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Abstract
We present a systematic theoretical study on the angular distribution and linear polarization of x-ray line emissions of neon-like ions following the electron-impact excitation from the ground state to the excited levels [(2p5)1/23d3/2]J=1, [(2p5)3/23d5/2]J=1, [(2p5)3/23d3/2]J=1, and [(2p5)1/23s]J=1. The cross sections are calculated by using the flexible atomic code under configuration-interaction plus many-body perturbation theory method. The angular distribution and linear polarization are obtained based on density matrix theory. Emphasis has been placed on the effect of the configuration mixing on the angular distribution and polarization. It has been proved that the strong mixing of configuration [(2p5)3/23d3/2]J=1 with configuration [(2p5)1/23s]J=1 can result in the abrupt change of Z-dependence of angular distribution and polarization. It indicates that angular distribution and polarization can be expected to serve as a tool for investigation of configuration mixing effect.
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1. Introduction
Neon-like ions exist widely in various hot plasmas owing to their stable closed-shell structure. From hot astrophysical objects [1–5] to magnetic confinement fusion [6–8] and laser-produced plasmas [9–11], neon-like ions exist spanning a broad range of plasma temperatures and densities. The characteristic spectral lines of neon-like ions indicates important informations of plasmas, so neon-like ions are of great significance in the diagnostics of plasmas state [12]. Apart from diagnostic unity, neon-like ions also play a crucial role in fundamental research of various atomic processes and physical effects [13–15].
Electron-impact excitation (EIE) is one of the most fundamental atomic processes in astrophysical and laboratory plasmas [16]. As the main electron-ion collision process in hot plasmas, the EIE seriously affects the energy level population. For example, the well-known $3C$ $([(2 p^{5})_{1 / 2} 3 d_{3 / 2}]_{J=1} \rightarrow (2 p^{6})_{J=0})$ and $3D$ $([(2 p^{5})_{3 / 2} 3 d_{5 / 2}]_{J=1} \rightarrow (2 p^{6})_{J=0})$ lines of Ne-like Fe$^{16+}$ ions are mainly populated by EIE [17]. Then, thus, it also strongly alters the spectra emitting from the hot plasmas.
On the other hand, EIE of atoms or ions generally results in non-statistical populations of magnetic sublevels of excited states [17–19]. It has been confirmed that the radiation decay from the excited states with non-statistical magnetic sublevels is polarized and anisotropic [20,21]. As the fact that the polarization and angular distribution of x ray emitting from excited atoms or ions only depend on the populations of magnetic sublevels, which results from the electron-ion collision, the polarization and angular distribution has been developed as a probe to explore the structure of the atoms (or ions) and dynamics of electron-ion collision. This novel probe has been used to study the Breit interaction [21–25], multipole mixing of radiation fields [26,27], hyperfine interaction [28,29], and et al. Obviously, the angular and polarization properties of x-ray lines are qualified for more studies on the atomic or ionic structure and various physical effects.
Generally, an atomic or ionic state cannot be represented by a single electronic configuration, but by a linear combination of a set of configurations with same symmetry. This effect is known as configuration mixing [30]. When two or more configurations have almost identical energy, the mixing become strong and important, accompanying some interesting and anomalous effects. The configuration mixing usually results in avoided crossing of energy levels [14,31]. As an example, it has been demonstrated that the $3D$ and $3F$ $([(2 p^{5})_{1 / 2} 3 s]_{J=1} \rightarrow (2 p^{6})_{J=0})$ level of neon-like ions appear avoided crossing at $Z\sim$ 54 due to strong configuration mixing [30]. On the other hand, the configuration mixing change the oscillator strengths and collision strengths. Recently, we found that the strong mixing of $3D$ with $3F$ apparently depresses the collision strength of the $3D$ line, resulting in the abrupt change of the $Z$-dependence of the $3C/3D$ line ratio [18]. In addition, Li et al. [32] investigated the effect of configuration mixing on the opacity of aluminum plasma. Beiersdorfer et al. [14] found that the QED contributions to energy levels are anomalously large and the accuracy of calculation of energy levels are compromised near the crossing. Most recently, Koike et al. [33] studied the effect of configuration mixing on the extreme ultraviolet (EUV) spectra emitting from galliumlike lanthanide atomic ions. In brief, the configuration mixing or configuration interaction is of great importance and it attracts a lot of interest. However, to our best knowledge, there is no report on its effect on the angular distribution and polarization of spectral lines.
In this work, we present a theoretical investigation on the polarization and angular distribution of x-ray emission of neon-like ions following the electron-impact excitation,
Special attention is paid to the effect of configuration mixing on the polarization and angular distribution of the radiated 3$C$, 3$D$, 3$E$, 3$F$ lines. The rest of this paper is structured as follows. In the next, the theoretical method is presented in Section 2 for calculating the EIE cross sections as well as the angular distribution and linear polarization of spectral lines of Ne-like ions. In Section 3, the $Z$-dependence of the alignment parameters of the excited levels and the angular distribution as well as linear polarization are discussed. In the end, a brief summary of the present work is given in Section 4.
2. Calculation
In the present work, the energy levels and EIE cross sections are calculated by using the FAC based on the relativistic configuration interaction (CI) plus many-body perturbation theory (MBPT) method [34]. The calculation is the same as our previous work [18], please refer to Ref. [18] for details. We will only provide a brief description here.
In this method, the atomic processes involved are described with basis wavefunctions. These wavefunctions are derived from a local central potential, which is self-consistently determined to represent electronic screening of the nuclear potential. An atomic-state function (ASF) with parity $\mathit {\Pi }$, total angular momentum $J$, magnetic quantum number $M$, and additional quantum numbers $\gamma$ for a unique specification of the state is expressed as a linear combination of configuration-state functions (CSFs) with the same $\mathit {\Pi } JM$ as follows,
Here, $n$ is the number of CSFs used for constructing the ASF. $c_r$ denote the configuration mixing coefficients, which are obtained by diagonalizing the model Hamiltonian matrix. The CSFs are constructed as an antisymmetrized sum of the products of one-electron Dirac spinors. In the present calculations, the ground $1s^2 2l^8$ and all single-excitation $1s^2 2l^7 nl ~(n=3,\ldots,6)$ configurations are included in the CI expansion. Such a CI expansion is sufficient to guarantee the convergence according to previous studies [15,18]. The potential is optimized separately for the $1s^2 2l^8$ and $1s^2 2l^7 3l$ configuration groups to improve the accuracy [15,34].
It is noted that relativistic effects are fully taken into account by using the Dirac-Coulomb-Breit Hamiltonian [35,36]
The EIE cross sections $\sigma$ from the initial state $\psi _0$ to the final state $\psi _1$ of target ions can be expressed in term of the collision strength $\Omega$ as [37],
where $g_0$ is the statistical weight of the initial state, $k_0$ is the kinetic momentum of the incident electron, and the collision strength $\Omega$ can be expressed as [37]In this expression, $[J_T]=2J_T+1$, $\kappa _0$ and $\kappa _1$ are the relativistic angular quantum numbers of the incident and scattered electrons. $J_T$ and $M_T$ indicate the total angular momentum coupling the target ion state with the continuum orbital of impact electron and its projection, respectively.
Based on the EIE cross sections, the relative population of the magnetic substates of the excited energy levels can be fully determined. Within the density matrix theory [38,39], alignment parameters are used to describe the population of the magnetic sublevels. For an excited level with total angular momentum $J$ and magnetic quantum number $M = -J, -J+1, \ldots, J$, its alignment parameter can be given as [20,38],
Here, $k$ is the rank, $\sigma _{M}$ is the normalized partial cross section of the sublevel $\left | J M \right \rangle$, and $\left (\begin {array}{c}{\ldots} \\{\ldots} \end {array}\right )$ denotes the Wigner $3j$ symbol. Specially, if an excited level possesses the total angular momentum of 1, only the second-order alignment parameter ${\cal A}_{2}$ is required and Eq. (10) can be reduced as [22]
With the alignment parameters available, the polarization and angular distribution of the spectral lines emitted in the radiative decay of the excited ions can be fully determined. Usually, the polarization is observed in the direction perpendicular to the propagation direction of impact electron beam. In this direction, the degree of linear polarization of the lines can be given by [22,38]
In addition to the polarization, the angular distribution can also be expressed in term of alignment parameters as [22,38],
where $P_2(\cos \theta )$ is the second-order Legendre polynomial and $\theta$ is the polar angle of the emitted photons with respect to the impact electron beam.It should be noted that Eqs. (11) to (13) are appropriate only for the excited levels with a total angular momentum of 1. Fortunately, the excited levels of all spectral lines considered, i.e., 3$C$, 3$D$, 3$E$, and 3$F$, have a total angular momentum of 1. It simplifies the calculation of the polarization and angular distribution of these spectral lines.
3. Results and discussion
3.1 Cross section
In our previous work [18], we have certified the reliability and accuracy of the calculations by comparing the calculated results with other calculations [40,41] and experimental measurements [18,42–46]. Considering that the present calculation is the same as the previous work, we will only show and analyze the present calculation results here and not further illustrate the accuracy of the calculation.
In Fig. 1, we show the presently calculated partial EIE cross sections for the excitations from the ground state to the excited levels of Ne-like isoelectronic sequence ions with impact energies of 1.56 times the 3$C$ excitation threshold for each ion. For Xe$^{44+}$ ion, this impact energy is equal to the electron beam energy used in the previous experiment [18]. The considered excitations, i.e., $[(2p^{5})_{1/2}3d_{3/2}]_{J=1}$, $[(2p^{5})_{3/2}3d_{5/2}]_{J=1}$, $[(2p^{5})_{3/2}3d_{3/2}]_{J=1}$, and $[(2p^{5})_{1/2}3s]_{J=1}$, all possess the identical total angular momenta 1. Therefore, these states have three substates, $|M_f=+1\rangle$, $|M_f=0\rangle$, and $|M_f=-1\rangle$. It should be noted that the partial EIE cross sections $\sigma _+$ for magnetic substates $|M_f=+1\rangle$ are fully equal to the cross sections $\sigma _-$ for $|M_f=-1\rangle$ due to the spatial symmetry of the EIE process considered. As seen from the figure, the normalized partial cross sections corresponding to 3$C$ and 3$D$ lines have similar $Z$-dependence. Except for the partial cross sections corresponding to 3$D$ line slight varying at around $Z=54$, the partial cross sections corresponding to both 3$C$ and 3$D$ lines hardly vary with atomic number $Z$. Moreover, the excited levels corresponding to 3$C$ and 3$D$ lines have nearly equal partial cross sections, which means that these two levels have nearly the same magnetic sublevel population. In contrast, for the partial cross sections of excited levels corresponding to 3$E$ and 3$F$ lines, they strongly depend on the atomic number $Z$. Let’s illustrate this with partial cross sections $\sigma _0$ of substate $|M_f=0\rangle$. For 3$E$ line, it first declines slowly and then increases with the increase of atomic number $Z$ as a whole, but there is a jump around $Z = 51$. For 3$F$ line, there is a downward trend as a whole, starting relatively flat, then an abrupt change around $Z = 51$, and finally a sharp decline after $Z = 52$. The $Z$-dependence of excited levels corresponding to both 3$E$ and 3$F$ lines shows an abrupt change around $Z=51$.
Fig. 1. The normalized partial EIE cross sections for the excitations from the ground state to the excited levels corresponding to 3$C$, 3$D$, 3$E$, and 3$F$ lines of Ne-like isoelectronic sequence ions. The electron impact energy is equal to 1.56 times the 3$C$ excitation threshold for each ion.
3.2 Alignment parameter
After knowing the partial EIE cross sections, the alignment parameters of the excited levels can be obtained according to Eq. (11), which are shown in Fig. 2. From the figure, one can see that the alignment parameters of the excited level of $3C$ line hardly change with the atomic number $Z$. The $Z$-dependent behavior of the alignment parameters of excited level of $3D$ line is similar to that of $3C$ line except that there is a minor change around $Z=54$. Since the alignment parameters are fully determined by the relative populations of their corresponding magnetic sublevels, this change is resulting from the variation of the partial EIE cross sections of $3D$ magnetic sublevels. The populations-determination of the alignment parameter is particularly evident in the excited levels corresponding to the $3E$ and $3F$ lines. It can be seen that the alignment parameters of the excited levels of 3$E$ and 3$F$ lines change more obviously with the atomic number $Z$, which clearly reflects the abrupt change of the corresponding magnetic sublevel population. The superposition of the cross sections of the magnetic sublevels $|M_f=\pm 1\rangle$ and $|M_f=0\rangle$ with the opposite change trend of the atomic number $Z$ brings more obvious $Z$-dependence, so that the “abrupt change" becomes more obvious.
Fig. 2. The alignment parameters of the excited levels corresponding to 3$C$, 3$D$, 3$E$, and 3$F$ lines.
Examining the alignment parameters for the excited levels corresponding to these four spectral lines, it becomes evident that only the excited level corresponding to the $3E$ line shows positive alignment within the considered atomic number range. Conversely, the other three excited levels exhibit negative alignment, which directly reflects the relative population of the magnetic energy levels. As demonstrated in Fig. 1, the magnetic sublevel $|M_f=\pm 1\rangle$ for the 3$E$ line has a greater share of population in the considered atomic number range than $|M_f=0\rangle$, contrary to the other three excited levels. It is this feature that makes its alignment different from the other three excited levels.
Additionally, the alignment parameters have a direct impact on the angular distribution and polarization of spectral lines. It can be predicted that alterations to atomic number $Z$ will have minimal effect on the angular distribution and polarization of 3$C$ and 3$D$ lines, whereas 3$E$ and 3$F$ lines are notably influenced by $Z$. Exciting phenomena are anticipated to take place near $Z$=51.
3.3 Angular distribution and linear polarization
Having the alignment parameters available, they can be used to calculate the angular distributions and polarization of the spectral lines. Figure 3 shows the currently calculated typical angular distributions of 3$C$, 3$D$, 3$E$, and 3$F$ line emissions for neon-like isoelectronic sequence ions In$^{39+}$, Sn$^{40+}$, Sb$^{41+}$, and Te$^{42+}$. The horizontal axis $\theta$ is the angle between the photon emission direction and the electron beam propagation direction, and the vertical axis is the angular distribution. It can be seen from the figure that the emission of the 3$C$, 3$D$ and 3$F$ lines is mainly concentrated in the direction of 90$^\circ$ with respect to the electron beam direction, while the 3$E$ line radiate dominantly in the forward directions (0$^\circ$ to the electron beam direction) and back directions (180$^\circ$ to the electron beam direction) for all ions considered. Furthermore, 3$C$ and 3$D$ lines have almost identical angular distributions over the entire atomic number range considered, especially for low $Z$ ions.
Fig. 3. The typical angular distribution of the 3$C$, 3$D$, 3$E$, and 3$F$ lines of Ne-like In$^{39+}$, Sn$^{40+}$, Sb$^{41+}$, and Te$^{42+}$ ions.
To exhibit the $Z$-dependent angular distribution obviously, we display the angular distribution coefficient $W(90^\circ )$ at the direction of 90$^\circ$ in Fig. 4. This coefficient, with a value of 1, indicates isotropic x-ray emission, whereas there is an anisotropic emission of x-ray lines when the coefficient deviates from 1. It can be found that the angular behavior of both 3$C$ and 3$D$ lines is not sensitive to the atomic number $Z$. In contrast, the angular distribution of 3$E$ and 3$F$ lines have a clear $Z$-dependence. The anisotropy of the 3$E$ line increases gradually as the atomic number increases from 26 to 46, decrease sharply from 46 to 50, and then increases again in the range from 51 to 52, followed by another decrease after $Z$ exceeds 52. The 3$F$ line exhibits decreasing anisotropic behaviour across the entire atomic number range studied, except for an abrupt change observed around $Z$=51. For both 3$E$ and 3$F$ lines, their angular distribution show an abrupt change around $Z$=51.
Fig. 4. The angular distribution coefficient $W(90^\circ )$ of the 3$C$, 3$D$, 3$E$, and 3$F$ lines in the direction perpendicular to the impact electron beam.
The polarization of the spectral lines can also be obtained after determining the alignment parameters. We calculated the degree of linear polarization of the 3$C$, 3$D$, 3$E$, and 3$F$ lines emitted by the neon-like isoelectronic sequence ions with atomic number range from 26 to 57 following electron impact excitation. The results are presented in Fig. 5 with the available theoretical result from Wu et al. [17] The figure reveals that, as expected, the polarization of the 3$C$ and 3$D$ lines remains mostly constant as the atomic number $Z$ increases. However, the polarization of the 3$E$ and 3$F$ lines have remarkable $Z$ dependence. For 3$F$ line, it decreases slowly as the atomic number $Z$ increases up to $Z$ = 51, where it drops abruptly and then rises immediately again. Finally, it decreases with a more significant trend as $Z$ increases. For 3$E$ line, the polarization initially decreases as the atomic number $Z$ increases, but it increases at $Z$ = 48 before suddenly falling back at $Z$ = 51 and ultimately increasing with the growing atomic number. It has been observed that in the relatively high-$Z$ region, the trend of change in the polarization of the 3$E$ line is opposite to that of the 3$F$ line. Additionally, both of their inflection points are situated near $Z$ = 51. As mentioned above, the polarization of the 3$E$ and 3$F$ lines is highly dependent on the atomic number due to the significant change in the alignment of the excited levels of their corresponding transitions. This holds true for the angular distribution as well. Thus, it can be seen that the Z-dependence of the polarization is highly consistent with that of the angular distribution coefficients in the $90^\circ$ direction.
Fig. 5. The degree of linear polarization of the 3$C$, 3$D$, 3$E$, and 3$F$ lines. Other available results for the linear polarization of the 3$C$ line of Fe$^{16+}$ ion from Wu et al. [17] are also shown for comparison.
3.4 Mixing coefficient
While the alignment parameters of excited levels determine the angular distribution and polarization of emitted lines, the $Z$-dependence of the angular distribution and polarization of both 3$E$ and 3$F$ lines change abruptly at $Z$=51, suggesting a possible correlation between 3$E$ and 3$F$ lines.
To examine what factors cause this correlation, we calculated the configuration mixing coefficient of the excited levels corresponding to the 3$E$ and 3$F$ lines, which are presented in Fig. 6 and Fig. 7, respectively. The square of the configuration mixing coefficient represents the contribution of each configuration component to the corresponding energy level. One can see from Fig. 6 that $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ and $[(2p^5)_{3/2}3d_{5/2}]_{J=1}$ are the two primary components contributing to the upper energy level of line 3$E$. Moreover, the contribution of the $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ component keeps increasing along with the atomic number $Z$, while the $[(2p^5)_{3/2}3d_{5/2}]_{J =1}$ component contributes less and less. At $Z$=51, the contribution of $[(2p^5)_{1/2}3s]_{J=1}$ sharply increases from almost 0 to 14.1%, which causes the contribution of $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ drop significantly. Examining the upper level of line 3$F$, as shown in Fig. 7, we see that in the region where $Z\leq$ 49, the 3$F$ energy level consists predominantly of configuration component $[(2p^5)_{1/2}3s]_{J=1}$. The contribution of other components is almost negligible, and there is almost no obvious change with increasing atomic number. Nevertheless, as the atomic number reaches $Z$ = 51, the proportion of $[(2p^5)_{1/2}3s]_{J=1}$ reduces dramatically, while the contribution of $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ suddenly increases significantly, from almost 0 to 15.8%, as does the contribution of $[(2p^5)_{1/2}3s]_{J=1}$ to the upper level of line 3$E$. As the atomic number $Z$ continues to increase, the contribution of $[(2p^5)_{1/2}3s]_{J=1}$ falls sharply from 96.2% at $Z$ = 52 to 63.7% at $Z$ = 55 and then rises suddenly to 96.4% at $Z$ = 57, showing another abrupt change around $Z$ = 55. Correspondingly, the contribution of $[(2p^5)_{3/2}3d_{5/2}]_{J=1}$ rises sharply from 1.9% at $Z$ = 52 to 33.6% at $Z$ = 55 and then falls rapidly to 3.3% at $Z$ = 57.
Fig. 6. Square of mixing coefficients of main configuration components involved in the upper level of 3$E$ line, which gives rise to the contribution (in percentage) of the corresponding components.
By combining Figs. 6 and 7, a strong configuration mixing between the upper level $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ of line 3$E$ and the upper level $[(2p^5)_{1/2}3s]_{J=1}$ of line 3$F$ is observed when $Z$ = 51. The contribution of configuration component $[(2p^5)_{1/2}3s]_{J=1}$ to the upper level of line 3$E$ is considerably enhanced. Simultaneously, the contribution of component $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ to the upper level of line 3$F$ is also significantly enhanced. This strong configuration mixing is the cause of the abrupt change of $Z$-dependence of the polarization and angular distribution of the 3$E$ and 3$F$ lines around $Z$ = 51. To be specific, at $Z$=51, the strong configuration mixing between the upper level of line 3$E$ ($[(2p^5)_{3/2}3d_{3/2}]_{J=1}$) and the upper level of line 3$F$ ($[(2p^5)_{1/2}3s]_{J=1}$) alter the population of the magnetic sublevels corresponding to 3$E$ and 3$F$ lines. The alignment parameters are then adjusted and affects the polarization and angular distribution of 3$E$ and 3$F$ lines finally.
What is the origin of the strong configuration mixing at $Z$=51? As introduced in section 1, two or more configurations with almost identical energy can lead to strong configuration mixing. The calculated energies of 3E and 3F are plotted in Fig. 8. It can be clearly seen that the two levels get close to each other but repel around $Z$=51. This shows that they are coupled by strong configuration mixing.
Fig. 8. The calculated energies of levels $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ and $[(2p^5)_{1/2}3s]_{J=1}$, denoted as 3E and 3F, respectively.
Besides the configuration mixing between the upper level of line 3$E$ ($[(2p^5)_{3/2}3d_{3/2}]_{J=1}$) and the upper level of line 3$F$ ($[(2p^5)_{1/2}3s]_{J=1}$) at $Z$=51, another strong mixing between the upper level of line 3$D$ ($[(2p^5)_{3/2}3d_{5/2}]_{J=1}$) and the upper level of line 3$F$ ($[(2p^5)_{1/2}3s]_{J=1}$) can be found when $Z$=55. The latter leads to an abrupt change in the $3C/3D$ intensity ratio at $Z$ = 55, which we discussed in detail in our previous work [18].
4. Conclusion
In summary, the electron-impact excitation of neon-like isoelectronic sequence ions from their ground state to the excited levels as well as the subsequent radiative decays have been investigated within the framework of configuration-interaction plus many-body perturbation theory method and density matrix theory. Special attention has been paid to the angular and polarization behaviors of the spectral line emissions. Also, the effect of the configuration mixing on the angular distribution and polarization of spectral lines has been discussed in details. It has been shown that both the angular distribution and linear polarization of the 3$E$ and 3$F$ lines are sensitive to the configuration-mixing effect. To be specific, the mixing of configuration $[(2p^5)_{3/2}3d_{3/2}]_{J=1}$ with configuration $[(2p^5)_{1/2}3s]_{J=1}$ is very strong at around $Z=51$, which lead the upper levels of 3$E$ and 3$F$ lines less aligned, resulting in the decrease of anisotropy and polarization of these two lines finally. Based on such sensitivity, the angular distribution and linear polarization can be employed to study the configuration mixing effect.
Funding
National Natural Science Foundation of China (12304276, 12075163, 12074352, 12374234, 12304275); Dazhou Science and Technology Program (22ZDYF0047); Scientific Research Foundation of Sichuan University of Arts and Sciences (2022GCC13Z).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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