Potential for $\alpha$-induced nuclear scattering, reaction and decay, and a resonance-pole-decay model with exact explicit analytical solutions

The decay of $\alpha$ particle from a nucleus is viewed as a quantum resonance state of a two-body scattering process of the $\alpha$+daughter nucleus pair governed by a novel nucleus-nucleus potential in squared Woods-Saxon form. By the application of the rigorous optical model (OM) potential scattering ($S$-matrix) theory the genuineness of the potential for the system is established by giving a good explanation of the elastic scattering and reaction cross sections data of the $\alpha$+nucleus pair. From the pole position in the complex momentum ($k$) plane of the $S$ matrix of the real part of the OM potential defined above, the energy and width of the resonance state akin to the decaying state of emission of $\alpha$ particle are extracted and from this width, the result of the $\alpha$-decay half-life is derived to account for the experimental result of the half-life in the cases of a large number of $\alpha$ emitters including heavy and superheavy nuclei. The $S$ matrix of the real OM potential is replaced by an analytical function expressed in terms of exact Schrödinger solutions of a global potential that closely represents the real Coulomb-nuclear interaction in the interior and the pure Coulomb wave functions outside, and the resonant poles of this $S$ matrix in the complex momentum plane are used to give satisfactory results of decay half-lives of $\alpha$ coming out from varieties of nuclei.

Figure 1

Plot of potential $V\left(r\right)$ as a function of radial distance for the $\alpha +{}^{208}\mathrm{Pb}$ system. The dashed curve represents the sum of the real nuclear potential expressed by Eq. (10) with parameters $V_0=22$ MeV, $a_s=0.62\mathrm{fm}$, $r_s=1.27\mathrm{fm}$, $r_v=0.66\mathrm{fm}$, $\delta =3.5$, and the Coulomb potential given by Eq. (12) with $r_C=1.2\mathrm{fm}$. The thick solid curve represents the potential expressed by Eq. (16) in the text with parameters $r_0=1.07\mathrm{fm}$, $a_g=0.63\mathrm{fm}$, $H_0=1\mathrm{MeV}$, ${\xi }_1=-100$, and ${\xi }_2=V_B=20.27$. The arrow indicates barrier position $R_B=10.75\mathrm{fm}$. The dotted curve represents the double folded potential with nuclear densities obtained using RMF theory [14, 19]. The thin solid curve represents the double folding potential with experimental density distributions for $\alpha$ and ${}^{208}\mathrm{Pb}$. The dash-dot-dashed curve represents the usual Woods-Saxon potential $V_{\mathrm{WS}}\left(r\right)=-V_R{[1+exp\left(\frac{r-R_R}{a_R}\right)]}^{-1}$, $V_R=96.44\mathrm{MeV}$, $R_R=r_R{\left(208\right)}^{1/3}$, $r_R=1.376\mathrm{fm}$, $a_R=0.625\mathrm{fm}$ combined with Coulomb potential.

Figure 2

Angular variation of elastic differential scattering cross section ${\sigma }_{Sc}$, relative to Rutherford ${\sigma }_{\mathrm{Ru}}$, of $\alpha$ particles scattered by ${}^{208}\mathrm{Pb}$ at 19, 20, and 22 Mev laboratory energies. The solid curves represent the results of present optical model calculations using the parameters: $V_0=22\mathrm{MeV}$, $r_s=1.27\mathrm{fm}$, $a_s=0.62\mathrm{fm}$, $r_v=0.66\mathrm{fm}$, $\delta =3.5$, $r_C=1.2\mathrm{fm}$, $W_0=5\mathrm{MeV}$, $a_I=0.28\mathrm{fm}$, $r_I=1.29\mathrm{fm}$ for 19 MeV, $r_I=1.3\mathrm{fm}$ for 20 MeV, and $r_I=1.37\mathrm{fm}$ for 22 MeV. The dotted curves represent the results calculated using $r_v=0.8\mathrm{fm}$ and $\delta =1$ with other parameters remaining the same as above. The experimental data shown by solid circles are obtained from [28].

Figure 3

Plot of reaction cross section ${\sigma }_R$ as function of laboratory energy for the $\alpha +{}^{208}\mathrm{Pb}$ collision. The solid curve represents the results of present optical model calculation with energy dependent $r_I$. The experimental data shown by solid circles are taken from [28].