculating overlap matrix elements [11][12][13][14][15][16]. In particular, by means of Fermion coherent states and the Grassmann integral, one can derive an alternative approach to calculate the rotated matrix element for general qp states [17], which serves as a theoretical framework to extend the PSM model space [18].…”