Relativistic calculations of angle-dependent photoemission time delay

Angular dependence of photoemission time delay for the valence $n{p}_{3/2}$ and $n{p}_{1/2}$ subshells of Ar, Kr, and Xe is studied in the dipole relativistic random phase approximation. Strong angular anisotropy of the time delay is reproduced near respective Cooper minima while the spin-orbit splitting affects the time delay near threshold.

Figure 1

Top: The angular anisotropy parameters ${\beta }_{3p_{1/2}}$ (green dashed line) and ${\beta }_{3p_{3/2}}$ (blue dotted line) from the RRPA calculation and ${\beta }_{3p}$ from the RPAE calculation (red solid line). The experiment (error bars) is from Ref. [20]. Other panels: The Wigner time delay ${\tau }_{\mathrm{W}}$ of Ar $3p_{1/2}$ and $3p_{3/2}$ (same line styles). The nonrelativistic RPAE results from Ref. [16] and RPAE-2 from Ref. [4] are shown by the red solid line and red filled circles, respectively. The total atomic time delay ${\tau }_a$ from the ECS-RPA calculation of Ref. [5] is plotted with open circles. The second top panel: $\theta =0^{\circ }$, third panel: $\theta ={45}^{\circ }$, bottom panel: angular average. The experimental data from Ref. [21] are shown on the bottom panel with error bars.

Figure 2

The time delay of Kr $4p_{1/2}$ (green dashed line) and $4p_{3/2}$ (blue dotted line) is plotted versus the photon energy in the fixed emission directions $\theta =0^{\circ }$ and $\theta ={45}^{\circ }$. The nonrelativistic RPAE result from Ref. [16] is shown by the red solid line. The angular-averaged experimental data from Ref. [21] are shown with error bars.

Figure 3

The time delay of Xe $5p_{1/2}$ (green filled circles) and $5p_{3/2}$ (blue open circles) is plotted versus the photon energy. The nonrelativistic RPAE result from Ref. [16] is shown by the red solid line. Regions of the autoionization series $5s^{1}np$ [23] and $4d^{9}np$ [24] are marked by vertical arrows.

Figure 4

The atomic time delay in the Kr $4p$ subshell. Top: The Wigner time delay from the $4p_{1/2}$ and $4p_{3/2}$ subshells is shown with filled green circles and open blue circles, respectively. Bottom: The time delay difference ${\tau }_a^{3/2}-{\tau }_a^{1/2}$. The raw RRPA calculations at $\theta =0^{\circ }$ and ${45}^{\circ }$ are shown with the red solid and purple dashed lines, respectively. The CLC correction from Ref. [10] is visualized with a thin dotted line. The RRPA result at $\theta =0^{\circ }$ corrected by the CLC is shown with a thick solid line.

Figure 5

Same as Fig. 4 for Xe $5p$ subshell. The experimental data from Ref. [7] are plotted with error bars.