Wigner time delay in photodetachment

Using ${\mathrm{Cl}}^{-}$ as a test case, Wigner time delay in the photodetachment process has been investigated theoretically, along with the photoionization of the isoelectronic Ar atom, for the outer $3p$ shell using the relativistic-random-phase approximation (RRPA). Time delay was probed in these systems from threshold to 80 eV, to investigate threshold effects, the centrifugal barrier shape resonance, and the Cooper minimum region. This study focuses on ${\mathrm{Cl}}^{-}$ because, for negative ions, the phase of the photoemission process is not dominated by the Coulomb phase as it is in photoionization. The results show significant differences, both qualitative and quantitative, between the time delays for ${\mathrm{Cl}}^{-}$ and Ar photoemission at low photoelectron energy, but they are rather similar in the Cooper minimum region, where the Coulomb phase is small. In particular, the Wigner time delay in ${\mathrm{Cl}}^{-}$ exhibits a dramatic energy dependence just above threshold, and a rapidly increasing time delay in the vicinity of the shape resonance. A strong angular dependence of time delay has also been found near the threshold region for the ${\mathrm{Cl}}^{-}$ case, and is absent in the case of the photoionization of Ar. The origin of these phenomenologies is explained and a prospectus for future work is presented.

Figure 1

Photoemission cross sections calculated for ${\mathrm{Cl}}^{-}$ (top panel) and Ar (bottom panel) showing $3p_{1/2}$ (dashed line), $3p_{3/2}$ (dotted line), and the sum, $3p$ (solid line), along with available experiments (solid asterisk): ${\mathrm{Cl}}^{-}$ [41, 42], Ar [43]. Note that the Ar data are for the sum of $3s$ and $3p$ cross sections. Thresholds are indicated as vertical dashed lines.

Figure 2

The absolute values of the complex transition matrix elements $\left|D\right|$ as a function of photon energy for all the three relativistically split $3p\to d$ photoemission channels: $3p_{1/2}\to d_{3/2}$ (dashed line), $3p_{3/2}\to d_{3/2}$ (dotted line), and $3p_{3/2}\to d_{5/2}$ (solid line), in ${\mathrm{Cl}}^{-}$ (top panel) and Ar (bottom panel). Thresholds are indicated as vertical dashed lines.

Figure 3

Phases of the complex matrix elements for ${\mathrm{Cl}}^{-}$ (upper left) and Ar (upper right) as functions of energy for all five relativistic $3p$ photoemission channels: $3p_{1/2}\to d_{3/2}$ (thick dashed line), $3p_{3/2}\to d_{3/2}$ (thick dotted line), $3p_{3/2}\to d_{5/2}$ (thick solid line), $3p_{1/2}\to s_{1/2}$ (thin solid line), and $3p_{3/2}\to s_{1/2}$ (thin dashed line). The corresponding time delays are shown below each phase in attoseconds (as) (same line convention as in phase plot used). Thresholds are indicated as vertical dashed lines.

Figure 4

Phase of the photodetachment amplitude for the $3p_{3/2}$ initial state of ${\mathrm{Cl}}^{-}$ in the direction of the photon polarization. The inset shows a closeup of the threshold region.

Figure 5

Wigner time delay for the photodetachment of the $3p$ state of ${\mathrm{Cl}}^{-}$, averaged over the $3p_{1/2}$ and $3p_{3/2}$ states.

Figure 6

Wigner time delay in the photodetachment of the $3p$ state of ${\mathrm{Cl}}^{-}$ (left panel) and in the photoionization of the $3p$ state of Ar (central and right panels), averaged over the $3p_{1/2}$ and $3p_{3/2}$ states shown for different photoelectron ejection angles ($0^{\circ }$: solid line; ${30}^{\circ }$: dashed line; ${60}^{\circ }$: dotted line; ${90}^{\circ }$: dashed-dotted line,) with respect to the direction of polarization. In the first two panels the time delay evaluated only in the direction of polarization ($0^{\circ }$ angle) has been shown in the inset on a different scale for the $3p$ states of ${\mathrm{Cl}}^{-}$ and Ar, respectively; present results (solid line), Ref. [25] (dashed line). In the right panel a comparison with the angular-resolved atomic time delay of Ar $3p$ from Ref. [48] is shown in the region of the Cooper minimum.