Quadrupole collective inertia in nuclear fission: Cranking approximation

A collective mass tensor derived from the cranking approximation to the adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) approach is compared with that obtained in the Gaussian overlap approximation (GOA) to the generator coordinate method. Illustrative calculations are carried out for one-dimensional quadrupole fission pathways in ${}^{256}$Fm. It is shown that the collective mass exhibits strong variations with the quadrupole collective coordinate. These variations are related to the changes in the intrinsic shell structure. The differences between collective inertia obtained in cranking and perturbative cranking approximations to ATDHFB, and within GOA, are discussed.

Figure 1

(a) Total SkM${}^{*}$+HFB energy. (b) Neutron (n), proton (p), and total (tot) pairing energies. (c) Particle-hole energy calculated along the static fission pathway for ${}^{256}$Fm. At $Q_{20}\approx 130$ b, a reflection-asymmetric path (aEF) branches away from the symmetric pathway (sEF). The bifurcation point is marked by a vertical dashed line.

Figure 2

Quadrupole mass parameter $B\left(Q_{20}\right)$ in ATDHFB-C (triangles) along the static fission pathway of ${}^{256}$Fm calculated in SkM${}^{*}$+HFB as a function of the mass quadrupole moment. The ATDHFB-C values are compared with those obtained in the (a) perturbative cranking approximation ATDHFB-C${}^{\mathrm{p}}$ and perturbative-canonical cranking approximation ATDHFB-C${}^{\mathrm{pc}}$, (b) canonical cranking approximation ATDHFB-C${}^{\mathrm{c}}$, and (c) Gaussian overlap approximation ATDHFB${}^{\mathrm{GOA}}$. The sEF-aEF bifurcation point is marked by a vertical dashed line. The diabatic jumps between various energy sheets around this point have been disregarded when computing the collective inertia.

Figure 3

Comparison between HFB and HF+BCS variants of calculations: (a) ATDHFB-C and ATDBCS-C and (b) ATDHFB-C${}^{\mathrm{p}}$, ATDHFB-C${}^{\mathrm{pc}}$, and ATDBCS-C${}^{\mathrm{p}}$. The sEF-aEF bifurcation point is marked by vertical dashed lines and arrows.