Experimental Uncertainty

Because of the importance of EC cross sections in various fields, careful analysis of the experimental uncertainty that determines the accuracy of EC cross section measurements is required. Several aspects were considered to evaluate the experimental uncertainties in the measurements of the EC cross section, including the accuracy of the determination of the effective collision length in the gas cell; efficiency and homogeneity of the detector; temperature and pressure fluctuations of the gas cell during the measurement procedure; stability of the ion beam in the EC cross section measurement; and statistical uncertainty.

Figure 2a shows the dimensions of the gas cell, which had a length of 60.8 mm and a diameter of 38.1 mm. The cell has an inlet aperture with a diameter of 1 mm and an outlet aperture with a diameter of 3 mm. The target gas diffuses from the gas cell owing to the apertures on both sides, leading to uncertainty in the effective collision length. To better understand the effective collision length, a numerical simulation using the finite element method was performed to analyze the pressure in the gas cell based on a molecular flow model. The calculated results for the target gas pressure along the cell axis are shown in Fig. 2b, where the pressure inside the gas cell is set to 0.02 Pa. The simulation showed the diffusion of the molecular flow through the small apertures at both ends, causing a decrease in pressure near the apertures inside the cell and an increase in pressure outside the gas cell. Because of the small apertures relative to the size of the gas cell and the conical shape design, the diffusion caused by the outflow of gas from the inlet and outlet apertures effectively decreased. The resulting correction factor $\alpha \approx 0.99$ obtained by integrating the pressure along the cell axis to the boundary of the diffusion model, is smaller than the analytical result $\alpha \approx 1.03$ [32]. The effective length is $l_{\text{eff}}\approx 0.99l$, where $l$ is the actual length of the gas cell from entrance to exit. Considering the machining precision of the gas cell and the inaccuracy of the calculation, the error in the effective collision length was conservatively estimated to be 2%, which is a three-fold improvement over our previous EC cross section measurement [28].

Fig. 2Full-screen View

(a) Schematic view of the gas cell and (b) the pressure data along the cylindrical symmetry gas cell axis. The pressure inside the gas cell is set to 0.02 Pa.

Additionally, the detector efficiency of a PSD based on MCP is generally not perfectly homogeneous. Therefore, it is necessary to test the homogeneity of the detector efficiency. Without the injection of incident ions and target gases, the detector area was divided into 90 equally sized rectangular regions, and the background counts generated by the cosmic rays in each region were recorded. The deviation between the distribution of particle numbers in the selected detector area and the average value was calculated as the uniformity coefficient according to Ref. [33]: $CU=\left(1-\frac{{\displaystyle {\sum }_{i=1}^n\left|h_i-\overline{h}\right|}}{{\displaystyle {\sum }_{i=1}^n{h}_i}}\right)\times 100,$ (4) where $\overline{h}$ is the average number of particles in the detection area, $h_i$ is the number of particles in each region, and n is the number of regions. The results show that the detector’s uniformity is 94.2%; thus, the inhomogeneity of the detector efficiency is 5.8%. This measurement was repeated two times during the entire experimental period for a reasonable evaluation.

As shown in Eq. (3), the fluctuation in temperature and pressure in the gas cell results in uncertainty in the measurement of the EC cross section. During the measurements, the temperature and pressure were continuously recorded and their relative fluctuations were below 1% and 3%, respectively. Moreover, the ion beam instability was conservatively estimated to be less than 4% by measuring the intensity of the primary ion beam without a gas target. The statistical errors depend mainly on the experimental setup, including the type of ion beam and target gas, the cross section of the EC, and the gas pressure. Normally, the statistics of the DEC are several times worse than those of the SEC because of the relatively smaller cross section. For instance, after sufficient data acquisition time, the statistical uncertainties of the SEC $N_{q-1}$ and DEC $N_{q-2}$ were 1.8% and 3.3%, respectively, for the Ne⁸⁺-O₂ collision.

A summary of the sources and magnitudes of the experimental errors is presented in Table 1. After a detailed analysis of the effective collision length and a dedicated test for detection efficiency, the uncertainties of SEC and DEC were 8.2% and 8.7%, respectively, for the Ne⁸⁺-O₂ collision. Additionally, the experimental uncertainty of the SEC was 8.2% for Ne⁸⁺ collisions with N₂ and CH₄. The DEC uncertainties for N₂ and CH₄ are 8.8% and 8.6%, respectively.

Data evaluation

For measurements of the absolute EC cross section, in addition to the systematic uncertainties, the gross error can also significantly affect the accuracy of the experimental measurements. One way to reduce the gross error is to repeat the measurements many times and obtain the mean value. Another method that can be used to identify gross errors and reconcile data is based on Chauvenet's criterion [34,35].

In practice, a measure is rejected if $\left|y_i-\overline{y}\right|>\omega \cdot s$, where $y_i$ is the measured value, $\overline{y}$ is the mean value, and $s$ is the standard deviation. $\omega$ can be obtained from the criterion table. In general, Chauvenet's criterion is used to identify gross errors in a Gaussian-distributed dataset. To extend this criterion for diagnosing gross errors in the presented EC cross section measurements, it is assumed that the tendency of the measured cross sections can be predicted well by a fitting model. Therefore, the deviations in the experimental data from this model should follow a Gaussian distribution. In this case, Chauvenet's criterion can be applied to identify gross errors in the measurements of the EC cross section.

For instance, the measurements of the SEC cross section of the Ne⁸⁺-O₂ collision are shown in Fig. 3. The cross section decreases with a gentle slope as a function of the incident energy, which can be fitted using a quadratic polynomial function (red line in Fig. 3). The standard deviation $s$ of each data point deviating from the fitted model is calculated as follows: $s=\sqrt{\frac{{\displaystyle {\sum }_{i,j=1}^n{\left(y_i-y_j\right)}^2}}{n-1}},$ (5) where $y_i$ is the experimental data, and $y_j$ is the value from the fitted line. Then, the confidence bound can be obtained by $y_j\pm s\times \omega$, as shown by the blue dotted line in Fig. 3. Finally, it is simple to identify the outliers located outside the confidence bound as gross errors. In Fig. 3a, the experimental data with an incident energy of 28.8 keV/u are considered as the gross error; thus, this measurement was repeated. After data reconciliation, the results in Fig. 3b of the SEC cross section satisfy Chauvenet's criterion and show a narrower confidence bound, which also proves that the gross error has been eliminated. It should be noted that the method of extending Chauvenet's criterion in estimating gross errors for EC cross section measurements works if the tendency of the cross sections is smooth and can be simply modeled.

Fig. 3Full-screen View

(Color online) Data evaluation for SEC cross sections in Ne⁸⁺ colliding with O₂ between 2.8 and 40 keV/u.

⁴He²⁺-He

The significance of EC processes has been demonstrated in both fundamental physics and applied fields [36]. Helium is a fusion reactor product that appears in different ionization states with various relative energies in fusion plasma. The precise measurement of EC cross sections involving ionized helium and molecules is important for plasma modeling and is crucial for the effective removal of helium ash from the divertor region in fusion devices [24]. The measurements of the SEC and DEC absolute cross sections for ⁴He²⁺-He were performed with an incident energy of 3.5 to 50 keV/u. The present results of the SEC and DEC cross section measurements of the ⁴He²⁺-He collision (black circle) are shown in Fig. 4, with experimental uncertainties of 8.5% and 8.9% for SEC and DEC, respectively.

Fig. 4Full-screen View

(Color online) Measured absolute a SEC and b DEC cross sections for ⁴He²⁺ colliding with He. a Experiments: filled black circle, current results; hollow blue diamond, results of Rudd et al. [40]; filled red diamond, results of DuBois et al. [39]; hollow orange circle, results of Shah et al. [37]; filled red inverted triangle, results of Mawhorter et al. [32]; hollow black square, results of Kusakabe et al. [39]. Theories: solid line, results of Fritsch. [41]; dotted line, results of Kimura. [42]. b Experiments: filled black circle, current results; filled red diamond, results of DuBois et al. [38]; hollow blue diamond, results of Rudd et al. [40]; hollow orange square, results of Berkner et al. [43]. Theories: solid line, results of Fritsch. [41]; dotted line, results of Kimura [42].

In Fig. 4a, the SEC cross section increases monotonically from 4.3×10^-17cm² to 2.9×10^-16cm² as the incident energy increases from 3.5 keV/u to 30 keV/u. When the collision energy is above 30 keV/u, the SEC cross section starts to decrease. The present results are consistent with previous experimental SEC cross sections measured by Shah et al. [37] and DuBios et al. [38]. Below 4.65 keV/u, the cross sections of our experiment deviate from the results reported by Mawhorter et al. [32] and Kusakabe et al. [39] but agree with the experimental results of Rudd et al. [40]. The semi-classical close-coupling traveling atomic orbital (AO) theory by Fritsch [41] also displayed good agreement with our measurement of the SEC cross section. Theoretical calculations using a traveling molecular orbital (MO) method within a semiclassical formalism by Kimura [42] are still in good agreement with the present results for energies below 10 keV/u but are approximately 17% lower than the experimental values above 10 keV/u. Figure 4b shows the measurement results of the DEC cross sections for ⁴He²⁺-He collisions. The cross sections decrease monotonically as the incident energy increases, which behaves differently from the SEC. The present results show good consistency with the experimental measurements of Berkner et al. [43], Rudd et al. [40], and DuBois et al. [38], as well as the theoretical calculations of Fritsch [41] and Kimura [42] in this energy range.

In general, the present results are in good agreement with previous experimental results, demonstrating the reliability of the experimental system, which includes the experimental platform operations, collision parameter optimizations, data acquisition systems, and data analysis methods. At low-impact energies, the cross section of the DEC is larger than that of the SEC. A similar result has also been reported for the He²⁺ -He [44] and C⁴⁺ -He collisions [45]. A possible reason for this is the resonant processes of the DEC explained by Zapukhlyak et al. [46]. To further understand the DEC cross section over the SEC in the low collision energy range, more work is needed from both experiments and theory to explain the physical mechanism.

²⁰Ne⁸⁺-O₂, N₂, CH₄

As one of the constituents of the solar wind, it is necessary to estimate the total cross sections of neon ions to simulate ion-neutral collision processes in astrophysical environments and to understand the interactions between the solar wind and the Earth's magnetosphere. Neon ions are also important in the study of fusion reactions and have been used as diagnostic tools to measure plasma temperature and density. The measurement results of the SEC and DEC cross sections for ²⁰Ne⁸⁺ colliding with O₂, N₂, and CH₄ with incident energy from 2.8 to 40 keV/u are shown in Fig. 5 and Table 2. In Fig. 5, the measured absolute cross sections for SEC and DEC both diminish with a gentle slope as the incident energy increases from 2.8 keV/u to 40 keV/u. Qualitatively, the EC cross sections for these targets do not show a typical dependence on the collision energy over the present energy range. The ratios of the DEC to SEC cross sections for the O₂ and N₂ targets were almost constant, with average values of 0.31 and 0.30, respectively, whereas the cross section ratio for the CH₄ target decreases by 21% with increasing collision energy from 0.33 at 2.8 keV/u to 0.26 at 40 keV/u. The experimental results show that for the same collision energy, there is only a slight variation in the absolute SEC cross sections of ²⁰Ne⁸⁺ ions when colliding with different target gases. This is because the capture process depends primarily on the initial binding energies of the electrons in the target. The first ionization energies of N₂, O₂, and CH₄ are 15.6 eV, 12.1 eV, and 12.6 eV, respectively.

Fig. 5Full-screen View

(Color online) Measured absolute SEC (filled blue square) and DEC (filled red circle) cross sections for ²⁰Ne⁸⁺ colliding with CH₄, N₂, and O₂. The errors are given at the 1σ level.

Currently, there is a relative scarcity of experimental and theoretical studies on collision systems involving ²⁰Ne⁸⁺ ions, atoms, or molecules. To date, experimental SEC cross sections of ²⁰Ne⁸⁺ have only been measured for collisions with He or H₂, as reported by Iwai et al. [47] and Bonnet et al. [48]. The present study reports precise measurements of SEC and DEC cross sections in the keV energy range for ²⁰Ne⁸⁺-O₂, N₂, and CH₄ collisions. Owing to the lack of available data on these collision systems, the SEC cross sections were evaluated by comparison with those estimated using the classical OBM [49]. According to the model, the SEC cross section is affected by only two factors: the charge state of the incident ion, which is treated as a bare nucleus, and the ionization potential of the target atom, which is treated as a one-electron atom with an appropriate effective charge. The principal quantum $n$ of the captured electron is predicted to be the largest integer satisfying the inequality $n\le q{\left[2I_{\text{P}}\left(1+\frac{q-1}{2\sqrt{q}+1}\right)\right]}^{-1/2},$ (6) where $I_{\text{P}}$ is the ionization potential of the target gas and $q$=8 is the charge state of Ne⁸⁺. The crossing distance $$. for collision systems is then given by $R_{\text{C}}=\frac{q-1}{q^2/2n^2-I_{\text{P}}},$ (7) and the resulting SEC cross section in the OBM is obtained from ${\sigma }_{q,q-1}=\pi R_{\text{C}}{}^2$. The values for $n$ and $R_{\text{C}}$ are calculated using the ionization potential of N₂ (15.6 eV or 0.57 au), O₂ (12.1 eV or 0.44 au), and CH₄ (12.6 eV or 0.46 au) from the NIST database [50]. For the same incident ion charge state and close ionization potential, the capture states $n$ of the different targets were all equal to five. The different calculation results of the SEC cross section determined for N₂, O₂, and CH₄ are 8.6×10^-15 cm², 6.2×10^-15 cm², and 6.5×10^-15 cm², respectively, only from the difference in ionization potential. As a relatively simple theoretical model, the OBM can only estimate the SEC cross sections in magnitude, as it does not consider the ion incident energy and the $l$ state of the captured electron. Other advanced theoretical approaches, such as the classical trajectory Monte Carlo method [51], the quantum-mechanical molecular-orbital close-coupling method [52,53], and the atomic-orbital close-coupling method [54,55], are considered more accurate for dealing with collision processes in multibody problems. Therefore, the present measurements of the EC cross section can provide experimental verification of these theoretical calculations.