STORAGE RING MEASUREMENT OF ELECTRON IMPACT IONIZATION FOR Mg⁷⁺ FORMING Mg⁸⁺

Cosmic plasmas driven by electron impact ionization (EII) are formed in many astrophysical sources. Interpreting the thermal and ionization properties of these objects requires an accurate understanding of the charge state distribution (CSD) of the sources (Brickhouse 1996; Landi & Landini 1999; Bryans et al. 2006). The CSD calculations in turn depend on the balance between EII and electron–ion recombination and much work has been carried out in deriving the atomic data needed for reliable models (reviewed in Bryans et al. 2009).

Unfortunately, a major limitation concerning the EII data has been the difficulty of providing unambiguous experimental results due to the challenge of producing well-characterized ion beams. Most ion sources generate beams with significant, and usually unknown, metastable fractions. This complicates and limits the ability both to generate reliable EII data and to benchmark theory. Experimental approaches that are particularly prone to this problem include crossed beams methods or merged electron–ion beams in a single-pass geometry (Müller 2008). In each arrangement, the metastable ions have insufficient time to radiatively relax to the ground state before the EII measurements are performed.

Additionally, a second issue arises from the need for EII cross sections for every astrophysically relevant ion, typically defined as all non-bare charge states of every element from H to Zn (e.g., Dere 2007). Limited experimental resources and personnel preclude being able to carry out all the required measurements.

In order to address these two issues, we have initiated a series of EII measurements employing an ion storage ring using a merged beams geometry. Utilizing a storage ring addresses the first issue by allowing for the ions to be stored long enough for typically all or nearly all metastable levels to radiatively relax before beginning data acquisition. To address the second issue, we have adopted the approach of measuring EII for at least one ion in every isoelectronic sequence. Our results can then be used to benchmark theory, which in turn can be used to calculate the needed EII data for other ions in the same isoelectronic sequence.

Boron-like was the first isoelectronic sequence selected for study. Here, we have investigated EII for B-like Mg⁷⁺ forming Be-like Mg⁸⁺. Mg⁷⁺ peaks in abundance in an electron-ionized plasma at a temperature of about 8 × 10⁵ K (Bryans et al. 2009). Spectral lines from this ion are observed in the solar corona (Curdt et al. 2004) and stellar spectra (Rasseen et al. 2002). The metastable lifetime of Mg⁷⁺ is several seconds. With storage times approximately an order of magnitude longer, this ion is particularly well suited for storage ring measurements of ground state EII. We then use our results here to benchmark theory.

Cross section measurements covered the center of mass energy range from 200 to 2000 eV. This energy range includes the following direct ionization channels:

$$\begin{equation} {{ e}^{-} + {\rm Mg}^{7+}} (1s^2\,2s^2\,2p) \rightarrow \left\lbrace \begin{array}{@{}l} \mathrm{Mg^{8+}}(1s^2\,2s^2) + 2{e^{-}} \\ \ms \mathrm{Mg^{8+}}(1s^2\,2s\,2p) + 2{e^{-}} \\ \ms \mathrm{Mg^{8+}}(1s\,2s^2\,2p) + 2{e^{-}}\end{array} \right. \!. \end{equation} \tag{ 1 }$$

The energy thresholds for these channels are 265.96 eV, 283.37 eV, and 1587.32 eV, respectively (Ralchenko et al. 2008). In addition, above about 1291 eV there are excitation-autoionization (EA) channels from 1s 2s² 2p nl excitation, leading to the first two configurations in Equation (1) (Safronova & Shlyaptseva 1996). Concerning ionization forming Mg⁸⁺(1s 2s² 2p), the third configuration in Equation (1), this state radiatively stabilizes with a probability of <5% and otherwise autoionizes giving a net double ionization to form Mg⁹⁺ (Gorczyca et al. 2003). Here, we detect only the single ionization channel.

This paper is organized as follows. In Section 2, we describe the experimental setup. The data analysis is presented in Section 3 and uncertainties in Section 4. Experimental results and comparison to theory are presented in Section 5. A brief summary is given in Section 6.

EII measurements were performed at the TSR heavy-ion storage ring of the Max-Plank-Institut für Kernphysik in Heidelberg, Germany. The experiments basically followed the procedure of previous experiments (Kilgus et al. 1992; Linkemann et al. 1995a; Kenntner et al. 1995; Schippers et al. 2001; Lestinsky et al. 2009). Here, we describe additional details relevant to the present work.

A beam of 80 MeV $\mathrm{^{24}Mg^{7+}}$ ions was injected into TSR in a series of 1–3 pulses, typically 0.5 s apart. In one of the straight sections of TSR, the ion beam was merged with a beam of electrons in the electron cooler. The two beams co-propagated and interacted for a distance of ≈1.5 m. During injection the electron energy was 1816.6 eV, accounting for space charge corrections based on the electron current and the geometry of the cooler as described in Kilgus et al. (1992). This energy is denoted as the cooling energy. It resulted in a small relative velocity between the electrons and ions, which allowed the ion beam to be cooled efficiently by the electrons and set the ion velocity to that of the electrons at cooling. The resulting center of mass energy spread at cooling was well described by a flattened Maxwellian distribution with temperatures in the perpendicular and parallel directions of k_BT_⊥ = 13.5 meV and k_BT_∥ = 180 μeV, respectively. These were determined from the widths of resonances in the dielectronic recombination (DR) spectrum (M. Lestinsky et al. 2010, in preparation). After injection the ions were cooled for 4.5–6 s, allowing the Mg⁷⁺ metastable states to decay. The 2s 2p²⁴P_{{1/2,3/2,5/2}} levels decay with lifetimes of <0.1 ms and contribute to the population of the 2s² 2p ²P_3/2 level, which has a lifetime of ≈3.1 s. Based on the transition rates and branching ratios, and assuming an initial Boltzmann-distributed level population within the n = 2 shell, the average metastable fraction during measurement was estimated to be about 3% (M. Lestinsky et al. 2010, in preparation). The stored ion current after cooling was 2–8 μA.

Ionization and recombination were measured simultaneously using separate detectors located behind the first dipole magnet downstream of the electron cooler. Ions that underwent ionization were deflected more strongly than the Mg⁷⁺ beam, striking a converter plate, and generating secondary electrons which were then detected by a channel electron multiplier (CEM; Rinn et al. 1982; Linkemann et al. 1995b). Similarly, ions that underwent recombination were deflected less strongly than the Mg⁷⁺ beam and collected by a scintillation crystal (Miersch et al. 1996). The detectors were positioned for maximum signal collection by stepping each unit horizontally and vertically in small increments across each product beam. From these scans, we also determined that the cooled ion beam was about 1 mm in diameter, which is small compared to the 10 mm circular diameter opening of the ionization detector or to the 20 mm × 20 mm area of the recombination detector. Using a beam profile monitor (Hochadel et al. 1994), we observed that the beam diameter increases to about 4 mm during measurement as the beam warms up. This can cause a small decrease in the detection efficiency as is discussed in Section 4.

A suitable CEM discriminator level was determined by measuring the pulse height distribution. The level was set so that the detection efficiency of ionization events was essentially unity. The total ionization detector count rate was kept below ≲60 kHz by adjusting the electron beam current. The dead time of the ionization detector was estimated from the length of the CEM pulse to be about 30 ns, giving an upper limit for the dead-time-induced loss of 0.2%.

Data collection was performed by repeating a sequential cycle consisting of injection, cooling, and then energy scans. Each energy scan consisted of ∼50 pairs of steps, one step at measurement and one at reference. The laboratory electron energies were always chosen to be higher than the cooling energy and were between 3200 and 8100 eV. The higher energy ranges required large changes in the acceleration voltage between the cooling step and the energy scan. A fast high-voltage amplifier with a ±1000 V dynamic range was used for the energy scan in combination with a slower principal electron beam power supply, which was used to lift the fast amplifier into the range of the energy scan following the cooling phase. At the beginning of each scan step a short delay, typically 50–100 ms, allowed the power supplies to settle. Data at each step were then collected for 50–100 ms. The measurement energy was scanned over a range of about 600 V in the laboratory frame. A typical scan required 10–20 s.

The background count rate was determined by measuring the count rate at a fixed relative energy during the reference step. For lower energy scans the reference step energy was set below the EII threshold so that the background ionization count rate was due only to stripping of electrons off the residual gas (RG). However, at higher energies it was not possible to set the reference energy below threshold. This is because of the limited range of the fast high-voltage amplifier used to set the electron beam energy. Instead the reference step was set to an energy where the EII cross section could be determined from the lower energy scans. In this way, the EII contribution to the reference count rate could be removed in order to determine the appropriate background count rate for subtraction. We describe in detail in the next section the steps necessary to derive the appropriate background count rate, valid at the measurement step energy, from the reference rate at the fixed energy.

After an energy scan was complete a new injection cycle was initiated and the process repeated. Scans over a particular energy range were typically repeated for 1 hr to ensure good statistical accuracy. The energy range for each scan was chosen to maintain at least 50% overlap with other scans.

The energy-dependent ionization cross section at each measured energy σ_I(E_m) was obtained from the measured ionization rate coefficient $\left< \sigma _{\rm I}v_{\rm {rel}} \right>$ which is averaged over the energy spread of the experiment. Because the relative velocity spread is very small, $\Delta v_{\rm {rel}} \ll v_{\rm {rel}}$, the averaged rate coefficient can be divided by the relative velocity giving

$$\begin{equation} \sigma _{\rm I}(E_{\rm m}) = \frac{1}{v_{\rm {rel}}} \left[\frac{ R_{\rm I}^{\rm m}(E_{\rm m}) - R_{\rm I}^{\rm b}(E_{\rm m}) }{(1 - \beta _{\rm i} \beta _{\rm e}) n_{\rm e}^{\rm m}N_{\rm {i}}^{\rm m}L} + \left< \sigma _{\rm I}v_{\rm {rel}} \right>(E_{\rm r})\frac{n_{\rm e}^{\rm r}}{n_{\rm e}^{\rm m}} \right]. \end{equation} \tag{ 2 }$$

Here, R^m_I denotes the total ionization count rate at the measurement step; R^b_I denotes the background ionization count rate at the measurement step, which is proportional to the count rate at the reference step R^r_I, as described below; L = 1.5 m is the length of the interaction region; the factor (1 − β_iβe) is a relativistic correction, where β_i and β_e are the ion and electron velocities normalized by the speed of light, v_i/c and v_e/c, respectively; and E_m and E_r are the center of mass energies at measurement and reference, respectively. The electron density n_e is calculated from the measured electron current and the geometry of the electron cooler by assuming a uniform electron density (Kilgus et al. 1992). The electron density was typically on the order of 10⁷ cm^-3. The total number of stored ions per unit length N_i is calculated from the measured ion current. The EII rate coefficient at the reference energy $\left< \sigma _{\rm I}v_{\rm {rel}} \right>(E_{\rm r})$ is included in order to remove the EII contribution to the background count rate R^b_I when the reference energy is above the EII threshold. The factor n^r_e/n^m_e accounts for the different electron densities at reference and measurement.

The electron and ion beams also interact in the merging and demerging sections on either side of the straight section. In this section, the ion and electron beams meet at an angle and the center of mass energy is greater than in the section where the beams are collinear. These toroidal effects from the merging and demerging of the beams are accounted for using the method described by Lampert et al. (1996). For a given collision energy the toroidal correction procedure requires knowledge of the cross section at energies up to 140 eV higher. At the highest measured energies an extrapolation was needed to supply cross section values so the correction could be applied. The extrapolation was obtained by fitting the experimental data using the Lotz formula (Lotz 1969). Over the entire scan range, this procedure corrects what would otherwise be an overestimate of the EII cross section by about 20% on average. This is mainly because the toroidal sections increase the effective length of the interaction region by about 20%.

The total ionization detector signal at the measurement step is made up of EII and electron stripping (ES) of the ions off the RG atoms,

$$\begin{equation} R_{\rm I}^{\rm m}(E_{\rm m}) = \left< \sigma _{\rm I} v_{\rm {rel}} \right> n_{\rm e}(E_{\rm m})N_{\rm {i}}^{\rm m}L + \left< \sigma _{\rm {ES}}v_{\rm {i}} \right> n_{\rm {RG}}(E_{\rm m}) N_{\rm i}^{\rm m}D. \end{equation} \tag{ 3 }$$

Here, $n_{\rm {RG}}$ is the density of the RG, D is the length along the beam from which ions created by ES can reach the ionization detector, and $\sigma _{\rm {ES}}$ is the cross section for ES. To calculate the EII cross section, the background rate

$$\begin{equation} R_{\rm I}^{\rm b}(E_{\rm m}) = \left< \sigma _{\rm {ES}}v_{\rm {i}} \right> n_{\rm {RG}}(E_{\rm m})N_{\rm i}^{\rm m}D \end{equation} \tag{ 4 }$$

must be subtracted from R^m_I(E_m). This background rate can be determined by measuring the count rate at a reference energy. We first consider the case where the reference energy E_r is below the EII threshold. To solve for R^b_I(E_m), we note that in this case $\sigma _{\rm {I}}(E_{\rm r}) = 0$ and the reference count rate results exclusively from stripping giving

$$\begin{equation} R_{\rm I}^{\rm r}(E_{\rm r}) = \left< \sigma _{\rm {ES}}v_{\rm {i}} \right> n_{\rm {RG}}(E_{\rm r})N_{\rm i}^{\rm r}D. \end{equation} \tag{ 5 }$$

Taking the ratio of Equations (4) and (5) and solving for R^b_I(E_m), one obtains

$$\begin{equation} R_{\rm I}^{\rm b}(E_{\rm m}) = \left[\frac{n_{\rm {RG}}(E_{\rm m})}{n_{\rm {RG}}(E_{\rm r})} \frac{N_{\rm i}^{\rm m}}{N_{\rm i}^{\rm r}}\right] R_{\rm I}^{\rm r}(E_{\rm r}). \end{equation} \tag{ 6 }$$

This equation corrects for the fact that the count rate at reference R^r_I(E_r) is not exactly equal to the background rate at the measurement step R^b_I(E_m). The RG density $n_{\rm {RG}}$ and the number of stored ions per unit length N_i can differ between each step. In particular, increasing the electron energy increases the electron current which can change the gas pressure in the vacuum system. Also, the number of ions in the interaction region varies between the measurement and reference steps because of the exponential decay of the stored ion beam, which here had a characteristic decay time of ≈30 s. This implies an offset between the reference rate and the background rate at the measurement step measured 50–100 ms later (or earlier) of about 0.5%. Both these effects can be a source of systematic error and are hence corrected, as far as possible, as described below.

For energy ranges with a reference E_r where $\sigma _{\rm {I}} = 0$, we can determine the prefactor on the right-hand side of Equation (6) by simultaneously measuring electron–ion recombination. At the high energies of interest for EII, the recombination signal at both measurement R^m_R and reference R^r_R is dominated by electron capture (EC) off the RG with a component due to electron–ion recombination in the cooler. That is,

$$\begin{equation} R_{\rm R}^{\rm m}(E_{\rm m}) = \left< \sigma _{\rm {rec}}v_{\rm {rel}} \right>n_{\rm e}(E_{\rm m})N_{\rm i}^{\rm m}L + \left< \sigma _{\rm {EC}}v_{\rm {i}} \right> n_{\rm {RG}}(E_{\rm m})N_{\rm i}^{\rm m}D \end{equation} \tag{ 7 }$$

and

$$\begin{equation} R_{\rm R}^{\rm r}(E_{\rm r}) = \left< \sigma _{\rm {rec}}v_{\rm {rel}} \right>n_{\rm e}(E_{\rm r}) N_{\rm i}^{\rm r}L +\left< \sigma _{\rm {EC}}v_{\rm {i}} \right> n_{\rm {RG}}(E_{\rm r})N_{\rm i}^{\rm r}D. \end{equation} \tag{ 8 }$$

Here, $\sigma _{\rm {EC}}$ is the cross section for EC and $\sigma _{\rm {rec}}$ is the cross section from the sum of radiative recombination and DR. The electron–ion component of the total recombination signal is removed using separate measurements of $\sigma _{\rm {rec}}$ made with the reference energy chosen using the criterion that the electron–ion contribution at that point be close to zero (M. Lestinsky et al. 2010, in preparation). Using Equations (7) and (8), the ratio of gas densities and ion numbers can now be expressed in terms of these quantities measured in the recombination channel,

$$\begin{equation} \frac{n_{\rm {RG}}(E_{\rm m})}{n_{\rm {RG}}(E_{\rm r})} \frac{N_{\rm i}^{\rm m}}{N_{\rm i}^{\rm r}}=\frac{ R_{\rm R}^{\rm m}(E_{\rm m}) - \left< \sigma _{\rm {rec}}v_{\rm {rel}} \right>n_{\rm e}(E_{\rm m})N_{\rm i}^{\rm m}L}{R_{\rm R}^{\rm r}(E_{\rm r})-\left< \sigma _{\rm {rec}}v_{\rm {rel}} \right>n_{\rm e}(E_{\rm r}) N_{\rm i}^{\rm r}L}. \end{equation} \tag{ 9 }$$

An example is shown in Figure 1. Substituting this expression into Equation (6) gives

$$\begin{equation} R_{\rm I}^{\rm b}(E_{\rm m})= \frac{ R_{\rm R}^{\rm m}(E_{\rm m}) - \left< \sigma _{\rm {rec}}v_{\rm {rel}} \right>n_{\rm e}(E_{\rm m})N_{\rm i}^{\rm m}L}{R_{\rm R}^{\rm r}(E_{\rm r})-\left< \sigma _{\rm {rec}}v_{\rm {rel}} \right>n_{\rm e}(E_{\rm r}) N_{\rm i}^{\rm r}L} R_{\rm I}^{\rm r}(E_{\rm r}). \end{equation} \tag{ 10 }$$

This correction ensures that the background level from ES is proportional to the energy-dependent RG pressure and the ion current during the measurement step. An example of this correction is shown in Figure 2 and its effect on the calculated EII cross section is illustrated in Figure 3.

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The correction outlined here cannot be applied to measurements having a reference energy where $\sigma _{\rm {I}}(E_{\rm r}) \ne 0$ due to the need to include the EII rate in Equation (5). However, Equation (9) continues to be valid and we found experimentally that at these higher energies the ratio on the right-hand side of Equation (9) was compatible with a constant to within the statistical errors. Moreover, the ratio N^m_i/N^m_i is always constant since the ion beam decays exponentially and the time interval between steps is fixed. Observing no significant variation of the ratio with energy and considering the fact that the only source of such a variation would be RG pressure fluctuations, we conclude that the RG pressure has saturated and we perform no correction. Hence, for those measurements only the effect of the ion beam decay was corrected for using direct measurements of the ion current. For the higher-energy scans with $\sigma _{\rm {I}}(E_{\rm r}) \ne 0$, the EII rate contribution included in the background rate R^b_I(E_m) has to be removed by the second term in Equation (2), which in this experiment has been taken from the cross section determined in the scans at lower energies.

Table 1 summarizes the errors arising from counting statistics and systematic sources. Because of the long times over which data were accumulated, the 1σ counting statistics results in uncertainties of less than 1%. The systematic error is primarily due to the 15% uncertainty of the stored ion beam current and uncertainties introduced by the heating of the ion beam phase space. The stored ion current is measured non-destructively with a DC beam current transformer (Unser 1981). The systematic error of the ion current, caused by these instrumental fluctuations, applies to each energy scan separately. Since the scans of higher energy are linked to the lower energy ones by the procedure described at the end of Section 3, the relative comparison of cross sections at collision energies strongly different from each other can also be affected by an error of similar size as the ion current measurement; hence, in spite of a much better statistical error, we conservatively apply a 15% error band on the functional form of the cross section. The beam could not be cooled after the initial injection and cooling period because of the limited dynamic range (±1000 V) of the fast high-voltage amplifier used to scan the electron energy. Thus, after the initial cooling period, the ion beam warms up. Using a beam profile monitor (Hochadel et al. 1994) we found that the diameter expands within a few seconds from about 1 mm to about 4 mm as measured during the experiment. Because the beam expansion is time varying and may not be isotropic, we estimated the uncertainty rather than attempt to correct for it. By assuming a Gaussian beam and comparing with the diameter of the ionization detector we estimated the uncertainty in the detection efficiency due to the increase in beam size to be ≈2%. The uncertainty arising from inaccurate knowledge of the overlap length L is strongly reduced by the toroidal correction as described in Lampert et al. (1996). The expansion of the ion beam introduces a spread of ≲1% in the effective overlap length experienced by ions at different transverse locations. Treating the systematic errors as Gaussian uncertainties and adding them in quadrature, we estimate the 1σ total systematic uncertainty in the cross section to be 15%. The total error bar from adding the above uncertainties in quadrature is 15% and is dominated by the systematic uncertainty.

Our experimental results are shown in Figure 4 for the full center of mass energy range from 200 to 2000 eV (corresponding to laboratory electron energies of 3200–8100 eV), while a more detailed view near the EII threshold energy is shown in Figure 5. In Figure 4, the error bars at selected points indicate the total 1σ experimental uncertainty of 15%. In Figure 5, the smaller error bars indicate the statistical uncertainty, which is more significant when the EII cross section is small because the count rates are low. The larger error bars in Figure 5 indicate the total 1σ experimental uncertainty. The thresholds for the different direct ionization channels described in Equation (1) are indicated by arrows in each figure. The observed EII threshold appears near the predicted value of 265.96 eV. The absence of any ionization signal below this threshold is an indication that the metastable fraction in the experiment is small. The threshold for the 2s ionization channel at 283.37 eV could not be observed experimentally.

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For B-like ions the EA contribution to the ionization process is expected to be only a few percent of the direct ionization contribution to the cross section (Sampson & Golden 1981). The expected threshold for this indirect ionization mechanism is about 1291 eV (Safronova & Shlyaptseva 1996). Our experimental measurement did not resolve the EA contribution. We do observe an apparent difference in the shape of the curve at high energies, but this is well within the experimental uncertainty. Our results are consistent with calculations and experimental measurements for other B-like ions which showed the EA contribution to be small (Yamada et al. 1989; Bannister 1997; Duponchelle et al. 1997; Chongyang et al. 1998; Loch et al. 2003). Additionally, we did not resolve contributions from indirect processes involving excitation of inner shell electrons by dielectronic capture followed by autoioinization, such as resonant excitation double autoionization or resonant excitation auto-double-ionization.

Also shown in Figures 4 and 5 are theoretical results from two different distorted wave calculations. One calculation was performed using the gipper code from the Los Alamos Atomic Physics Code suite (Magee et al. 1995). This total cross section is the sum of the three direct ionization channels in Equation (1). The other theoretical cross section was reported by Dere (2007) and was calculated using the Flexible Atomic Code (fac) code (Gu 2003). These results are the sum of the three direct ionization channels plus EA and have been incorporated into the chianti database (Dere et al. 2009). To within the experimental uncertainties, our measured cross section is in agreement with both of these calculations. This agreement increases the confidence in the EII data for Mg⁷⁺ based on these codes and used in astrophysics.

We have measured the EII cross section of Mg⁷⁺ using the TSR ion storage ring to produce an ion beam with an average estimated metastable fraction of ≈3%. The measured cross section agrees with theoretical distorted wave calculations to within the experimental uncertainties. This adds confidence to astrophysical measurements that rely on the atomic EII data for this ion.

The most significant limitation on these measurements is a systematic uncertainty in the ion current measurement. We plan to improve this situation in future measurements. To this end, we are now developing new methods to mitigate the effect of this uncertainty by correcting for fluctuations between energy scans, which will remove distortions of the functional shape of the cross section.

We gratefully acknowledge the efficient support by the accelerator and TSR groups during the beamtime. This work was supported in part by the NASA Astronomy and Physics Research and Analysis program and the NASA Solar Heliospheric Physics program.