Fine structure of $\alpha$ decay from the variational principle

Starting from the variational principle, the time-dependent pairing equations are generalized by including the Landau-Zener effect and the Coriolis coupling. A system of microscopic equations of motion for configuration mixing is deduced, allowing the determination of quantities that have the same meaning as the preformation factors of the $\alpha$ particle. These equations are solved in order to reproduce the hindrance factors of the $\alpha$ decay of an odd-$A$ mass nucleus. The $\alpha$ decay of ${}^{211}\mathrm{Po}$ is treated as a superasymmetric fission process, by following the rearrangement of the nuclear orbitals from the parent ground state up to the scission configuration. The probabilities of finding the excited states of the daughter at scission are obtained from the microscopic equations of motion. The intensities of the transitions to the excited states of the daughter were evaluated theoretically. The experimental data were compared with the theoretical findings. A very good agreement was obtained. A mean value of the tunneling velocity of about $2\times {10}^4$ fm/fs was extracted.

Figure 1

The deformation energy $V$ as function of the internuclear distance $R$ for $\alpha$ decay. The height of the barrier is renormalized by taking into account a zero-point vibration energy of 0.5 MeV.

Figure 2

The Woods-Saxon potential $V_{WS}$ for the $\alpha$-particle emission as a function of the axial cylindrical coordinate $z$ is displayed for several internuclear distances $R$. The internuclear distances are marked on the plots. The corresponding nuclear shapes are also represented at the top of each panel.

Figure 3

Neutron single-particle energies as a function of the distance between the centers of the fragments $R$. The single-particle levels of the spherical parent nucleus are labeled with their spectroscopic notations on the left side. After the scission produced at $R\approx 10\mathrm{fm}$, the single-particle levels of the daughter nucleus are superimposed on the single-particle level belonging to the $\alpha$ particle, labeled on the right side with the spectroscopic notation $1{\mathrm{s}}_{1/2}$.

Figure 4

Proton single-particle energies. The single-particle levels of the parent are labeled on the left with their spectroscopic notations. The single-particle level pertaining to the $\alpha$ particle is marked on the right.

Figure 5

The variations of several single-particle levels ${\epsilon }_{\mathrm{\Omega },m}$ located around the Fermi energy as a function of the internuclear distance $R$ are displayed in panels (a), (d), and (g). The spin projection of these levels is $\mathrm{\Omega }=1/2$. In each panel two adjacent single-particle levels are selected and are plotted with a thick line. They are labeled with their spectroscopic notations corresponding to the parent on the left and to the daughter on the right. The differences in energies $\mathrm{\Delta }\epsilon ={\epsilon }_{\mathrm{\Omega },m}-{\epsilon }_{\mathrm{\Omega },m-1}$ are plotted on the panels (b), (e), and (h) for the two adjacent single-particle levels selected in the panels (a), (d), and (g), respectively. Minimum values of these differences should locate the avoided crossing regions. These minima are marked with arrows. The total intrinsic spins $j_{\mathrm{\Omega },m}$ of the single-particle levels selected in the panels (a), (d), and (g) are displayed in the panels (c), (f), and (i), respectively. The total intrinsic spin corresponding to the superior single-particle level is plotted with a thick line. The arrows indicate the locations of the avoided level crossing regions. The values of $j_{\mathrm{\Omega },m}$ intersect in the same regions in panels (f) and (i).