Numerical calculation of supercritical Dirac resonance parameters by analytic continuation methods

The spectrum of the Dirac equation for hydrogenlike systems with extended nuclei becomes complicated when the nuclear charge exceeds a critical value $Z\approx 170$, since the lowest bound state becomes a resonance in the negative energy continuum. We address the problem of computing the resonance parameters by extending the mapped Fourier grid method to incorporate either complex scaling of the radial coordinate, or alternatively a complex absorbing potential. The method is tested on the case of quasimolecular collisions in the monopole approximation.

Figure 1

(Color online) The projections $\mid ⟨\varphi \left(E_{\nu }\right)\mid {\psi }_{1S\sigma }⟩{\mid }^2∕(E_{\nu }-E_{\nu -1})$ of a bound ${\psi }_{1S\sigma }$ state onto the supercritical U-Cf quasicontinuum states $\varphi \left(E_{\nu }\right)$ (calculated for a $N=3900$ and $s=6000$ basis) are shown as crosses $(+)$. The subcritical bound ${\psi }_{1S\sigma }$ was obtained for a separation of $R=45\mathrm{fm}$, while the supercritical basis is for $R=20\mathrm{fm}$. The line represents a Breit-Wigner fit to the data with $E_{\mathrm{res}}=-1.75795m{c}^2$ and $\Gamma =4.12\mathrm{keV}$ for the U-Cf system at $R=20\mathrm{fm}$.

Figure 2

(Color online) The density of states Eq. (7) from a Hermitian matrix diagonalization for the U-Cf $(R=20\mathrm{fm})$ Hamiltonian obtained with a basis with $N=3900$ states and $s=6000$, is shown by crosses $(+)$. The line shows a fit with Eq. (8), and yields $E_{\mathrm{res}}=-1.75794m{c}^2$ and $\Gamma =4.17\mathrm{keV}$.

Figure 3

(Color online) The width $\Gamma \left(1S\sigma \right)$ as a function of $r_c$ (in units of $\hslash ∕mc$) for a linear $(\times )$ and a quadratic $(+)$ CAP, as obtained in a basis specified by $N=2000$ and $s=6000$ for the U-Cf $(R=20\mathrm{fm})$ system.

Figure 4

(Color online) The width $\Gamma \left(1S\sigma \right)$ and mean energy $E_{\mathrm{res}}\left(1S\sigma \right)$ for the U-Cf $(R=20\mathrm{fm})$ system as a function of the mapping parameter $s$ [cf. Eq. (2)] with $N=2500$ points for the CS method $(\times )$ and $N=2000$ points for the CAP method $(+)$. The lines indicate values from Ref. 5, the phase-shift result is given as a solid line, and the truncated potential method result as a dashed line.

Figure 5

(Color online) The width $\Gamma \left(1S\sigma \right)$ as a function of mesh size $N$ using the CS $(\times )$, and quadratic CAP $(+)$ methods with mapping parameter $s=1000$ for the U-Cf $(R=20\mathrm{fm})$ system.

Figure 6

(Color online) The width $\Gamma \left(1S\sigma \right)$ as a function of $E_{\mathrm{res}}\left(1S\sigma \right)$ calculated by different methods and using different supercritical potentials. The U-U and U-Cf data from Ref. 5 were obtained for the distances $R={16}^{*}$, 16, 20, 25, $30\mathrm{fm}$, while the single-center CAP/CS data from the present work were obtained for $Z=\mathrm{195,193,190},\mathrm{187,184,181},178$. The leftmost U-U and U-Cf points marked by $R={16}^{*}\mathrm{fm}$ were calculated with pointlike nuclei.