$\alpha$ decay in the complex-energy shell model

Background: Alpha emission from a nucleus is a fundamental decay process in which the alpha particle formed inside the nucleus tunnels out through the potential barrier.

Purpose: We describe alpha decay of ${}^{212}$Po and ${}^{104}$Te by means of the configuration interaction approach.

Method: To compute the preformation factor and penetrability, we use the complex-energy shell model with a separable $T=1$ interaction. The single-particle space is expanded in a Woods-Saxon basis that consists of bound and unbound resonant states. Special attention is paid to the treatment of the norm kernel appearing in the definition of the formation amplitude that guarantees the normalization of the channel function.

Results: Without explicitly considering the alpha-cluster component in the wave function of the parent nucleus, we reproduce the experimental alpha-decay width of ${}^{212}$Po and predict an upper limit of $T_{1/2}=5.5\times {10}^{-7}$ sec for the half-life of ${}^{104}$Te.

Conclusions: The complex-energy shell model in a large valence configuration space is capable of providing a microscopic description of the alpha decay of heavy nuclei having two valence protons and two valence neutrons outside the doubly magic core. The inclusion of proton-neutron interaction between the valence nucleons is likely to shorten the predicted half-live of ${}^{104}$Te.

Figure 1

$u_{\mathrm{diff}}\left(r\right)=|u\left(r\right)-\tilde{u}\left(r\right)|$ for the neutron core states in ${}^{208}$Pb for $\Delta R=0.5$ fm, $M=30$, and $R_{\max }=15$ fm.

Figure 2

Eigenvalues of the norm kernel (25) for ${}^{212}$Po for $R_{\max }=13$ fm for different values of $\Delta R$.

Figure 3

Similar as in Fig. 2 except for ${}^{104}$Te and $R_{\max }=10$ fm.

Figure 4

Formation amplitude $g\left(R\right)$ for ${}^{212}$Po in the M4 model space expanded in eigenfunctions of the norm kernel for $\Delta R=0.500$ fm and $R_{\max }=13$ fm.

Figure 5

Convergence of $\mathcal{S}$ for ${}^{212}$Po (model space M4) as a function of $R_{\max }$ of the normalized SGB for different values of $\Delta R$ (with $n_{\min }=0.001$.)

Figure 6

Similar as in Fig. 5 but as a function of the step size $\Delta R$ for different values of $R_{\max }$.

Figure 7

Similar as in Fig. 5 but as a function of $n_{\min }$ for different values of $R_{\max }$ and $\Delta R=0.57$.

Figure 8

Formation amplitude $g\left(R\right)$ for ${}^{212}$Po obtained in this work in the model spaces M0 to M4 as defined in Table . The imaginary part of the formation amplitude in M4 (dotted line) is also shown.

Figure 9

Dependence of the absolute alpha-decay width (1) of ${}^{212}$Po on the $R$-matrix channel radius $R$ for three different model spaces M0, M3, and M4.

Figure 10

Modified formation amplitude $G\left(R\right)$ of Eq. (24) in the extended model space M4 with $n_{\min }=0.001$, $\Delta R=0.56$ fm, and $R_{\max }=11.76$. Unlike $g\left(R\right)$, $G\left(R\right)$ properly accounts for the normalization and antisymmetrization of the decay channel. The asymptotic behavior of $G\left(R\right)$ is given by the Coulomb-Hankel function at the alpha-decay energy $Q_{\alpha }=8.986$ MeV (dashed line).

Figure 11

Absolute width from $R$-matrix expression (1) calculated in the M4 model space using the modified formation amplitude $G\left(R\right)$ of Fig. 10. At $R>9$ fm, the result obtained by assuming $G\left(R\right)\propto H_0^{+}(\eta ,kR)$ is marked by a dotted line.

Figure 12

Single-particle width of ${}^{212}$Po from current expression (20).

Figure 13

Ground-state alpha-decay width (left scale) and half-life (right scale) in ${}^{104}$Te as functions of the decay energy.