Exact estimate of the $\alpha$-decay rate and semiclassical approach in deformed nuclei

We compare the quantum mechanical procedures to estimate the total $\alpha$-decay width from deformed nuclei in the laboratory and intrinsic systems of coordinates. Our analysis shows that the total half-life estimated in the intrinsic frame by neglecting the rotational motion of the core (adiabatic approach) is one order of magnitude smaller at ${\beta }_2=0.3$ than the corresponding value in the spherical case. A similar calculation in the laboratory system of coordinates by considering the core motion (giving the correct theoretical estimate) predicts a reduction by only a factor of 2. The widely used “angular WKB” (Wentzel-Kramers-Brillouin) semiclassical procedure provides decay widths which are comparable to the adiabatic approach. We propose a new and very simple semiclassical “angular momentum WKB” procedure to evaluate the decay width in deformed nuclei. It provides decay widths very close to the ones obtained by the exact laboratory coupling channels procedure.

Figure 1

(a) Fragmentation potential $V(R,\theta )-Q_{\alpha }$ versus the $\alpha$-daughter radius at $\theta =0^0$ (solid line), $\theta ={45}^0$ (dashed line), and $\theta ={90}^0$ (dot-dashed line) for ${\beta }_2=0.2$. The decay process is ${}^{230}\mathrm{U}\to {}^{226}\mathrm{Th}+\alpha$. (b) As in (a) plus a repulsive core reproducing the experimental $Q$ value.

Figure 2

(a) The radial components of the lowest resonant wave function in the laboratory frame versus radius for $J=0$ (solid line), $J=2$ (dashed line), and $J=4$ (dot-dashed line) for ${\beta }_2=0.2$. (b) The multipole components of the fragmentation potential versus radius for $\lambda =0$ (solid line), $\lambda =2$ (dashed line), and $\lambda =4$ (dot-dashed line). The decay process is ${}^{230}\mathrm{U}\to {}^{226}\mathrm{Th}+\alpha$.

Figure 3

Total half-life computed in the laboratory system (solid line) and its “angular momentum WKB” estimate given by Eq. (17) normalized to the laboratory value at ${\beta }_2=0$ (dashed line) versus the quadrupole deformation. Total half-life computed in the intrinsic system of coordinates (dot-dashed line) and its “angular WKB” estimate given by Eq. (15) (dotted line) versus the quadrupole deformation. The decay process is ${}^{230}\mathrm{U}\to {}^{226}\mathrm{Th}+\alpha$.

Figure 4

Square root of the ratio between sum of diagonal and sum of all matrix elements of the interaction squared versus radius for laboratory (solid line) and intrinsic system of coordinates (dashed line) for ${\beta }_2=0.2$. The decay process is ${}^{230}\mathrm{U}\to {}^{226}\mathrm{Th}+\alpha$.

Figure 5

Partial decay widths given by the “angular momentum WKB” (17) for $J=0,2,4$ and the total value divided by the sperical decay width versus the quadrupole deformation in the laboratory system of coordinates (dashed lines) for the decay process ${}^{230}\mathrm{U}\to {}^{226}\mathrm{Th}+\alpha$. By solid lines are given the corresponding coupled-channel values.

Figure 6

Total half-life computed in the laboratory system (solid lines) and its “angular momentum WKB” estimate given by Eq. (17) (dashed lines) versus the quadrupole deformation for three decays processes.