Coincidence spectroscopy of molecular normal Auger decay by ultrashort x-ray pulses

When a core-shell electron is ionized by coherent ultrashort x-ray pulses with varied duration, different nuclear wave packets of a cationic molecule are formed, which can be effectively studied by detecting the photoelectron-Auger electron-ion fragment as it is proposed in the present paper. We develop the theory for the photoelectron, i.e., Auger electron, ion fragment three-body coincidence spectroscopy, where dissociative Auger final states are considered. Simulations to display the HF molecule show that the coincidence spectra encode the detailed information of coherent nuclear wave-packet dynamics. Our work paves the way for using currently available coherent ultrashort x-ray pulses to investigate and manipulate molecular nuclear wave packets. Moreover, we discuss possibilities for high-resolution coincidence experimental techniques.

Figure 1

Illustration of the normal Auger process following x-ray photoionization. The molecule in the ground state $V_G$ is photoionized to the core-hole cation state $V_I$ by an ultrashort x-ray pulse with central frequency $\omega$, accompanied by the release of a photoelectron with energy ${\varepsilon }_{\mathrm{ph}}$; Auger decay happens and releases an Auger electron with energy ${\varepsilon }_{\mathrm{Aug}}$ when the system decays to the final dissociative dication state $V_F$. Ion fragments are also released carrying kinetic energies in the dissociative dication state. The coherent vibrational wave packet in the state $V_I$ is created by the ultrashort pulse, changing the subsequent Auger decay dynamics and Auger electron spectra.

Figure 2

Potential energy curves of the ground ($V_G$), core-ionized ($V_I$), and nine final dicationic states ($V_{F1},\cdots ,V_{F9}$) of the HF molecule, modified from Ref. [56]. The ground-state equilibrium geometry is marked by the pink dashed line. The final dication states are labeled in order of increasing energy at the ground-state equilibrium geometry; the labels $V_{F1},\cdots ,V_{F9}$ correspond to the $1{}^1\mathrm{\Delta }$, $1{}^1{\mathrm{\Sigma }}^{+}$, $1{}^1\mathrm{\Pi }$, $2{}^1{\mathrm{\Sigma }}^{+}$, $1{}^3\mathrm{\Pi }$, $1{}^3{\mathrm{\Sigma }}^{+}$, $2{}^1\mathrm{\Pi }$, $3{}^1\mathrm{\Pi }$, and $3{}^1{\mathrm{\Sigma }}^{+}$ states, respectively

Figure 3

Photoelectron (${\varepsilon }_{\mathrm{ph}}$)–Auger electron (${\varepsilon }_{\mathrm{Aug}}$) joint energy spectra ${\sigma }_{\mathrm{ph}\text{−}\mathrm{Aug}}^I({\varepsilon }_{\mathrm{ph}},{\varepsilon }_{\mathrm{Aug}})$, triggered by (a) 1-, (b) 4-, and (c) 8-fs pump pulses (lower panels). The upper panels are the PES obtained by integrating ${\sigma }_{\mathrm{ph}\text{−}\mathrm{Aug}}^I({\varepsilon }_{\mathrm{ph}},{\varepsilon }_{\mathrm{Aug}})$ over Auger electron energy ${\varepsilon }_{\mathrm{Aug}}$; the vertical dashed lines in (b) shows three relative photoelectron energies ($\mathrm{\Delta }{\varepsilon }_I=0.0$, 0.26, and 0.44 eV) used for the joint energy spectra simulations in Figs. 4 and 5. The relative Franck-Condon factors are also given as a red histogram in (c). (d) The AES, obtained by integrating ${\sigma }_{\mathrm{ph}\text{−}\mathrm{Aug}}^I({\varepsilon }_{\mathrm{ph}},{\varepsilon }_{\mathrm{Aug}})$ in (a)–(c) over $\mathrm{\Delta }{\varepsilon }_I$ (see the legend). The pink line with a square is the AES computed in the SA model [56].

Figure 4

Auger electron–ion fragment KER joint energy spectra ${\sigma }_{\text{KER}}({\varepsilon }_{\mathrm{ph}},{\varepsilon }_{\mathrm{Aug}},E)$ for (b) $\mathrm{\Delta }{\varepsilon }_I=0.0\mathrm{eV}$, (c) $\mathrm{\Delta }{\varepsilon }_I=0.26\mathrm{eV}$, and (d) $\mathrm{\Delta }{\varepsilon }_I=0.44\mathrm{eV}$ with a pulse duration of 4 fs. The black solid, red dashed, and green dash-dotted lines in (a) correspond to AESs obtained by integration over the KER of the joint energy maps in plots (b), (c), and (d), respectively. The approximate positions of the nine final states in their respective energy regions are marked with numbers (same as in Fig. 2) in (b). The insets in (b)–(d) show a close-up of the coincidence spectral line corresponding to the first state $1{}^1\mathrm{\Delta }$.

Figure 5

Same as in Fig. 4 but only the lowest final state $1{}^1\mathrm{\Delta }$ is shown. The lower and upper dashed lines in (b)–(d) represent the spectra generated from $\nu =0$ and 1, respectively, of the core-ionized state ${\text{HF}}^{+}\left(1s^{-1}\right)$.

Figure 6

Auger electron–ion fragment KER joint energy spectra ${\sigma }_{\text{KER}}({\varepsilon }_{\mathrm{ph}},{\varepsilon }_{\mathrm{Aug}},E)$ of only the lowest final state $1{}^1\mathrm{\Delta }$ for $\mathrm{\Delta }{\varepsilon }_I=0.26\mathrm{eV}$ [as in Fig. 5] computed for three different pulse durations of (b) 1 fs, (c) 8 fs, and (d) 16 fs.