Issue 
A&A
Volume 639, July 2020



Article Number  A25  
Number of page(s)  11  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/202037794  
Published online  03 July 2020 
Multiconfiguration DiracHartreeFock calculations of Landé gfactors for ions of astrophysical interest: B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II^{⋆}
^{1}
Department of Materials Science and Applied Mathematics, Malmö University, 20506 Malmö, Sweden
email: wenxian.li@mau.se
^{2}
Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio Av. 3, 10222 Vilnius, Lithuania
^{3}
Division of Mathematical Physics, Lund University, Post Office Box 118, 22100 Lund, Sweden
^{4}
Department of Physics, Çanakkale Onsekiz Mart University, Çanakkale, Turkey
^{5}
Hebei Key Lab of Opticelectronic Information and Materials, The College of Physics Science and Technology, Hebei University, Baoding, 071002, PR China
^{6}
Shanghai EBIT Lab, Key Laboratory of Nuclear Physics and Ionbeam Application, Institute of Modern Physics, Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, PR China
Received:
21
February
2020
Accepted:
4
May
2020
Aims. The Landé gfactor is an important parameter in astrophysical spectropolarimetry, used to characterize the response of a line to a given value of the magnetic field. The purpose of this paper is to present accurate Landé gfactors for states in B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II.
Methods. The multiconfiguration DiracHartreeFock and relativistic configuration interaction methods, which are implemented in the generalpurpose relativistic atomic structure package GRASP2K, are employed in the present work to compute the Landé gfactors for states in B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II. The accuracy of the wave functions for the states, and thus the accuracy of the resulting Landé gfactors, is evaluated by comparing the computed excitation energies and energy separations with the National Institute of Standards and Technology (NIST) recommended data.
Results. All excitation energies are in very good agreement with the NIST values except for Ti II, which has an average difference of 1.06%. The average uncertainty of the energy separations is well below 1% except for the even states of Al I; odd states of Si I, Ca I, Ti II, Zr III; and even states of Sn II for which the relative differences range between 1% and 2%. Comparisons of the computed Landé gfactors are made with available NIST data and experimental values. Analysing the LScomposition of the wave functions, we quantify the departures from LScoupling and summarize the states for which there is a difference of more than 10% between the computed Landé gfactor and the Landé gfactor in pure LScoupling. Finally, we compare the computed Landé gfactors with values from the Kurucz database.
Key words: atomic data / magnetic fields
Tables 5–23 are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/cat/J/A+A/639/A25
© ESO 2020
1. Introduction
Magnetic fields play a fundamental role in astrophysical systems, and thus in the evolution of the Universe. Measurement of the polarization of light as a function of wavelength, known as spectropolarimetry, is the most powerful tool for the accurate determination of magnetic fields in astrophysics. Highly accurate atomic data (e.g. excitation energies, transition rates, oscillator strengths, and Landé gfactors) are essential for interpreting and modelling the spectropolarimetric observations (Judge 2017). The need for atomic data have increased tremendously over the last 20 years with the development of new instrumentation like the Daniel K. Inouye Solar Telescope (DKIST, formerly ATST; Keil et al. 2009). The next generation groundbased solar telescopes will also offer spectropolarimetric capabilities covering a broad wavelength range from the visible into the nearinfrared, the latter of which is largely unexplored spectroscopically.
When an atom or ion is placed in a magnetic field, level splitting occurs that breaks the degeneracy of the energy levels for the different magnetic quantum numbers. This splitting, known as the Zeeman effect, is caused by the interaction between the magnetic moment of the atom and an external magnetic field (Cowan 1981) and is expressed in terms of the Landé gfactor. Accordingly, the effective Landé gfactor of a spectral line, which can be expressed in terms of the Landé gfactors of the lower and upper levels, is an important parameter in astrophysical spectropolarimetry used to characterize the response of the line to a given value of the magnetic field (Landi Degl’Innocenti 1982; Landi Degl’Innocenti & Landolfi 2004). The effective Landé gfactor, , and the second order effective Landé gfactor, , are respectively related to the circular and the linear polarization of a spectral line produced by the Zeeman effect (Landi Degl’Innocenti & Landolfi 2004). For this reason, detailed investigations of the magnetic fields require knowledge of accurate Landé gfactors.
There are a number of measurements of Landé gfactors. In the first half of 20th century massive efforts were made by atomic physicists to establish the energy level structures of atoms and ions from the observed spectral lines. The experimental Landé gfactors were derived from the analysis of Zeeman patterns in individual spectral lines produced in the magnetic fields. These gfactors were collected and made available in the critical compilation by Moore (1949). However, there was a surprising scarcity of reliable data on observed Zeeman patterns among the spectra of the light elements. For example, among the elements studied in this work, the observed gfactors compiled in Moore (1949) are available only for a few states in Ca I and Ti II, which reveals a glaring need of further observations. Later on, Lott et al. (1966) studied the Zeeman effect using strong pulsed magnetic fields and derived the gfactors for a number of states in B I, C I, C III, O II, O III, Mg I, Mg II, Si I, Si III, Si IV, Ca II, and Cu II. Li (1972) measured the Zeeman effect of P II using the electrodeless discharge tubes operated in a field of 32 215 G. The Landé gfactor of the Sn II 5s5p^{2}^{4}P_{1/2} level was measured by David et al. (1980) by direct magnetic resonance. As of today there are, to the knowledge of the authors, no experimental efforts to cover the needs of Landé gfactors, and thus they have to be calculated.
If there are no experimental or calculated data available, the Landé gfactors in pure LScoupling are sometimes used (see Sect. 2). While in many cases this is a good approximation, there are many cases where this fails, thus giving erroneous polarization profiles. One example is the Fe I transition at 7389.398 Å, where the circular polarization is produced by the Zeeman effect due to the nonzero experimental Landé gfactor value, which is instead missing under the LScoupling scheme because of the zero Landé gfactor (Li et al. 2017). More accurate values of the Landé gfactors are obtained in the intermediate coupling approximation, as described in Sect. 2 below. Using the Cowan code in the intermediate coupling approximation, Biémont et al. (2010) calculated Landé gfactors for elements along the sixth row of the periodic table. These data were collected in the DESIRE database^{1}. In this context we should also mention the MCHF/MCDHF database of Froese Fischer for which the Landé gfactors are provided for a few collections^{2}. Fully relativistic calculations of Landé gfactors were pioneered by Cheng & Childs (1985) for states of the 4f^{N}6s^{2} configurations in rareearth elements. More recently relativistic gfactor calculations have been performed for states in Ne I and Ne II (Fischer et al. 2004) and Si IX (Brage et al. 2000). A full set of gfactors was also calculated for the n = 2 states in beryllium, boron, carbon, and nitrogenlike ions (Verdebout et al. 2014).
The purpose of the present work is to compute accurate Landé gfactors for states in B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II within the fully relativistic scheme. Looking at the ions and states studied in this work, the National Institute of Standards and Technology (NIST) database (Kramida et al. 2019) reports Landé gfactors for only 3 out of 100 states for C I, 76 out of 106 for P II, 15 out of 99 for Ti II, and 1 out of 22 for Sn II (see Table 1 for a summary).
Summary of ions, the number of computed energy levels N_{cal − levels}, and the number of Landé gfactors in NIST N_{NIST − gJ}.
2. Theory
We start the theory section with a brief discussion of the Breit–Pauli and intermediate coupling approximations, which provide the necessary background for understanding the validity and limitations of the often used pure LScoupling approximation of the Landé gfactors. The Breit–Pauli and intermediate coupling approximations also provide the theoretical background for the labelling and description of states by the LScomposition, for example as done in the NIST Atomic Spectra Database (Kramida et al. 2019). After this brief discussion we present the fully relativistic theory and show how it links to the Breit–Pauli and intermediate coupling approximations.
2.1. Multiconfiguration wave functions
In the nonrelativistic multiconfiguration Hartree–Fock (MCHF) approach the wave function Ψ for a state labelled γLS, where L and S are the total orbital and spin angular quantum numbers and γ represents the configuration and other quantum numbers needed to specify the state, is expanded in terms of configuration state functions (CSFs) with the same LS term:
The CSFs are constructed from products of oneelectron spin orbitals. The radial orbitals and the expansion coefficients of the CSFs are determined by iteratively solving a set of coupled differential equation resulting from the stationary condition of the energy functional of the nonrelativistic Hamiltonian (Fischer et al. 2016). Once radial orbitals have been obtained, Breit–Pauli configuration interaction (CI) calculations can be performed where the wave function is expanded in LS Jcoupled CSFs:
In the CI calculation the expansion coefficients, c_{j}, are obtained by diagonalizing the Hamiltonian interaction matrix with respect to the Breit–Pauli operators. This is the intermediate coupling or LS J approximation. If the interaction matrix is ordered according to LS terms the interaction has a block structure. Diagonal blocks represent interaction within CSFs of a given LS and offdiagonal blocks between CSFs of different LS terms. If offdiagonal interactions occur for a specific J we say that the LS terms interact and as a result of this interaction the terms mix in the wave function expansion.
In the fully relativistic multiconfiguration DiracHartreeFock (MCDHF) approach the wave function Ψ for a state labelled γJ is expanded in terms of jjcoupled CSFs:
The CSFs, Ψ(γJ), are constructed from products of relativistic oneelectron spin orbitals. The radial orbitals and the expansion coefficients of the CSFs are determined by iteratively solving a set of coupled differential equations resulting from the stationary condition of energy functional of the relativistic DiracCoulomb Hamiltonian (Grant 2007; Fischer et al. 2016). Once the radial orbitals have been obtained, relativistic configuration interaction (RCI) calculations can be performed where the Breit interaction and quantum electrodynamic (QED) effects can be added to the Hamiltonian. Relativistic wave functions are given in terms of jjcoupled CSFs. In order to have a labelling that is consistent with the one from the intermediate coupling approximation, CSFs are transformed from jjcoupling to LS Jcoupling using the methods developed by Gaigalas et al. (2003, 2017).
2.2. Zeeman effect
The Zeeman effect is caused by the interaction between the magnetic moment of the atom and an external magnetic field. The operator representing the interaction is given by
where μ is the magnetic moment of the electrons and B is the magnetic field. If the external magnetic field is weak such that the magnetic interaction energy is small compared to the fine structure separations, the interaction can be treated in firstorder perturbation theory with the wave functions from the Breit–Pauli approximation or from the fully relativistic theory as zeroorder functions.
In the Breit–Pauli approximation the magnetic moment can be written as
where μ_{B} is the Bohr magneton and g_{s} ≈ 2.00160 is the gfactor of the electron spin corrected for QED effects. Using the Wigner–Eckart theorem to relate the matrix elements of L + g_{s}S with the matrix element of J, it can be shown that the magnetic moment is proportional to J, i.e.
where the factor of proportionality, g_{γJ}, is the Landé gfactor. Choosing the direction of the external field as the zdirection the operator for the interaction can, using tensoroperator notation, be written as
Inserting the wave function expansion from Eq. (2) and computing the reduced matrix elements of the interaction gives the Landé gfactor in intermediate coupling, i.e.
The matrix elements between the LS Jcoupled CSFs can be analytically evaluated to give
where
is the Landé gfactor in pure LScoupling (Cowan 1981). Summing up the contributions from the different LS terms, we have
where w(LS) is the accumulated squared expansion coefficients for the CSFs with the specified LS term (Jönsson & Gustafsson 2002). The set of w(LS) determine the LS Jcomposition of the wave function. The Landé gfactor in intermediate coupling thus provides a valuable probe of the coupling conditions in the atom (Fawcett 1990). To summarize, the full sum in Eq. (11) gives the Landé gfactor, g_{γJ}, in the intermediate coupling approximation. Truncating the sum to a single dominating LS term, often the one used to label the state, gives the Landé gfactor, g_{J}(LS), in pure LScoupling.
In the relativistic theory the interaction between the magnetic moment of the atom and an external field can be written as
where
is an operator of the same tensorial form as the magnetic dipole hyperfine operator (Cheng & Childs 1985). Just as in the Breit–Pauli approximation, we express the operator ℋ_{M} in terms of J and the Landé gfactor, i.e.
Inserting the wave function expansion from Eq. (3) and evaluating the matrix elements of the interaction gives
In the relativistic Dirac theory the electron gfactor is exactly 2. The QED corrections to this factor lead to a correction of the Landé gfactor. Defining the operator ΔN by
the correction to the Landé gfactor is given by
3. Computational scheme
The accuracy of the computed Landé gfactors depends on the quality of the wave functions. From Eq. (11) we see that the gfactors require the mixing of LSterms in the wave functions to be accurately determined (Fischer et al. 2004). This in turn depends on the CSFs expansions, what electron correlation effects are captured, and how well the resulting wave functions reproduce measured energy separations. For the studied atoms and ions the CSF expansions, aimed at producing accurate energies, are based on the multireferencesingledouble (MRSD) approach (Fischer et al. 2016). In the MRSD approach, the CSF expansions are obtained first by defining a set of important configurations referred to as the MR and then by allowing SD substitutions, according to some rules, of the orbitals in the MR configurations to orbitals in an active set (AS) (see Olsen et al. 1988; Sturesson et al. 2007; Fischer et al. 2016). Depending on the rules, substitutions for the CSF expansion will account for valence–valence (VV), core–valence (CV), and core–core (CC) electron correlation effects. The CSF expansions are systematically enlarged by increasing the active set along with the MR. A number of studies show that expansions accounting for VV and CV effects and based on reasonably large MR and active orbital sets often are sufficient for reproducing energy separations with high accuracy (Jönsson et al. 2017). The Breit interaction and leading QED effects (e.g. vacuum polarization and selfenergy) can be accounted for in the following RCI calculations.
The computational schemes, as well as the evaluation of the wave functions and atomic data for each atomic system, are described in detail in Wang et al. (2018) (for B II), Papoulia et al. (2019a) (for C IIIIV), Papoulia et al. (2019b) (for Al I−II), Pehlivan Rhodin et al. (2019) (for Si I−II), Atalay et al. (2019) (for Si IIIIV), Rynkun et al. (2019a) (for P II), Rynkun et al. (2019b) (for S II, Cl III and Ar IV), and Rynkun et al. (2020) (for Zr III). The corresponding manuscripts of C IC IV by Li et al., Ti II by Li et al., and Sn II by Atalay et al. (in prep.). The ions, as well as the details of the computational schemes and correlation effects (e.g. targeted configuration states, MR for RCI calculations, definition of core orbitals, correlation model for final RCI calculations, AS, and the number of generated CSFs) are summarized in Table 2.
Summary of the computational schemes of B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II.
All calculations of the wave functions were done using the MCDHF and RCI programs (Grant 2007; Fischer et al. 2016), which are parts of general relativistic atomic structure package GRASP2K (Jönsson et al. 2013; Fischer et al. 2019). The evaluation of the Landé gfactors was then done with the HFSZEEMAN programs (Andersson & Jönsson 2008; Li et al. 2020).
4. Evaluation of data
The accuracy of the Landé gfactors is to a large extent determined by the accuracy of the energy separations. In this section we evaluate the accuracy of the calculated energy levels by comparing them with the NIST recommended data. We then present the results for the Landé gfactors, g_{γJ}, and compare them with the Landé gfactors in pure LScoupling, g_{J}(LS). Finally, we compare the Landé gfactors with values from Kurucz’s atomic database (Kurucz 2017).
4.1. Energy levels
The computed excitation energies and wave function composition in LScoupling of the targeted atomic states in the B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II ions are displayed in Tables 5−23, respectively, and are available at the CDS. In the calculations the labelling of the eigenstates is determined by the LS Jcoupled CSF with the largest coefficient in the expansion resulting from the transformation from jjcoupling to LS Jcoupling using the methods by Gaigalas et al. (2017).
One of the quality indicators of calculations is the ability to reproduce the energy structure. Therefore, the accuracy of the wave functions from the calculations can be evaluated by comparing the calculated energy levels with data from the NIST database (Kramida et al. 2019). Here we define the average percentage difference between the present calculations and NIST as “Av. accuracy” to indicate the accuracy of the calculations. In Table 3 a summary of the Av. accuracy is presented for the targeted atoms and ions. As seen from Table 3, all energies are in very good agreement with the NIST recommended values. In particular, the Av. accuracy values are less than 0.1%; they are 0.089%, 0.088%, 0.044%, 0.004%, 0.05%, and 0.09%, respectively, for the Be II, C II−IV, Si III, and Si IV ions. The Av. accuracy values are less than 0.68% for C I, Al I−II, Si I−II, P II, S II, Cl III, Ar IV, Ca I, Zr III, and Sn II. For Ti II the average difference is larger, about 1.06%. The excellent agreement of the excitation energies with the NIST recommended values allows us to infer that the corresponding wave functions are very accurate (see the references given in Sect. 3 for each atom or ion for more details on how to estimate the accuracy.)
Comparison of computed energy levels in the present work with data from the NIST database.
As we have already discussed, the Landé gfactor depends on the mixing of different LS terms, which in turn depends on the separation of these terms. The accuracy of the energy separation of the terms is thus a more proper measure of the accuracy of the Landé gfactors than the excitation energies. To evaluate the accuracy of the energy separation of the terms, we define a new average accuracy parameter by (i) classifying the states into different blocks by Jvalues and parity, one for each symmetry block; (ii) computing the energy separation relative to the lowest state of each block, E_{S} = E_{i} − E_{min}, where E_{i} is the excitation energy and E_{min} is the lowest energy of each block; (iii) computing the relative difference with the NIST values, dE_{S} = , where E_{S − NIST} is the energy separation from the NIST database; and (iv) averaging the difference, = , where N_{ES} is the number of the energy separations. The results are shown in Table 4 for even and odd states. Generally, the uncertainties of the energy separations are larger than those of the excitation energies, especially when the energy separations are very small. However, for most of the levels, dE_{S} is well below 1%. For a few levels for which E_{S} is small, dE_{S} is higher than 5% and these data have been excluded to obtain the average difference values shown in Table 4 (one level for C I, one for C III, one for Si I, one for Si II, one for Si III, one for Ar IV, one for Ca I, and five levels for Ti II). Table 4 shows that the average uncertainty is well below 1%, except for even states of Al I; odd states of Si I, Ca I, Ti II, Zr III; and even states of Sn II. For these levels the relative difference is between 1% and 2%. The good agreement of the energy separations of the terms with the NIST data confirms the reliable values of the mixing between the relativistic CSFs, which is a good indicator of the quality of the produced Landé gfactors.
Comparison of computed energy separations with NIST data.
4.2. Landé gfactors
Tables 5−23 display the Landé gfactors, g_{γJ}, for the lowest states (shown in Table 1) in the B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II ions, respectively.
In Table A.1, the computed g_{γJ} values are compared with the available experimental values for C I (Lott et al. 1966), C III (Lott et al. 1966), P II (Li 1972), Si I (Lott et al. 1966), Si III (Lott et al. 1966), Si IV (Lott et al. 1966), Ti II (Moore 1949), and Sn II (David et al. 1980). The corresponding g_{J}(LS) values are also displayed in the third column for a comparison. From Eq. (11) it is clear that there is a significant change in g_{γJ} only when there is a strong mixing between terms with greatly different g_{J}(LS) values. For C I, C III, Si I, Si III, Si IV, and Sn II the relative differences between g_{γJ} and g_{J}(LS) are rather small, within 0.7%, meaning that the mixing between terms is either small or occurs between terms with nearly the same g_{J}(LS), which will not change g_{γJ} appreciably (Fischer et al. 2004). The experimental values for the C and Si ions, displayed in the last column, are obtained from observations of atomic Zeeman patterns using strong pulsed magnetic fields (Lott et al. 1966). For most of the levels the computed and the experimental g_{γJ}values agree within the experimental errors, i.e. 1%−3% for C I, C III, Si I, and Si III, and a factor of two higher for Si IV due to the broad spectra lines. The good agreement between theory and experiment, and with the g_{J}(LS), indicates that these states of C I, C III, Si I, Si III, and Si IV are well described in LScoupling. One exception is level 7 of Si IV for which the LScomposition (see Table 15) is dominated by one term, giving g_{γJ} = 0.66583 in close agreement with the value g_{J}(LS) = 0.66667 in pure LScoupling. These values differ by more than 8% from the measured value g_{γJ} = 0.72. Since there is excellent agreement with the NIST recommended data, for the excitation energies and for energy separations, we are confident in our value and suggest a remeasurement for this level.
Some states of P II are strongly mixed in LScoupling. Out of the 76 levels in P II for which experimental Landé gfactors are available, 23 levels have relative differences between g_{γJ} and g_{J}(LS) greater than 3%. We especially note a 44% difference for levels 27 and 50, a 30% difference for levels 29 and 54, a 29% difference for level 94, and a 22% difference for level 57. The departure from LScoupling is quantified in Table 16, for example level 27, labelled ^{3}D_{1}°, has the composition 45% 3s^{2} 3p ^{2}P 3d ^{3}D, 26% 3s^{2} 3p ^{2}P 3d ^{3}P°, and 8% 3s ^{2}S 3p^{3}(D) ^{3}D°. The g_{J}(LS) of ^{3}D_{1} has the relatively small value of 0.50000, whereas the value for ^{3}P_{1} is 1.50000, and so the mixing results in an appreciable increase in the smaller value. The experimental Landé gfactors of P II are from measurements of the Zeeman effect using the electrodeless discharge tubes operated in a given magnetic field (Li 1972). Comparisons between the computed g_{γJ} and measured values show that there is a good agreement for most of the levels within the experimental uncertainties of 0.01 (0.02 for most of the g_{γJ} values of 3p5d levels), except for levels 27 and 29 for which the relative difference is about 5%, and for levels 49 and 51 for which the relative difference is about 4%.
For Ti II there are 15 levels for which experimental data are available. Of these 15 levels, 7 have relative differences between g_{γJ} and g_{J}(LS) greater than 1%, especially 18% for level 18 and 23% for level 57. From the LSpercentage composition shown in Table 21, level 18 has the composition 61% 3d^{3}(P) ^{2}P_{1/2}, 26% 3d^{2}(P) ^{3}P 4s ^{2}P_{1/2}, and 3% 3d^{2}(P) ^{3}P 4s ^{4}P_{1/2}. The g_{J}(LS) for ^{2}P_{1/2} has a relatively small value of 0.66607, whereas the value for ^{4}P_{1/2} is 2.63809 and the mixing results in a 17% increase in the smaller value. Level 57 has the composition of 51% 3dD) ^{1}D 4p ^{2}D, 29% 3dD) ^{1}D 4p ^{2}P°, and 5% 3dP) ^{3}P 4p ^{2}D°. The g_{J}(LS) for ^{2}D_{3/2} has a relatively small value of 0.8, whereas the value for ^{2}P_{3/2} is 1.33333, and the mixing results in an appreciable change in the 3d^{2}4p ^{2}D value to 0.98131. Compared with the NIST recommended Landé gfactors of Ti II (Corliss & Sugar 1979; Sugar & Corliss 1985; Saloman 2012), levels 18, 45, 56, and 57 have relative differences of 15%, 15%, 5%, and 23%, respectively. The suggested LSpercentage compositions are from the calculations of Huldt et al. (1982). The experimental gvalues were determined by Catalán from the Zeeman patterns observed by King and Babcock and quoted by Russell (1927). They were published by Moore (1949). For level 18, NIST gives similar values of the leading compositions with present calculation, 62% 3d^{3}(P) ^{2}P_{1/2} and 24% 3d^{2}(P) ^{3}P 4s ^{2}P_{1/2}, but not for that of 3d^{2}(P) ^{3}P 4s ^{4}P_{1/2}, which mainly contributes to the changes in g_{γJ}. The same happens for level 45. The labelling of level 56 identifies the dominant component of the composition with 94% 3dP) ^{3}P 4p ^{2}S, which indicates a good description in LScoupling. The weak mixing results in g_{γJ} = 1.99657 in relative to g_{J}(LS) = 2.00000. However, NIST suggests 99% ^{2}S, but gives an even larger g_{γJ} value of 2.09. For level 57, NIST suggests 48% 3d^{2}(D) ^{1}D 4p ^{2}D and 36% 3d^{2}(D) ^{1}D 4p ^{2}P° and g_{γJ} = 1.21 in relative to the computed value g_{γJ} = 0.98131. As stated in Russell (1927), due to the limitations inherent in old laboratory analyses, very few patterns have been resolved which resulted in large uncertainties of the observed gvalues. It is highly desirable to redo the measurements by using the current highresolution instruments and techniques. For level 3 of Sn II, LScoupling is a good approximation and the computed g_{γJ} = 2.65984 agrees well with the measured value of 2.66085 within the experimental uncertainty of 5% (David et al. 1980).
In Fig. 1, we compare the computed g_{γJ} values with g_{J}(LS), and with available results in Kurucz’s atomic database (Kurucz 2017) for all the atoms and ions presented in this work. The good agreement between g_{γJ} and g_{J}(LS) in B II, C II, C III, C IV, Al I, Al II, Si II, Si III, Si IV, and Ca I, indicates that these atoms and ions are well described in LScoupling approximation. For some states of the rest of the atom and ions, the strong mixing of LSterms results in large differences between g_{γJ} and g_{J}(LS) by > 10%. Kurucz’s atomic data (Kurucz 2017), which are widely adopted by the solar scientists for the spectropolarimetric modelling, are either taken from the experimental results or, when no experimental values are available, from semiempirical values. From Fig. 1 we see that there is good agreement between the computed g_{γJ} values and Kurucz’s data, except for a number of energy levels in C III, Si I, Ti II, and Zr III with relative differences > 10%. In Table A.2, we display all these states with a relative difference between g_{γJ} and g_{J}(LS) of more than 10%. Additionally, we present the g_{γJ} values in the last column for comparison. Except for the levels of P II and level 18 and 57 of Ti II, for which the Kurucz’s data are from experimental results (see discussion above), the others are semiempirical determinations of the Landé gfactor in intermediate coupling. The computed g_{γJ} using the fully relativistic MCDHF approach differ from the semiempirical values by more than 10% for several cases, e.g. level 82 of C I; levels 40, 41, 68, 83, 138, and 144 of Si I; level 87 of S II; levels 18 and 22 of Cl III; and levels 18 and 101 of Ar IV. The data for Sn II are absent in Kurucz’s database.
Fig. 1. Comparison of computed Landé gfactors, g_{γJ}, in the present work with values in LScoupling, g_{J}(LS) (red cross sign) and with Kurucz’s data (black plus sign). Differences ((g_{other}g_{γJ})/g_{γJ}) are given in percentage. The dashed lines indicate the −10% and 10% deviations. 

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5. Summary
In the present work Landé gfactors are computed for the B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II ions, all of which are of astrophysical interest. The MCDHF and RCI methods, which are implemented in the generalpurpose relativistic atomic structure package GRASP2K, are used in the present work. The accuracy of the present calculations is validated by extensive comparisons of the excitation energies and energy separations with the NIST recommended data. All excitation energies are in good agreement with the NIST values. The Av. accuracy values are within 0.1%, 0.089%, 0.088%, 0.044%, 0.004%, 0.05%, and 0.09%, respectively, for the Be II, C II−IV, Si III, and Si IV ions, and are less than 0.68% for the C I, Al I−II, Si I−II, P II, S II, Cl III, Ar IV, Ca I, Zr III, and Sn II ions. For Ti II, the average difference is about 1.06%.
The Landé gfactor depends on the mixing of LS terms, which in turn, depends on the separation of these terms. The accuracy of the energy separations in each symmetry block is thus a more proper measure of the accuracy of the Landé gfactors than the excitation energies. The average accuracy of the energy separations is well below 1% except for even states of Al I; odd states of Si I, Ca I, Ti II, Zr III; and even states of Sn II, all of which show a relative difference of between 1% and 2%.
The computed g_{γJ} values are compared with available experimental values, and with the values in pure LScoupling. The differences with the values in LScoupling are explained by analysing wave function LScompositions. The observed and theoretical gvalues differ by a small percentage in some cases, which may be due to the limitations in old laboratory analyses. It is highly recommended to redo the measurements for these cases. We summarize the levels with a difference of more than 10% between the g_{γJ} and g_{J}(LS) values, and make a comparison with the semiempirical values from the Kurucz’s database. The present calculations provide a substantial amount of critically evaluated Landé gfactors that are useful for modelling and diagnostics of astrophysical plasmas.
Acknowledgments
This work is supported by the Swedish research council under contracts 201504842 and 201604185. Betül Atalay acknowledges financial support from the Scientific and Technological Research Council of Turkey (TUBITAK) – BIDEB 2219 International PostDoctoral Research Fellowship Program.
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Appendix A: Additional tables
Comparison of computed Landé gfactors, g_{γJ}, with their LScoupling values g_{J}(LS) and experimental values (Exp.).
Summary of levels with a relative difference ≥10% with respect to g_{J}(LS).
All Tables
Summary of ions, the number of computed energy levels N_{cal − levels}, and the number of Landé gfactors in NIST N_{NIST − gJ}.
Summary of the computational schemes of B II, C I−IV, Al I−II, Si I−IV, P II, S II, Cl III, Ar IV, Ca I, Ti II, Zr III, and Sn II.
Comparison of computed energy levels in the present work with data from the NIST database.
Comparison of computed Landé gfactors, g_{γJ}, with their LScoupling values g_{J}(LS) and experimental values (Exp.).
All Figures
Fig. 1. Comparison of computed Landé gfactors, g_{γJ}, in the present work with values in LScoupling, g_{J}(LS) (red cross sign) and with Kurucz’s data (black plus sign). Differences ((g_{other}g_{γJ})/g_{γJ}) are given in percentage. The dashed lines indicate the −10% and 10% deviations. 

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