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Sci Rep. 2017; 7: 11589.

Published online 2017 September 14. doi: 10.1038/s41598-017-11823-8

PMCID: PMC5599530

Rui Jin,^{1,}^{2} Xiao-Ying Han,^{3} Xiang Gao,^{}^{2} De-ling Zeng,^{2,}^{4} and Jia-Ming Li^{1,}^{4,}^{5}

Xiang Gao, Email: nc.ca.crsc@oagx.

Received 2017 May 30; Accepted 2017 August 30.

Copyright © The Author(s) 2017

An extended atomic data base with sufficiently high precision is required in astrophysics studies and the energy researches. For example, there are “infinite” energy levels in discrete energy region as well as overlapping resonances in autoionization region. We show in this paper the merits of our relativistic eigenchannel R-matrix method R-R-Eigen based on the analytical continuation properties of scattering matrices for the calculations of the energy levels, overlapping resonances and the related transitions. Using Ne^{+} as an illustration example, the scattering matrices of Ne^{+} in both discrete and continuum energy regions are calculated by our R-R-Eigen method directly. Based on our proposed projected high dimensional quantum-defect graph (symmetrized), one can readily determine the accuracies of the calculated scattering matrices using the experimental energy levels in a systematical way. The calculated resonant photoionization cross sections in the autoionization region are in excellent agreement with the benchmark high resolution experiments. With the scattering matrices checked/calibrated against spectroscopy data in both discrete and continuum energy regions, the relevant dynamical processes should be calculated with adequate accuracies. It should then satisfy the needs of the astrophysical and energy researches.

In astrophysical and fusion energy researches, the atomic data such as energy levels, collision cross sections, photoionization cross sections, dielectronic recombination rates and transition rates of atoms with sufficient precision are needed^{1–8}. For instance, the optical recombination lines (ORLs) and collisional excitation lines (CELs) are both used to determine the abundances of metal elements (such as Oxygen) in Planetary nebular (PNe)^{1–3}. While the metal abundances determined from the ORLs are much higher than that determined by CELs sometimes^{1}, showing the strong dependence on the precision of related atomic data, especially the dielectronic recombination rates with appropriate cascading correction. It is generally accepted that the R-matrix type methods are good candidates to obtain the required atomic data^{9–15}. But for the conventional R-matrix method: 1) it’s hard to assure the accuracies for each level/resonance^{16, 17}; 2) a very fine energy grid is needed to guarantee all the levels/resonances without missing any lines; 3) it’s not trivial to determine and assign all the levels and resonances precisely.

Based on the analytical continuation properties of the short range scattering matrices, there exist intimate relations between atomic energy levels and the related electron-ion collision processes^{18–33}. According to this property, we have proposed a scenario to provide such large scale atomic data with enough physical precisions which can be comparable with spectroscopic accuracies^{18–23}. In our scenario, the short range scattering matrices (i.e., physical parameters in multi-channel quantum defect theory (MQDT)^{24–33} as well as corresponding wave functions in both bound and continuum energy regions can be calculated directly with high accuracy by our recent developed codes, i.e., R-Eigen code (Eigen-channel based on non-relativistic R-matrix method) and R-R-Eigen code (Eigen-channel based on relativistic R-matrix method)^{18–23}. By applying the MQDT, one can calculate and predict all energy level positions up to fine-structures in discrete energy regions (i.e., bound state energy regions) without missing any lines “semi-analytically”. On the other hand, the short range scattering matrices in the discrete energy regions can be examined stringently by precise spectroscopic data experimentally. Therefore, accuracies of the MQDT parameters (i.e., scattering matrices) in bound energy regions can be readily ascertained. Through analytical properties of short range scattering matrices, the scattering matrices in continuum energy regions can be obtained with desired accuracies. One can then obtain relevant cross sections for electron-ion collision with similar accuracy. Note that the short range scattering matrices vary smoothly with energy because of their analytical continuation property. Therefore, one only need to calculate the short range scattering matrices in a few sample energy grids over the energy regions of interest, which is one merit of the R-R-Eigen method. The details will be given in the next section.

In present paper, we exhibit the merits of our R-R-Eigen method in the complex overlapping resonances for Ne^{+} photoionization processes (i.e., the inverse process of the dielectronic recombinations). Using all available precision spectroscopic data to calibrate our calculated scattering matrices and the corresponding dipole transition matrix elements in the discrete energy region, the resonant photoionization cross sections are calculated in the autoionization region. They are in excellent agreement with the benchmark high resolution experiments conducted at the synchrotron radiation light source, i.e., the Advanced Light Source (ALS) at Lawrence Berkeley National Laboratory. It should be noted that the conventional R-matrix method results reported therein^{34} are only in a fair agreement with the experiments. The origins of all the overlapping resonances in the experimental energy regions are assigned at the same time. Furthermore, in the calibration processes of the scattering matrices, we proposed a graphical method, i.e., projected high dimensional quantum-defect graph (symmetrized), to compare the theoretical energy levels with all the spectroscopic data in a systematical way readily for general multi-thresholds (more than two) cases. This is an extension of the Lu-Fano plot^{35, 36} valid only for two-thresholds cases and can be applied for any general atoms. With this method, one should be able to provide various accurate atomic data such as photoionization rates, dielectronic recombination rates for any atoms (ions). Hopefully, with all necessary atomic data calculated with the method, the R-R-Eigen method should be indispensable in the study of basic dynamic processes in astrophysics and laboratory plasmas.

The N+1 electron system consisting of an N-electron target atom and an excited electron can be calculated using the R-matrix type method^{9–15}, which has been successfully developed as an ab initio method for treating a variety of dynamic processes in atomic physics. Let’s briefly review the relativistic eigenchannel R-matrix method (i.e., the R-R-Eigen code)^{18}, which mainly differs from the traditional R-matrix method^{9–15} by the definitions of physical (ionization) channels, including the opened channels and some relevant closed channels(i.e. *n*
_{p}). Other closed channels for much higher thresholds with deep negative orbital energies and the *N*+1 bound type configurations are defined as closed computational channels(i.e. *n*
_{com}), which are included in our calculations to assure electron correlations taken into account adequately. More specifically, with *n*
_{p} physical (ionization) channels and *n*
_{com} computational channels for symmetry block with total angular momentum *J* and parity *π*, the *n*
_{p}×*n*
_{p} short-range reaction matrices

1

For the *i*
^{th} physical (ionization) channel associated with the target state in Φ_{i}, the relativistic regular and irregular Coulomb wave functions *f*
_{i}(*r*, *E*), and *g*
_{i}(*r*, *E*), cover the entire energy ranges of one-electron orbitals, i.e., *ε*
_{l}=*E*−*I*
_{i}>−*q*
^{2}/*l*
^{2} (except for *l*=0,

2

where the *i*, *j* denote the physical (ionization) channel indexes and the *U*
_{iα} can be represented by *n*_{p}(*n*_{p} − 1)/2 Euler-type angles *θ*
_{k}
^{27}.

In the Ne^{+}(

In order to demonstrate how to calibrate our results of R-R-Eigen calculations with the available precise spectroscopic data, we start with the eigenchannel wave-functions Ψ_{α} outside the reaction zone (i.e., *r*≥*r*
_{0})^{18, 25–33},

3a

In the discrete energy region for the spectroscopic energy levels, the energy eigen wave-function (also for auto-ionization states) can be expressed as a superposition of eigenchannel wave-functions:

3b

where *A*
_{α} are determined by the asymptotic boundary conditions^{18, 25–33}. The bound state asymptotic boundary conditions (i.e., *N*
_{c}=*n*
_{p}, where all channels are closed) lead to,

4a

with the *N*
_{th} effective principal quantum numbers *ν*
_{i} corresponding to *N*
_{th} number of different thresholds *I*
_{i} and spanning a *N*
_{th}-dimensional space, defined as,

4b

with *q*=2 for the Ne^{+} system. The existence of nontrivial *A*
_{α} requires the vanishing of determinant of coefficient matrix:

5

To illustrate various features in the calibration process clearly, we’ll discuss two methods of the solution of Eqs (^{4b}) and (^{5}) in the following part of the section, i.e., 1) recursive projection method and 2) projected high dimensional quantum-defect graph (symmetrized).

From the geometric view, Eqs (^{4b}) and (^{5}) can be represented as a one-dimension curve and a (N_{th}−1)-dimension surface {*ν*_{i}(*i* = 1, ..., *N*_{th})} in a N_{th} dimensional space respectively. For cases with *N*_{th} > 2, which are somewhat different from the case associated with only two thresholds^{19–21, 25–28, 35, 36}, where the solutions of the Eqs (^{4b}) and (^{5}) can be represented graphically in a plot with the only two effective principal quantum numbers. More specifically, with only two thresholds, there are two sets of effective principal quantum numbers, *n*
_{1} number of *ν*
_{1} corresponding to the first threshold *I*
_{one} and *n*
_{2} number of *ν*
_{2} corresponding to the second threshold *I*
_{two} with *I*
_{one}<*I*
_{two}. In the (*ν*
_{1}, *ν*
_{2}) plot, it is so called the Lu-Fano plot^{35, 36}, where the *ν*
_{2} can be regarded as known variables to scan the energy according to Eq. (^{4b}) and the *ν*
_{1} can be determined by solving the Eq. (^{5}). In such a plot, there will be avoid-crossing curves basically consisting of *n*
_{1} number of “horizontal” curves with *n*
_{2} number of “vertical” resonances. The final solutions are then the crossing points of the curves and the one-dimensional energy curve according to Eq. (^{4b}). With more than two thresholds, in order to semi-analytically and graphically calculate the discrete levels and compare with the experimental spectroscopy levels, the (*N*
_{th}−1)-dimensional surface should be projected onto a two-dimensional plots. According to physical requirements, we can select a pair of any two adjacent thresholds *I*
_{one} and *I*
_{two} with *I*
_{one}<*I*
_{two}. Therefore there are two sets of effective principal quantum numbers: *n*
_{1} number of *ν*
_{i} forming a vector

Let’s return to the Ne^{+} (*J*^{π} = 1/2^{+}) case as an illustration example. There are five effective principal quantum numbers (*ν*_{3P2}, *ν*_{3}_{P1}, *ν*_{3}_{P0}, *ν*_{1}_{D2}, *ν*_{1}_{S0}) associated with five thresholds of Ne^{2+} (2*p*
^{4 3}
*P*
_{2,1,0}, ^{1}
*D*
_{2} and ^{1}
*S*
_{0}). Because of the first two appearing strongly perturbed Rydberg series,

Graphical representation to solve Eq. (^{4b}) and Eq. (^{5}). (**a**) A projected two-dimensional graph . Effective eigen-quantum-defects are shown as one (black) and four (black, blue, violet, green) branches of colored curves with sharp resonances for two-channel **...**

From the point of view of effective eigenchannels, the wavefuncitons of discrete energy levels below the first threshold can also be expressed as the superposition of the effective eigenchannel wavefunctions. From the asymptotic boundary conditions of these discrete energy level wavefunctions, an equation similar to Eq. (^{5}) is derived, which can be solved to obtain all energy levels below the first threshold. More specifically, in the present four effective physical channel region, there are four effective eigen-quantum-defects *τ*_{ρ} shown in Fig. 2(a) and the corresponding 4×4 effective transformation matrices *T*_{i′ρ}
^{18, 27, 28, 33}. The boundary conditions now lead to,

6

Because the effective four physical channels are just associated with

and , we select , and here. Then and vectors will simply become two scalars of and respectively. The Eq. (In summary, these resonances belong to one isolated resonance and the six series of resonances with the corresponding six eigenchannel characters as shown in Fig. 2(b), i.e., one isolated resonance

marked as a light-magenta cross and the six series of resonances such as one periodic red vertical curves , one locally isolated resonances dark-green vertical curve , two pairs of orange and magenta curves , and two quasi-periodic curves ( as black vertical curves and as violet vertical curves). Note that the auxiliary abscissasTherefore, in order to obtain all the energy levels more clearly and conveniently, we propose a new projection method with additional constrain conditions. For the present Ne^{+} (*J*^{π} = 1/2^{+}) case with

7

As shown in Fig. 3, because of the constrain, two blue and green branch curves converging to the threshold

go up-tilted as energy increases, the other two black and violet branch curves associated with the threshold appear as “horizontal” and smoothly cross theIn the autoionization region, there are *N*
_{c} number of eigenchannels with negtive orbital energy and *N*
_{o} number of eigenchannels with positive orbital energy, which satisfy closed channel boundary condition and open channel boundary condition respectively. The asymptotic boundary conditions require^{18, 25–33},

8

Here *τ*
^{ρ} is the effective eigenchannel quantum defects (i.e., collisional eigenchannel phase shifts), which correspond to the effective eigenchannel quantum defects in discrete energy region. The oscillator strength density *df*/*dE* can be obtained as,

9

with the reduced dipole matrix elements *D*_{α}=*ψ*
_{α}| |** D**| |Ψ

With the MQDT parameters (i.e., {*μ*
_{α}, *U*
_{iα}} as well as the dipole matrix elements *D*
_{α}) checked/calibrated with the precise experimental data in both discrete and continuum energy ranges, one can study various dynamic processes such as the electron-atomic ion collisions. We return to examine photoionization processes of Ne^{+} from both the

Effective eigenchannel quantum defects (collisional eigenchannel phase shifts) and the oscillator strength densities from different initial state to final channel symmetries. Because the abscissa is photon energy, the resonances from 1/2^{−} initial **...**

Photoionization cross section of Ne^{+}. (**a**) Normalized experimental photo-ion Yields of Ne^{+} photoionization from the ; (**b**) Red: the R-R-Eigen calculation of total cross section (convoluted with effective energy resolution Δ*E*=13 **...**

Figure 5(b) displays our calculated total cross section as the red line with effective energy resolution Δ*E*=13*meV*, by which the major experimental width and the minor radiative width are included along with the autoionization width having been taken into account adequately in our calculations. As a comparison, the R-Matrix calculation^{34} are plotted as blue line in Fig. 5(b) with Δ*E*=11*meV*. Our calculation agrees well with the normalized experimental photo-ion yields of Ne^{+} photoionizations shown in Fig. 5(a), which have been observed at the synchrotron radiation light source ALS^{34} with the line width Δ*E*=11*meV*. In Fig. 5(c) we decompose our R-R-eigen calculated total cross section shown in Fig. 5(b) into partial cross sections from both

We would like to conclude by the following remarks. With the R-R-Eigen code, we calculated the short-range scattering matrices (i.e., the MQDT parameters {*μ*
_{α}, *U*
_{iα}}) and the corresponding eigenchannel dipole matrix elements *D*
_{α} with good analytical properties in the whole energy regions for 1/2^{+}, 3/2^{+} and 5/2^{+} symmetry block of Ne^{+} respectively. Note that one should choose an appropriate set of *n*
_{p} physical channels to guarantee the analytical continuation property of scattering matrices as shown in Fig. 1. If one would choose an inappropriate set of *n*
_{1}<*n*
_{p} physical channels as the *n*
_{1} effective physical channels, one then could calculate the *n*
_{1} effective eigenchannels scattering matrices {*τ*
^{ρ}, *T*
_{i}′_{ρ}} with R-R-Eigen code as shown in Fig. 2(a). It would take greater efforts to scan all these resonant structures with a much finer energy grid. The calculated *τ*
^{ρ} are equivalent to effective eigen-quantum-defects obtained from MQDT procedure^{23} based on scattering matrices from full physical channel calculations. For the multi-thresholds problem shown in this work, we proposed a pro** j**ected

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

This work is supported by the National Key Research and Development Program of China (Grant No. 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos 11274035, 11371218, 11474031 and U1530401), the National High-Tech ICF Committee in China, Institute of Applied Physics and Mathematics, Beijing, China and National Basic Research Program of China (2013CB922200). We would like to acknowledge Prof. Xiaowei Liu at Peking University, Prof. Feilu Wang at National Astronomical Observatories, Chinese Academy of Sciences, and Prof. Luyou Xie at Northwest Normal University for helpful discussions.

Author Contributions

R.J., X.G. and J.M.L. developed the self-adaptive energy gridding in MQDT solving program, and conceived the projected high dimensional quantum-defect graph (symmetrized); X.Y.H. noticed the problem and R.J. conducted the theoretical calculations. D.L.Z. helped to test the codes and verify the results. X.G. and J.M.L. supervised the study. R.J., X.G. and J.M.L. drafted the manuscript. All authors participated in the discussion and contributed to the manuscript writing.

The authors declare that they have no competing interests.

**Publisher's note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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