Electron-impact double and triple ionization of Se³⁺

Merger of two neutron stars or neutron star with black hole leads to production of heavier than iron elements through the rapid neutron capture (r-proces) nucleosynthesis [1]. Modeling based on the fully general relativistic approach with neutrino transport simulation demonstrates a large abundance for elements with atomic mass number A ∼ 90 [2]. Selenium is one of the elements produced in the merging process with the large probability. Also, this element was detected in various astrophysical nebulae and metal poor stars [3–8]. Selenium is potentially useful in studies for nucleosynthetic models of stellar populations. Accurate atomic data for processes such as photoionization, autoionization, dielectronic recombination as well as electron-impact excitation and ionization are needed for these studies. Ion fractions in the plasma have to be known since the emission line fluxes strongly depend on the ion concentration. The ionization and recombination processes determine charge-state distribution in plasma. The electron-impact single ionization (SI) is the strongest among the ionization processes. Nevertheless, the multiple ionization can play a significant role in various environments with an abundance of energetic electrons [9, 10]. The study of multiple ionization processes is quite complicated [11, 12] as one has to deal at least with four body Coulomb problem. Database of electron-impact multiple-ionization cross sections for all elements from He–Zn was produced using the proposed semiempirical formulae [13]. However, selenium was not presented.

The first and to date the only experimental studies on electron-impact single, double and triple ionization (TI) for Se ions were performed at the Multicharged Ion Research Facility of the University of Nevada in Reno using the dynamic-crossed-beams technique [14]. Absolute single to TI cross sections for the Se²⁺ ions from their ionization thresholds up to 500 eV [15] and for the Se³⁺ ions from their ionization thresholds up to 1 keV [16] were measured. For comparison with the experimental values, the Lotz semi-empirical formula for the direct ionization [17] was applied. Significant differences were observed, which implied large contribution from the indirect processes. For double ionization (DI) of Se²⁺ [15], a least-squares fit of the experimental data was obtained by using the semi-empirical formula [18].

Ab initio investigation for electron-impact single and DI cross sections for the ground configurations of the Se²⁺ and Se³⁺ ions was performed using the semi-relativistic configuration-average distorted-wave (CADW) method [19]. Recently, electron-impact SI of the Se³⁺ ion was studied by performing level-to-level calculations [20]. The theoretical study for electron-impact TI of the Se²⁺ ion was also presented [21]. It was demonstrated that the TI process is formed by direct double ionization with subsequent autoionization (DDI-AI) as well as Auger cascade following creation of the vacancy in the 3p shell. This study provided a good agreement with experimental measurements for TI of the Se²⁺ ion. Unfortunately, no theoretical investigations are so far available for TI process in the Se³⁺ ions.

The aim of this paper is to investigate electron-impact DI and TI processes for the Se³⁺ ion by performing level-to-level calculations. The influence of ionization-autoionization (IA), excitation double-autoionization (EDA), DDI, and resonant-excitation triple-autoionization (RETA) processes to DI is analyzed. Similarly as in the case of the Se²⁺ ion [21], in this paper, the electron-impact TI of Se³⁺ is studied as a result of (i) the DDI-AI process and (ii) Auger cascade following SI of the inner shell of the Se³⁺ ion. In addition, influence of the correlation effects to the formation of the Se⁶⁺ ions after the 3p shell ionization is investigated. The DDI process is studied using a few-step approach [22] which involves ionization–ionization (II), excitation-ionization–ionization (EII), and ionization-excitation-ionization (IEI) processes. The contribution from different atomic shells can be important for modeling non-equilibrium plasma. Therefore, we present cross sections and rates for the different processes and different shells.

The rest of the paper is structured as follows. An overview of the theoretical approach used to investigate double and TI cross sections is given in section 2; in section 3, double and TI cross sections for the Se³⁺ ion are presented and compared with experimental measurements; a brief summary with some final conclusions and directions for future work are provided in section 4.

Energy levels, radiative and Auger transition probabilities, electron-impact excitation and ionization cross sections are studied using the Flexible Atomic Code [23] which implements the Dirac–Fock–Slater approach. Continuum orbitals of incident and scattered electrons are evaluated in the potential of the ionized ion since this approach provides a better agreement with experimental measurements. Electron-impact excitation and SI cross sections are studied using the distorted-wave (DW) approximation. Single configuration approximation is used in calculations except the 3p shell ionization.

Direct and indirect DI processes contribute to the total DI cross sections for the electron-impact ionization:

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{DI}}(\varepsilon )={\sigma }_{{if}}^{\mathrm{DDI}}(\varepsilon )+{\sigma }_{{if}}^{\mathrm{IDI}}(\varepsilon ),\end{eqnarray} \tag{ 1 }$$

here ${\sigma }_{{if}}^{\mathrm{DDI}}(\varepsilon )$ is the cross section of DDI from the initial level i of Se³⁺ to the final level f of Se⁵⁺ at the incident electron energy ε. Two- and three-step processes (II, IEI, and EII) were used before to describe DDI [21, 22, 24]. It should be noted that the three-step processes played the significant role in explaining TI in the Se²⁺ ion [21]. In a few step approach, the first electron-impact ionization process produces an ejected electron. The atomic system can also be excited before the ionization. Thus, the scattered or/and ejected electrons participate in the further step or steps and interact with the remaining bound electrons of the atomic system. Two limiting cases of electron energy distribution after the first ionization process are investigated in this work. In one case, the scattered and ejected electrons share the excess energy equally. In other case, one of the electrons takes all the excess energy.

The equation for the DDI(II) process from the level i to the level f is expressed by

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{DDI}(\mathrm{II})}(\varepsilon )=\displaystyle \sum _{j}{\sigma }_{{ij}}^{\mathrm{CI}}(\varepsilon )\displaystyle \frac{{\sigma }_{{jf}}^{\mathrm{CI}}({\varepsilon }_{1})}{4\pi {\bar{R}}_{{nl}}^{2}},\end{eqnarray} \tag{ 2 }$$

here ε₁ is the energy of the scattered or ejected electron. The probability of the second ionization by electron impact equals to $\tfrac{{\sigma }_{{jf}}^{\mathrm{CI}}({\varepsilon }_{1})}{4\pi {\bar{R}}_{{nl}}^{2}}$, ${\bar{R}}_{{nl}}$ is the mean distance from the nucleus for the electrons in the nl shell from which the second electron is kicked off. ${\sigma }_{{jf}}^{\mathrm{CI}}({\varepsilon }_{1})$ is the electron-impact collisional ionization (CI) cross sections from the intermediate level j to the final level f.

The DDI process for the EII path can be written as

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{DDI}(\mathrm{EII})}(\varepsilon )=\displaystyle \sum _{{kj}}{\sigma }_{{ik}}^{\mathrm{CE}}(\varepsilon )\displaystyle \frac{{\sigma }_{{kj}}^{\mathrm{CI}}({\varepsilon }_{1})}{4\pi {\bar{R}}_{{nl}}^{2}}\displaystyle \frac{{\sigma }_{{jf}}^{\mathrm{CI}}({\varepsilon }_{2})}{4\pi {\bar{R}}_{n^{\prime} l^{\prime} }^{2}},\end{eqnarray} \tag{ 3 }$$

where ${\varepsilon }_{1}=\varepsilon -{\rm{\Delta }}{E}_{{ik}}$, ${\rm{\Delta }}{E}_{{ik}}$ is a transition energy and ε₂ is the energy of the scattered or ejected electron. ${\sigma }_{{ik}}^{\mathrm{CE}}(\varepsilon )$ is the cross section for the collisional excitation from the level i to the level k.

The DDI cross sections for the IEI part are given by

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{DDI}(\mathrm{IEI})}(\varepsilon )=\displaystyle \sum _{{kj}}{\sigma }_{{ik}}^{\mathrm{CI}}(\varepsilon )\displaystyle \frac{{\sigma }_{{kj}}^{\mathrm{CE}}({\varepsilon }_{1})}{4\pi {\bar{R}}_{{nl}}^{2}}\displaystyle \frac{{\sigma }_{{jf}}^{\mathrm{CI}}({\varepsilon }_{2})}{4\pi {\bar{R}}_{n^{\prime} l^{\prime} }^{2}},\end{eqnarray} \tag{ 4 }$$

here $\tfrac{{\sigma }_{{kj}}^{\mathrm{CE}}({\varepsilon }_{1})}{4\pi {\bar{R}}_{{nl}}^{2}}$ is the excitation probability from the level k to the level j.

The indirect DI processes, studied in this work, corresponds to IA, EDA as well as RETA:

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{IDI}}(\varepsilon )={\sigma }_{{if}}^{\mathrm{IA}}(\varepsilon )+{\sigma }_{{if}}^{\mathrm{EDA}}(\varepsilon )+{\sigma }_{{if}}^{\mathrm{RETA}}(\varepsilon ).\end{eqnarray} \tag{ 5 }$$

The IA process is initiated by electron-impact ionization from the level i to the intermediate level j of the Se⁴⁺ ion:

$$\begin{eqnarray}&&{\sigma }_{if}^{{\rm{I}}{\rm{A}}}(\varepsilon )=\displaystyle \sum _{j}{\sigma }_{ij}^{{\rm{C}}{\rm{I}}}(\varepsilon ){B}_{jf}^{a}.\end{eqnarray} \tag{ 6 }$$

The autoionization branching ratio is expressed by the equation

$$\begin{eqnarray}&&{B}_{{jf}}^{a}=\displaystyle \frac{{A}_{{jf}}^{a}}{{\displaystyle \sum }_{m}{A}_{{jm}}^{a}+{\displaystyle \sum }_{n}{A}_{{jn}}^{r}},\end{eqnarray} \tag{ 7 }$$

where A^a and A^r are the Auger and radiative transition probabilities from the level j, respectively.

The cross sections produced by the EDA process are given by

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{EDA}}(\varepsilon )=\displaystyle \sum _{{jm}}{\sigma }_{{ij}}^{\mathrm{CE}}(\varepsilon ){B}_{{jm}}^{a}{B}_{{mf}}^{a},\end{eqnarray} \tag{ 8 }$$

where summation is performed over the autoionizing levels j of the Se³⁺ ion and the autoionizing levels m of the Se⁴⁺ ion. Excitations from the 3p shell up to shells with the principal quantum number $n\leqslant 15$ and orbital quantum number $l\leqslant 5$ are analyzed.

The RETA process starts with capture of the incident electron by the atomic system and promotion of the bound shell electron to the higher shell. The formed autoionizing state of the Se²⁺ ion decays through radiative and Auger cascades leading to ions in various ionization stages. The decay which ends in the Se⁵⁺ ion contributes to the DI. The cross sections for the RETA process are expressed by the equation:

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{RETA}}(\varepsilon )=\displaystyle \sum _{{lkj}}{\sigma }_{{il}}^{\mathrm{DC}}(\varepsilon ){B}_{{lk}}^{a}{B}_{{kj}}^{a}{B}_{{jf}}^{a},\end{eqnarray} \tag{ 9 }$$

where summation is performed over the autoionizing levels l of the Se²⁺ ion, the levels k of Se³⁺, and the levels j of Se⁴⁺. ${\sigma }_{{il}}^{\mathrm{DC}}(\varepsilon )$ is the cross section of the dielectronic capture (DC) process [25].

The first step of the RETA process corresponds to DC with excitation from the 3p shell:

$$\begin{eqnarray}&&3{p}^{6}3{d}^{10}4{s}^{2}4p+{e}^{-}\to 3{p}^{5}3{d}^{10}4{s}^{2}\,4p\,{n}_{1}{l}_{1}{n}_{2}{l}_{2}.\end{eqnarray} \tag{ 10 }$$

The autoionizing states with the principal and orbital quantum numbers n₁ = 4, l₁ < 4 and ${n}_{2}\leqslant 12$, l₂ < 5 are considered. Capture to the higher shells is not studied as their contribution would be much smaller compared to the presented one. The DC process which includes the 3s shell is not investigated as its contribution to the DI cross sections is much smaller compared to the excitation from the 3p shell. Furthermore, DC which involves the excitation from the 3d shell results only in SI which is not considered in this work.

Recently, it has been demonstrated for Se²⁺ that the autoionizing levels of the Se⁴⁺ ion produced by the DDI process can decay further to Se⁵⁺ resulting in TI [21]. The same investigation is performed here for the Se³⁺ ion. The TI cross sections from the II, IEI, and EII processes of DDI with subsequent autoionization are obtained from equations

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{DDI}(\mathrm{II})-\mathrm{AI}}(\varepsilon )=\displaystyle \sum _{j}{\sigma }_{{ij}}^{\mathrm{DDI}(\mathrm{II})}(\varepsilon ){B}_{{jf}}^{a},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{DDI}(\mathrm{EII})-\mathrm{AI}}(\varepsilon )=\displaystyle \sum _{j}{\sigma }_{{ij}}^{\mathrm{DDI}(\mathrm{EII})}(\varepsilon ){B}_{{jf}}^{a},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{DDI}(\mathrm{IEI})-\mathrm{AI}}(\varepsilon )=\displaystyle \sum _{j}{\sigma }_{{ij}}^{\mathrm{DDI}(\mathrm{IEI})}(\varepsilon ){B}_{{jf}}^{a}.\end{eqnarray} \tag{ 13 }$$

Here, summation is performed over the autoionizing levels j of the Se⁵⁺ ion. The level f belongs to the Se⁶⁺ ion.

The ionization from the inner shell with double autoionization (IDA) also leads to TI. The cross sections of the IDA process are given by

$$\begin{eqnarray}&&{\sigma }_{{if}}^{\mathrm{IDA}}(\varepsilon )=\displaystyle \sum _{{kj}}{\sigma }_{{ik}}^{\mathrm{CI}}(\varepsilon ){B}_{{kj}}^{a}{B}_{{jf}}^{a}.\end{eqnarray} \tag{ 14 }$$

The Maxwellian rate coefficients are calculated using the electron-impact ionization cross sections

$$\begin{eqnarray}&&{\alpha }_{{ij}}({T}_{e})={\left(\displaystyle \frac{1}{{k}_{B}{T}_{e}}\right)}^{\tfrac{3}{2}}{\left(\displaystyle \frac{8}{{m}_{e}\pi }\right)}^{\tfrac{1}{2}}{\int }_{0}^{\infty }\epsilon \,{\sigma }_{{ij}}(\epsilon )\,\exp \left(-\displaystyle \frac{\epsilon }{{k}_{B}{T}_{e}}\right)d\epsilon .\end{eqnarray} \tag{ 15 }$$

Here k_B is the Boltzmann constant, m_e is the electron mass, and σ_ij() is the cross section for the transition from the level i to the level j.

The configuration interaction strength [26–28] is used to define the admixed configurations showing the strongest mixing with the Se⁴⁺ $3{s}^{2}3{p}^{5}3{d}^{10}4{s}^{2}4p$ configuration. The single and double electron excitations from the 3s, 3p, 3d, 4s, and 4p shells up to shells with the principal quantum number n = 8 for single and n = 5 for double excitations and orbital quantum number l < n are analyzed for the configuration to produce the basis of interacting configurations. The configuration interaction strengths are calculated between the considered configuration and several hundreds of configurations from the produced basis. The same approach was used previously to study electric [29] and magnetic [30–32] dipole transitions as well as Auger cascades [33–35].

Energy levels of the main configurations describing double and TI for the Se³⁺ ion as well as single, double, and TI thresholds are presented in figure 1. The SI threshold for the ground state of the Se³⁺ ion equals to 42.35 eV which is in a good agreement with the NIST value of 42.95 eV [36]. The theoretical SI threshold is within the error bars of the experimental value determined to be 42.2 ± 1.8 eV [16]. The DI threshold corresponds to 108.21 eV while the NIST provided value amounts to 111.25 eV. The theoretical TI threshold of 189.33 eV is slightly below the NIST value of 193.08 eV.

Zoom In Zoom Out Reset image size

The contribution from the different processes to DI of the Se³⁺ ion is presented in the figure 2. All calculations correspond to data obtained in the single configuration approach. The DDI process dominates at the peak of the DI cross sections. To demonstrate the influence of the DDI-AI process, the DDI contribution is not corrected by possible decay of populated levels of Se⁵⁺ through Auger transitions to the Se⁶⁺ ion. It should be noted that two- and three-step processes are presented in the study. Only the two-step process in Se³⁺ was analyzed before using the CADW approach [19].

Zoom In Zoom Out Reset image size

The ionization from the 3p shell with subsequent autoionization provides the largest influence to the DI cross sections at the higher energies of the incident electron (figure 2). The threshold of the process is 164.48 eV. The decay of the produced $3{p}^{5}3{d}^{10}4{s}^{2}4p$ configuration leads only to formation of the Se⁵⁺ ion. The same process was also studied previously using the CADW approach [19]. Our values are slightly below the CADW calculations. At the peak our cross section reaches 1.44 Mb while the CADW produces about 1.54 Mb [19]. The similar tendency has been observed also for other ions [37–39]. The possible reason of the differences can be related to the fact that continuum orbitals of the incident and scattered electrons are studied in the potential of the ionizing ion and those of the ejected electrons are obtained in the potential of the ionized ion for the CADW data [19]. Our CI investigation is performed in the potential of the ionized ion.

The large contribution to the DI cross sections is observed from the excitations of the 3p shell up to shells with the principal quantum number $n\leqslant 15$ and orbital quantum number $l\leqslant 5$. The produced configurations decay through Auger cascade. The contribution of the excitations from the 3p shell to shells with the principal quantum number n = 4 amounts to about 50% of the total value of the process. It should be noted that influence of this process to the DI cross sections of the Se³⁺ has not been studied before. It is seen from figure 2 that the process contributes about 0.5 Mb to the total DI cross sections at their peak value.

In addition to the three above mentioned processes, we have estimated the contribution of DC with subsequent Auger cascade resulting in Se⁵⁺. This means that triple-autoionization is studied in this case. The influence of this process to electron-impact DI cross sections also has not been investigated before for the Se³⁺ ion.

It can be seen from figure 2 that the theoretical DI cross sections are well below the experimental values at the lower energies of the incident electron while the experimental data are overestimated at the higher energies. However, the presented DDI cross sections do not include the possible decay of the populated levels to the next ionization stage through the Auger transitions.

Figure 3 demonstrates contribution from the two- and three-step processes to the DDI cross sections. The II path produces about 57% of the total cross sections for the case when one of the ejected or scattered electrons takes all the excess energy after the first ionization process. The strongest branch of the II process corresponds to the sequential ionization from the 3d shell. This leads to the Se⁵⁺ $3{d}^{8}4{s}^{2}4p$ configuration with the energy levels above the TI threshold. At the peak of the II cross sections the population of the energy levels for the configuration amounts to 69%. Unfortunately, this configuration cannot decay through Auger transitions to the Se⁶⁺ $3{d}^{9}4s$ and $3{d}^{9}4p$ configurations. The energy levels of these configurations are above the ones of the $3{d}^{8}4{s}^{2}4p$ configuration. Other configurations produced by the II process at the peak of the II cross sections are Se⁵⁺ $3{d}^{9}4s4p$ (17%), $3{d}^{9}4{s}^{2}$ (14%), $3{d}^{10}4s$ (0.5%), $3{d}^{10}4p$ (0.2%). However, all these configurations have energy levels below the TI threshold. Therefore, the II process does not provide contribution to the TI of Se³⁺.

Zoom In Zoom Out Reset image size

Surprisingly, the EII process plays a significant role in the formation of the DDI cross sections (figure 3). It constitutes about 34% of the total DDI cross sections at the peak value. The input from the IEI process amounts to 8%.

For the case when electrons share the excess energy, the DDI cross sections at the lower energies of the incident electron are below the DDI cross sections obtained when one of the electrons takes all the excess energy (figure 3). The situation is different for the higher energies. The cross sections for the case when the electrons share the excess energy are higher compared to the case when one of the electrons takes all the excess energy. Previous studies demonstrated that the situation when one of the electrons takes all the excess energy provides better agreement with measurements for the DDI cross sections at the lower energies [22]. On the other hand, the better agreement with experimental data at the higher energies is produced when the scattered and ejected electrons share the excess energy equally.

Figure 4 presents the DDI cross sections when Auger decay of the excited levels of Se⁵⁺ to the next ionization stage is taken into account. Contribution of the different DDI processes drastically changes compared to the situation shown in figure 3. The cross sections produced by the II part amounts to about 91% of the total data. Contribution from the EII process drastically decreases as many produced excited levels of the Se⁵⁺ ion decay to Se⁶⁺. The levels of the Se⁵⁺ $3{d}^{8}4s4{p}^{2}$ configuration has the relative population of 55% at the peak of the EII cross sections. The configuration is produced by the $4s\to 4p$ excitation with the subsequent double CI from the 3d shell. The followed decay through the Auger transitions leads to the Se⁶⁺ $3{d}^{9}4s$ (54%) and $3{d}^{9}4p$ (1%). There are some levels of the $3{d}^{8}4s4{p}^{2}$ configuration below the levels of the $3{d}^{9}4s$ and $3{d}^{9}4p$ configuration (figure 1). However, the population of these levels is negligible when compared to the population transfer to the next ionization stage.

Zoom In Zoom Out Reset image size

The $3{d}^{8}4{s}^{2}4d$ configuration has the second largest population of 22% produced by the EII process at the peak of cross sections. The $4p\to 4d$ excitation with the subsequent double CI by electron impact from the 3d shell determines the formation of the configuration. The Auger transitions from the levels of the configuration populate the $3{d}^{9}4s$ configuration.

The Se⁴⁺ $3{p}^{5}3{d}^{10}4{s}^{2}4p$ configuration produced by the electron impact ionization from the 3p shell has energy levels in the range from 209.67 to 218.57 eV which is above the TI threshold for Se³⁺. The Auger transitions to the $3{d}^{8}4{s}^{2}4p$ configuration from the $3{p}^{5}3{d}^{10}4{s}^{2}4p$ are forbidden since the energy levels of the final configuration are above the energy levels of the initial configuration. On the other hand, the energy levels of the Se⁵⁺ $3{d}^{9}4{s}^{2}$, $3{d}^{9}4s4p$, $3{d}^{10}4p$, and $3{d}^{10}4s$ configurations are below the TI threshold. Therefore, the electron impact ionization from the 3p shell does not result in Se⁶⁺. However, the situation drastically changes when correlation effects are taken into account. Configuration interaction strength is used to determine the admixed configurations for the Se⁴⁺ $3{p}^{5}3{d}^{10}4{s}^{2}4p$ configuration. The basis of interacting configurations includes the $3{d}^{8}4{s}^{2}4p\ 4f$, $3{d}^{8}4{s}^{2}4p\ 5f$, $3{d}^{8}4{s}^{2}4p\ 5p$, $3{p}^{5}3{d}^{10}4{p}^{3}$, $3{p}^{5}3{d}^{10}4s\ 4p\ 4d$, $3{d}^{8}4{s}^{2}4p\ 6f$, $3{d}^{8}4{s}^{2}4p\ 7f$, $3{d}^{8}4{s}^{2}4p\ 6p$, $3{d}^{8}4{s}^{2}4p\ 8f$, $3{d}^{8}4{s}^{2}4p\ 9f$, $3{d}^{9}4s\ 4{p}^{2}$, $3{d}^{9}4s\ 4{p}^{2}$, $3{d}^{8}4{s}^{2}4{p}^{2}$, $3{p}^{5}3{d}^{10}4s\ 4p\ 5d$, and $3s3{p}^{6}3{d}^{10}4s\ 4{p}^{2}$ configurations. This results in diminishing of the total cross sections for the electron impact DI produced by the 3p shell ionization with subsequent autoionization. The change by 9% is observed for the cross sections at the peak value.

The total DI cross sections with the corrected DDI and IA values are presented in figure 5. The partial and total cross sections for the DI process are also listed in tables 1–8. The better agreement with measurements is obtained at the peak of the cross sections in this case. Calculations are slightly above experimental error bars at the higher energies. Unfortunately, theoretical values are still well below the measurements at the lower energies of the incident electron. The similar result was observed in the previous study [19], and the reason of disagreements is still unclear.

Zoom In Zoom Out Reset image size

The main part of the TI cross sections is provided by the DDI-AI process (figure 6). At the peak value of the total TI, the contribution of the ionization from the 3s and 3p shells is about 12% and 9%, respectively. The contribution from the DDI-AI process diminishes at the higher energies. The theoretical values are above measurements at the peak of the total cross sections. The experimental data are slightly above the calculations at the lower and higher energies of the incident electron. A disagreement among the theoretical and experimental values can be related to the correlation effects that can be important for the DDI process. However, these effects would require a separate study. The partial and total cross sections for the TI process are presented in tables 9–14. The corresponding Maxwellian rate coefficients are presented in table 15.

Zoom In Zoom Out Reset image size

The contribution of the three-step DDI processes to the TI cross sections is presented in figure 7. The main role in the formation of TI cross sections plays the EII process with the subsequent autoionization from the excited levels. The process amounts to about 83% for the peak value of DDI-AI for the case when one of the electrons takes all the excess energy after the first ionization process. The obtained results show that the main part of population (79%) at the peak of cross sections for EII resides in the Se⁶⁺ $3{d}^{9}4s$ configuration. The second populated configuration is Se⁶⁺ $3{d}^{9}4p$ (4%). The population of the ground configuration of Se⁶⁺ equals to 3%.

Zoom In Zoom Out Reset image size

Electron impact double and TI cross sections are analyzed for the ground level of the Se³⁺ ion. Influence of the DDI, DC, the 3p shell ionization and excitation processes are studied for DI. It is shown that excitation from the 3p shell with subsequent double autoionization provides large contribution to the formation of the Se⁵⁺ ions. On the other hand, DC which involves excitation from the 3p shell leads to the lowest contribution among the considered processes. This process plays the main role at the lower energies of the incident electron. However, there are still differences among theoretical values and measurements for the DI cross sections in this energy range. Future experiments could help to resolve this problem.

Two- and three-step DDI processes are studied to determine the contribution to the DI data. It is demonstrated that the II process plays the significant role in the formation of the cross sections for the DDI process.

The significant part of the excited levels of Se⁵⁺ produced by the EII and IEI processes decays through Auger transitions to the next ionization stage. This is the main mechanism of the TI for the Se³⁺ ion. Furthermore, these transitions diminish the DI cross sections at the peak and higher energy side where theoretical values overestimate the measurements. The studies which involve only the two-step DDI process would not be able to explain the total formation of the Se⁶⁺ ions from Se³⁺.

The electron-impact ionization from the 3p shell with the subsequent double autoionization contributes to the TI cross sections when the correlation effects are taken into account. This leads to reduction of the DI cross sections produced by the electron-impact ionization from the 3p shell with the subsequent single autoionization at the peak and higher energy side.

The electron-impact ionization from the 3s shell leading to Auger cascade from the autoionizing levels produces the Se⁶⁺ ions. Good agreement with experimental cross sections for the TI process is found when contributions from the DDI-AI as well as electron-impact ionization from the 3s and 3p shells are taken into account. Several obtained discrepancies can be explained by the importance of correlation effects for the DDI-AI process.

Finally, the DW DDI calculations for atom or singly charged ion against a very large scale non-perturbative close-coupling calculation

Part of computations was performed on resources at the High Performance Computing Center HPC Saulėtekis in Vilnius University Faculty of Physics.