Attosecond time delay in the photoionization of Mn in the region of the $3p\to 3d$ giant resonance

Initial insight into time delay in Mn photoionization in the region of the $3p\to ,3d$ giant autoionization resonance is gained in the framework of the “spin-polarized” random-phase approximation with exchange. The dramatic effect of the giant autoionization resonance on the time delay of photoemission from the $3d$ and $4s$ valence subshells of the Mn atom is unraveled. Strong sensitivity of the time delay of the $4s$ photoemission to the final-state term of the ion remainder $[{\mathrm{Mn}}^{+}\left(4s^1{,}^{5}S\right)$ vs ${\mathrm{Mn}}^{+}(4s^1,{}^{7}S)]$ is discovered. It is shown that photoionization time delay in the autoionizing resonance region is explicitly associated with the resonance lifetime, which can thus be directly measured in attosecond time-delay experiments. Similar features are expected to emerge in photoionization time delays of other transition-metal and rare-earth atoms with half-filled subshells that possess giant autoionization resonances as well.

Figure 1

Feynman diagrammatic representation of the SPRPAE (RPAE) equation for the photoionization amplitude $\langle k\left|\widehat{D}\right|i\rangle$ of the $i\text{th}$ subshell into the $k\text{th}$ final state [30]. Here, the time axis is directed from the left to right, the lines with arrows to the left (right) correspond to holes (electrons) in the atom, a dotted line represents an incoming photon, a dashed line represents the Coulomb interaction $V\left(r\right)$ between charged particles, and a shaded circle marks the effective operator $\widehat{D}$ for the photon-atom interaction which accounts for electron correlation in the atom.

Figure 2

(a) Solid line: present calculated SPRPAE data for the Mn $3d\uparrow$ photoionization cross sections ${\sigma }_{3d}$ (in units of megabarns); dots: experimental data on an absolute scale taken from Table 3 of Ref. [25]; dashed line: SPRPAE ${\sigma }_{3d}$ calculated previously [28] on the basis of the Fano autoionization formalism in the region of the $3p\downarrow \to 3d\downarrow$ giant resonance. (b) Solid line: present calculated SPRPAE angular-asymmetry parameter ${\beta }_{3d}$; dots: experimental data from Table 3 of Ref. [25]. (Note that former SPRPAE results for ${\beta }_{3d}$ [28], obtained on the basis of the Fano formalism in the region of the $3p\downarrow \to 3d\downarrow$ giant resonance, practically coincide with the present data and are not plotted). (c) Solid line: calculated SPRPAE phase ${\phi }_{3d}\left(\omega \right)$ (in units of radians) of the $3d\uparrow$ photoionization amplitude $T_{3d}$ [Eq. (2)]; dashed line: the same as above but calculated with the help of the single-channel Fano formula [Eq. (3)]. (d) Solid line: calculated SPRPAE time delay ${\tau }_{3d}\left(\omega \right)$ (in units of femtoseconds); dashed line: ${\tau }_{3d}\left(\omega \right)$ calculated with the help of the single-channel Fano formula [Eq. (4)].

Figure 3

(a) SPRPAE real $\left(\mathrm{Re}{T}_{3d\to f}\right)$ and imaginary $\left(\mathrm{Im}{T}_{3d\to f}\right)$ parts (in atomic units) of the dominant $T_{3d\to f}$ photoionization amplitude [Eq. (2)], calculated in the region of the giant $3p\downarrow \to 3d\downarrow$ resonance. (b) SPRPAE real $\left(\mathrm{Re}{T}_{3d}\right)$ and imaginary $\left(\mathrm{Im}{T}_{3d}\right)$ parts (in atomic units) of the total amplitude $T_{3d}$, calculated as the weighted sum of the two matrix elements $\langle Ef\parallel D\parallel 3d\rangle$ and $\langle Ep\parallel D\parallel 3d\rangle$ [Eq. (2)] in the region of the giant $3p\downarrow \to 3d\downarrow$ resonance.

Figure 4

Calculated SPRPAE (present work) and experimentally measured [26] (open circles and triangles) Mn $4s$ photoionization cross sections (in units of megabarns) ${\sigma }_{4s\uparrow }{(}^{5}S)$ and ${\sigma }_{4s\downarrow }{(}^{7}S)$, as well as calculated SPRPAE phase shifts (in units of radians) ${\phi }_{4s\uparrow }{(}^{5}S)$ and ${\phi }_{4s\downarrow }{(}^{7}S)$ of the corresponding photoionization amplitudes [Eq. (2)] and time delays (in units of femtoseconds) ${\tau }_{4s\uparrow }{(}^{5}S)$ and ${\tau }_{4s\downarrow }{(}^{7}S)$, as marked. Relative experimental data [26] were normalized to the maxima in the calculated SPRPAE ${\sigma }_{4s\uparrow }{(}^{5}S)$ and ${\sigma }_{4s\downarrow }{(}^{7}S)$ and were shifted by $0.28$ eV towards higher energies to match the position of the maxima in the calculated cross sections.