Triaxial quadrupole dynamics and the inner fission barrier of some heavy even-even nuclei

Background: Inner fission barriers of actinide nuclei have been known for a long time to be unstable with respect to the axial symmetry. On the other hand, taking into account the effect of the relevant adiabatic mass parameter reduces or even may wash out this instability. A proper treatment of the dynamics for both axial and triaxial modes is thus crucial to accurately determine the corresponding fission barriers. This entails in particular an accurate description of pairing correlations.

Purpose: We evaluate the potential energies, moments of inertia, and vibrational mass parameters in a two-dimensional relevant deformation space (corresponding to the usual $\beta$ and $\gamma$ quadrupole deformation parameters) for four actinide nuclei (${}^{236}\mathrm{U}, {}^{240}\mathrm{Pu}, {}^{248}\mathrm{Cm}$, and ${}^{252}\mathrm{Cf}$). We assess the relevance of our approach to describe the dynamics for a triaxial mode by computing the low energy spectra (exploring thus mainly the equilibrium deformation region). We evaluate the inner fission barrier heights releasing the axial symmetry constraint.

Method: Calculations within the Hartree-Fock plus BCS approach are performed using the SkM* Skyrme effective interaction in the particle-hole channel and a seniority force in the particle-particle channel. The intensity of this residual interaction has been fixed to allow a good reproduction of some odd-even mass differences in the actinide region. Adiabatic mass parameters for the rotational and vibrational modes are calculated using the Inglis-Belyaev formula supplemented by a global renormalization factor taking into account the so-called Thouless-Valatin corrections. Spectra are obtained through the diagonalization of the corresponding Bohr collective Hamiltonian.

Results: The experimental low energy spectra are qualitatively well reproduced by our calculations for the considered nuclei. Inner fission barrier heights are calculated and compared with available estimates from various experimental data. The reproduction of the data is better for ${}^{236}\mathrm{U}$ and ${}^{240}\mathrm{Pu}$ (up to about 300 keV) than for ${}^{248}\mathrm{Cm}$ and ${}^{252}\mathrm{Cf}$ (up to about one MeV).

Conclusions: While these results are encouraging, they call for, in particular, a better treatment of pairing correlations, especially as far as the particle number conservation is concerned. Besides, these results could provide a basis for the determination of the least action trajectories which would generate better grounds for the evaluation of fission half lives.

Figure 1

Potential energy surface of the ${}^{236}\mathrm{U}$ nucleus. The plotted energies are relative to the energy of the equilibrium point (indicated by the black dot) which is equal to $-1778.23$ MeV.

Figure 2

Same as Fig. 1 for the ${}^{240}\mathrm{Pu}$ nucleus. The energy of the equilibrium point is equal to $-1800.40$ MeV.

Figure 3

Same as Fig. 1 for the ${}^{248}\mathrm{Cm}$ nucleus. The energy of the equilibrium point is equal to $-1848.52$ MeV.

Figure 4

Same as Fig. 1 for the ${}^{252}\mathrm{Cf}$ nucleus. The energy of the equilibrium point is equal to $-1867.30$ MeV.

Figure 5

Experimental (Exp.) [33] and calculated (Calc.) low energy spectra in ${}^{236}\mathrm{U}$.

Figure 6

Same as Fig. 5 for ${}^{240}\mathrm{Pu}$. The experimental spectrum is taken from Ref. [34].

Figure 7

Same as Fig. 5 for ${}^{248}\mathrm{Cm}$. The experimental spectrum is taken from Ref. [35].

Figure 8

Same as Fig. 5 for ${}^{252}\mathrm{Cf}$. The experimental spectrum is taken from Ref. [36].

Figure 9

Section of the potential energy surface of ${}^{236}\mathrm{U}$ as a function of $Q_{22}$ (in barns), for a given value of $Q_{20}$ (in barns), in the inner saddle point region. Energies are given in MeV.

Figure 10

Same as Fig. 9 for ${}^{240}\mathrm{Pu}$.

Figure 11

Same as Fig. 9 for ${}^{248}\mathrm{Cm}$.

Figure 12

Same as Fig. 9 for ${}^{252}\mathrm{Cf}$.