Relativistic extension of the complex scaled Green function method

Resonances play a critical role in the formation of many physical phenomena. The complex scaled Green function method provides a powerful tool for the exploration of resonances. In the paper, we extend this method to the relativistic framework. With ${}^{120}\mathrm{Sn}$ as an example, we present the details of the scheme and seek resonant states in real nuclei. The results are compared, and are in satisfactory agreement with those from frequently used methods. In particular, the present method gathers the advantages of the complex scaling method and Green function method, and can be used to determine more accurately resonance parameters such as energy and lifetime of decay.

Figure 1

The density of states ${\rho }_{\theta }^N\left(\varepsilon \right)$, the density of continuum states ${\rho }_{\theta }^{0N}\left(\varepsilon \right)$, and the continuum level density $\mathrm{\Delta }\rho \left(\varepsilon \right)$ for the state $j_{15/2}$ from RMF-CGF calculations with NL3. Here, the black solid line, red dashed line, and blue dot-dashed line in each subfigure represent respectively the results obtained with the complex rotation angles $\theta =6^{\circ }, 9^{\circ }$, and ${12}^{\circ }$, and the main shell number of the harmonic oscillator basis is $N=100$.

Figure 2

The continuum level density $\mathrm{\Delta }\rho \left(\varepsilon \right)$ with different values of $\theta$ for the state $j_{15/2}$ in the RMF-CGF calculations with NL3. Here, the main shell number of the harmonic oscillator basis is $N=100$.

Figure 3

The same as Fig. 1, but with PK1.

Figure 4

The same as Fig. 2, but with PK1.

Figure 5

The density of states ${\rho }_{\theta }^N\left(\varepsilon \right)$, the density of continuum states ${\rho }_{\theta }^{0N}\left(\varepsilon \right)$, and the continuum level density $\mathrm{\Delta }\rho \left(\varepsilon \right)$, which are denoted respectively with the black solid line, red dashed line, and blue dot-dashed line in each subfigure for the states $i_{11/2}$ and $i_{13/2}$ in the RMF-CGF calculations with NL3. A closer focus is shown in the inset for the states $i_{13/2}$. Here, the complex rotation angle $\theta ={10}^{\circ }$, and the main shell number of the harmonic oscillator basis is $N=100$.

Figure 6

The continuum level density $\mathrm{\Delta }\rho \left(\varepsilon \right)$ with different main shell numbers for the states $i_{11/2}$ and $i_{13/2}$ in the RMF-CGF calculations with NL3 and the complex rotation angle $\theta ={10}^{\circ }$. Here, the black solid line, red dashed line, and blue dot-dashed line in each subfigure represent respectively the results obtained with the main shell numbers of the harmonic oscillator basis $N=$ 80, 90, and 100.

Figure 7

The continuum level density $\mathrm{\Delta }\rho \left(\varepsilon \right)$ for the states $g_{9/2}$ in the RMF-CGF calculations with NL3. The left peak corresponds to the bound state $1g_{9/2}$ and the right peak corresponds to the resonant state $2g_{9/2}$. Here, the main shell number of the harmonic oscillator basis is $N=100$.