Complements to the Theoretical Treatments of the Electron Collision with Co³⁺: An R-Matrix Approach

Abstract

This paper presents new refined results from the theoretical treatment of electron collision with the Fe-peak element Co³⁺. We have investigated the relevance of relativistic effects on the accurate representation of the target electron wave functions within the Breit-Pauli R-matrix approach. The calculated values for fine-structure levels are compared with the available experimental data in Atomic Structure Database of the National Institute for Standards and Technology. The agreement between the calculated and the experimental data is reasonably good, the energy difference average percentage of the low-lying levels usually agreeing to within 3.6% of each other. For completeness, we summarize herein the existing theoretical R-matrix treatments intended specifically to explore the role of including configuration interaction wave functions both in the target-state expansion, and in the (𝑁+1)-electron quadratically integrable function expansion. To the best of our knowledge, the work reported herein describes for the first time a detailed calculation for this atomic system, and the results are relevant to the laboratory and astrophysical plasmas.

1. Introduction

The present work aims to provide atomic data and electron collision data for the Fe-peak element Co IV and to supply with additional information the studies on astrophysical opacities. The electron collision data for Fe, Ni, and Co elements, and their ions are important in line identification in stellar objects or in tokamak plasmas, since the collisional rate of deexcitation for metastable levels may be lower than the decay rate by magnetic dipole or electric quadrupole transitions, which become observable. Due to a lack of precise abundance values these data are of importance in the analysis of many astronomical spectra. The needed atomic data can be obtained in various approximations that differ in their demands on resources by several orders of magnitude. Ranges of validity of expansions and applicability of perturbation treatments must be established in order to obtain reliable data with the most economical methods. The R-matrix approach has enabled vast amounts of accurate electron and photon collision data which have had wide applications.  The recent volume by Burke [1] provides both a general review of the R-matrix method as applied to atomic and molecular collision processes and a very extensive bibliography. The R-matrix treatments [2–9] have provided results for electron collision with most of the ions of Fe, Ni and Co such as: Fe I, Fe II, Fe III, Fe IV, Fe V, Fe VI, Fe VII, Co V, Co VI, Co II, Ni II, Ni III, Ni V, Ni VI, and Ni VII.  Apart from our recent work [10], there is no other R-matrix calculation on Co IV ion. In the absence of experimental data on the electron-impact excitation cross-sections for Co IV, the need for accurate collision data can be accomplished only through detailed and accurate target description and atomic level energy data calculation. The calculation reported here is part of a general investigation which started with studies of collision strengths for the electron-impact excitation of forbidden transitions between 136 terms arising from 3𝑑⁶, 3𝑑⁵4𝑠, and 3𝑑⁵4𝑝 configurations of Co³⁺ [10]. In particular, for Co IV ion, 1𝑠²2𝑠²2𝑝⁶3𝑠²3𝑝⁶3𝑑^6 5D ground configuration, the energies of the 3𝑑⁵4𝑝 terms are lying between 3𝑑⁶ and 3𝑑⁵4𝑠 states but overlapping both. Therefore, in our earlier work, the accuracy of a series of models for the target terms was considered which forms the basis of the R-matrix collision calculations. It was found that one could obtain a better representation for the 136 Co IV levels arising from the 3𝑑⁶, 3𝑑⁵4𝑠, and 3𝑑⁵4𝑝 manifolds by allowing double-electron promotions from the 3p-shell into the 3d-shell and single-electron promotion into the 4s and 4p-shell. In order to explore the effect of including of additional states, we have performed additional collision calculation including a further 136 states which arise from the inclusion of the fourth 3𝑑⁵4𝑑 configuration in the R-matrix expansion. Section 2 gives a summary of our previously reported results. At this level of accuracy, the key conclusion is that the interaction between the lowest even configurations and the perturbation on odd parity configurations play an important role in such calculations. Therefore, we have decided to study the relativistic effects on the accurate representation of target electron wave-functions. Section 3 gives the results from semirelativistic Breit-Pauli R-matrix computation of the fine-structure splitting in Co³⁺. The calculated theoretical data are compared with the available experimental data in Atomic Structure Database (ASD) of the National Institute for Standards and Technology [11]. The ASD data is based on the compilation of Co ions by Sugar and Corliss [12]. In their work, all energy levels are given in units of cm⁻¹, beginning with a value of zero for the ground levels. Although uncertainties are not provided with these extrapolated values, the levels uncertainty is presumed ±0.5 cm⁻¹. Section 4 gives our concluding remarks and the future directions of research.

2. The Electron Collision with Co³⁺: An R-Matrix Approach

In our earlier work [10] we have initiated a study of electron-impact excitation of Co³⁺, 1𝑠²2𝑠²2𝑝⁶3𝑠²3𝑝⁶3𝑑⁶ (⁵D^e) ground configuration, considering the forbidden transitions which arise between the 136 terms of the 3𝑑⁶, 3𝑑⁵4𝑠 and 3𝑑⁵4𝑝 configurations. The following excitation ways have been considered: (1)

In the first stage of the calculation, we used CIV3 program [13] to obtain accurate target state energies and wave functions which are used in the following stages of the R-matrix calculations. The target wave functions and the (𝑁+1)-electron quadratically integrable functions were constructed with a common set of one-electron spin-orbital functions. In the second stage of calculation we used RMATRXI program by Berrington et al. [14] to solve the electron-atom collision problem in the 𝑅-matrix internal region. In the asymptotic region, the radius is propagated to a new distance, chosen large enough that the radial functions which represent the colliding electron can be accurately represented by an asymptotic expansion. In the third stage of calculation we used FARM [15] to solve the electron-atom collision problem in the 𝑅-matrix external and asymptotic regions. These programs enable the 𝐾-matrix, 𝑆-matrix, and hence collision strengths to be determined. The 𝐾-matrix elements and the 𝑆-matrix elements, and therefore, the collision strengths Ω_𝑖𝑗 can be calculated by matching the solution of the inner and outer regions at the 𝑅-matrix boundary radius 𝑟=𝑎. For completeness, the expression for the electron scattering cross section is given herein. The expressions for the 𝑆-matrix and 𝑇-matrix as related to the 𝐾-matrix are 𝑆=(1+𝑖𝐾)(1−𝑖𝐾)⁻¹,𝑇=𝑆−1, (2) where these matrices have particular total 𝐽 and 𝜋 values. The partial collision strength which contributes to the cross-section for a transition of the target from state 𝑟 to a state 𝑠 is a summation over the channels 𝛼_𝑟𝐽^𝑡_𝑟𝑗_𝑟 to 𝛼_𝑠𝐽^𝑡_𝑠𝑗_𝑠 that are coupled to them. 𝛼 represents all the other quantum numbers that uniquely identify a target state. The total collision strength is a sum of the partial collision strengths for each symmetry as follows: Ω^𝐽𝜋_𝑟𝑠=𝑔2_{𝑗𝑟𝑗𝑠}𝑇_𝑟𝑠²,Ω_𝑟𝑠=_𝐽𝜋Ω^𝐽𝜋_𝑟𝑠, (3) where 𝑔=2𝐽+1 for 𝑗𝑗 coupling, and Ω_𝑟𝑠 should be symmetric. Then, the total cross section for this transition is given by 𝜎_𝑟→𝑠=𝜋𝑎²₀𝑘²_𝑟𝑔_𝑟Ω_𝑟𝑠, (4) where 𝑔_𝑟=2𝐽^𝑡_𝑟+1 and 𝑎₀ is the Bohr radius to convert from atomic units.

In our preliminary work, three target-model calculations have been developed: (a) a three-LS-coupled R-matrix calculation including 3𝑑⁶, 3𝑑⁵4𝑠, and 3𝑑⁵4𝑝 states in the N-electron-wave function expansion; (b) a six-target-model calculation, where electron correlation effects were explored by carrying out separate calculations with six configurations 3𝑑⁶, 3𝑑⁵4𝑠, 3𝑑⁵4𝑝, 3𝑝⁴3𝑑⁸, 3𝑝⁴3𝑑⁷4𝑠, and 3𝑝⁴3𝑑⁷4𝑝 in the target state expansion and the configurations 3𝑑⁷, 3𝑑⁶4𝑠, 3𝑑⁶4𝑝, 3𝑑⁵4𝑠², 3𝑑⁵4𝑠4𝑝, 3𝑑⁵4𝑝², 3𝑝⁵3𝑑⁸, 3𝑝⁴3𝑑⁹, 3𝑝⁵3𝑑⁸, 3𝑝⁴3𝑑⁹, 3𝑝⁴3𝑑⁸4𝑠, and 3𝑝⁶3𝑑⁵4𝑠² in the (𝑁+1)-electron quadratically integrable function expansion; and (c) a nine-target-model calculation. Starting with the 136-level model, we have included in the R-matrix expansion all 184 LS-coupled states which arise from nine-target configurations 3𝑑⁶, 3𝑑⁵4𝑠, 3𝑑⁵4𝑝, 3𝑝⁴3𝑑⁸, 3𝑝⁴3𝑑⁷4𝑠, 3𝑝⁴3𝑑⁷4𝑝, 3𝑝⁵3𝑑⁷, 3𝑝⁵3𝑑⁶4𝑠, and 3𝑝⁵3𝑑⁶4𝑝. The 136 LS target states arising from the three, six, and nine basis configurations above, were optimally represented by configuration interaction type expansions in terms of eight orthogonal basis orbitals. Figure 1 shows the graphical abstraction of the 𝑅-matrix H-file as output from HBrowse, a Graphical 𝑅-matrix Atomic Collision environment, GRACE, tool [16]. This H-file was used to convey information from the inner-region to the outer-region in the six-target model calculation. In order to have a consistent set of wave functions for the 𝑁- and (𝑁+1)-electron wave functions, the (𝑁+1)-electron configuration data have been obtained by adding one electron to the 𝑁 electron configurations in all possible ways. Hence, we can include these (𝑁+1)-electron configurations in the collision wave function by including the configuration interaction wave functions with configuration 3𝑝⁵3𝑑⁷ in the basis set used in the CI expansion of the 𝑁-electron target states.

Figure 1 

The graphical abstraction of the R-matrix H-file as given from HBrowse code, a Graphical R-matrix Atomic Collision environment: six LS-coupled states are included into the R-matrix expansion: 3𝑝63𝑑6, 3𝑝63𝑑54𝑠, 3𝑝63𝑑54𝑝, 3𝑝43𝑑8, 3𝑝43𝑑74𝑠, and 3𝑝43𝑑74𝑝, total angular quantum number 0 < L < 9, 2 < (2S + 1) < 8, maximum number of channels 18, R-matrix radius 15 a.u, Hamiltonian size 290, and scattering symmetry 4Fe.

For collision calculations, a number of systematic checks on Co³⁺ have been performed. We have investigated the position of 3𝑑⁵4𝑑 and 3𝑑⁴4𝑠² terms including the configurations 3𝑑⁵4𝑠², 3𝑑⁵4𝑝², 3𝑑⁵4𝑑², 3𝑑⁵4𝑠4𝑝, and 3𝑑⁵4𝑝4𝑑 in the (𝑁+1)-electron quadratically integrable function expansion. The 3𝑑⁵4𝑑 terms are lying between 3𝑑⁵4𝑝 and 3𝑑⁴4𝑠² states but overlapping both. We have carried out a first calculation in which 272 LS terms of the four configurations 3𝑑⁶, 3𝑑⁵4𝑠, 3𝑑⁵4𝑝, and 3𝑑⁵4𝑑 with a maximum 841 channels which includes the 3𝑝⁶3𝑑⁵4𝑑 configuration, are included in the R-matrix expansion. Figure 2 presents a comparison between the collision strengths as outputs from two different calculations where 136 terms arising from the 3𝑑⁶, 3𝑑⁵4𝑠, and 3𝑑⁵4𝑝 manifolds [10], and 272 terms arising from 3𝑑⁶, 3𝑑⁵4𝑠, 3𝑑⁵4𝑝, and 3𝑑⁵4𝑑 manifolds are included into the 𝑅-matrix expansion, respectively. The inclusion of additional states affects the resonance structure in the collision strengths. The collision-strength results correspond to the transition from the ground state 3𝑑⁶(⁵D^e) to the first excited state 3𝑑⁶(³P^e), ⁴F^e symmetry. As shown in Figure 2, the resonance positions are pushed to lower scattered electron energies.

Figure 2 

Collision strengths for the 5De-3Pe transition of Co3+. The full curve corresponds to the calculation where 136 states arising from the 3𝑑6, 3𝑑54𝑠, and 3𝑑54𝑝 manifolds are included into the scattering process. The broken curve corresponds to the calculation where 272 states arising from 3𝑑6, 3𝑑54𝑠, 3𝑑54𝑝, and 3𝑑64𝑑 configurations are included into the scattering process. Scaled units are used for the collision energy: for positively charged ions energies are scaled by a factor 1/𝑍2, with 𝑍 = 3 for Co3+ ion.

3. Semirelativistic Breit-Pauli 𝑅-Matrix Calculation of the Fine-Structure Splitting

The 𝑅-matrix theory commences by partitioning configuration space into three regions describing the scattering of an electron by an 𝑁-electron atom or ion: an internal region, an external region, and an asymptotic region. The solutions obtained in each of these regions are related by the 𝑅-matrix which corresponds to the inverse of the logarithmic derivative of the wavefunction on the boundary of the two regions. One important point to note is that, in the 𝑅-matrix method, the inner region solution is obtained only once, and then cross-sections for any number of energy points are readily available. In the asymptotic region, the radius is propagated to a new distance, chosen large enough that the radial functions which represent the colliding electron can be accurately represented by an asymptotic expansion. The RMATXI package of codes [14] includes relativistic effects in the Breit-Pauli approximation (BPRM) [17, 18]. The Hamiltonian is written as follows: 𝐻^BP_𝑁+1=𝐻^NR_𝑁+1+𝐻^mass_𝑁+1+𝐻^DAR_𝑁+1+𝐻^so_𝑁+1, (5) where 𝐻^NR_𝑁+1 is the nonrelativistic Hamiltonian, 𝐻^mass_𝑁+1 is the one-body mass-velocity term, 𝐻^DAR_𝑁+1 is the Darwin term, and 𝐻^so_𝑁+1 is the spin-orbit term. This last term breaks LS symmetry leading to fine-structure levels Jπ of total angular momentum quantum number 𝐽 at parity π. In relativistic BPRM calculations, sets of collisional symmetry SLπ are recoupled to obtain the states of the (𝑁+1) atomic system with total Jπ, followed by diagonalization of the (𝑁+1)-electron Hamiltonian. Details of diagonalizing 𝐻^BP_𝑁+1 at the 𝑅-matrix boundary are given in [17], as is the outward propagation. In our BPRM calculations, we have constructed an eigenfunction expansion over the three configurations 3𝑑⁶, 3𝑑⁵4𝑠, and 3𝑑⁵4𝑝 of Co³⁺, yielding 136 fine-structure levels corresponding to 43 LS terms. Eight orthogonal one-electron orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s, and 4p were used both in the definition of the target states and also for the (𝑁+1)-electron quadratically integrable functions. In all of the models, the 1s, 2s, 2p, 3s, 3p, and 3d radial functions have been taken from the Hartree-Fock ground state 1𝑠²2𝑠²2𝑝⁶3𝑠²3𝑝⁶3𝑑^6 5D, given in the tables of Clementi and Roetti [19]. Using the nonrelativistic Schrödinger Hamiltonian, we have established the wave functions by optimizing further three orbitals on various L = 2, 3, and 4 states. Full details of the three-target model calculation have been reported in our previous work [10]. The one- and two-electron radial integrals are computed by STG1 of the BPRM codes using one-electron target orbitals. The calculation considers all possible bound levels for 0≤𝐽≤8 with 0≤𝐿≤7, and (2S + 1) = 1, 3, 5, 7 even and odd parities. The intermediate coupling calculations are carried out on recoupling the LS symmetries in a pair-coupling representation in stage RECUPD. The (electron + core) Hamiltonian matrix is diagonalized for each resulting Jπ in STGH. The number of coupled channels was 1024 and the Hamiltonian matrix size was 20502. Table 1 gives results for the 47 lowest even parity energy levels and the comparison with ASD data. Agreement between theory and experiments is reasonably good, the energy difference average percentage of the low-lying levels usually agreeing to within 3.6% of each other.

Table 1 

Comparison between the calculated theoretical levels energy and the experimental data from ASD, in Rydberg units.

4. Conclusions

We have carried out the first detailed calculation on the electron scattering with Fe-peak element Co IV where 272 LS terms arising from configurations 3𝑑⁶, 3𝑑⁵4𝑠, 3𝑑⁵4𝑝, and 3𝑑⁵4𝑑 have been included in the 𝑅-matrix expansion. Given detailed calculation provided here and comparison with our earlier work [10], we estimate that the accuracy of our data is within 20% for transitions within the 3𝑑⁶ manifold. Due to the lack of experimental data for electron-impact excitation on this complex ion, there is no way to fully assess the accuracy of our earlier and present calculations. We expect that these data on Co IV and that from our previous study [10] on this system will be of fundamental value in gauging the accuracy of future calculations for electron impact excitation on this Fe-peak element.