EXPERIMENTAL INVESTIGATIONS OF ION CHARGE DISTRIBUTIONS, EFFECTIVE ELECTRON DENSITIES, AND ELECTRON–ION CLOUD OVERLAP IN ELECTRON BEAM ION TRAP PLASMA USING EXTREME-ULTRAVIOLET SPECTROSCOPY

In order to obtain the dispersion function of the spectrometer, 20 well-known lines emitted from N iv–N vi and O v–O vii ions including some of their second order of diffraction were used, thus encompassing nearly the whole wavelength range of interest. The centroid of those lines was determined using Gaussian fit. A full width at half-maximum (FWHM) of the observed lines is found to be 5 pixels, corresponding to ∼0.6 Å which is larger than the theoretical resolution limit of the present grating (∼0.003 Å at 200 Å). The line is believed to be broadened mainly due to the ion cloud size at the trap center.

The standard wavelengths for the calibration lines were adopted from the NIST database⁶ which claims to be accurate within 0.005 Å for most of the lines used here. A cubic polynomial with the form

$$\begin{eqnarray} \lambda (x) & = & A + B(x-dx) + C(x-dx)^2 + D(x-dx)^3 \end{eqnarray} \tag{ 1 }$$

is used to represent the observed line position (x: channel number)–wavelength (λ: Å) relation. Here dx indicates a possible slight shift of the peak positions among different measurements over a long time span. Such a shift can be caused by mechanical and thermal instabilities of the whole spectrometer system and variations of the electron as well as the ion beam size and position. The mean values obtained over several days for the dispersion function in Equation (1) are found to be A = 358.18(13), B = −0.12986(4), C = 9.77(11) × 10⁻⁶, and D = 3.15(29) × 10⁻¹⁰. The overall calibration uncertainty is estimated to be in the order of 0.01 Å (rms) over the 107–353 Å range. Taking a particular peak position as a reference, the relative peak shifts dx for other lines were determined. The maximum shift observed in these spectra was found to be usually less than 3 pixels (= 0.27 Å, at the pixel position of ∼1885 channel ≅170 Å), i.e., less than 40 μm in space. Sixteen independent calibrations show that these polynomial coefficients remain constant despite of slight shifts dx. Therefore, the polynomial fitting above can be used for the calibration of the observed spectrum simply by shifting the whole curve by dx.

The observed line intensities in the spectrum depend not only on the emission strengths and the populations of ions in the observing region but also on the spectrometer efficiencies including the grating response and the detector efficiency. The grating response curve was adopted from Edelstein et al. (1984) and Poletto et al. (2001). The detector quantum efficiency curve for the CCD camera was taken from the manual (DO436) provided by Andor Technology company.⁷ Furthermore, the absorption edge features at ∼100 Å were taken into account with more detail by calculating the transmission through the topmost, 900 Å thick SiO₂ layers on the CCD using data by Henke et al. (1993). Yet the assumed overall efficiency in other regions of the spectrum was not modified. The response for the second-order diffraction lines was determined from the direct comparison of the emission lines observed in the first and second orders. As shown in Figure 2, we have estimated the overall spectral efficiency of the whole spectrometer system within an accuracy of about ±10% in 100–220 Å photon range, its uncertainty slowly increasing up to ±30% at 340 Å region. However, such relatively large uncertainties did not affect seriously the present analysis and results because the relative intensities are compared only over narrow spectral region where the spectral sensitivity is more or less constant.

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In the following, 26 Fe spectra with very small pixel shifts (dx in Equation (1)) were analyzed in detail. The strong, isolated resonance lines at 284.164 Å (NIST value) due to the 2p⁶3s3p¹P₁–2p⁶3s^2 1S₀ transition of Fe xv and at 171.073 Å due to the 3s²3p⁵3d¹P₁–3s²3p^6 1S₀ transition of Fe ix were used to monitor the pixel shift. Using the mean parameters (A, B, C, and D) given above and the estimated shift, the wavelength scales for an individual spectrum were determined. Once the wavelength scale was fixed with this procedure, nearly 200 emission lines were found and fitted locally using Gaussian profiles. One of these spectra taken under the electron beam of E_e = 5.64 keV/I_e = 320 mA is shown in Figure 3.

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To reproduce the observed spectra including the relative line intensities and to identify the emitted lines, a series of the collisional-radiative simulations was carried out. Here we assume that the electronic impact excitation and de-excitation, as well as the radiative decay rates among levels of each ion (see Table 1) are the main mechanism for level population. The population of excited levels from ionization and recombination processes is assumed to be small in comparison with those due to the direct electronic excitation and the following cascades. The population of a particular level is determined by solving a system of coupled rate equations given by

$$\begin{eqnarray} \frac{dN_j^q}{dt} & = & \sum _{i<j}N_i^qn_{\rm e}C_{i\rightarrow j}+\sum _{k>j}N_k^q(n_{\rm e}C_{k\rightarrow j}+A_{k\rightarrow j})\nonumber\\ && -N_j^q\sum _{i<j}(n_{\rm e}C_{j\rightarrow i}+A_{j\rightarrow i}), \end{eqnarray} \tag{ 2 }$$

where N^q_j is the population density of state j of ions with charge of q, n_e is the electron density and C_i→j is the excitation/de-excitation rate from the state i to j. A_j→i is the spontaneous decay (transition) rate from j to i. The first term on the right-hand side represents the excitation to the state j from lower excited state, the first part of the second term represents the de-excitation into j from higher excited states and the second part represents the spontaneous decay to j, and the third term represents the de-excitation and its spontaneous decay to lower excited state from j, respectively. Using the data from FAC (Gu 2008) and quasi-stationary approximation for excited level (dN^q_j/dt = 0) for each Fe ionic state ranging from Fe vii to Fe xxiv, we obtain the level population and, furthermore, calculate the line intensity defined as

$$\begin{eqnarray} I^q_{j\rightarrow i} & =& N_j^q A_{j\rightarrow i}. \end{eqnarray} \tag{ 3 }$$

The calculated energies for ions of the present interest have been found to be in agreement with experimental values, which are available from the NIST version 3.0 (see footnote 6), within 1%. The corresponding theoretical wavelengths were corrected with these available experimental data. The detailed assessment of the atomic data will not be discussed here as it exceeds the scope of the present work. The atomic data of Fe vii–Fe xv have been used to analyze the EUV spectra from EBIT at low electron impact energies (Liang et al. 2009). The resultant line intensities (N^q_iA_ij where N^q_i indicates the level population of the ith energy level of an ion in the charge state q and A_ij refers to the decay rate of the i → j transition) are then convolved with a Gaussian profile with a line width of 0.6 Å. Then, the simulated spectra at E_e = 5.64 keV and I_e = 320 mA with n_e = 1.0 × 10¹² cm⁻³ is normalized by the measured spectra by a single known emission line of each ionic state (see top arrows in each panel of Figure 3). The sum spectrum (blue solid curve) is the addition of all the theoretical spectra over Fe vii–Fe xxiv ions without higher order diffraction spectra. By using the second-order efficiency in Figure 2, we deduce the second-order diffraction spectra in wavelength range of 200–353 Å (see dark-red step curve in Figure 3).

Figure 3 demonstrates that the present simulations satisfactorily reproduce the measured spectra. This supports the following discussions on the effective electron density and electron–ion cloud overlapping. For some emission lines at longer wavelengths (>220 Å), significant deviations were observed and found to be caused by the second-order diffraction (see marks with (2) in Figure 3(b)). Based upon the present simulated spectra and the recorded emission lines between 90 and 120 Å, we further note several third-order diffraction lines (see marks with (3) in Figure 3). Several contaminations from highly charged C v, O iv, and O v ions have been identified with the aid of the NIST list (see footnote 6).

The nonthermal unidirectional electrons in the present EBIT experiment will result in an anisotropy of emitted line radiations (Gu et al. 1999; Savin et al. 1996; Zhang et al. 1990; Shlyaptseva et al. 1998). This anisotropy results from the unidirectionality of the electron beam which can cause a population imbalance of the magnetic sublevels (m_(−J)) and(m_(+J)) leading to a possible linear polarization of the EUV emission. The observed line intensity, I(90°) (emitted at 90° to the electron beam direction), has a relation to the total emitted line intensity I in 4π steradians as follows:

$$\begin{eqnarray} I(90^{\circ }) & = & \frac{3 I}{3-P}, \end{eqnarray} \tag{ 4 }$$

where P is the polarization of the transition of interest.

Using the FAC, electron impact collision strengths among the magnetic sublevels of Fe vii–Fe xxiv ions have been calculated. By solving the extended rate equation (see Equation (1) in the work of Liang et al. (2009)), we obtain the magnetic sublevel population, and further the polarization P.

In Figure 4, we compare the synthetic spectra for monoenergetic electrons (E_e = 5.64 keV with n_e = 1.0 × 10¹² cm⁻³) with and without polarization effect for Fe xxi and Fe xi–Fe xiv ions. The polarization reduces some observed line intensities, e.g., the 2s2p^3 3D₁ → 2s²2p^2 3P₀ transition of Fe xxi at 128.755 Å by 13%; the 3s²3d²D_5/2→3s²3p²P_3/2 of Fe xiv at 219.130 Å by 24%, where P < 0. We also note some observed line intensities are enhanced when the polarization effect (P > 0) is included, e.g., the 3s3p^2 2P_3/2 → 3s²3p²P_3/2 transition line of Fe xiv at 264.790 Å by 20%. This reveals that the polarization plays an important role on the observed line intensity for some emission lines at high electron energies. Therefore, we should be cautious when determining the line intensity ratios and the corresponding effective electron density (n^eff_e) of the EBIT plasma using the EUV line ratio technique.

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The observed line intensities of ions in various charge states provide information on their population distribution (see Section 4) and the plasma parameters in the trap. In particular, the electron density, a critically important parameter in all kinds of plasmas, can be determined with the help of EUV spectroscopy. In the EBIT, the electron density depends on a number of its running parameters such as the electron energy and current, the magnetic field strength, the vacuum, the trapping conditions, etc.

The allowed EUV transitions follow excitation so quickly (in a matter of ps) that the light emission pattern basically reflects the ion–electron beam overlap region (r_e) in which the excitation actually takes place. On the other hand, ions emitting photons through the decay of long-lived metastable electronic states can leave the electron beam region before the radiative transition occurs. The actual radius (r_i) of ion cloud in the EBIT can be, under certain conditions, larger than the electron beam radius r_e, as the artist's representation shown in Figure 6. A real physical simulation of an ion orbit in EBITs is given in the work of (Gillaspy et al. 1995, see Figure 2). Then, the effective electron density (n^eff_e) depends on the geometric overlap ($f_{q_{\rm i}} = \big(\frac{\bar{r_{\rm q_i}}}{\bar{r_{\rm e}}}\big)^2$ for $\bar{r_{\rm q_i}} > \bar{r_{\rm e}}$) between the spatial distributions of the monoenergetic electron beam and those of the ion cloud in the trap region. Indeed, the effective electron density (n^eff_e) governs basically the actual rates of all excitation and ionization processes in the EBIT.

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The geometrically averaged electron density (n^g_e) can be estimated from the electron beam size:

$$\begin{eqnarray} n_{\rm e}^{\rm g} & = & I_{\rm e}/\big({\rm e} \pi r_{\rm e}^2v_{\rm e}\big), \end{eqnarray} \tag{ 5 }$$

where e, I_e, r_e, and v_e are the electronic charge, the electron beam current, radius, and velocity, respectively. We estimate the geometrical electron density n^g_e of 1.6 × 10¹³ cm⁻³ under the present conditions (E_e = 5.64 keV, I_e = 320 mA, r_e = 30 μm).

However, we have no information on the spatial distributions of ions in the present experiment and only those forbidden lines can give a true representation of the ion spatial distributions, since the long elapsing time between excitation and radiative decay essentially decouples the light emission spatial pattern from the geometry of the narrow excitation region defined by the electron beam. Then, we have to infer the effective electron density n^eff_e by a suitable spectroscopic diagnostic method, and then derive information on the ion–electron beam overlap factor for each charge state.

5.1.1. Fe xxi Ion Lines

The diagnostic potential of EUV lines from Fe ions for determining n^eff_e had been pointed out previously. The intensity ratios of several partner lines of the carbon-like Fe xxi ion are known to be sensitive to the electron density but rather insensitive to the electron temperature (Ness et al. 2004). This feature follows from the fact that the upper level is populated mainly through collisional excitation from a density-sensitive lower metastable level. As shown in Figure 7, the 2s2p^3 3D₂ level of this ion has a high excitation rate from and decay rate to the metastable 2s²2p^2 3P₁ level, which is forbidden to decay to the ground state. By increasing the electron density, a significant fraction of the ³P₁ population can be re-excited to the ³D₂ state, which relaxes to the ³P₁ state and, therefore, the line intensity at 142.148 Å increases. On the other hand, through depopulation of the ground state ³P₀ due to the shelving of electrons into the metastable level (if production of the ³P₀ level of Fe xxi from lower charge states is too slow), the excitation to the ³D₁ state and therefore the first resonance line (³D₁–³P₀) intensity at 128.755 Å decreases at high electron densities, as illustrated in Figure 8(a).

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It is important to note that the 2s2p^3 3D₁ state can also decay to the metastable ³P₁ state with a low branching ratio. As this transition line (at 142.278 Å) cannot be resolved from the 142.148 Å line due to the limited spectrometer resolution, we have to take this blending effect into account. Since the 2s2p^3 3D₁ level is directly populated from the ground state, the intensity of 142.278 Å line has a dependence on n_e which is similar to that of the 128.755 Å resonance line, and its relative contribution to the total line intensity becomes dominant below the electron density of n_e = 1.0 × 10¹² cm⁻³. On the other hand, at higher electron densities, the 142.148 Å line becomes more intense. As shown in Figure 8(a), the line intensities, I(142.148 Å), I(142.278 Å), and I(128.755 Å), are calculated for a plasma excited by monoenergetic electrons (E_e = 5.64 keV) and also for thermal electrons (T_e = 1.0 × 10⁷ K) as a function of the electron density. These results indicate that the intensity of 142.148 Å line is greatly affected by the electron energy distributions (see the dashed red line in Figure 8(a)). The monoenergetic electron beam populates predominantly the high-lying levels (³D₁), whereas thermal energy electrons preferentially feed states with lower excitation energies. These effects result in a difference of the line intensity ratio in the two cases as shown in Figure 8(b). With increasing electron density, the deviations get weaker.

By applying the collisional-radiative model described above, we calculated the intensities of the lines of interest for different electron densities for both monoenergetic and thermal electrons. Figure 8(b) demonstrates the intensity ratios of $\frac{I(142.148 \;{\rm \AA})+ I(142.278 \;{\rm \AA})}{I(128.755 \;{\rm \AA})}$ as a function of the electron density for both cases. The line intensity ratios for monoenergetic electron beam are depicted with three separate lines (blue: 4.5 keV, red: 5.0 keV, and black: 5.64 keV). The hatched bands (gray) indicate the ratio obtained for thermal electrons with the temperatures ranging from 5.0 × 10⁶ to 5.0 × 10⁷ K, showing that the line ratio is not too sensitive to the electron temperature. A noticeable difference of this ratio can be seen between monoenergetic and thermal electrons below the electron density of 3.0 × 10¹² cm⁻³. This different behavior between two modes of excitation is due to the fact that the population of the 2s²2p^2 3P₁ level, a major excitation channel to the upper ³D₂ level which emits the 142.148 Å line, is greatly influenced by the electron energy distributions, as shown in Figure 8(a), resulting from features of electron collision processes that the excitation efficiencies of low energy electrons are higher than those of high energy electrons.

By comparing the calculated line intensity ratio $\frac{I(142.148 \;{\rm \AA})+ I(142.278 \;{\rm \AA})}{I(128.755\;{\rm \AA})}$ with the present experimental result (0.15 ± 0.01 at 5.64 keV/320 mA, the error in line intensities results from a quadrature sum of the errors from fitting the line shapes and spectrometer response uncertainty shown in Figure 2), the effective electron density was determined to be n^eff_e = 2.60^+0.4_−0.39 × 10¹² cm⁻³. At slightly different electron beam conditions (taken on different occasions), namely 5.0 keV/400 mA and 4.5 keV/197 mA, we have obtained 2.30 ± 0.42 × 10¹² cm⁻³ and 7.8^+4.0_−3.9 × 10¹¹ cm⁻³, respectively. The relatively large uncertainties are mainly due to the fact that the observed ratios are already close to the lower density limit (∼10¹¹ cm⁻³). When the polarization effect is taken into account, the effective electron densities are estimated to 1.40^+0.35_−0.32 × 10¹² cm⁻³ (5.64 keV/320 mA), 1.12^+0.37_−0.34 × 10¹² cm⁻³ (5.00 keV/400 mA), and an upper limit 1.9 × 10¹¹ cm⁻³ (4.50 keV/197 mA), respectively, indicating that exact information on the polarization is important.

5.1.2. Fe x Ion Lines

We have tried to check our EUV spectroscopy technique for estimating the electron densities using a different ion charge, namely Fe x ions, produced simultaneously under the same EBIT conditions. The upper level (3s²3p⁴3d²D_5/2) of resonance transition line at 174.526 Å are populated dominantly by direct excitations from the ground state (3s²3p^5 2P_3/2), as shown in Figure 9(a). The population of the excited level ²D_3/2 is dominantly populated by excitation from the metastable level (²P_1/2, 51%). So the line intensity of the transition (3s²3p⁴3d²D_3/2 → 3s²3p^5 2P_1/2, with a wavelength of 175.265 Å) is sensitive to the electron density. The upper level (²P_1/2) of the transition line at 175.475 Å is also dominantly populated by excitation from the metastable level (25%). Its intensity (relative to that of the resonance line at 174.526 Å) is also sensitive to the electron density. In the present measurement, this transition line cannot be resolved from the line at 175.265 Å. By taking this blending effect into account, we calculate the line intensity ratio $\frac{I(175.265 \;{\rm \AA})+ I(175.475 \;{\rm \AA})}{I(174.526 \;{\rm \AA})}$ of Fe x as a function of the electron density for monoenergetic electron beam (see Figure 9(b)).

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Based upon this curve and from the present observed ratios, we have determined the effective electron densities to be 1.3^+0.5_−0.3 × 10¹¹, 4.6^+1.0_−0.8 × 10¹⁰, and 1.5^+0.7_−0.4 × 10¹¹ cm⁻³ under 4.5 keV/197 mA, 5.0 keV/400 mA, and 5.64 keV/320 mA electron beams, respectively. Polarization effect on this line ratio is found to be negligible. Note that the errors are much smaller than those in Fe xxi because the observed ratios in Fe x ions are in the most density-sensitive region, compared with those in Fe xxi (see Figure 8).

5.1.3. Other Ion Lines

Other EUV line intensity ratios have been used to determine the electron density in solar corona, EBIT plasma, etc. (Keenan et al. 2005b—Fe xi; Del Zanna & Mason 2005—Fe xii; Yamamoto et al. 2008; Watanabe et al. 2009—Fe xiii). Here, we selected n_e-sensitive lines without blending from emission lines of other ionic states to further estimate the electron density in the EBIT plasma under the electron beam energy and current of E_e = 5.64 keV and I_e = 320 mA, respectively. For example, the line intensity ratios (see Figure 10) with nearby wavelengths of Fe xi: $\frac{I(179.763 \;{\rm \AA})}{I(181.801 \;{\rm \AA})+I(182.176 \;{\rm \AA})}$, Fe xii: $\frac{I(191.049 \;{\rm \AA})}{I(195.119 \;{\rm \AA})+I(195.179 \;{\rm \AA})}$, Fe xiii: $\frac{I(213.768 \;{\rm \AA})}{I(200.021 \;{\rm \AA})}$, Fe xiv: $\frac{I(219.130 \;{\rm \AA})}{I(211.317 \;{\rm \AA})}$, Fe xxii: $\frac{I(156.020\;{\rm \AA})}{I(13.5812 \;{\rm \AA})}$. The corresponding transitions are listed in Table 2. The resultant effective electron densities are listed in Table 3. Note that the polarization correction is critical in some cases. We note that there is significant difference in the electron densities estimated from different ions under the completely same EBIT running conditions. It seems that there is a trend that higher effective electron densities are obtained from higher ionic charge under the same operation conditions. However, the result from Fe xiii slightly deviates from the trend.

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Table 2. Transition Lines Used in Estimation of the Effective Electron Density n^eff_e

λ(Å)	Transition		Ion
174.526	3s23p43d2D5/2	→3s23p5 2P3/2	Fe x
175.265	3s23p43d2D3/2	→3s23p5 2P1/2	Fe x
175.475	3s23p43d2P3/2	→3s23p5 2P1/2	Fe x
179.763	3s23p33d1F3	→3s23p4 1D2	Fe xi
181.801	3s23p33d3P0	→3s23p4 3P1	Fe xi
182.176	3s23p33d3D2	→3s23p4 3P1	Fe xi
191.049	3s23p23d2D5/2	→3s23p3 2P3/2	Fe xii
195.119	3s23p23d4P5/2	→3s23p3 4S3/2	Fe xii
195.179	3s23p23d2D3/2	→3s23p3 2D3/2	Fe xii
213.768	3s23p3d3P2	→3s23p2 3P3	Fe xiii
200.021	3s23p3d3D2	→3s23p2 3P1	Fe xiii
264.789	3s3p2 2P3/2	→3s23p2P3/2	Fe xiv
274.203	3s3p2 2S1/2	→3s23p2P1/2	Fe xiv
128.755	2s2p3 3D1	→2s22p2 3P0	Fe xxi
142.148	2s2p3 3D2	→2s22p2 3P1	Fe xxi
142.278	2s2p3 3D1	→2s22p2 3P1	Fe xxi
135.812	2s2p2 2D3/2	→2s22p2P1/2	Fe xxii
156.020	2s2p2 2D5/2	→2s22p2P3/2	Fe xxii

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Table 3. Effective Electron Densities (cm⁻³) for Iron Ions With Different Charges in the EBIT Plasma

	Effective Electron Density
Ion	neffe	neffe(Correction for Polarization)
Fe x	1.5+0.7−0.4 × 1011	1.5+0.7−0.4 × 1011
Fe xi	3.0+0.8−0.7 × 1011	3.4+1.1−0.9 × 1011
Fe xii	4.03+0.48−0.47 × 1011	3.98+0.47−0.43 × 1011
Fe xiii	6.1+0.7−0.3 × 1010	1.7+0.3−0.9 × 1011
Fe xiv	3.03+0.43−0.35 × 1011	3.03+0.43−0.35 × 1011
Fe xxi	2.60+0.40−0.39 × 1012	1.4+0.35−0.32 × 1012
Fe xxii	1.10+0.23−0.23 × 1013	1.12+0.22−0.23 × 1013

Note. Effective electron densities (cm⁻³) for iron ions with different charges in the EBIT plasma under E_e = 5.64 keV/I_e = 320 mA, estimated from line intensity ratios without and with polarization corrections (see Figures 8–10) of Fe x–Fe xxii.

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Watanabe et al. (2009) performed the electron density diagnostic for the solar active region with Fe xiii line ratios and Hinode⁸ EUV imaging spectrometer (EIS) spectra, in which many line ratios were presented (see Figure 3 in their work). However, most emission lines of Fe xiii are blended by lines of other ionic charge states in our present measurement. In this work, we try to use ratios of lines without contamination. So the two contamination-free density-sensitive lines are used here (see Figures 3 and 11), which differs from other cases, e.g., Fe x and Fe xxi, one resonance line and one density-sensitive line (the population of the upper level is dominated by excitations of the metastable level). Due to the different competitions for level population, the line intensity ratio between them also shows a good sensitivity to the electron density. As explored by Watanabe et al. (2009), different models show a slight discrepancy using different atomic parameters. Generally, the atomic parameters for forbidden transitions (decays from metastable levels) show a larger uncertainty than those for resonance transitions. Hence, the ratio between two density-sensitive lines shows a slightly larger deviation than other ratios between the density-sensitive line and the resonance line.

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At first glance, it is quite surprising that the effective electron densities n^eff_e observed from the transition lines are so small (less than 2 orders of magnitude), compared with simple geometrical electron densities n^g_e (see Equation (5)) and also much different between differently charged ions observed under the same EBIT conditions. In order to understand such features, it is interesting to compare the observed "effective" electron densities n^eff_e with the "geometrical" electron densities n^g_e calculated from Equation (5). These results are summarized in Table 4 demonstrating that the n^eff_e/n^g_e ratios vary by almost 2 orders of magnitude. Low charged ions show low ratios and higher charged ions show higher ratios.

Similar observations of the effective electron densities under quite different electron beam conditions (1–3 keV and 1–10 mA) in an EBIT at Livermore were performed based upon soft X-ray spectroscopy of helium-like N vi ions (Chen et al. 2004). Silver et al. (2000) and Matranga et al. (2003) also performed similar diagnostic of the electron densities for the NIST EBIT plasmas using K-shell X-ray spectroscopy. The lifetimes of the levels observed are much shorter than the present metastable levels of Fe ions. A linear behavior between the electron densities and [I_e/E^1/2_e] was indeed suggested under a fixed electron beam radius by Chen et al. (2004) (see Figure 6 in their work) based upon their limited experimental data. Considering that most of EBITs all over the world have similar (designed) electron beam radii, we combined the data available so far into our present analysis as shown in Figure 12. The relative normalized electron density n^eff_e/[I_e/E^1/2_e] clearly deviates from such expectation. Particularly such deviations at high [I_e/E^1/2_e] values with high electron current (like in the present work) region are significant, compared with those at low [I_e/E^1/2_e] values (like in Livermore experiment by Chen et al. 2004). This trend can be understood: at high electron currents, the ions tend to be more heated and thus their radii get large, resulting in a lower effective electron density. As the effective electron density depends on the overlap between ion cloud and electron beam, the deviation observed indicates a significant change of the ion–electron beam overlap f which depends on [I_e/E^1/2_e], and also the axial trap voltage applied on the end cap drift tubes on both sides of the central drift tube as explored by Porto et al. (2000). The significant deviations for lower charged iron ions in the present results mean a smaller overlap than previously found at high electron energy/high current results. By taking the polarization effect into account, we re-analyze our previous measurements with low energies and currents (E_e = 70–500 eV and I_e = 2.1–8.0 mA) at the Heidelberg FLASH-EBIT (Liang et al. 2009). Significant deviation appears again as illustrated by dark-green square symbols in Figure 12, indicating a smaller overlap for the lower charged ions. The large scattering in the present results is due to the overlap f (or r²_i/r²_e) dependence on the ionic charge as will be explained in the following.

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The ratios, r_i/r_e, of the ion cloud radius over the electron beam radius obtained from the observed n^eff_e/n^g_e ratios are illustrated in the last column of Table 4. The range of ion radius ratio against the real electron radius thus estimated show that the ion clouds significantly expand far outside (∼10 times) the geometrical electron beam radius r_e for lower charged ions in the Heidelberg FLASH-EBIT, particularly for Fe x and Fe xiii (see Table 4 and Figure 13) in the present measurement. This large expansion of the ion could be understood due to the axial trap of 360 V applied on the end cap trap tubes. Porto et al. (2000) experimentally demonstrated that the width of ion cloud (Ar¹³⁺) would increase by a factor of ∼5 from an open trap to a deep trap with the static trapping voltage of 300 V (see Figure 6 in their work). However, for higher charged ions (Fe xxi and Fe xxii), the ion cloud was found to be less expanded. Thus, it is concluded that the expansion of ion clouds has a dependence on the ionic charge. Though the present data are limited, there is a clear trend that ions with higher charge (Fe²⁰⁺ and Fe²¹⁺) have less expanded. Assuming the diameter of electron beam to be ∼60μm in the work of Silver et al. (2000), Matranga et al. (2003), and Liang et al. (2009), we further derive the ion expansion in their measurements. Though these measurements were performed with different trap conditions at different EBITs, a quantitative relation between the expansion of ion clouds and the effective charge of ions can be concluded. Figure 13 demonstrates that the expansion (r_i/r_e) of iron ion cloud has a relation with the effective charge of ions 1/(q_i/Z)^α with 1 < α ⩽ 2. It is noted that the Levine's model (Levine et al. 1988) of the inverse exponential expansion of ion cloud size seems to be slightly off from the general trend including all the possible ion charge data available. At comparable values of the effective charge of ions, the expansion of Ne⁸⁺ ion cloud in works of Matranga et al. (2003) and Silver et al. (2000) shows a similar expansion as Fe²¹⁺ ion cloud extracted in the present measurement though they are generated at different EBITs with different conditions. Measurements by Silver et al. (2000) at slightly different conditions for N⁵⁺ and O⁶⁺ (2.0 keV/35.9 mA), as well as Ne⁸⁺ (2.2 keV/39.5 mA) also show such dependence on the effective ion charge. The data from Chen et al. (2004) demonstrate the dependence of the expansion of ion could (N⁵⁺) on the operation conditions of EBIT as revealed in Figure 12.

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There are only very few direct observations on such large expanded ion clouds. NIST group has reported their results for ion cloud sizes in their EBIT by measuring lines from metastable state ions (Porto et al. 2000). Under their EBIT conditions, these ions have been found to spread out up to 400 μm radius, compared with their estimated electron radius (35 μm). Their results for Ar¹³⁺ ions provide some clues and support in understanding the present results. This is generally in agreement with a recent observation based upon visible photon spectroscopy at MPI. Another observation shows that the Ar¹³⁺ ion cloud radius is observed to reach 250 μm (Draganic 2007) under lower energy electrons. These measurements support our present estimations based upon the EUV spectroscopy.

In the following, we present an analytical representation for the dependence of the overlap f on the charge of ions for an EBIT plasma. We assume a radial Boltzmann distribution for the ion cloud of charge state q_i

$$\begin{eqnarray} n_{q_{\rm i}}(r) & = & n_{q_{\rm i}}(0){\rm exp}(-{\rm e} q_{\rm i}\phi (r)/ kT_{q_{\rm i}}) \end{eqnarray} \tag{ 6 }$$

and a Gaussian shape for the electron beam density radial distribution

$$\begin{eqnarray} n_{\rm e}(r) & = & n_{\rm e}(0){\rm exp}\left(-\frac{r^2}{r^2_{\rm e}}{\rm ln 2}\right), \end{eqnarray} \tag{ 7 }$$

where r_e refers to the electron beam radius estimated by Herrmann's theorem (Herrmann 1958), $T_{q_{\rm i}}$ is the temperature of thermalized ions with charge of q_i. The potential function ϕ(r) = ϕ_e(r) + ϕ_i(r) accounts for the space charge potential of the compressed electron beam and radial screening by ions, respectively. ϕ_e(r) is expressed as follows by Gillaspy (2001):

$$\begin{eqnarray} \phi _{\rm e} (r \le r_{\rm e}) & = & \frac{I_{\rm e}}{4\pi \epsilon _0 v_{\rm e}}\left[ \left(\frac{r}{r_{\rm e}}\right)^2+{\rm ln}\left(\frac{r_{\rm e}}{r_{\rm dt}}\right)^2-1\right] \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \phi _{\rm e} (r \ge r_{\rm e}) & = & \frac{I_{\rm e}}{2\pi \epsilon _0 v_{\rm e}}{\rm ln}\left(\frac{r}{r_{\rm dt}}\right), \end{eqnarray} \tag{ 9 }$$

where r_dt represents the radius of the central drift tube of the EBIT, ₀ is permittivity and v_e is the electron velocity. The ion charge density n_i and the potential are connected by Poisson's equation, which can be solved numerically:

$$\begin{eqnarray} -\frac{\epsilon _0}{r}\frac{\rm d}{{\rm d}r}r\frac{{\rm d}\phi (r)}{{\rm d}r} & = & -{\rm e}n_{\rm e}(r)+\sum {\rm e}q_{\rm i}n_{q_{\rm i}}(r). \end{eqnarray} \tag{ 10 }$$

From Equations (6)–(10), we can see that the overlap factor $f_{q_{\rm i}} = \big(\frac{\bar{r_{\rm q_i}}}{\bar{r_{\rm e}}}\big)^2$ has a strong dependence on the ionic charge.