The Model

In order to make a comparison between various methods and models transparent, in the present calculations we only consider one collective variable $q$, an average axial quadrupole moment of the mass distribution in the nucleus:

$$q = Q_{20}\equiv \langle\phi\vert\hat Q_{20 }\vert\phi\rangle,$$ (3)

where

$$\hat Q_{20 } = \sum_{i=1}^A\sqrt{\frac{16\pi}{5}}r_i^2Y_{20}(\theta_i,\phi_i)= \sum_{i=1}^A(2z_i^2-x_i^2-y_i^2)\,.$$ (4)

To account for pairing correlations, we applied the BCS approximation to the density-dependent $\delta$-pairing interaction (DDDI):

$$v_{\rm pair}(\vec r_1-\vec r_2) = v_{0t}\delta(\vec r_1-\vec r_2) \left[1-\frac{\rho(\vec r_1)}{\rho_0}\right],$$ (5)

where $t$=p/t for protons/neutrons, $\rho=\rho(\vec r)$ is the isoscalar nucleonic density, and $\rho_0$=0.16 fm$^{-3}$. The coupling constants $v_{0n}$=842 and $v_{0p}$=1020 MeVfm$^3$ were fitted to the experimental pairing gaps in $^{252}$Fm: $\Delta_n=0.696$ and $\Delta_p=0.803$MeV. The pairing-active space consisted of the lowest $Z$ ($N$) proton (neutron) single-particle states. We have also performed calculations with the seniority pairing force (with the constant strength parameters) given by

$$\begin{array}{l} G_{n}=\left[24.70-0.108 \left( N-Z \right)\... ..._{p}=\left[14.76+0.241 \left( N-Z \right)\right]/A. \end{array}$$ (6)

The calculations were performed using the code HFODD (v2.19l)[10,11,12] that allows for an arbitrary symmetry breaking. For the basis, we took the lowest 1140 single-particle states of the deformed harmonic oscillator. This corresponds to 14 oscillator shells at the spherical point. Based on the HFODD self-consistent wave-functions, the collective mass tensor components and ZPE corrections were computed. In principle, all the collective-tensor components $\lambda\mu$ with multipolarities $\lambda=2, 3, \dots, 9$ can be calculated.

In this study, only specific static fission paths are considered; they have been obtained in the calculations presented in Ref.[13]. Specifically, for the fermium isotopes considered, one predicts both reflection-symmetric (s) and reflection-asymmetric (a) fission valleys. There are two kinds of symmetric paths predicted; namely, the valley that corresponds to elongated fission fragments (E), and that with more compact (C) fragments that resemble spherical $^{132}$Sn clusters when approaching $^{264}$Fm. (The shorthand notation sEF, used in the following, means symmetric, elongated-fragments fission path; similarly aEF and sCF.)