Fine structure of $\alpha$ decay from the time-dependent pairing equations

Abstract

The $\alpha$ decay half-lives and the fine structure phenomenon are investigated with fission-like models. A superasymmetric fission path in a configuration space spanned by five degrees of freedom is determined in accordance with the least action principle. The deformation energy is evaluated within the macroscopic–microscopic approach while the inertia is obtained in the framework of the cranking model. The single particle levels schemes are calculated connecting the ground state of the parent nucleus and the asymptotic configuration of two separated nuclei. The probabilities to find different seniority configurations are obtained by solving time-dependent pairing equations generalized by including the Landau–Zener effect and the Coriolis coupling. The microscopic equations of motion for even numbers of particles are deduced, those for odd-nuclear systems being obtained in previous works. The models used in the calculations are reviewed within a detailed description. The microscopic equations of motion are solved by starting from the ground state configuration and arriving at the scission point. A description of all the possible configurations at scission together with their realization probabilities is given. By fitting the inter-nuclear velocity, the best agreement between experimental and theoretical hindrance factors is retained. The theoretical results for the $\alpha$ decay half-lives for $^{211,212}$Po and $^{211}$Bi are compared with experimental data showing discrepancies ranging over three orders of magnitude. The accuracy of the model concerning the calculations of the half-lives for different channels is discussed. The connections between the classical theories concerning the preformation of the $\alpha$ particle and the fission-like descriptions are highlighted.

Appendix A: Matrix elements

Using the property

$$\begin{aligned} I_{\pm }{{\mathcal {D}}}^I_{M\varOmega }(\omega )= \sqrt{(I\pm \varOmega )(I\mp \varOmega +1)}{{\mathcal {D}}}^I_{M\varOmega \mp 1}(\omega ), \end{aligned}$$

(A.1)

the relevant matrix elements needed to develop the functional (46) can be obtained. They are given by the following identities:

$$\begin{aligned}&\begin{array}{ll} &{}\left\langle \sum \nolimits _Ic^I_{0}{{\mathcal {D}}}^I_{M0}\prod \nolimits _{k}\left( u_{k(0)}+v_{k(0)}a_{k}^{+}a_{{\bar{k}}}^{+}\right) \right. \\ &{}\qquad \times \left| \sum \nolimits _{k}\epsilon _{k}\left( a_{k}^{+}a_{k}+a_{{\bar{k}}}^{+}a_{{\bar{k}}}\right) - G\sum \nolimits _{k,l}a_{k}^{+}a_{{\bar{k}}}^{+}a_{l}a_{{\bar{l}}}\right| \\ &{}\qquad \times \left. \sum \nolimits _{I'} c^{I'}_{0}{{\mathcal {D}}}^{I'}_{M0}\prod \nolimits _{k'}\left( u_{k'(0)}+v_{k'(0)}a_{k'}^{+}a_{{\bar{k}}'}^{+}\right) \right\rangle \\ &{}\quad = \sum \nolimits _I\frac{8\pi ^2}{2I+1}|c^I_{0}|^{2}\left[ 2\sum \nolimits _{k}\mid v_{k(0)}\mid ^{2}\epsilon _{k}\right. \\ &{}\qquad \left. +G\mid \sum \nolimits _{k}u_{k(0)}v_{k(0)}\mid ^{2}-G\sum \nolimits _{k}\mid v_{k(0)}\mid ^{4}\right] \delta _{II'}, \end{array} \end{aligned}$$

(A.2)

$$\begin{aligned}&\begin{array}{ll} &{}\left\langle \sum \nolimits _I\sum \nolimits _{j,l\ne j}c^I_{jl}{{\mathcal {D}}}^I_{M(\varOmega _j-\varOmega _l)} a_{j}^{+}a_{{\bar{l}}}^{+}\right. \\ &{}\qquad \left. \times \prod \nolimits _{k\ne j,l} \left( u_{k(jl)}+v_{k(jl)}a_{k}^{+}a_{{\bar{k}}}^{+}\right) \right. \\ &{}\qquad \times \left| \sum \nolimits _{k}\epsilon _{k}(a_{k}^{+}a_{k}+a_{{\bar{k}}}^{+}a_{{\bar{k}}}) - G\sum \nolimits _{k,l}a_{k}^{+}a_{{\bar{k}}}^{+}a_{l}a_{{\bar{l}}}\right| \\ &{}\qquad \times \sum \nolimits _{I'}\sum \nolimits _{j',l'\ne j'}c^{I'}_{j'l'}{{\mathcal {D}}}^{I'}_{M(\varOmega _j'-\varOmega _l')}a_{j'}^{+}a_{{\bar{l}}'}^{+} \\ &{}\qquad \left. \times \prod \nolimits _{k'\ne j',l'} \left( u_{k'(j'l')}+v_{k'(j'l')}a_{k'}^{+}a_{{\bar{k}}'}^{+}\right) \right\rangle \\ &{}\quad =\sum \nolimits _I\frac{8\pi ^2}{2I+1} \sum \nolimits _{j,l\ne j}\mid c^I_{jl}\mid ^{2}\left[ 2\sum \nolimits _{k\ne j,l} \mid v_{k(jl)}\mid ^{2}\epsilon _{k}\right. \\ &{}\qquad +\epsilon _{j} +\epsilon _{l} +G\left| \sum \nolimits _{k\ne j,l}u_{k(jl)}v_{k(jl)}\right| ^{2}\\ &{}\qquad \left. - G\sum \nolimits _{k\ne jl}| v_{k(jl)}|^{4}\right] \delta _{II'}\delta _{(\varOmega _j-\varOmega _l) (\varOmega _{j'}-\varOmega _{l'})}, \end{array} \end{aligned}$$

(A.3)

$$\begin{aligned}&\begin{array}{ll} &{}\left\langle \sum \nolimits _{I}c^{I}_{0}{{\mathcal {D}}}^{I}_{M0}\prod \nolimits _{k}(u_{k(0)}+v_{k(0)}a_{k}^{+}a_{{\bar{k}}}^{+})\left| {\partial \over \partial t} \right| \right. \\ &{}\qquad \left. \times \sum \nolimits _{I'} c^{I'}_{0}{{\mathcal {D}}}^{I'}_{M0} \prod \nolimits _{k'}\left( u_{k'(0)}+v_{k'(0)}a_{k'}^{+}a_{{\bar{k}}'}^{+}\right) \right\rangle \\ &{}\quad =\sum \nolimits _I\frac{8\pi ^2}{2I+1}\left[ c_{0}^{I*}\dot{c}^{I}_{0}\right. \\ &{}\qquad \left. +|c_{0}^I|^{2}\sum \nolimits _{k}\left( u_{k(0)}\dot{u}_{k(0)}+ v_{k(0)}^{*}\dot{v}_{k(0)}\right) \right] \delta _{II'}, \end{array} \end{aligned}$$

(A.4)

$$\begin{aligned}&\begin{array}{ll} &{}\langle \sum \nolimits _{I}\sum \nolimits _{j,l\ne j}c^I_{jl}{{\mathcal {D}}}^I_{M(\varOmega _j-\varOmega _l)}a_{j}^{+}a_{{\bar{l}}}^{+}\\ &{}\qquad \times \prod \nolimits _{k\ne j,l} (u_{k(jl)}+v_{k(jl)}a_{k}^{+}a_{{\bar{k}}}^{+})| {\partial \over \partial t} |\\ &{}\qquad \times \sum \nolimits _{I'}\sum \nolimits _{j',l'\ne j'}c^{I'}_{j'l'}{{\mathcal {D}}}^{I'}_{M(\varOmega _j'-\varOmega _l')}a_{j'}^{+}a_{{\bar{l}}'}^{+}\\ &{}\qquad \times \prod \nolimits _{k'\ne j',l'} (u_{k'(j'l')}+v_{k'(j'l')}a_{k'}^{+}a_{{\bar{k}}'}^{+})\rangle \\ &{}\quad =\sum \nolimits _I\frac{8\pi ^2}{2I+1}\sum \nolimits _{j,l\ne j}[c_{jl}^{I*}\dot{c}_{jl}^I\\ &{}\qquad + |c_{jl}^I|^{2}\sum \nolimits _{k\ne j,l}(u_{k(jl)}\dot{u}_{k(jl)}+ v_{k(jl)}^{*}\dot{v}_{k(jl)})]\\ &{}\qquad \times \delta _{II'} \delta _{(\varOmega _j-\varOmega _l)(\varOmega _{j'}- \varOmega _{l'})}, \end{array} \end{aligned}$$

(A.5)

$$\begin{aligned}&\begin{array}{ll} &{}\langle \sum \nolimits _I c^I_{0}{{\mathcal {D}}}^I_{M0}\prod \nolimits _{k}(u_{k(0)}+v_{k(0)}a_{k}^{+}a_{{\bar{k}}}^{+})\\ &{}\qquad \times | \sum \nolimits _{i,j\ne i}h_{ij}\alpha _{i(0)}\alpha _{{\bar{j}}(0)}\\ &{}\qquad \times \prod \nolimits _{k\ne i,j}\alpha _{k(0)}a_{k}^{+}a_{k}\alpha ^{+}_{k(ij)} |\\ &{}\qquad \times \sum \nolimits _{I'}\sum \nolimits _{j',l'\ne j'}c^{I'}_{j'l'}{{\mathcal {D}}}^{I'}_{M(\varOmega _{j'}-\varOmega _{l'})}a_{j'}^{+}a_{{\bar{l}}'}^{+}\\ &{}\qquad \times \prod \nolimits _{k'\ne j',l'} (u_{k'(j'l')}+v_{k'(j'l')}a_{k'}^{+}a_{{\bar{k}}'}^{+})\rangle \\ &{}\quad =\sum \nolimits _I\frac{8\pi ^2}{2I+1} \sum \nolimits _{j,l\ne j}h_{jl}c_{0}^{I*}c_{jl}^I\delta _{II'}\delta _{0(\varOmega _j-\varOmega _l)}, \end{array} \end{aligned}$$

(A.6)

$$\begin{aligned}&\begin{array}{ll} &{}\langle \sum \nolimits _I\sum \nolimits _{j,l\ne j}c_{jl}{{\mathcal {D}}}^{I}_{M(\varOmega _j-\varOmega _l)} a_{j}^{+}a_{{\bar{l}}}^{+}\\ &{}\qquad \times \prod \nolimits _{k\ne j,l} (u_{k(jl)}+v_{k(jl)}a_{k}^{+}a_{{\bar{k}}}^{+})| \sum \nolimits _{i,j\ne i}h_{ij}\alpha _{i(0)}^{+}\alpha _{{\bar{j}}(0)}^{+}\\ &{}\qquad \times \prod \nolimits _{k\ne i,j}\alpha _{k(ij)}a_{k}^{+}a_{k}\alpha ^{+}_{k(0)} |\\ &{}\qquad \times \sum \nolimits _{I'} c_{0}{{\mathcal {D}}}^{I'}_{M0}\prod \nolimits _{k'}(u_{k'(0)}+v_{k'(0)}a_{k'}^{+}a_{{\bar{k}}'}^{+})\rangle \\ &{}\quad =\sum \nolimits _I\frac{8\pi ^2}{2I+1}\sum \nolimits _{j,l\ne j}h_{jl}c_{0}^Ic_{jl}^{I*}\delta _{II'}\delta _{0(\varOmega _j-\varOmega _l)}, \end{array} \end{aligned}$$

(A.7)

$$\begin{aligned}&\begin{array}{ll} &{}\langle \sum \nolimits _I c^I_{0}{{\mathcal {D}}}^I_{M0}\prod \nolimits _{k}(u_{k(0)}+v_{k(0)}a_{k}^{+}a_{{\bar{k}}}^{+}) |I_{\pm }j_{\mp }|\\ &{}\qquad \times \sum \nolimits _{I'}\sum \nolimits _{j',l'\ne j'}c^{I'}_{j'l'}{{\mathcal {D}}}^{I'}_{M(\varOmega _{j'}-\varOmega _{l'})}a_{j'}^{+}a_{{\bar{l}}'}^{+}\\ &{}\qquad \times \prod \nolimits _{k'\ne j',l'} (u_{k'(j'l')}+v_{k'(j'l')}a_{k'}^{+}a_{{\bar{k}}'}^{+})\rangle \\ &{}\quad =\sum \nolimits _I\frac{8\pi ^2}{2I+1}\sqrt{[I\mp (\varOmega _j{-}\varOmega _l)][I{\pm }(\varOmega _j{-}\varOmega _l){+}1]}c_{0}^{I*}c_{jl}^{I}\\ &{}\qquad \times [u_{j(0)}v_{l(0)}^*\langle a_l^+|j_{\mp }|a_j^+\rangle \\ &{}\qquad +u_{l(0)}v_{j(0)}^*\langle a_{\bar{j}}^+|j_{\mp }|a_{\bar{l}}^+\rangle ]P_{0jl} \delta _{\pm 1(\varOmega _j-\varOmega _l)}, \end{array} \end{aligned}$$

(A.8)

and

$$\begin{aligned}&\langle \sum \nolimits _{I}\sum \nolimits _{j,l\ne j}c^{I}_{jl}{{\mathcal {D}}}^{I}_{M(\varOmega _{j}-\varOmega _{l})}a_{j}^{+}a_{{\bar{l}}}^{+}\nonumber \\&\qquad \times \prod \nolimits _{k\ne j,l} (u_{k(jl)}+v_{k(jl)}a_{k}^{+}a_{{\bar{k}}}^{+})\nonumber \\&\qquad \times |I_{\pm }j_{\mp }|\sum \nolimits _{I'} c^{I'}_{0}{{\mathcal {D}}}^{I'}_{M0}\prod \nolimits _{k'}(u_{k'(0)}+v_{k'(0)}a_{k'}^{+}a_{{\bar{k}}'}^{+})\rangle \nonumber \\&\quad =\sum \nolimits _I\frac{8\pi ^2}{2I+1}\sqrt{[I\mp \varOmega _0][I\pm \varOmega _0+1]}c_{jl}^{I*}c_{0}^{I}\nonumber \\&\qquad \times {[}u_{l(0)}v_{j(0)}\langle a_{{{\bar{l}}}}^+|j_{\mp }|a_{{\bar{j}}}^+\rangle \nonumber \\&\qquad +u_{j(0)}v_{l(0)}\langle a_{{j}}^+|j_{\mp }|a_{{l}}^+\rangle ]P_{jl0} \delta _{\mp 1(\varOmega _j-\varOmega _l)}. \end{aligned}$$

(A.9)

For the seniority-0 configuration in the ground state, $I=$ 0 and $\varOmega _0=$ 0. In the previous relations, the overlaps between the Bogoliubov wave functions are denoted by:

$$\begin{aligned} P_{0jl}=P_{jl0}^*=\prod _{k\ne j,l}(u_{k(jl)}u_{k{0}}+v_{k(jl)}v_{k(0)}^*). \end{aligned}$$

(A.10)

Also, the absolute values related to the Lagrange multiplier are written as:

$$\begin{aligned} \begin{array}{ll} &{}\left\langle \varphi _{IM}\left| -\lambda | N_2{{\hat{N}}}_1-N_1{{\hat{N}}}_2|\right| \varphi _{IM} \right\rangle \\ &{}\quad =-s\lambda \left[ |c_0|^2\left( \sum \nolimits _{k_1} |v_{k_1(0)}|^2N_{2}-\sum \nolimits _{k_2} |v_{k_2(0)}|^2N_{2}\right) \right. \\ &{}\qquad \left. +|c_{jl}|^2\left( \sum \nolimits _{k_1\ne j,l} |v_{k_1(jl)}|^2N_{2}\right. \right. \\ &{}\qquad \left. \left. -\sum \nolimits _{k_2\ne j,l} |v_{k_2(jl)}|^2N_{1}+N_j+N_l\right) \right] , \end{array} \end{aligned}$$

(A.11)

where $N_{j,l}=-N_1$ or $N_2$, depending on the location of the levels j or l in the second or the first well after scission, respectively. The sign $s=\mp 1$ ensures that the expression in the right side of the equation is negative.

Other matrix elements are trivial or obsolete. The derivatives of the BCS occupation amplitudes can be obtained by using the time-dependent pairing equations (55). The following energy terms are obtained:

$$\begin{aligned} \begin{array}{ll} &{}T_{k(jl)}={i\hbar \over 2}(v_{k(jl)}^{*}\dot{v}_{k(jl)}-\dot{v}_{k(jl)}^{*}v_{k(jl)})\\ &{}\qquad =2|v_{k(jl)}|^2(\epsilon _{k}-s_k\lambda )-2G|v_{k(jl)}|^{4}\\ &{}\quad \qquad +\mathfrak {R}\left\{ \varDelta ^*_{0} \left( \frac{|v_{k(jl)}|^4}{u_{k(jl)}v^*_{k(jl)}}-u_{k(jl)}v_{k(jl)}\right) \right\} , \end{array} \end{aligned}$$

(A.12)

where $\varDelta _{k(0)}=G|\sum \nolimits _{k\ne j,l}u_{k(jl)}v_{k(jl)}|^2$. With the previous identities, the expected value of the functional (54) is then obtained by introducing the many-body energies (49) and (50).