Near K-edge Photoionization and Photoabsorption of Singly, Doubly, and Triply Charged Silicon Ions

The Si^q+ ion beams contained ground-level ions and possibly also unknown fractions of long-lived metastable ions with lifetimes longer than the ions' flight time (a few microseconds) through the apparatus. For aluminum-like Si⁺, where we assume a statistical population of the experimentally unresolved two fine-structure components of the 3s² 3p² P ground term, the long-lived metastable levels are the 3s 3p^{2 4} P_J levels with excitation energies of ∼5.3 eV and lifetimes in the range 0.13–1.2 ms (Froese Fischer et al. 2006). Magnesium-like Si²⁺ has long-lived 3s 3p³ P_J excited levels with excitation energies of ∼6.6 eV and lifetimes of 58 μs (J = 1), 7.8 s (J = 2), and ∞ (J = 0) with regard to single-photon emission (Froese Fischer et al. 2006). Sodium-like Si³⁺ does not have long-lived singly excited metastable levels. However, the core-excited 2p⁵ 3s 3p⁴ D_7/2 level is metastable against autoionization with a lifetime in the 1–10 μs range (Howald et al. 1986). Its excitation energy is 106 eV (Schmidt et al. 2007).

The fractional populations of metastable excited levels depend on their excitation energies and on the plasma conditions in the ion source. In outer-shell photoionization experiments, their presence is often revealed by characteristic threshold and resonance features in the measured photoionization cross sections. By comparing experimental and theoretical photoionization cross sections Kennedy et al. (2014) inferred a 3s 3p^{2 4} P fraction of about 10% for a Si⁺ ion beam that was also produced with an ECR ion source. Similarly, Mosnier et al. (2003) arrived at a 2%–3% 3s 3p³ P fraction of their Si²⁺ ion beam. In the present inner-shell photoionization measurements we did not recognize any specific cross-section features that can be attributed to the presence of metastable levels. Nevertheless, such metastable levels cannot be excluded for the present Si⁺ and Si²⁺ beams similar to in the previous experiments by Kennedy et al. (2014) and Mosnier et al. (2003).

In view of its high excitation energy, one does not expect a significant population of the autoionizing Si³⁺(2p⁵ 3s 3p⁴ D_7/2) metastable level. However, even a tiny contamination of the Si³⁺ beam can lead to a noticeable production of Si⁴⁺ ions via autoionization. In the present experiment, this created a large background in the Si³⁺ single-ionization channel, such that no meaningful experimental result could be obtained for the production of Si⁴⁺ by photoionization of Si³⁺. We conclude that, apart from this limitation, metastable primary ions do not play a significant role in the present study.

The photon-energy scale was calibrated by a separate measurement of the krypton L₃ and L₂ absorption edges at about 1677 and 1730 eV (Figure 1) using a combination of a gas jet and a photoelectron spectrometer (see Müller et al. 2017, 2018, for details). The recommended values for the Kr L₃ and L₂ threshold energies are 1679.07(39) and 1730.90(50) eV (Deslattes et al. 2003). Unfortunately, it is not clear where to read these values off the measured absorption curves. Therefore, we have applied energy shifts to our measured data such that the presently measured krypton absorption edges line up with the corresponding absorption curves of Wuilleumier (1971) as displayed in Figure 1.

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Wuilleumier (1971) provided wavelengths in Cu x units referring to the Cu K α₁ line. For the conversion of these units to electronvolts, the CODATA 2018 set of fundamental physical constants was used (Tiesinga et al. 2021). The resulting conversion factor agrees with the one provided earlier by Deslattes et al. (2003) within its negligible (in the present context) uncertainty.

Absorption at the Kr L₃ and L₂ edges was also measured by others. The results of Kato et al. (2007) differ from those of Wuilleumier (1971) by ∼1.25 and ∼0.55 eV, respectively. Kato et al. (2007) calibrated their energy scale to the Kr $2{p}_{3/2}^{-1}5p$ excitation energy in the vicinity of the L₃ threshold as determined by Ibuki et al. (2002). These latter authors did not provide an estimate for the uncertainty of their energy scale. The same holds for the Kr L₃ absorption measurements of Schmelz et al. (1995). Nagaoka et al. (2000) and Okada et al. (2005) calibrated their energy scales by referring to the work of Wuilleumier (1971) as is also done here.

The energy shifts applied to our absorption curves were 4.65 and 4.7 eV for the L₃ edge and the L₂ edge, respectively. We attribute a read-off uncertainty of 0.05 eV to this calibration method. For the extrapolation to higher energies we used a linear fit through the two calibration points. By this extrapolation, the read-off error propagates to energy uncertainties of ±0.2 eV at 1840 eV and of ±0.3 eV at 1920 eV (Figure 1(c)). Taking the 0.5 eV uncertainty of the recommended L₂-threshold energy (Deslattes et al. 2003) and a 0.2 eV uncertainty associated with imperfections of the photon beamline (see Section 4.3) additionally into account, we arrive at an uncertainty of ± 1 eV of the present calibrated photon-energy scale for the photon energies in the current experimental range of 1835–1900 eV.

In calibrating the photon-energy scale of the silicon ions, we have additionally considered the Doppler shift that is caused by the unidirectional movement of the ions in the Si^q+ ion beams (see, e.g., Schippers et al. 2014, for details). This does not introduce any significant additional uncertainty of the photon-energy calibration.

Figure 2 provides an overview of the measured and calculated cross sections σ_q,r for multiple photoionization of Si^q+ ions (see Equation (1)) with primary charge states q = 1, 2, 3 together with the cross-section sums

$$\begin{eqnarray}&&{\sigma }_{q,{\rm{\Sigma }}}=\sum _{r}{\sigma }_{q,r}.\end{eqnarray} \tag{ 2 }$$

The experimental energy ranges comprise the thresholds for direct ionization of one K-shell electron. The experimental photon-energy spread was ΔE ≈ 3 eV. This is sufficient for resolving the individual 1s → np resonance groups for n = 3 and n = 4. For Si³⁺, even the 1s → 5p resonance group can be discerned.

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The logarithmic cross-section scales cover up to 4 orders of magnitude. For Si⁺, the main ionization channel is triple ionization. For Si²⁺, triple ionization dominates only for energies above ∼1875 eV where direct K-shell ionization becomes energetically possible. At lower energies, double ionization is stronger. For Si³⁺, double ionization is the dominating ionization channel in the entire experimental photon-energy range. The single-ionization channel could not be measured as explained above (Section 2.1). Other Si³⁺ ionization channels were scrutinized, but no detectable signal was found. This indicates that the associated cross sections are small, as is also corroborated by our theoretical calculations.

Our theoretical ab initio cross sections are in remarkable agreement with the measured ones considering the simplifications that had to be applied in order to make the calculations of the complex deexcitation cascades tractable. We like to point out that the calculations were performed independently of the measurements.

The experimental and theoretical resonances line up if energy shifts of −1.7, −1.8, and −1.2 eV are applied to the theoretical Si⁺, Si²⁺, and Si³⁺ cross sections, respectively. These differences between our theoretical and experimental resonance positions are less than a factor of 2 larger than the 1 eV uncertainty of the experimental energy scale. Our theoretical calculations do not account for any of the metastable primary ions discussed in Section 2.1. The comparison between the theoretical and the experimental resonance structures suggests that metastable ions do indeed not play a significant role in the present study as already concluded above.

Only the theoretical and experimental cross sections σ_3,5 can be directly compared on an absolute scale. The mutual agreement is within the ± 15% experimental uncertainty over nearly the entire experimental energy range. The largest discrepancy between the experimental and present theoretical cross sections concerns the (small) Si⁺ double-ionization cross section, which is underestimated by a factor of ∼2. At energies between 1845 and 1858 eV, slight discrepancies occur, which are due to a limited ability of the theory to correctly describe the experimentally observed resonance structure. There are no such obvious discrepancies for Si⁺ and Si²⁺.

We conclude that the present theoretical approach is capable of reliably predicting the multiple photoionization cross sections of low-charged silicon ions. Similar accuracy can be expected for neighboring elements from the periodic table. The benchmarking by our experimental results leads to rather small corrections of the theoretical resonance energies, which are only slightly larger than the uncertainty of the present experimental photon-energy scale.

For Si⁺ and Si²⁺, the experimental cross-section sums σ_q,Σ were put on absolute scales by multiplying all individual cross sections σ_q,r for a given primary charge state q by the same factor such that the sums line up with the respective theoretical absorption cross sections of Verner et al. (1993, Figures 3(b)–(d)). We have applied the same approach already earlier to the photoabsorption of negatively charged Si⁻ ions (Perry-Sassmannshausen et al. 2021) where we used the theoretical absorption cross section for neutral silicon by Henke et al. (1993) as a reference. These data are shown in Figure 3(a) for comparison. Apparently, the cross sections for nonresonant absorption are nearly independent of the primary charge state. At 1890 eV the cross sections for Si⁰ (Henke et al. 1993) and for Si⁺, Si²⁺, and Si³⁺ (Verner et al. 1993) all amount to about 0.15 Mb.

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For energies where there are no signatures from resonant processes, our theoretical cross-section sums for Si⁺, Si²⁺, and Si³⁺ agree with the results of Verner et al. (1993), which do not account for resonant photoabsorption. The same is true for the more recent theoretical results of Witthoeft et al. (2011), who also included resonant photoabsorption in their calculations. However, their resonance positions deviate significantly by up to ∼9 eV from the measured ones. Our present energy-shifted (by less than 2 eV, see Section 4.1) theoretical results fit much better to the experimentally observed resonance positions than the results of Witthoeft et al. (2011).

Another significant discrepancy between the present and the previous results concerns the thresholds for direct K-shell photoionization. Our theoretical results are 1860, 1874, and 1891 eV for Si⁺, Si²⁺, and Si³⁺, respectively, with the abovementioned energy shifts applied. As can be seen from Figure 3, the threshold energies as predicted by Verner et al. (1993) are lower by more than 20 eV. This shows that our experimental benchmarks are vital for arriving at an accurate positioning of the various absorption features. From the comparison with the theoretical absorption cross sections we conclude that, within the present experimental uncertainties, the experimental cross-section sums σ_q,Σ represent the Si^q+ photoabsorption cross sections.

The most prominent resonance feature in each absorption cross section is the lowest-energy resonance group, which is associated with 1s → 3p excitation (Figure 3). Its relative strength increases with increasing primary charge state. This also holds for the resonances associated with 1s excitation to higher np subshells. In order to provide accurate resonance data we have measured the most prominent resonance features at lower photon-energy spreads. This was achieved at the expense of photon flux by narrowing the monochromator exit-slit.

Figure 4 displays high-resolution measurements of the Si⁺(1s⁻¹ 3s² 3p²) resonance group, which consists of the four LS terms ²S, ²P, ⁴P, and ²D resulting from the coupling of the single 1s electron to the three 3p² parent terms ¹S, ³P, and ¹D. For these measurements the dominant triple-ionization channel (σ_1,4) was chosen. The experimental resolving power increases when going from panel (a) to panel (d). The numerical values of the respective photon-energy spreads ΔE were obtained by a simultaneous fit of four Voigt line profiles to the four measured data sets. The resonance parameters that resulted from the fit, i.e., the resonance energies E_r , the Lorentzian line widths Γ, and the resonance strengths S_1,4 are listed in Table 2.

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In the fit, an overall energy shift was used as an additional fit parameter individually for each data set (see the caption of Figure 4). The maximum shift amounts to −0.19 eV, which is much less than the 1 eV uncertainty of the experimental photon-energy scale. Ideally, one would expect that the resonance positions were independent of the width of the monochromator's exit-slit. We attribute the observed shifts to a slight asymmetry of the photon-energy distribution and to mechanical imperfections of the photon-beamline's mechanics. A pertaining 0.2 eV uncertainty has already been considered in the error budget of our photon-energy calibration (see Section 2.2).

In analogy to the work of Schlachter (2004) for carbon, we note that the 1s⁻¹ 3s² 3p^{2 2} P and 1s⁻¹ 3s² 3p^{2 4} P levels, which are populated by 1s → 3p excitation of Si⁺, can also be prepared by the removal of a 1s electron from neutral ground-term neutral Si(1s² 3s² 3p^{2 3} P). Therefore, the widths of the 1s⁻¹ 3s² 3p^{2 2} P and 1s⁻¹ 3s² 3p^{2 4} P levels from Table 2 correspond to the core-hole lifetime τ of initially neutral silicon. Both widths agree within their experimental uncertainties. From the less uncertain value we obtain τ = h/Γ = 1.4 ± 0.5 fs in agreement with the present calculations and with the value of 1.47 fs calculated by Palmeri et al.(2008).

In Figure 4(d), the present theoretical results for σ_1,4 are compared with the experimental high-resolution results. For the comparison, the theoretical resonances were shifted in energy by −1.7 eV and convoluted with a Gaussian with an FWHM of 0.3 eV. This resolution is a factor of 10 higher as compared to the one in Figure 3 and reveals some discrepancies between experiment and theory, which are missed when looked at with larger photon-energy spread. The calculated splittings between the various 1s⁻¹ 3s² 3p² terms are larger than what is found experimentally.

The computed resonance strengths of individual terms contributing to σ_1,4 (see Figure 4(d)) are larger than the experimental ones by about 50%. Since the present theoretical and experimental absorption cross sections σ_1,Σ agree with one another within the experimental uncertainty (Figure 3(b)) it must be concluded that the mismatch for σ_1,4 is due to the overestimation of the corresponding branching ratios for the 1s⁻¹ 3s² 3p² resonances.

The theoretical branching ratios for the triple-ionization channel are almost the same for all of the 1s⁻¹ 3s² 3p² resonances, i.e., their differences are smaller than the uncertainties of the individual S_1,4 values. This allows us to scale the fitted individual resonance strengths to the absorption cross section. From a fit to the 1s⁻¹ 3s² 3p² peak in Figure 3(b) we obtain a strength of 3.15(17) Mb eV for the sum of the four tabulated resonances. The sum of the S_1,4 values in Table 2 amounts to 2.02(17) Mb eV. Using these values we have calculated the tabulated absorption resonance-strengths as S_Σ = S_1,4 × 3.15/2.02. In the same way we have calculated S_Σ separately for the 1s⁻¹ 3s² 3p [factor 4.30(3)/3.20(22)] and 1s⁻¹ 3s² 4p [factor (1.12(4)/1.03(21)] resonances of the Si²⁺ absorption spectrum (Table 3). For Si³⁺, the absorption spectrum is essentially identical with σ_3,5 and, thus, S_Σ = σ S_3,5 for all resonances in Table 4.

Table 4. Results of the Resonance Fits to the Si³⁺ High-resolution Data (σ_3,5) for the 1s⁻¹ 3s np Resonance Groups with n = 3, 4, 5 (Figure 6)

Designation	Er	Γ	S3,5	SΣ
	(eV)	(eV)	(Mb eV)	(Mb eV)
$1{s}^{-1}\,3s\,3p{{(}^{3}P)}^{2}P$	1849.64(01)	0.45(02)	5.20(09)	5.20(09)
$1{s}^{-1}\,3s\,3p{{(}^{1}P)}^{2}P$	1853.24(01)	0.35(04)	1.23(06)	1.23(06)
$1{s}^{-1}\,3s\,4p{{(}^{3}P)}^{2}P$	1871.91(02)	0.34(05)	1.19(08)	1.19(08)
$1{s}^{-1}\,3s\,4p{{(}^{1}P)}^{2}P$	1872.96(04)	0.45(12)	0.56(08)	0.56(08)
1s−1 3s 5p2 P	1880.20(05)	1.00(44)	0.79(06)	0.79(06)
1s−1 3s 6p2 P	1883.86(04)	0.23(12)	0.25(04)	0.25(04)

Note. For the explanation of the notation see Table 2.

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The fits to the Si²⁺ and Si³⁺ high-resolution data are shown in Figures 5 and 6, respectively. The agreement between experiment and the present theory (shifted by −1.8 eV for Si²⁺ and by −1.2 eV for Si³⁺, see Figure 3) is remarkable, particularly for σ_2,4 (Figure 5). There is less agreement for Si³⁺, as already noted in the low-resolution data in Figures 2 and 3. The too large separation of the first two peaks is probably caused by subtleties concerning the atomic fine structure associated with the angular momentum coupling of three open atomic subshells.

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Figure 7 shows a comparison between the strongest Si K-shell absorption features for Si⁺, Si²⁺, and Si³⁺ (panel (a)) with the absorption by some silicon-containing minerals that potentially occur in interstellar dust (panel (b); Zeegers et al. 2019). In order to be able to discriminate between the different forms of silicon in X-ray absorption spectra, the energies of the absorption features must be known with sufficient precision. The systematic uncertainty of the present experimental energy scale is ±1 eV (Section 2.2). Zeegers et al. (2019) did not quantify the uncertainty of their energy scale. Instead, for their energy calibration they referred to previous works (Li et al. 1995; Nakanishi & Ohta 2009), which, in turn, are based on earlier investigations. The energy scale of Nakanishi & Ohta (2009) can be traced back to the work of Wong et al. (1999) who mention that their energy calibration varies by 2–3 eV depending on the operating conditions of their synchrotron light source. This suggests that the uncertainty of the energy scale of (Zeegers et al. 2019; Figure 7(b)) is at least ±1.5 eV.

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Different minerals exhibit different chemical shifts of the Si K-shell absorption features, which vary by up to 1.5 eV (Li et al. 1995). This is still less than the combined energy uncertainties of the present energy scale and the one of Zeegers et al. (2019). Therefore, we conclude that, in general, X-ray absorption spectroscopy can discriminate between absorption by gaseous silicon and absorption by silicon-containing minerals. In particular, the Si⁺ absorption featured in Figure 7(a) appears at an unambiguous location (∼1841.5 eV). This is also true for the corresponding absorption feature of Si⁻, which occurs at an even lower energy of 1838.4 eV (Figure 3(a); Perry-Sassmannshausen et al. 2021). To the best of our knowledge, there are no such experimental data for gaseous neutral silicon. In view of the findings for Si⁻ and Si⁺ its dominating absorption feature can be expected close to 1840 eV.

Gatuzz et al. (2020) investigated the gaseous component of the ISM using the theoretical absorption data of Witthoeft et al. (2011). In their data, the 1s → 3p resonance group for neutral silicon is located at ∼1839.5 eV in accord with the above considerations. For the other charge states there are significant discrepancies as already noted above. The deviations between the theoretical resonance positions of Witthoeft et al. (2011) and the present experimental ones are 6.7, 4.4, and 9.1 eV for Si⁺, Si²⁺, and Si³⁺, respectively (Figure 3). All these differences are substantially larger than the uncertainty of the experimental energy scale and they are as large or larger than the energy differences between adjacent charge states (Figure 7(a)). This will lead to a wrong assignment of astronomically observed resonance features if their analysis is based on these theoretical absorption cross sections.