Outer-shell photodetachment of the metastable ${\mathrm{Mg}}^{-}$ $\left[\text{core}\right]3s3p^2$ and ${\mathrm{Ca}}^{-}$ $\left[\text{core}\right]4s4p^2$ ${}^{4}P^e$ states

We report on calculated outer-shell photodetachment cross sections from the metastable $\left[\text{core}\right]nsn{p}^2{}^{4}P^e$ states in ${\mathrm{Mg}}^{-}$ $(n=3)$ and ${\mathrm{Ca}}^{-}$ $(n=4)$ negative ions in the photon energy range $\omega =0–10\mathrm{eV}$. Double-ionization thresholds for $\mathrm{Mg}\mathrm{II}$ and $\mathrm{Ca}\mathrm{II}$ as well as doubly excited thresholds of the residual atoms $\mathrm{Mg}\mathrm{I}$ and $\mathrm{Ca}\mathrm{I}$ are located within the photon energy range considered and thus doubly and triply excited states of both ${\mathrm{Mg}}^{-}$ and ${\mathrm{Ca}}^{-}$ are reached. We use a complex scaled configuration interaction (CSCI) method for the three active electrons supplemented with a sophisticated model potential as developed by Laughlin [Phys. Scr. 45, 238 (1992)] to account for core-valence interactions. The CI calculations are based on ${\mathrm{Mg}}^{+}$ and ${\mathrm{Ca}}^{+}$ one-electron orbitals expanded in terms of $B$-spline basis set. We compare our cross sections with the only calculation available up to a photon energy $0.25\mathrm{Ry}$ $(\omega \sim 3.4\mathrm{eV})$ [Zeng et al., Phys. Rev. A 62, 022713 (2000)] performed by using the $R$-matrix method, in which only one ${}^{4}P^o$ resonance for ${\mathrm{Mg}}^{-}$ and two (${}^{4}P^o$ and ${}^{4}D^o$) for ${\mathrm{Ca}}^{-}$ were predicted. In this work, by analyzing both the cross sections and the locations of the $\mathcal{S}$-matrix poles in the complex plane, we are able to predict six ${}^{4}P^o$, four ${}^{4}D^o$, and four ${}^{4}S^o$ resonant states for ${\mathrm{Mg}}^{-}$, and three ${}^{4}P^o$ and two ${}^{4}D^o$ resonant states in ${\mathrm{Ca}}^{-}$.

Figure 1

(Color online) (a) Model potential given by Eq. (6) with energy positions of bound and resonant states above the continuum threshold. (b) Complex eigenvalue spectrum for this model potential computed with UCS with rotation angle $\theta =20°$ and several ECS computations with the same rotation angle and varying $r_0$.

Figure 2

(Color online) Calculated partial photodetachment cross section to ${\mathrm{Mg}}^{-}$ ${}^{4}P^o$, ${}^{4}D^o$, and ${}^{4}S^o$ final symmetries from the metastable ${\mathrm{Mg}}^{-}$ $\left[\text{core}\right]3s3p^2{}^{4}P^e$ state. (Black) solid line, our CSCI result with a rotation angle $\theta =15°$; (red) dashed line, $R$-matrix result by Zeng et al. [21]. The vertical (blue) thick lines indicate our $\mathrm{Mg}\mathrm{I}$ thresholds $\left[\text{core}\right]n\ell n^{\prime }{\ell }^{\prime }{}^{3}L^{\pi }$ quoted in Table ; the vertical long thick dotted lines indicate the positions of the $\mathrm{Mg}\mathrm{II}$ double-ionization thresholds ${}^{2}S^e$ and ${}^{2}P^o$, enclosing a region of triply excited states. Resonance positions are labeled as $Rn$ and the arrows point to the maximum cross sections of the CSCI calculation.

Figure 3

(Color online) Complex eigenvalue spectrum of ${\mathrm{Mg}}^{-}$ ${}^{4}P^o$, ${}^{4}D^o$, and ${}^{4}S^o$ symmetries for different values of the rotation angles from $\theta =10°$ to 20°. The eigenvalues fall into the lower half of the complex plane with an angle $2\theta$. The eigenvalues accumulated at fixed points—not affected by the complex rotation—show a resonance behavior and they are labeled as $Rn$. (Blue) vertical lines indicate the position of our $\mathrm{Mg}\mathrm{I}$ thresholds $\left[\text{core}\right]nl{n}^{\prime }{l}^{\prime }{}^{3}L^{\pi }$ included in Table .

Figure 4

(Color online) Calculated partial photodetachment cross section to ${\mathrm{Ca}}^{-}$ ${}^{4}P^o$, ${}^{4}D^o$, and ${}^{4}S^o$ final symmetries from the metastable ${\mathrm{Ca}}^{-}$ $\left[\text{core}\right]4s4p^2{}^{4}P^e$ state. (Black) solid line, our CSCI result type A with a rotation angle $\theta =15°$ (blue) dash-dotted line, our CSCI result type B with $\theta =15°$; (red) dashed line, $R$-matrix result by Zeng et al. [21]. The vertical (blue) thick lines indicate $\mathrm{Ca}\mathrm{I}$ thresholds $\left[\text{core}\right]n\ell n^{\prime }{\ell }^{\prime }{}^{3}L^{\pi }$ taken from the NIST database quoted in Table ; the vertical long thick dotted lines indicate the positions of the $\mathrm{Ca}\mathrm{II}$ double-ionization thresholds. Resonance positions are labeled as $Rn$ and the arrows point to the maximum cross section of the CSCI calculation.

Figure 5

(Color online) Complex eigenvalue spectrum of ${\mathrm{Ca}}^{-}$ ${}^{4}P^o$, ${}^{4}D^o$, and ${}^{4}S^o$ symmetries. (Black) squares, CSCI result type A for a rotation angle $\theta =15°$; (blue) circles, CSCI result type B for six different values of the rotation angle from $\theta =10°$ to 20°. Vertical lines indicate the position of our $\mathrm{Ca}\mathrm{I}$ thresholds $\left[\text{core}\right]n\ell n^{\prime }{\ell }^{\prime }{}^{3}L^{\pi }$. Upper thresholds (in blue) correspond to a calculation type B and lower thresholds (in black) to the model potential calculation type A (see text for further explanation). The latter are those quoted in Table .

Figure 6

(Color online) Total photodetachment cross section $({}^{4}P^o+{}^{4}D^o+{}^{4}S^o)$ from the metastable ${\mathrm{Mg}}^{-}$ (a) and ${\mathrm{Ca}}^{-}$ (b) ${}^{4}P^e$ state. (Black) solid line, CSCI result; (red) dashed line, $R$-matrix result by Zeng et al. [21].