Summary

The primary motivation of the present work has been to assess various approximations to the collective mass for fission. The collective mass plays a crucial role in determining the adiabatic collective motion of the nucleus and strongly impacts predicted half-lives. In the majority of previous studies, cranking approximation to collective mass has been employed, in which the time-odd fields are ignored and the collective momenta (i.e., derivatives with respect to collective coordinates) needed in the evaluation of the ATDHFB mass are calculated using the perturbation theory.

In our study, we performed the full ATDHFB cranking treatment of quadrupole inertia. The numerical evaluation of the derivatives appearing in ATDHFB mass expression poses a serious computational challenge as the accurate self-consistent HFB solutions need to be obtained at several neighboring points around every deformation along the fission pathway. By comparing three- and five-point approximations, we conclude that the three-point Lagrange formula provides a reasonable description of collective derivatives.

The main conclusions of this work can be summarized as follows.

• The collective masses obtained in non-perturbative treatment of derivatives show more variations due to shell structure changes along the fission path as compared to the perturbative approximation and GOA.
• The collective mass in full ATDHFB-C is very close to that obtained in ATDHFB-C$^{\rm c}$ and ATDBCS-C. This means that the diagonal approximation (40) for the HFB energy matrix is a very reasonable one.
• The ATDHFB-C$^{\rm p}$ and ATDHFB$^{\rm GOA}$ inertias exhibit very similar pattern, with the ATDHFB-C$^{\rm p}$ mass being systematically larger. A similar conclusion has been reached for the HF+BCS case.
• The main difference between HFB and HF+BCS calculations shows up in the perturbative treatment: the collective masses calculated in ATDBCS-C$^{\rm p}$ and ATDHFB-C$^{\rm p}$ are sometimes fairly different.
• Considering the differences between exact cranking results and ATDHFB-C$^{\rm p}$ and ATDHFB$^{\rm GOA}$ variants, we conclude that the perturbative treatment of derivatives cannot be justified.

The present work deals with the cranking approximation to ATDHFB in which only time-even mean fields have been kept when evaluating the collective inertia. The discussion of the full ATDHFB treatment, including the time-odd response that is expected to play a significant role in the description of collective dynamics [34], will be the subject of a forthcoming study.

This work was supported in part by the National Nuclear Security Administration by the National Nuclear Security Administration under the Stewardship Science Academic Alliances program through DOE Grant DE-FG52-09NA29461; by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), and DE-FC02-09ER41583 (UNEDF SciDAC Collaboration); by the NEUP grant DE-AC07-05ID14517 (sub award 00091100); by the Polish Ministry of Science under Contracts Nos. N N202 328234 and N202 231137; and by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme. Computational resources were provided by the National Center for Computational Sciences at Oak Ridge National Laboratory and the National Energy Research Scientific Computing