Wavelengths and Energy Levels of the Upper Levels of Singly Ionized Nickel (Ni ii) from 3d 8(3 F)5f to 3d 8(3 F)9s

Using high-resolution spectra of Ni ii recorded using Fourier transform (FT) spectroscopy of continuous, nickel–helium hollow cathode discharge sources in the region 143–5555 nm (1800–70,000 cm−1, the analysis of 1016 Ni ii lines confirmed and optimized 206 previously reported energy levels of the (3 F) parent term, from 3d 8(3 F)5f to 3d 8(3 F)9s, lying between 122,060 and 138,563 cm−1. The uncertainties of these levels have been improved by at least an order of magnitude compared with their previously reported values. With the increased resolution and spectral range of the FT measurements, compared to previously published grating spectra, we were able to extend our analysis to identify and establish 33 new energy levels of Ni ii, which are reported here for the first time. Eigenvector compositions of all revised and newly established energy levels were calculated using the orthogonal operator method. In addition, an improved ionization energy of 146,541.35 ± 0.15 cm−1 for Ni ii, using highly excited levels of the 3d 8(3 F)5g, 3d 8(3 F)6g, and 3d 8(3 F)6h configurations, has been derived.


Introduction
As most lines observed in stellar spectra are blended and without accurate wavelengths, either obtained experimentally or derived as Ritz wavelengths from experimentally determined energy levels, it is impossible to disentangle these blends.As such, accurate and extensive measurements of energy levels and transition wavelengths are crucial for the interpretation of astrophysical spectra.
The iron group elements are of particular importance in stellar spectroscopy due to their combination of rich, dense spectra, arising from their complex atomic structure (with partially filled 3d shells) and their high relative cosmic abundances.As such, iron group elements are responsible for the majority of opacity observed in stellar spectra.Singly ionized members of the iron group are the dominant ionization stage in the atmospheres of spectral classes of stars from B to G, and singly ionized nickel (Ni II) has been observed in a variety of these stellar spectra (e.g., Jaschek & Jaschek 1995;Klein et al. 2011).Ni II has also been observed in many other astrophysical phenomena from evolved stars (Richardson et al. 2011) to gamma-ray bursts (Vreeswijk et al. 2007) and luminous blue variables (Hartman et al. 2004) to quasi-stellar objects (Fynbo et al. 2010) making it an important element in the study of many astrophysical objects.
This paper is a continuation of the extensive analysis of the lower-lying energy levels of Ni II, carried out by Clear et al. (2022) using Fourier transform spectroscopy (FTS), and extends the analysis of the energy level structure of Ni II to the higher-lying energy levels of the ( 3 F) parent term.Only this parent term saw excitations in the measured spectra to levels beyond 3d 8 6s and these are termed "high-lying" levels in this work.Accurate data for high-lying levels are especially important for astrophysical spectra in the infrared (IR), where many of the strongest transitions from these levels are to be found.
The last major analysis of the higher-lying levels of Ni II was carried out by Shenstone (1970) who measured the spectrum of nickel-helium hollow cathode lamps (HCLs) using prism and grating spectrometers between 10,000 and 137,500 cm 1 (72.7-1000nm).For the high-lying levels, Shenstone provided values for 292 energy levels beyond 3d 8 ( 3 F)6s in the ( 3 F) parent term of Ni II, with series observed in 3d 8 ( M L)ns, np, nd, nf, and ng, up to 9s.However, Shenstone was only able to determine a small number of levels belonging to the 6p and 7p configurations due to a lack of measurements in the IR where the strong transitions to and from these levels lie.All levels determined by Shenstone were labeled using the LS coupling scheme exclusively.Brault & Litzén (1983) extended the analysis of Ni II into the IR with the first FTS measurements in this region.They recorded the spectra of nickel-helium HCLs in the region 2000-10,000 cm 1 (1000-5000 nm), observing two groups of strong Ni II transitions in this region.The first were transitions between the 3d 8 ( 3 F)5f and 3d 8 ( 3 F)6g energy levels, which had been previously identified by Shenstone, and the second were between 3d 8 ( 3 F)5g and the new 3d 8 ( 3 F)6h levels identified by Brault and Litzén.These high-lying levels formed closely spaced pairs and Brault and Litzén concluded that all Ni II levels with higher l values (l 3) were better described by the jK coupling scheme than the LS scheme used by Shenstone (1970).
In their 1985 compilation of energy levels, which was previously the most recent source of Ni II published data for Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
high-lying levels, Sugar & Corliss (1985) (referred to as S+C from now on) relabeled Shenstoneʼs energy levels with l 3 (i.e., f, g, and h levels) using the jK coupling scheme.Shenstone (1970) did not include any energy level uncertainties in his work and S+C estimated these to be ±0.05cm 1 .Although the FTS measurements carried out by Brault and Litzén (1983) were of much higher resolution than those of Shenstone (1970), their energy level uncertainties were limited by the accuracy of Shenstoneʼs 3d 8 ( 3 F)5f and 3d 8 ( 3 F)5g levels that they connected to.The uncertainty for all previously measured energy levels is therefore ±0.05 cm 1 .These uncertainties are now too large to meet the requirements of modern, high-resolution astrophysical spectral analyses.

Experimental Details
The spectra used in this work and their wavenumber and intensity calibration are described in detail by Clear et al. (2022), and only a brief outline is given here for reference.Nine Fourier transform (FT) spectra were recorded from the vacuum ultraviolet (VUV) to the near-IR with two additional archival spectra extending the spectral range further into the IR.The spectra were recorded using three instruments: the VUV FT spectrometer at Imperial College London (Thorne et al. 1987) in the region 15,798-75,000 cm −1 (133-633 nm), the 2 m FT spectrometer at NIST in the region 9000-23,000 cm −1 (435-1111 nm), and two archival spectra in the region 1550-990 cm −1 (1010-6450 nm) that were recorded in the 1980s by J. Brault using the Kitt Peak 1 m FT spectrometer and were extracted from the National Solar Observatory archives. 5hese spectra were all taken of nickel-helium plasmas generated in HCLs with varying currents and gas pressures.Table 1 of Clear et al. (2022) gives details of the measurement parameters of each spectrum.Helium was chosen as the HCL carrier gas due to the observed increase in signal-to-noise ratio (S/N) for Ni II lines measured in nickel-helium plasmas compared to those in nickel-neon and nickel-argon.The enhancement of lines in nickel-helium spectra was also observed by Brault & Litzén (1983) who noted a 3-4× increase for most transitions, but an approximate enhancement of 50× for IR transitions from levels lying close to the ionization energy of He II.This increased enhancement is due to charge transfer between the helium and nickel ions, which results in an increased population of the high-lying levels of Ni II.This enhancement is of particular importance to the highlying energy levels analyzed in this work.
Voigt profiles were fitted to the observed spectral lines with a least-squares fitting routine using the analysis program XGREMLIN (Nave et al. 2015).The fitted profiles produced the wavenumber, S/N, line width as the FWHM and relative intensity of the lines.For lines with asymmetric profiles, which were usually low-S/N lines, a center-of-gravity fit was manually set to determine the line properties.Linelists of all observed lines across the nine nickel-helium spectra were compiled from the fitted profiles.The linelists were wavenumber calibrated onto an absolute scale using Ar II reference lines in the visible.The wavenumber calibration was extended to the VUV and IR using overlapping Ni lines in neighboring spectra.The spectral lines were intensity calibrated onto a relative scale using spectra of radiometrically calibrated standard lamps to determine instrument-response functions.The relative line intensities presented in this work should only be used as a rough guide and are not suitable for the calculation of branching fractions.Full details of the wavenumber and intensity calibration of the spectra used in this work are given in Clear et al. (2022).

Transition Line Identification
The HCLs recorded in the spectra used by this work produce transitions of neutral and singly ionized species of both the nickel cathode material and the helium carrier gas.Ni I lines were identified using Ritz wavenumbers determined from energy levels published by Litzén et al. (1993) and helium lines were identified by comparison with the NIST Atomic Spectra Database (Kramida et al. 2021).In addition, the NiHeCH spectrum contained a small number of low-S/N impurity lines of O I and N I (probably due to a small leak in the HCL) that were also identified using the NIST Atomic Spectra Database.In total, 1318 Ni I, 25 He II, 117 He I, and 131 impurity lines were identified, leaving an unclassified linelist containing 5090 lines.
The structure of the singly excited terms of Ni II can be described as a valence electron outside a core consisting of the 3d 8 ( M L) terms of Ni III, as shown in Figure 2 of Clear et al. (2022).The state of the core is described by the ground state terms of Ni III, which are known as parent terms.This work contains energy levels of the ( 3 F) parent term greater in energy than 3d 8 ( 3 F)6s.Figure 1 shows a schematic diagram of these upper levels of the ( 3 F) parent term of the singly excited term system of Ni II.The spacing of the upper levels of Ni II means that transitions between these levels lie in the IR.
The initial identification of unclassified lines as potential Ni II lines was performed by comparison with Ritz wavenumbers calculated from the energy levels of S+C.As we revised these energy level values and determined new levels during our analysis (this process is described in Section 5.1), new transitions were identified.In total, 615 lines were identified as transitions between the levels revised or newly discovered in this work and those previously published by Clear et al. (2022) and 401 lines were identified as Ni II transitions between the levels solely optimized in this paper.All of these lines are given in Table 1.
Column (1) of Table 1 gives the logarithm of the relative intensity of the line.Lines with wavenumbers <9032 cm 1 were recorded at KPNO and hence are not intensity calibrated.Columns (2) and (3) are the S/N and FWHM of the line, respectively.Columns (4)-( 7) give the observed and Ritz wavenumbers with their uncertainties.Columns (8)-( 10) give the air and vacuum Ritz wavelengths with their uncertainties.For lines between 200 nm and 2 μm, air wavelengths are given and were calculated from vacuum wavelengths using the fiveparameter conversion formula of Peck & Reeder (1972) (Equation (3) in their paper).Columns (11)-( 14) give the upper and lower energy level labels (in the form configuration, term, and J), and their energy values.The final column notes any blends or asymmetries with the observed line.

Calculations of Energy Levels and Level Eigenvector Compositions
The identification of energy levels, in combination with the assignment of physically realistic eigenvectors, requires accurate atomic structure calculations.Closely spaced complex spectra are most accurately described by a semiempirical approach, in which a model Hamiltonian is parametrically adjusted to fit eigenvalues as close to experimental energies as possible.To this end, the orthogonal operator method (Uylings & Raassen 2019; Uylings 2021) has been employed in this work.As an extension and refinement of the Slater-Condon theory of atomic spectra incorporated in Cowanʼs programs (Cowan 1981), the orthogonal operator method can be seen as a next step in the semiempirical description of complex spectra.
Orthogonal operators have the marked advantage that the parameters in a least-squares fit are as independent and stable as possible.Smaller interactions can therefore also be included, yielding a physically significant reduction of the standard error of the fit.The fitting process has to be preceded by a reasonably accurate ab initio calculation (Fischer et al. 2016;Jönsson et al. 2017;Froese Fischer et al. 2019) to provide starting parameters, especially for the smaller fine-structure effects; experience with neighboring spectra helps in this regard.
Nearly all known levels of the highly excited ¢ ¢ d n l 3 8 configurations of Ni II are based on the 3d 8 ( 3 F) parent.As a result, all 3d core parameters, except the average energy, are kept fixed on values interpolated between the 3d 8 4p configuration and the 3d 8 ground configuration of Ni III.For both predictions and a correct eigenvector composition, it is important to use a physically accurate core description.
Configuration interaction is much less important in these higher-lying levels than in the low even and odd configurations.Nevertheless, a seven-configuration model space of the even system including 3d 8 6d + 3d 8 7d + 3d 8 7s + 3d 8 8s + 3d 8 9s + 3d 8 5g + 3d 8 6g and a five-configuration model space of the odd system including 3d 8 6p + 3d 8 7p + 3d 8 5f + 3d 8 6f + 3d 8 6h was used, yielding overall mean deviations of the fit of 8.8 cm 1 and 25 cm 1 , respectively.As usual in an orthogonal operator set, the first order 3d-¢ ¢ n l Coulomb interaction is separated from the remaining higher-order, two-particle effects.For ¢ ¢ d n l 3 8 configurations with ¢ ¢ n l highly excited, and thus relatively close to ionization, the only 3d-¢ ¢ n l operator with an appreciable contribution to the structure is proportional to ).Even this parameter will usually be much less important than ζ 3d , while z ¢ ¢ n l is negligible.This indicates that for ¢ ¢ = n l f f f g g 4 , 5 , 6 , 5 , 6 , and 6h, jK coupling is the preferred coupling scheme to attain more unambiguous level assignments.In this work, the calculated energy eigenvalues and eigenvector compositions were an aid to searching for new energy levels and for assigning level quantum labels.
In a forthcoming article, the complete list of transition probabilities calculated by the orthogonal operator method with accurate intermediate coupling will be detailed and these intensities will be compared with the experimental FTS values for identified lines.

Energy Level Optimization
In combination with the program LOPT (Kramida 2011), observed Ni II transitions and uncertainties were used to optimize the energy levels.LOPT uses a least-squares fitting routine, with lines weighted by the inverse squares of their observed total wavenumber uncertainties.The optimization was performed in multiple stages, starting with the strongest transitions to the previously published lower Ni II levels of Clear et al. (2022).The levels of this initial optimization were then used to refine the identifications of further lines in the calibrated linelist.This process was performed multiple times, with weaker transitions and transitions to additional levels included.Any lines that had a large deviation from the Ritz wavenumber derived from the two energy levels of that fit were examined to check for possible misidentifications, blends, or asymmetries.In these cases, the weighting of the line was amended to lessen its influence on the LOPT fitting or the identification was discarded.The fitting process was repeated until no more previously published levels could be optimized.A search for new energy levels was then carried out using unclassified transitions in our linelist and our new energy level calculations as a guide.
The energy levels optimized during this term analysis are given in Table 2. Columns (1)-(3) provide the configuration, term, and J value of the energy level.Level energy and uncertainty are shown in the columns (4) and (5).Column (6) gives the number of observed lines that were used in the optimization fit.Columns (7)-( 14) show the calculated eigenvector component(s) and percentage contributions to the   Notes.The columns are as follows: (1) Int., the log 10 of the relative intensity of the line.
(2) S/N of the line.
(3) FWHM of the line in units of 0.001 cm −1 .(4) σ obs , the observed wavenumber of the line, in cm −1 .(5) The uncertainty in the observed wavenumber.(6) σ Ritz , the Ritz wavenumber of the line, derived from the optimized energy levels.(7) The uncertainty in the Ritz wavenumber.(8)-( 9) The observed wavelength of the line in air or vacuum, in nanometers.Air wavelengths were calculated using the five-parameter dispersion formula of Peck & Reeder (1972).( 10 (This table is available in its entirety in machine-readable form.)(This table is available in its entirety in machine-readable form.)energy level.The final column contains notes about the level such as "n" for new or "r" for redesignated from S+C.

Revision of Previously Known Energy Levels
In total, 206 of 292 high-lying Ni II energy levels in S+C have been confirmed and their energy values revised as a result of our analysis in this paper.The energy difference, E − E S+C , between our revised energy level values and those of S+C are shown in Figure 2. The 0.17 cm 1 offset from zero (shown as a dotted line in Figure 2) is due to a problem with the calibration of a photographic spectrum plate in the work of Shenstone (1970) and is discussed in detail in Clear et al. (2022).With our new measurements and energy level analysis, this error has now been corrected.
The majority of energy levels we were unable to revise are from the 3d 8 ( 3 F 3 )6f, 3d 8 ( 3 F 2 )6f, 3d 8 ( 3 F)7f, 3d 8 ( 3 F)7g, 3d 8 ( 3 F)8p, and 3d 8 ( 3 F)8d subconfigurations.The strongest transitions to these levels were either not seen in our FTS spectra, because they were either weak and lost in the noise or their upper energy levels were not populated in our HCL sources, or the transitions lie in the VUV beyond the wavelength limit of FTS.Work is planned to record grating spectra of the VUV region, which may enable these levels to be revised in the future.
The configuration, term, or J value of 37 previously published levels have also been revised from those given in S+C.Our new calculations of eigenvector components were crucial in determining where these revisions were needed.The newly revised level labels are given together with their previous designation in Table 3.

New Energy Levels
Following the optimization of previously published levels, we were able to use the remaining unclassified lines to find and identify 33 new energy levels of Ni II.These levels are identified with an "n" in the last column of Table 2.Many of the crucial connecting transitions to these new levels lie in the IR.Several of the newly found energy levels were found to lie very close to previously known levels and would not have been  Note.
a Energy level value from this work.resolvable with lower accuracy grating lines, but thanks to the high resolution of FTS, these levels were able to be fitted correctly.
5.2.1.New Levels of the ( 3 F)6p Subconfiguration Seventeen new ( 3 F)6p energy levels have now been found, fully completing the ( 3 F)6p subconfiguration.Strong transitions to ( 3 F)6s, 7s, and 6d levels in the IR enabled excellent fits in LOPT for all ( 3 F)6p levels (differences between observed and Ritz wavenumbers for all strong lines <0.0005 cm 1 ).A total of 148 newly classified IR lines have been identified as transitions from levels in the ( 3 F)6p subconfiguration.

New Levels of the ( 3 F)6d Subconfiguration
The ( 3 F)6d subconfiguration is now also complete following the determination of two new levels, 4 D 1/2 and 2 D 3/2 .Fits for these two new levels were enabled by strong transitions to the newly discovered ( 3 F)6p levels.In total, 12 newly classified IR lines have been identified as transitions from the new ( 3 F)6d levels.

New Levels of the ( 3 F)7p Subconfiguration
The completeness of the ( 3 F)7p subconfiguration has been extended with 13 new energy levels.Although transitions from ( 3 F)7p levels were generally weaker than transitions from ( 3 F)6p, due to lower populations for these higher-lying levels, strong lines in the IR enabled excellent fits with transitions to the well-established ( 3 F)7s and ( 3 F)8s levels.

New Level of the ( 3 F)7d Subconfiguration
One new ( 3 F)7d level was able to be established, with two transitions to the new 6p and 7p levels.As with levels of the ( 3 F)7p, 7d levels with low J values had weak lines in the IR which were not distinguishable from noise in our FT spectra, preventing the identification of the remaining missing levels of the ( 3 F)7d subconfiguration.

Ionization Energy
The best available value for the ionization energy of Ni II is given by Shenstone (1970) as the limit of the ng series at 146,541.56 ± 0.20cm 1 .We have now derived a new ionization energy based on highly excited high angular momentum levels, using the quadrupole polarization method of Schoenfeld et al. (1995).
For the levels 3d 8 ( 3 F)5g, 6g, and 6h, which are well described by the jK coupling scheme, the valence electron is only coupled weakly to the core and levels form three distinct groups, separated by the fine-structure splitting of the 3d 7 ( 3 F) Ni III term.Within these groups, levels form closely spaced jK pairs, with centers of gravity E(nlJ c K ) defined by where IE is the ionization energy, E(J c ) is the energy of the d F 3 4 .This plot for the 5g, 6g, and 6h configurations is shown in Figure 3.
Linear equations were fitted to each set of data to obtain values for Q and the intercept of each configuration.IE and α were derived by simultaneously solving the intercept equations for pairs of configurations.The results are given in Table 4. Uncertainties for the values in Table 4 were derived from the uncertainties of the linear fits to the polarization equations, which were much greater than the uncertainties in the energy level values.We have adopted the weighted average of the three derived values of IE, 146,541.35±0.15 cm −1 , as the ionization energy of Ni II.Our new value agrees well with the previously published ionization energy, within the combined uncertainties.

Summary
An extensive analysis of the spectrum of Ni II, using spectra previously measured by Clear et al. (2022) using FTS, has produced transitions with a wavelength accuracy over an order of magnitude better than previous measurements.These new FT wavelengths were then used to extend the analysis of the energy level system of Ni II to the upper levels of the ( 3 F) parent term.This analysis resulted in the revision of 206 known high-lying energy level values with an order of magnitude increase in accuracy over previously published values.Thirtyseven of these previously known energy levels have had their quantum number designation revised following our analysis.Transition lines in the IR, measured for the first time using FTS, enabled 33 new energy levels to be established and are reported here for the first time.
In total, 239 levels and 1016 classified lines of Ni II are presented in this work.The eigenvector components of all reported levels have been calculated by the orthogonal operator  146,541.35(15)method and a new ionization energy of 146,541.35 ± 0.15 cm 1 has been derived using highly excited levels.

Figure 1 .
Figure 1.Diagram showing (a) the subconfigurations of the singly excited (normal) system 3d 8 ( 3 F)nl of Ni II and (b) only the high-lying subconfigurations.The height of the boxes indicate the maximum energy range of the levels in each subconfiguration.The f, g, and h levels are each split into three groups by the J c values of the Ni III parent core (J c = 2, 3 or 4).

)
The uncertainty in wavelength.(11) The configuration, term, and J value of the lower energy level.(12) The configuration, term, and J value of the upper energy level.(13)-(14) Lower and upper energy level values respectively, in units of cm −1 .(15) Notes: (N) Line in areas of strong noise and/or ringing, (B) blended line or asymmetric line profile.
F rNotes.The columns are as follows: (1)-(3) The energy level label consisting of: (1) configuration, (2) term, and (3) J value.(4) Energy level value in cm −1 .(5) Energy level uncertainty in cm −1 (6) The number of lines contributing to the level fit.(7)-(14) Eigenvector components and percentage contribution to the level.(15) Notes about the level: n = new level, r = level label revised in this work.

Figure 2 .
Figure 2. Comparison between the revised energy levels of this work and values previously published by Sugar & Corliss (1985).The newly revised values minus the previously published values are plotted against the newly revised energies with error bars in red for previously published values and in black for our new energy levels.
of Ni III, R Ni is the Rydberg constant for Ni II, Z c is the effective charge of the core (4 for Ni II), 〈r −4 〉 nl and 〈r −3 〉 nl are hydrogenic radial expectation values of an nl electron (in atomic units) given by Equations (3) and (4) ofSchoenfeld et al. (1995), and A, B, and D are constants calculated from the angular momentum quantum numbers by Equation (5) inSchoenfeld et al. (1995).The three unknown quantities in Equation (1), the ionization energy, dipole polarisability (α), and quadrupole moment of the core (Q), are determined from experimental energy level values

Table 1
Classified Lines of Ni II

Table 2
Energy Levels of Ni II