High-resolution Laboratory Measurements of K-shell X-ray Line Polarization and
Excitation Cross Sections in Heliumlike S XV Ions

According to Beiersdorfer et al. (1996), the X-ray line intensity observed using a crystal spectrometer is

where $R_{\parallel}$ and $R_{\perp}$ are the integrated crystal reflectivities for X-rays polarized parallel and perpendicular to the plane of dispersion, and the X-ray line polarization is

As mentioned above, X-rays are recorded simultaneously using two crystal spectrometers, oriented in such a way that their planes of dispersion are parallel (vertical; V) and perpendicular (horizontal; H) to the electron beam propagation direction. The line intensity observed by each spectrometer can then be written as

and

The two crystals are identical, and the plane of dispersion of the vertical spectrometer is rotated by 90^∘, as compared to that of its horizontal counterpart; thus we can write $R_{\perp}\equiv R^{\mathrm{V}}_{\parallel}=R^{\mathrm{H}}_{\perp}$ and $R_{\parallel}\equiv R^{\mathrm{V}}_{\perp}=R^{\mathrm{H}}_{\parallel}$, i.e., the perpendicular and parallel reflectivities of the vertical spectrometer are interchanged, compared to that of the horizontal, when assuming the same reference frame for the orientation of $I_{\parallel}$ and $I_{\perp}$. Using this relation to solve Eqs. 8 and 9 for $I_{\parallel}$ and $I_{\perp}$, and substituting them in Eq. 7, we derive an expression for the degree of linear polarization of the X-rays, as a function of the observable intensities by the two crystal spectrometers:

where the ratio $R={R_{\perp}}/{R_{\parallel}}$ depends on the Bragg angle $\theta$, and can be estimated by $|\cos^{2}2\theta|$ for a mosaic crystal, and $|\cos 2\theta|$ for a perfect crystal. Real crystals typically have $R$ values between these two limits (Beiersdorfer et al., 1996); however, we believe that the crystals used in our experiment are close to perfect. We therefore calculated the $R$ values assuming perfect crystals, using X0h¹¹1https://x-server.gmca.aps.anl.gov/x0h.html, which calculates the integrated reflectivities for quartz(101) in the energies of interest (2.4–2.5 keV) using data from Henke et al. (1993). These values were further verified using the X-ray Oriented Program (XOP; Sánchez del R´ıo & Dejus, 2011) (see details in MacDonald et al. (2021)).

The diffracted X-rays are recorded with CCD cameras, stabilized at $-110~^{\circ}$C, ensuring low thermal noise and dark current during their operations. The CCD images were filtered for readout noise, cosmic rays, and other background events, following the method described by Pych (2004). Subsequent to these corrections, the sum of all data is projected onto the axis corresponding to the dispersion direction. Several images, each with an hour’s exposure, were obtained at each electron beam energy. The spectra from all exposures were added together in order to improve the signal-to-noise ratio.

Figure 3 shows an example of a summed spectrum at 6.5 keV of beam energy, with a medium charge-balance setting. The wavelength calibrations for both spectrometers were performed using the theoretical value of line $w$ from Drake (1986), and the experimentally measured value of line $z$ of S XV (Hell et al., 2016, relative to Drake (1986)). The observed spectral peaks are then fitted using multiple Voigt functions, i.e., a convolution of Lorentz and Gaussian functions. The forbidden line $z$ at 2430 eV has a very narrow natural line width ($\sim 9\cdot 10^{-10}$ eV, see Crespo López-Urrutia et al. (2006)) compared to our instrumental resolution. This line allows us to determine the spectral-line broadening due to the Doppler motion of the trapped ions and due to any contribution of an intrinsic spectrometer line-spread function. Based on the axial trap depth ($V_{0}\sim 50$ V), the expected initial ion temperature ($T_{i}\sim$140 eV), and the crystal spectrometer line shape (MacDonald et al., 2021), the observed Gaussian width, which is 0.47-eV FWHM at 2430 eV ($T_{i}\sim$200 eV), should be dominated by the Doppler width, while the dominant contribution to the Lorentzian width is due to the instrument, with a negligible part due to the natural linewidth. Thus, in our fitting procedure, the Gaussian width obtained from line $z$ is shared between all peaks, while Lorentzian widths, representing the sum of instrument and natural line widths, are allowed to vary, as are the peak centroids and amplitudes.

Any photon-energy independent differences between the two crystal spectra that may arise from differences in their geometry and efficiency can be corrected using an unpolarized line for cross-normalization. X-ray lines with total angular momentum in the upper level of $J\leq{1/2}$ are unpolarized (Balashov et al., 2000). In this experiment, we used an unpolarized line, $r$ (excited state $[1s\,2s\,2p_{1/2}]_{J={1/2}}$), from lithiumlike S XIV at 2437 eV to normalize the relative efficiencies of the vertical and horizontal spectrometers.

Finally, the normalized intensities and relative reflectivities, $R$, obtained from XOP are used in Eq. 10 to determine the degree of linear polarization of an X-ray line following the EIE. The uncertainty in the measured degree of polarization includes contributions from statistical fitting error of the line intensities, background removal, crystal reflectivities, and the relative efficiency normalization between the two crystal spectrometers inferred from the lithiumlike S XIV line $r$. The results obtained for $P_{\mathrm{EIE}}$ at a beam energy of 6.4 keV are shown in the top panel of Fig. 3. We followed a similar procedure for data sets with electron beam energies of 2.6, 3.6, and 8.9 keV. Given that lines $w$ and $z$ are well isolated, we checked the validity of our fitted line intensities by integrating counts over their respective energy ranges, obtaining degree-of-polarization results within the measurement uncertainty.

Table. 3 shows the measured degree of linear polarization of lines $w$ and $z$ at four different electron beam energies. Our measured values for lines $w$ and $z$ agree very well with the values predicted by the distorted-wave method, within uncertainty limits (Fig. 7). The resonance line, $w$, shows a positive degree of polarization, which suggests that it is polarized parallel to the quantization axis of the electron beam propagation direction axis. This is due to the fact that the $m_{j}=0$ magnetic sublevel of the $1s2p\,\,^{1}P_{1}$ upper state is predominantly populated relative to that of $m_{j}=\pm 1$ following direct excitation from the ground state, resulting in nonzero alignment (Inal & Dubau, 1987; Surzhykov et al., 2006).

Conversely, for direct excitation of the $1s2s\,\,^{3}S_{1}$ upper state, the populations of magnetic sublevels $m_{j}=0$ and $m_{j}=\pm 1$ are identical at any given electron-impact energy. Thus, the alignment is zero, and consequently, the degree of polarization for line $z$ is zero. Our basic two-level ($1s^{2}-1s2s$) calculation also predicts the same for line $z$ (Fig. 7, dashed-dotted line). However, we observed a negative degree of polarization. This is because the upper level of line $z$ is mostly populated by cascades from higher-lying levels (Beiersdorfer et al., 1996; Hakel et al., 2007; Chen et al., 2015). At 2.6 keV beam energy, only cascades from $n=2$ levels are possible, and they contribute significantly to the degree of polarization of line $z$. Above this beam energy, the high $1snl$ states are open for direct excitation from the ground state. Therefore, at 3.6, 6.4, and 8.9 keV beam energies, strong cascades from $n\geq 3$ further decrease the degree of polarization for both lines $w$ and $z$. The dashed curves in Fig. 7 show changes in X-ray polarization due to cascades from different $n$ levels, up $n=16$.

For beam energies above 3.14 keV, collisional inner-shell ionization of lithiumlike S XIV ions plays an important role, as it populates the $1s2s\,^{3}S_{1}$ level through ionization of the lithiumlike ground state, $1s^{2}2s\,^{2}S_{1/2}$ (Inal & Dubau, 1987). We used FAC to calculate the magnetic sublevel populations following collisional ionization, in order to check its effect on the polarization. We vary the fractional abundance of S XIV ions relative to that of S XV ions in these calculations. The solid curves in Fig. 7 show polarization predictions, including both cascades and CI, with a varying relative ion population. The contribution to line $z$ line from CI is essentially unpolarized, as both magnetic sublevels, $m_{j}=\pm{1/2}$, of lithiumlike ground state $\,{}^{2}S_{1/2}$ have identical values in an axially symmetric system (Mehlhorn, 1968). Therefore, the larger the fraction of the upper state of the line $z$ populated via CI of S XIV, the greater the reduction in the observed degree of polarization. This effect becomes more significant with increasing electron beam energy, as the relative importance of CI increases with increasing collision energy. In fact, our experiment with medium charge-balance settings shows excellent agreement with the theory when CI is accounted for using the charge balance measured from the strengths of the RR peak strengths and the inner-shell satellite lines $q$ and $r$. Conversely, the predicted polarization for the high charge-balance measurements has only a small correction from inner-shell ionization, and a model only including cascades agrees very well with our measurements, as the fractions of S XIV ions are relatively small compared to those of S XV.

The Breit interaction (Breit, 1929), a relativistic correction to the instantaneous Coulomb repulsion, is usually important for high-$Z$ ions (Fritzsche et al., 2009), but it may considerably alter the degree of polarization for low- and medium-$Z$ ions (Shah et al., 2015). Moreover, such relativistic effects become progressively more significant as the incident electron-impact energy increases (Reed & Chen, 1993). We considered these effects in our calculations, but we did not find a substantial change in the degree of linear polarization for lines $w$ and $z$. Moreover, two-photon $E1$ radiative decay can also affect the $1s2l$ line polarizations (Derevianko & Johnson, 1997; Surzhykov et al., 2010; Dipti et al., 2020). Our level-population calculations automatically take this into account, based on two-photon rates calculated by Drake (1986). The external magnetic and electric fields present in the trapping region of an EBIT could also modify the magnetic-sublevel population, thus the degree of polarization. However, in our experiment, the direction of the external magnetic field is the same as that of the electron beam, so the magnetic-sublevel populations should not be redistributed.

The EIE cross sections at an observed angle of 90^∘ are obtained by normalizing the EIE intensities with RR cross sections and RR intensity. Finally, the 90^∘ cross sections are corrected for measured polarization to extract the total line-emission cross sections. The uncertainties in the derived cross sections include contributions from all possible terms, including statistical fitting uncertainties on the RR ($\sim 4$%), and EIE intensities ($\sim 1$%), detector efficiency, and filter transmissions ($\sim 3$%), theoretical differential RR cross sections ($\sim 5$%), and, as explained in Sec. 3.1, the uncertainty associated with the measured degree of polarization ($\sim$5-7%). The overall uncertainty in the measured cross sections is on the order of $\sim 10$%.

In addition, we also checked background ions with ionization energies near to that of S XV ions, such as O VIII and N VII ions, which may contribute to the observed RR spectrum. In such a case, the S XV RR $n$=2 line can blend with RR $n$=1 lines from O VIII and N VII at any given electron beam energy. We cannot directly resolve these background lines in the RR spectrum, as the electron beam energy spread is much larger than the difference between the ionization potentials of background ions. If present, however, the fractional abundances of such ions can be estimated based on the observed and expected strengths of collisionally-excited hydrogenlike Lyman-$\alpha$ line. In order to estimate line strengths of contaminant ions, we have removed the $12.7~\mu\mathrm{m}$ Be filter from the beam path. In the case of the MC experiment, we observed only a minor enhancement over a continuum at $\sim 500$ eV N VII Ly$\alpha$ and $\sim 653$ eV O VIII Ly$\alpha$ energies. Having taken into account the ECS efficiency (Fig. 2), we find that the data indicates a negligible contribution to the RR spectrum. On the other hand, we see a strong oxygen line at 653 eV for the high charge-balance experiment. This may be due to the fact that we used a pulsed injection for HC, in contrast to a continuous injection of S for the MC experiment. In the latter case, the oxygen ion buildup in the trap may have been inhibited by the continuous flow of sulfur into the trap.

For HC data, the fractional abundance ratio between O VIII and S XV can be estimated using Eq. 1, i.e., by taking the ratio between $I(\mathrm{O\,Ly\alpha})/I(\mathrm{S}{w})$ and $\sigma_{90^{\circ}}(\mathrm{O\,Ly\alpha})/\sigma_{90^{\circ}}(\mathrm{S}{w})$. However, here we must also consider attenuation from molecular contaminants that may freeze on the filters. The main expected contaminants primarily contain hydrogen, oxygen, and nitrogen. As the photoelectric absorption cross section above threshold scales approximately as $E^{3}$, any such contaminant would produce the same transmission curve at energies above the oxygen $K$-edge, assuming a given optical depth at a given photon energy. We therefore treat the contaminant as a layer of oxygen atoms. We measured the optical depth decrement between two energies by taking the intensity ratio of the O Ly$\alpha$ and Ly$\beta$ lines, and comparing them to previously measured ratios using the same EBIT. The O Ly$\alpha$/Ly$\beta$ intensity ratio at high impact energies becomes constant (see Fig. 23 of Beiersdorfer (2003)). Thus, we took the weighted average of measured ratios in the range of 2–10 keV beam energies (6.13 $\pm$ 0.11). Based on the optical depth decrement inferred from the measured ratio, we estimated the areal density of oxygen contaminants to be $53\pm 6\,\mu$g cm^-2; the equivalent layer thickness was $0.53\pm 0.06\,\mu$m, assuming a fiducial density of 1 g cm^-3. We corrected the total filter transmission for the contaminant, and obtained the relative fractional abundance between O and S ions. We found these to be $\sim 21$%, $\sim 38$%, and $\sim 16$% for 2.6, 3.6, and 6.4 keV, respectively, for HC measurements. These oxygen fractions are used then to obtain the real RR intensity due to S XV ions only. An extra RR line, based on the ionization potential of O VIII, is added, and its flux is constrained to the inferred $n_{\mathrm{O}\textsc{viii}}/n_{\mathrm{S}\textsc{xv}}$ ratios in the fitting procedure, as explained in Sec. 3.1.2. The inferred S XV RR intensity and associated uncertainty from this step are then used to obtain the total cross sections for HC measurements.

The final cross section results are listed in Tab. 3 and shown in Fig. 8. Distorted wave predictions using FAC show excellent agreement with the measured cross sections, given that we account for necessary contributions from cascades and collisional inner-shell ionization, as also explained in Sec. 4.1.

The measured cross sections for the resonance line, $w$, show very good agreement with the theory. Cascades within $n=2$, and from high-$n$ levels, modify the $w$ cross sections by only a very small amount (6%–7%). In contrast, the cross sections for line $z$ are significantly altered by radiative cascades. For example, cascades from within $n=2$ increase the cross section of $z$ by $\sim 73$% at 2.7 keV beam energy, as compared to direct excitation from the ground (Fig. 9, panel (b)). As the electron-impact energy increases, cascades from high $1snl$ states further enhance the emission cross sections of line $z$. Figure 8 shows cascade contributions from different $n$ levels, up $n=16$, as dashed lines.

In addition, as explained earlier, collisional inner-shell ionization of lithiumlike S XIV ions starts to contribute to line $z$ above 3.14 keV beam energy. As shown in Fig. 9, panels (c) and (f), CI contributes as much as $\sim 36$% to the measured cross sections at a 3.6 keV beam energy, depending on the relative abundances of S XIV and S XV. This contribution increases significantly at high electron-impact energies, and dominates over the usual contributions from cascades and direct excitations. For example, our experiment shows that CI enhances the cross sections of line $z$ by $54^{+0.6}_{-0.7}$% at 6.4 keV beam energy, when the Li-like S ion concentration is increased from 24% to 80% in the trap. This agrees very well with distorted wave predictions of 53.4% enhancement. As depicted in Fig. 9, panels (d), (e), (g), and (h), CI completely dominates the cross sections at high beam energies of 6.4 and 8.9 keV, even if the S XIV fraction is very low compared to that of S XV. It is therefore imperative to take CI into account for the spectral modeling of high-temperature plasmas, not only those in equilibrium conditions, but particularly those that are out of equilibrium.

Our systematic measurements under two different charge-balance conditions clearly distinguish between predictions with different fractions of S XIV relative to that of S XV and show that both cascades and CI are essential for accurate predictions of emission cross sections for line $z$. Furthermore, experimental agreement between data taken at a 2.6 keV beam energy with two different charge balances indicates that the overall analysis, the extracted charge balance, as well as the theoretical treatment of cascades and CI, are reliable.

Line $z$ may also be populated via charge exchange (CX) and RR into H-like S XVI at higher electron beam energies, and this may modify the inferred polarization and cross sections. The cross sections for RR into $1s2s$ levels are on the order of $10^{-24}-10^{-25}$ cm² (see Tab. 1), which are at least 3-4 orders of magnitude smaller compared to line-formation cross sections due to direct excitation, cascades, and inner-shell ionization processes. On the other hand, the CX cross section are on the order of $\sigma_{\mathrm{CX}}=q\times 10^{-15}$ cm², where $q$ is the ionic charge (Phaneuf, 1983; Janev & Winter, 1985; Otranto et al., 2006). The CX rate can be represented as the product of $\sigma_{\mathrm{CX}}\,N_{\mathrm{i}}\,N_{\mathrm{n}}\,\nu_{\mathrm{in}}$, where $N_{\mathrm{i}}$ and $N_{\mathrm{n}}$ are the ion and neutral densities, and $\nu_{\mathrm{in}}$ is the collision velocity of ions and neutrals inside an EBIT, which is set by the ion temperature. Previously, we measured CX between S XVI – XVII and SF₆ using the same EBIT and ECS detector (Betancourt-Martinez et al., 2014). By comparing the count rate, populations of bare, hydrogenic, and heliumlike S ions, and gas injection pressure used in that experiment, as in this one, we estimated that the CX rate contributing to line $z$ is approximately three orders of magnitude smaller than the electron-impact excitation rate for our MC measurements. Furthermore, our HC measurements used a pulsed gas injector to introduce neutrals to the trap only at the beginning of the charge-breeding cycle. Given the very low background pressure ($\leq 10^{-11}$ Torr) at the center of the trap, and the lack of injection after the breeding part of the HC measurement cycle, X-rays from CX must be negligible. Data taken at 2.6 and 3.6 keV beam energies must be even less affected, due to lower production of H-like S ions, resulting in fewer targets for CX that can produce S K-shell emission lines. Thus, population of the $1s2s\,\,^{3}S_{1}$ level through RR and CX processes, is negligible for the present experiment. Furthermore, dielectronic recombination and resonance excitation are not relevant for the present experimental electron beam energies used here, and 3-body recombination contribution should be negligible, due to the low density of the EBIT plasma (Levine et al., 1988).