Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. V. Extension to fission barriers

Large-scale fission barrier calculations have been performed in the framework of the Skyrme-Hartree-Fock model. Our Hartree-Fock-Bogoliubov calculations restore broken symmetries such as translational invariance, particle-number conservation, parity, and, in a more approximate way, rotational invariance. Axial symmetry is imposed, but reflection asymmetry is allowed. The energy surface properties are analyzed with the flooding method. A large set of Skyrme interactions, which were fitted to all known masses under different specific constraints, is used to study the main effects influencing the energy surface and the barrier heights. The principal interaction used in the comparison with experimental barriers is BSk8, the force on which the HFB-8 mass table is based. We found that for nuclei with $92\le Z\le 98$ the agreement of our calculations with experimental data is excellent; the rms deviation on the primary barriers is 0.722 MeV. For lighter nuclei, however, the calculated primary barriers are always too high because of the existence of a third barrier at very high deformations. However, our calculated superheavy barriers appear to be too low, although they are consistent with previous calculations.

Figure 1

Energy and deformation energy convergence for ${}^{240}\mathrm{Pu}$ as function of dimensionality $n_{max}$: (upper panel) total energy (constrained HFB + PLN) at five deformations; (lower panel) deformation energy $\Delta E_{\mathrm{tot}}=E_{\mathrm{tot}}(Q_2)-E_{\mathrm{tot}}^{\mathrm{gs}}$, normalized to $\Delta E_{\mathrm{tot}}(n_{max}=21)$ (see text).

Figure 2

Same as Fig. 1 for ${}^{268}\mathrm{Po}$, situated close to the neutron drip line. Note that the neutron Fermi energy becomes positive for $Q_2\gtrsim 500$ b.

Figure 3

Projection energy of the positive-parity state (unprojected energy − projected energy) as a function of the asymmetry parameter $\tilde{\alpha }$ (left panel) and of the octupole moment $Q_3$ (right panel), for part of the ${}^{240}\mathrm{Pu}$ three-dimensional energy surface calculated with the HFBCS + PLN (BSk8) model: $1.55\le c\le 1.80$ and $-0.20\le h\le 0.15$.

Figure 4

Evolution [energy curves (in MeV) of ${}^{194}\mathrm{Pb}$] of the parity projection effect with respect to the octupole moment, at different quadrupole moments

Figure 5

Contour plot of the energy surface of ${}^{240}\mathrm{Pu}$ in the $(c,h)$ plane for left-right symmetry ($\alpha =0$). The contour lines are spaced by 1 MeV. Small tick marks along each contour point in the downhill direction. G refers to the ground state (${\beta }_2=0.276$), A to the inner saddle point (${\beta }_2=0.545$), M to the isomeric state (${\beta }_2=0.834$), and B to the left-right symmetric outer saddle point (${\beta }_2=1.515$); C denotes the position in the $(c,h)$ plane to which this saddle point shifts when left-right asymmetry is allowed (${\beta }_2=1.267,{\beta }_3=-0.398$). The rectangle containing B shows the limits in the $(c,h)$ plane taken to calculate the local 3D energy surface and to locate the saddle point C. The upper right-hand zone of the panel corresponds to the fission valley, which is however reached from the lower right-hand part.

Figure 6

Contour plot of the same energy surface as in Fig. 5, but in the $({\beta }_2,{\beta }_4){}_{{\beta }_3=0}$ plane.

Figure 7

Calculated inner and outer (left-right symmetric) barrier heights of some of the Np, Pu, Cm, and Cf isotopes with the HFB + PLN method; we use both the original BSk8 force and a variant in which the pairing cutoff is reduced to 9 MeV.

Figure 8

Calculated inner barrier heights of the Pu, Am, Cm, Bk, and Cf isotopes compared to experimental barriers (dotted line). The HFB + PLN method is used with the Skyrme force BSk8 with staggered pairing included (solid line) and excluded (dashed line).

Figure 9

Influence of the reflection asymmetry (including parity projection) on the symmetric outer HFB + PLN (BSk8) barrier heights for $88\le Z\le 98$. The upper panel gives both reflection-symmetric and reflection-asymmetric barrier heights; the lower panel gives the difference $B_o(\alpha =0)-B_o(\alpha >0)$. Note that the figure does not include the comparison for the outermost third barriers observed in subactinides (see text for more details).

Figure 10

Comparison with experiment of primary barriers calculated within the HFB + PLN + PP (BSk8) and ETFSI (SkSC4) frameworks. In each case we show the deviation $\Delta B$ (theory − experiment). The crosses in the lower left panel indicate the deviation $\Delta \hspace{1em}B$ between the second barrier and the experimental data whenever a triple-humped barrier is predicted.

Figure 11

Inner ($B_i$) and outer ($B_o$) barrier heights of Pu isotopes calculated with Skyrme force BSk8 with four methods: HFBCS (stars), HFBCS + PLN (full dots), HFB (crosses), and HFB + PLN (circles); reflection symmetry is imposed.

Figure 12

HFB + PLN (BSk8) inner and outer barrier heights compared to those of HFBCS (BSk6); $\Delta B_{i,o}=B_{i,o}$(BSk8)$-B_{i,o}$(BSk6) (in MeV).

Figure 13

Inner and outer reflection-symmetric barrier heights of even-even actinides ($Z=88\text{−}98$) with BSk8 ($J=28$ MeV) and BSk9 ($J=30$ MeV) within the HFB + PLN framework; the bottom panel gives $\Delta a_{\mathrm{sf}}(I)=a_{\mathrm{sf}}(I,\mathrm{BSk}8)-a_{\mathrm{sf}}(I,\mathrm{BSk}9)$, the difference between the effective surface-energy coefficients of both forces for all nuclei considered.

Figure 14

HFBCS (BSk6) inner and outer barrier heights compared to those of HFBCS (BSk2); $\Delta B_{i,o}=B_{i,o}$ (BSk6) $-B_{i,o}$ (BSk2).

Figure 15

Comparison of inner and outer BSk6 and BSk7 barrier heights calculated in the HFBCS approximation. The two Skyrme forces differ in the density dependence of the pairing interaction.

Figure 16

Inner and outer barrier heights obtained with BSk1 and BSk2 Skyrme forces that differ only in the cutoff prescription: $\hslash \omega$ for the former and ${\epsilon }_F+15$ MeV for the latter.