# Attosecond Time Delay in Photoionization of Noble-Gas and Halogen Atoms
## 3 Results and Discussion
### General Features  
Figures 1 and 2 shows the results for three outermost subshells of xenon, iodine, krypton, and bromine atoms, while Figure 3 shows that for two outermost subshells of argon and chlorine atoms. Our first impression is the great similarity between any two neighbor atoms on the periodic table, which is not a surprise due to their similar electron configurations. Another reason is that in the LDA approximation, the multiplet splitting of the halogen atoms is not included. The LDA potential is a mean-field potential only depending on the radial wave functions \(R_{nl}\), which indicates spherical symmetry for the target atoms. However, due to electron interactions, open-shell atoms (like halogen) don't have exact spherical symmetry and there will be an energy splitting among different ionic states. For example, the singly-ionized iodine configuration \(5s^{2}5p^{4}\) has three ionic states \({}^{3}P\), \({}^{1}D\), and \({}^{1}S\), whose energy splitting is shown in Table 1 of Ref. [60]. Even though the LDA potential can not resolve the energy splitting, it still provides valuable results for the halogen atoms. This is because for deeply-bound subshells, the energy splitting is much smaller than its binding energy, which suggests spherical symmetry is not a bad approximation. The partial photoionization cross sections can be seen as a total cross sections from all the ionic states, while the asymmetry parameters, phase shifts, and Wigner time delay can be treated as average values among the different ionic states. For noble-gas atoms, both their partial cross sections and asymmetry parameters presented in Figures 1-3 agree well with other theoretical and experimental data (Ref. [61] and references there in). For halogen atoms, our calculations also have good qualitative agreement with Ref. [62]. More comparison is available for iodine atom in Refs. [63; 64; 65; 60]. The asymmetry parameter \(\beta\) measures the channel interference effects for the photoionization process. In the central-potential model, \(\beta\) for photoionization of an electron with initial orbital angular momentum \(l_{0}\) is given by the Cooper-Zare formula [66]  
\[\beta =\frac{l_{0}(l_{0}-1)P_{l_{0}}^{2}+(l_{0}+1)(l_{0}+2)-6l_{0}(l_{0 }+1)P_{l_{0}}\cos\Delta}{(2l_{0}+1)\left[l_{0}P_{l_{0}}^{2}+(l_{0}+1)\right]},\] (19) \[P_{l_{0}} =\frac{\left|\left\langle R_{E_{l0}-1}(r)|r+\delta V(r;\omega)|R_ {R_{nl_{monotonically. We did not plot the Coulomb phase shift \(\sigma_{l}\) due to its known analytical form and trivial behavior. It increases quickly within 10 eV above the ionization threshold, then slowly approaches zero at higher photon energy. The dramatic increase of the Coulomb phase shift near the threshold, has a great impact on the Wigner time delay. As shown in the fourth row of all figures, Wigner time delays tend to be infinity near the threshold, because of the large positive energy derivative of the Coulomb phase shift. In all the graphs of Figure 1, there are spikes located roughly at 63 eV, 140 eV and 180 eV for Xe, and 50 eV, 120 eV and 165 eV for iodine. These are resonances generated from excited Rydberg states when the ionization channels are opened for the 4\(d\), 4\(p\) and 4\(s\) subshells. Since Rydberg resonances are not the focus of this paper, we have removed some of the resonance data points to smooth the curve. Spikes appearing in Figures 2 and 3 have the same origin. Figure 1: Left to right columns: calculations for xenon (**a**,**c**,**e**,**g**) and iodine (**b**,**d**,**f**,**h**) atoms. Top to bottom rows: partial cross section (**a**,**b**), asymmetry parameter (**c**,**d**), short-range plus induced phase shift (**e**,**f**), and Wigner time delay (**g**,**h**) for subshells 5\(p\), 5\(s\) and 4\(d\). Figure 2: Left to right columns: calculations for krypton (**a**,**c**,**e**,**g**) and bromine (**b**,**d**,**f**,**h**) atoms.