Results

The illustrative calculations were performed for the nucleus $^{256}$Fm by using the SkM$^*$ energy density functional [25] in the particle-hole (ph) channel. In the particle-particle (pp) channel we employed the density-dependent pairing interaction in the mixed variant of Refs. [26,27]:

$$V_\tau(\vec r) = V_{\tau 0}\left(1-\rho(\vec r)/2\rho_0\right)\delta(\vec r)\,,$$ (67)

where $\tau=n,p$ and $\rho_0=0.16\,$fm$^{-1}$. To test the accuracy of various approximations, we carried out both HF+BCS and HFB calculations. The pairing interaction strengths, which were adjusted to reproduce the neutron and proton ground-state pairing gaps in $^{252}$Fm, are (in $\mbox{MeV fm}^3$):

$$V_{n0}=-372.0\,,\quad V_{p0}=-438.0\,.$$ (68)

The fission pathways were studied in the previous Ref. [28] using the SkM$^*$-HF-BCS approach with the seniority pairing interaction. It has been found that the SkM$^*$ energy density functional favors the asymmetric fission pathway in $^{256}$Fm, and our HFB results are consistent with this result. The one-dimensional collective pathway, determined by the axial quadrupole moment $q=Q_{20}$, was obtained by means of the HFB solver HFODD [29].

Figure 1: (Color online) The quadrupole mass parameter (top: total; middle: neutron contribution; bottom: proton contribution) along the fission pathway of $^{256}$Fm calculated in SkM$^*$+HF+BCS as a function of the mass quadrupole moment. The ATDBCS-C results (triangles) are compared with those obtained in the perturbative cranking approximation (ATDBCS-C$^{\rm p}$, circles) and Gaussian overlap approximation (ATDBCS$^{\rm GOA}$, diamonds). See text for details.

Using the self-consistent solutions along the fission pathway, we calculate the collective quadrupole mass parameter using various approximations described in Sec. 3. First, we discuss results obtained within the HF+BCS formalism (dubbed ATDBCS). Figure 1 compares the results of the non-perturbative cranking approach (ATDBCS-C) with the perturbative cranking approximation (ATDBCS-C$^{\rm p}$) and Gaussian overlap approximation (ATDBCS$^{\rm GOA}$). In ATDBCS-C the derivatives of the density matrices and the mean-field potentials have been obtained using the Lagrange formula, which requires the knowledge of self-consistent solutions in several neighboring deformation points. We have evaluated the density matrices for quadrupole deformations ranging from $Q_{20}$= 0 to 320b in steps of 1b. The derivatives were obtained by using the 3-point Lagrange formula (53), and also the 5-point Lagrange formula [16]. The results for collective mass obtained with 3-point and 5-point expressions differ only in the third decimal place; hence, in the following, we shall stick to the 3-point Lagrange formula. It needs to be stressed that - in order to guarantee consistent labeling of canonical states - the underlying single-particle basis should be identical for all three points in Eq. (53) involved in the derivative evaluation. This has been achieved by performing HF+BCS calculations using the same basis deformation for all neighboring points.

As seen in Fig. 1, the total ATDBCS-C mass exhibits a rather irregular behavior characterized by the presence of several sharp maxima. Some of these peak-like structures, although considerably suppressed, also show up in ATDBCS-C$^{\rm p}$ and ATDBCS$^{\rm GOA}$.

Figure 2: (Color online) Total HF+BCS energy $E_{\rm tot}$ (top), pairing energies $E_{\rm pp}$ (total, proton, and neutron; middle), and the HF energy $E_{\rm ph}$ (bottom) calculated along the fission pathway for $^{256}$Fm. The borders between different self-consistent configurations are marked by vertical solid lines. The dotted lines mark positions of peaks in the collective ATDBCS-C mass parameter of Fig. 1.

To unravel the origin of the peak structures in the collective mass, the total energy of $^{256}$Fm is depicted in Fig. 2, together with corresponding pairing-energy $(E_{\rm pp})$ and HF energy $(E_{\rm ph})$ contributions. The one-dimensional total energy curve shows several discontinuities due to intersections of close-lying energy sheets (surfaces) with very different mean fields. The corresponding pathways can in fact be well separated when studied in more than one dimension of the collective manifold [28]. The diabatic jumps between various energy sheets have been disregarded when computing the collective inertia shown in Fig. 1. Indeed, in such cases the adiabatic theory is unable to provide a meaningful result for the collective mass.

There are also configuration changes within each pathway. Because of pairing correlations, these changes are adiabatic in character [30,13]. Still, they manifest themselves in the collective inertias through the appearance of peaks [31,32,33]. Figures 1 and 2 nicely illustrate this point: the peaks in the collective ATDBCS-C mass parameter appear in the regions of large local variations in $E_{\rm pp}$ and $E_{\rm ph}$ that are indicative of changes in the shell structure with elongation. (We note that the local variations in the total energy are much weaker than those in pairing and HF energies, due to the well-known anticorrelation between pairing and HF energies, clearly seen in Fig. 2.)

Two general conclusions can be drawn from the results of Fig. 1. First, the ATDBCS-C$^{\rm p}$ and ATDBCS$^{\rm GOA}$ inertia show fairly similar behavior, with the ATDBCS-C$^{\rm p}$ mass being systematically larger. Second, the exact treatment of derivative terms in ATDBCS-C gives rise to less adiabatic behavior in the corresponding collective mass.

The results obtained within the HFB framework are presented in Fig. 3. The results obtained in the canonical approximation ATDHFB-C$^{\rm c}$ (42), perturbative treatment of derivatives ATDHFB-C$^{\rm p}$ (57), and ATDHFB$^{\rm GOA}$ (62) were obtained by using the canonical HFB wave functions and employing the diagonal (``equivalent BCS'') ansatz. The ATDHFB-C calculations (34) were carried out in the full quasiparticle basis.

Figure 3: (Color online) The quadrupole mass parameter (top: total; middle: neutron contribution; bottom: proton contribution) along the fission pathway of $^{256}$Fm calculated in SkM$^*$+HFB as a function of the mass quadrupole moment. The ATDHFB-C results (triangles) are compared with those obtained in the canonical approximation (ATDHFB-C$^{\rm c}$, squares), perturbative cranking approximation (ATDHFB-C$^{\rm p}$, dots), and Gaussian overlap approximation (ATDHFB$^{\rm GOA}$, diamonds). See text for details.

The most interesting finding is that the collective mass in ATDHFB-C is very close to that obtained in ATDHFB-C$^{\rm c}$. Similar to the HF-BCS case, the ATDHFB-C$^{\rm p}$ and ATDHFB$^{\rm GOA}$ results follow each other with the ATDHFB-C$^{\rm p}$ mass being systematically larger. Again, the exact treatment of derivatives gives rise to less adiabatic behavior of collective mass that manifests itself through the presence of peaks.

In Ref. [9], the quadrupole collective mass was evaluated in the canonical basis and exhibited a singular behavior at certain deformation points. The primary reason for this singularity is due to the pairing collapse at certain deformations that results in unphysical phase transition and the presence of unavoided level crossings. In our work, the peak structures are present at nonzero pairing and are related to the shell structure changes along the fission pathways.

In order to highlight the differences between HFB and BCS treatments, Fig. 4 shows the quadrupole masses obtained in these approaches in the region of the ground-state minimum and the inner fission barrier of $^{256}$Fm ( $Q_{20}\in[20,100]$). This region plays a crucial role in the evaluation of fission half-lives. It is evident from Fig. 4 that the non-perturbative cranking masses ATDBCS-C and ATDHFB-C have very similar behavior. On the other hand, the masses calculated in the perturbative approximations, ATDBCS-C$^{\rm p}$ and ATDHFB-C$^{\rm p}$, are quite different for $Q_{20} < 40$. Furthermore, in this region, the perturbative masses appear to be quite large as compared to the cranking values.

Figure 4: (Color online) Similar as in Fig. 3 except for ATDHFB-C (filled triangles), ATDBCS-C (open triangles), ATDHFB-C$^{\rm p}$ (dots), and ATDBCS-C$^{\rm p}$ (circles) in the narrower region of 20b $\le$$Q_{20}$$\le$100b.

The high-frequency fluctuations of collective mass can be traced back to the imperfect numerical convergence of HFB calculations. In the present work, we assumed the accuracy of 0.001MeV for the total energy. This results in an uncertainty of about 0.002 $\hbar^2$/(MeV b$^2$) in the collective inertia. If required, the precision of these calculations can be increased at the expense of an appreciably higher CPU time.