Photoelectron spectroscopy of Rydberg excited states in singly charged molecular ions CS₂⁺ by NIR laser pulses

Figure 3 shows the energy levels of a neutral CS₂ molecule, singly charged ${{{\rm{CS}}}_{2}}^{+},$ and doubly charged ${{{\rm{CS}}}_{2}}^{2+}$ molecular ions. With the help of this energy diagram, we suggest the following sequential channels as the most probable processes induced by the XUV-FEL (ω_H = 20.9 eV). In the first step, the ionization of the 2π_g valence orbital occurs,

$$\begin{eqnarray}&&{{\rm{CS}}}_{{\rm{2}}}+{\omega }_{{\rm{H}}}\to {{\rm{CS}}}_{2}^{+}[2{\pi }_{{\rm{g}}}^{-{\rm{1}}}{\rm{X}}{}^{2}{\rm{\Pi }}_{{\rm{g}}}]+e(2{\pi }_{{\rm{g}}}).\end{eqnarray} \tag{ 1 }$$

Then, in the second step, the excitation of the 5σ_u valence orbital of the ${{{\rm{CS}}}_{2}}^{+}$ into the n Rydberg states by the XUV-FEL sequentially occurs,

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{CS}}}_{2}^{+}[2{\pi }_{{\rm{g}}}^{-{\rm{1}}}{\rm{X}}{}^{2}{\rm{\Pi }}_{{\rm{g}}}]+{\omega }_{{\rm{H}}}\to \\ \,\,{{\rm{CS}}}_{2}^{+}[5{\sigma }_{{\rm{u}}}^{-{\rm{1}}}2{\pi }_{{\rm{g}}}^{-{\rm{1}}}({}^{3,1}{\rm{\Pi }}_{{\rm{u}}})\,n{\rm{s}}{\sigma }_{{\rm{g}}}]\end{array}\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&\to {{\rm{CS}}}_{2}^{+}[5{\sigma }_{{\rm{u}}}^{-{\rm{1}}}2{\pi }_{{\rm{g}}}^{-{\rm{1}}}({}^{3,1}{\rm{\Pi }}_{{\rm{u}}})n{\rm{d}}{\pi }_{{\rm{g}}}].\end{eqnarray} \tag{ 2b }$$

In the photoexcitation processes expressed by Reactions (2a) and (2b), we suppose that the most probable Rydberg series are nsσ_g and ndπ_g, on the analogy of the excitation of the 1σ_u of a CO₂ molecule into the n Rydberg states [17]. Moreover, the nsσ_g Rydberg series is much stronger than the ndπ_g in [17], so that we mainly focus our discussions on the nsσ_g hereafter, unless otherwise stated. Since the ionization potential of a CS₂ molecule is 10.1 eV [18], the Rydberg state energies of the ${{{\rm{CS}}}_{2}}^{+}$ induced by the two XUV-FEL photons are located around 31.0 eV = 10.1 eV + 20.9 eV from the ground state of the CS₂, which corresponds to the energy region just below the double ionization thresholds of ${{{\rm{CS}}}_{2}}^{2+}$ (³Π_u), 31.6 eV and ${{{\rm{CS}}}_{2}}^{2+}$ (¹Π_u), 32.3 eV [19], see figure 3.

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To evaluate the range of the photoexcited Rydberg states, we take the pragmatic view that a Rydberg state is one which follows the Rydberg formula in atomic units;

$$\begin{eqnarray}&&{E}_{n}=IP-\displaystyle \frac{{Z}^{2}}{{(n-{\delta }_{{\ell }})}^{2}},\end{eqnarray} \tag{ 3 }$$

where E_n is the energy of the nth level, IP the ionization potential of the initial state, Z the effective charge, n the principal quantum number of the level considered, and ${\delta }_{{\ell }}$ the quantum defect for the azimuthal quantum number ${\ell }$ [20]. Referring to the quantum defects ${\delta }_{{\ell }=0}\sim 1.9$ and ${\delta }_{{\ell }=2}\sim 0.0$ for a neutral CS₂ [21] and the energies of ${{{\rm{CS}}}_{2}}^{2+}$ ³Π_u and ${{{\rm{CS}}}_{2}}^{2+}$ ¹Π_u [19], we obtain from the Rydberg formula of equation (3) that the n takes values from 10 to 16 for the nsσ_g series converging to the ³Π_u limit and from 8 to 9 for those to the ¹Π_u limit.

In order to pursue our study of photoabsorption processes to the higher Rydberg states, we make the following simulations. If the oscillator strength ${f}_{n}$ varies monotonically for large n, it can be regarded as a smooth function of n, ${f}_{n}=df/dn.$ Whereas, differentiating equation (3), we obtain

$$\begin{eqnarray}&&\displaystyle \frac{d{E}_{n}}{dn}=\displaystyle \frac{2{Z}^{2}}{{(n-{\delta }_{{\ell }})}^{3}}\approx \displaystyle \frac{2{Z}^{2}}{{n}^{3}}.\end{eqnarray} \tag{ 4 }$$

Thus, we can write

$$\begin{eqnarray}&&\displaystyle \frac{df}{dE}\approx \displaystyle \frac{1}{2{Z}^{2}}{n}^{3}{f}_{n}.\end{eqnarray} \tag{ 5 }$$

On the other hand, if the wave functions of the higher-lying Rydberg states are approximated as hydrogenic wave functions, then it follows that ${f}_{n}\propto {n}^{-3}$ [22]. Combining these expressions, we can reach the simplest result that the oscillator strength distribution, df/dE, is constant over the excitation photon energy of E for large $n$ [23].

The energy separations between adjacent levels of the higher-lying Rydberg states are smaller than the bandwidth 0.5 eV (FWHM) of the XUV-FEL. Therefore, a band of the n Rydberg states within the bandwidth are excited simultaneously. Both this and the constant oscillator strength distribution lead to the straightforward prediction that the population distribution $P(E)$ of the n Rydberg states is directly reflected by the spectrum distribution $S({\omega }_{{\rm{H}}})$ of the XUV-FEL pulse;

$$\begin{eqnarray}&&P(E)\approx c\times S({\omega }_{{\rm{H}}}).\end{eqnarray} \tag{ 6 }$$

As the third step, we consider the photoemission from the n Rydberg states of ${{{\rm{CS}}}_{2}}^{+}$ by the NIR laser (ω_L). They are expressed by

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{CS}}}_{2}^{+}[5{\sigma }_{{\rm{u}}}^{-{\rm{1}}}2{\pi }_{{\rm{g}}}^{-{\rm{1}}}({}^{3,1}{\rm{\Pi }}_{{\rm{u}}})n{\rm{s}}{\sigma }_{{\rm{g}}}]+{\omega }_{{\rm{L}}}\to \\ \,\,{{\rm{CS}}}_{2}^{2+}({}^{3,1}{\rm{\Pi }}_{{\rm{u}}})\,+\,e(n{\rm{s}}{\sigma }_{{\rm{g}}}\to \varepsilon l{\lambda }_{n}),\end{array}\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{CS}}}_{2}^{+}[5{\sigma }_{{\rm{u}}}^{-{\rm{1}}}2{\pi }_{{\rm{g}}}^{-{\rm{1}}}({}^{3,1}{\rm{\Pi }}_{{\rm{u}}})n{\rm{d}}{\pi }_{{\rm{g}}}]+{\omega }_{{\rm{L}}}\to \\ \,\,{{\rm{CS}}}_{2}^{2+}({}^{3,1}{\rm{\Pi }}_{{\rm{u}}})\,+\,e(n{\rm{d}}{\pi }_{{\rm{g}}}\to \varepsilon l{\lambda }_{n}),\end{array}\end{eqnarray} \tag{ 7b }$$

with Reaction (7a) being the most dominant processes, as mentioned above. The photoionization cross section ${\sigma }_{n}(\omega )$ of the n Rydberg state for large n takes the following form within the semiclassical approximation [22, 24];

$$\begin{eqnarray}&&{\sigma }_{n}(\omega )={\sigma }_{n}^{th}\times {\left(\displaystyle \frac{{\varepsilon }_{n}}{\omega }\right)}^{3},\end{eqnarray} \tag{ 8 }$$

where ${\varepsilon }_{n}$ is the binding energy of the n Rydberg state, ${\sigma }_{n}^{th}$ is the cross section at the threshold photon energy of $\omega ={\varepsilon }_{n}$ and is dependent on the charge of the ionic core Z. Since we are considering the Rydberg states of the ${{{\rm{CS}}}_{2}}^{+}$ ions, we choose Z = 2 in this case. The cross sections for the n Rydberg states of hydrogenic ions with Z = 2 as a function of photon energy are calculated using equation (8) and depicted in figure 4(a). Under a fixed photon energy, the cross section ${\sigma }_{n}(\omega )$ reduces to ${\sigma }_{n}\propto {n}^{-5},$ because ${\varepsilon }_{n}\propto {n}^{-2}$ and ${\sigma }_{n}^{th}\propto n$ [24–26]. This n-dependent cross section can be seen along the vertical line at ω_L in figure 4(a).

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Photoelectron spectra are usually measured as a function of photoelectron energy with the fixed photon energy. Hence, it is convenient that ${\sigma }_{n}\propto {n}^{-5}$ is transformed to the function of the energy ${\tilde{\varepsilon }}_{n}$ of photoelectrons emitted from the n Rydberg state, ${\tilde{\varepsilon }}_{n}=({\omega }_{{\rm{L}}}-{\varepsilon }_{n}).$ Along this line, we can get the following equation,

$$\begin{eqnarray}&&{\sigma }_{n}={\sigma }_{0}\times {(\omega -{\tilde{\varepsilon }}_{n})}^{5/2},\end{eqnarray} \tag{ 9 }$$

where ${\sigma }_{0}$ is a constant having the dimension of a cross section. The photoelectron spectrum for the n Rydberg states at ${\omega }_{{\rm{L}}}=1.56\,{\rm{eV}}$ is simulated by equation (9) and depicted in figure 4(b).

In the above discussions, we have introduced the expressions for a population distribution of the n Rydberg states, $P(E),$ in equation (6), and their photoionization cross sections of them, ${\sigma }_{n},$ in equation (9). As depicted in figure 5, combining these expressions, an analytical form of the photoelectron spectrum for each n Rydberg state, which was measured by the XUV-FEL pump and NIR laser probe experiment, is obtained;

$$\begin{eqnarray}&&I({\tilde{\varepsilon }}_{n})=c\times S({\omega }_{{\rm{H}}})\times {\sigma }_{0}\times {(\omega -{\tilde{\varepsilon }}_{n})}^{5/2}.\end{eqnarray} \tag{ 10 }$$

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Then, let us apply equation (10) to the present case of CS₂. In our experimental conditions, the double ionization thresholds ³Π_u and ¹Π_u of CS₂ are accessible via the XUV-FEL two-photon sequential ionization/excitation and probe NIR laser ionization. Therefore, the two nsσ_g Rydberg series converging to ³Π_u and ¹Π_u are the intermediate states of the present pomp-probe processes. Adopting the XUV-FEL central photon energy of 20.9 eV with the bandwidth of 0.5 eV and the NIR photon energy of 1.56 eV, the ionization energy 10.1 eV into the ground state ²Π_g of ${{{\rm{CS}}}_{2}}^{+}$ [18], and the double ionization energy 31.6 eV and 32.3 eV into the ³Π_u and ¹Π_u states of ${{{\rm{CS}}}_{2}}^{2+}$ [19], respectively, we finally obtain the simulated photoelectron spectra from the two nsσ_g series, which are shown in figure 6 by the vertical bars. Here, we assumed that excitation probability into the two nsσ_g Rydberg series converging to the ³Π_u and ¹Π_u limits are the same. In the figure, the simulation results overall reproduce the experimental spectrum though there are some discrepancies in the energy. The discrepancies in the energy might be due to spectral fluctuations of the XUV-FEL (∼0.3 eV), nonlinearity in the low velocity region of our VMI (∼0.4 eV), and uncertainties of the double ionization thresholds (∼0.3 eV). In fact, one can see better agreement between the experiment and simulation by shifting the experimental spectrum into the lower energy side.

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The overall agreement between the experimental and the calculated spectra implies that the outer ring component with the energies around 1.1 eV in figure 2 originates essentially from photoionization of the nsσ_g Rydberg states with principal quantum numbers n = 10–16 belonging to the ³Π_u dicationic state, and the low energy component with the energies of ∼0.1 eV corresponds to that of nsσ_g states with n = 8 belonging to the ¹Π_u state and that with a small additional component with n = 9.

As can be seen in Reaction (2a), the nsσ_g Rydberg states of the ${{{\rm{CS}}}_{2}}^{+}$ ions are due to the 5σ_u → nsσ_g transition induced by the linearly polarized XUV-FEL. Similarly, the ndπ_g Rydberg states of the ${{{\rm{CS}}}_{2}}^{+}$ ions are due to the 5σ_u → ndπ_g transition, see Reaction (2b). In the former, the axis distribution of the ensemble of the ${{{\rm{CS}}}_{2}}^{+}$ molecular ions is expressed by ${\cos }^{2}{\rm{\Theta }}=1+2{P}_{2}(\cos \,{\rm{\Theta }}),$ where ${\rm{\Theta }}$ is the angle between the molecular axis and the polarization vector of the XUV-FEL and ${P}_{2}(\cos \,{\rm{\Theta }})$ the 2nd Legendre polynomial, because the nsσ_g Rydberg states are excited by the σ_u → σ_g parallel transitions [27]. In the latter, the axis distribution of the ensemble is expressed by ${\sin }^{2}{\rm{\Theta }}=1-{P}_{2}(\cos \,{\rm{\Theta }}),$ because the ndπ_g Rydberg states are excited by the σ_u → π_g perpendicular transitions. In the present pump-probe experiments, the time-delay is zero within the timing jitter between the pump and the probe laser pulses, and the pulse width of the probe NIR laser is 490 fs. Since these time scales are much shorter than the rotational period of CS₂ molecules (153 ps) [28], we ignore the time evolution of the rotational wave packets after the resonance excitation of the n Rydberg states in the following discussion.

The laboratory frame PAD (LF-PAD) from the aligned molecules is expressed as a convolution of the molecular frame PAD (MF-PAD) with the laboratory frame molecular axis distribution expressed by an expansion in polarization moments [29]. Taking both the polarization vector of the XUV-FEL along the LF z-axis and cylindrical symmetry of the ${{{\rm{CS}}}_{2}}^{+}$ molecular ions into account, such LF-PAD may be expressed by,

$$\begin{eqnarray}&&\begin{array}{l}I(\theta )\propto \displaystyle \sum _{L,\,{L}_{\gamma },\,K}\displaystyle \sum _{M}(10,\,10| {L}_{\gamma }0){A}_{LM}^{{L}_{\gamma }}{(-1)}^{M}(L-M,\,{L}_{\gamma }M| K0)\\ \,\times \,(L0,\,{L}_{\gamma }0| K0)\sqrt{\displaystyle \frac{2L+1}{2K+1}}{a}_{K0}{P}_{L}(\cos \,\theta ),\end{array}\end{eqnarray} \tag{ 11 }$$

where the dynamical parameter ${A}_{LM}^{{L}_{\gamma }}$ is described by relevant dipole matrix elements [30], L the vector sum of the angular momenta of photoelectron partial waves in the MF, ${L}_{\gamma }$ the vector sum of photon angular momentum, ${a}_{K0}\sqrt{\tfrac{2K+1}{4\pi }}\equiv {a}_{K}$ the axis polarization moment, and $\theta$ the angle between the photoelectron momentum and the LF z-axis. Under the present experimental conditions, ${L}_{\gamma }=0,2$ and ${a}_{0}=1,$ ${a}_{2}=2$ (for parallel transitions) or ${a}_{2}=-1$ (for perpendicular transitions). Thus, after the summation over ${L}_{\gamma }$ and K, the LF-PAD formula of equation (8) is simply re-written by

$$\begin{eqnarray}&&I(\theta )\propto \displaystyle \sum _{\begin{array}{l}L=0\\ {\rm{step2}}\end{array}}^{4}{B}_{L}{P}_{L}(\cos \,\theta ).\end{eqnarray} \tag{ 12 }$$

The limitation of L in equation (12) comes from the non-zero condition $(L0,\,{L}_{\gamma }0| K0)\ne 0$ of the Clebsch–Gordan coefficients in equation (11).

In figure 7, the polar plot of the LF-PAD from the n Rydberg states is shown together with the fitted curve of equation (12). Although the fitted curve reproduces the experimental LF-PAD well, one cannot distinguish the Reactions (7a) and (7b) with the present analytical procedure. The reason of this is that the applicable LF-PAD formula, i.e. Equation (12), is common for the two different reactions. However, if one applies the present approach to an isolated resonance state, one may be able to derive interesting parameters on photoionization dynamics, i.e., amplitudes of the dipole matrix elements and their phase shifts, from the experimental data with the help of the analyses using equation (12).

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