Measurements of the Hyperfine Structure of Atomic Energy Levels in Co ii

For an atom with nucleus of spin quantum number I, individual fine structure atomic energy levels are split due to the interaction between the associated nuclear and total electronic magnetic dipole moments. Assuming no other perturbations, the energies of the hyperfine levels are shifted from the fine structure energy according to Kopfermann (1958):

$$\begin{eqnarray}&&{\rm{\Delta }}{W}_{{FJ}}=\displaystyle \frac{1}{2}{AK}+B\displaystyle \frac{(3/4)K(K+1)-J(J+1)I(I+1)}{2I(2I-1)J(2J-1)},\end{eqnarray} \tag{ 1 }$$

where ΔW_FJ is the change in energy from the fine structure level with angular momentum J to its corresponding HFS level with total atomic angular momentum F.

$$\begin{eqnarray*}&&F=I+J;I+J-1;...;| I-J| ,\end{eqnarray*}$$

and K is defined as

$$\begin{eqnarray*}&&K=F(F+1)-J(J+1)-I(I+1).\end{eqnarray*}$$

A and B are the magnetic dipole and electric quadrupole hyperfine interaction constants respectively. Within a fine structure transition (HFS pattern), the relative intensity of a hyperfine component transition, ${R}_{{JFJ}^{\prime} F^{\prime} }$, between the hyperfine levels with quantum numbers JF and $J^{\prime} F^{\prime}$ is given by

$$\begin{eqnarray}{R}_{{JFJ}^{\prime} F^{\prime} }=\displaystyle \frac{(2F+1)(2F^{\prime} +1)}{(2I+1)}{\left\{\begin{array}{ccc}J & F & I\\ F^{\prime} & J^{\prime} & 1\end{array}\right\}}^{2},\end{eqnarray} \tag{ 2 }$$

where the squared factor is a Wigner $6-j$ symbol, which includes the selection rules: ΔF = 0, ±1 but not $F=0\leftrightarrow F^{\prime} =0$ (Cowan 1981; Wahlgren et al. 1995).

For a transition between two fine structure levels, the hyperfine A and B constants of the levels determine the wavenumbers of all allowed hyperfine component transitions, relative to the center of gravity wavenumber of the transition.

Without any other forms of energy level splitting, the individual hyperfine component transitions observed from hollow cathode discharge lamps are emission spectral lines of a plasma, described by the Voigt profile (Thorne et al. 1988). Therefore, fine structure lines in the FT spectra of Co II were observed as blends of several Voigt profiles, defined by relative intensities and hyperfine interaction constants A and B. An example composition of an observed asymmetric line profile is shown in Figure 2.

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A Voigt spectral line profile results from the dominant Doppler-broadening and pressure-broadening effects in the gas, which are characterized by the Gaussian and Lorentzian width components respectively in the Voigt profile.

An observed line profile from an FT spectrometer is the convolution between its true profile and the instrumental profile. The instrumental profile is mostly dependent on source–aperture geometry and interferogram apodization procedures, but its functional form can be difficult to fully specify. For the purposes of fitting observed lines, apodizing the FT of the model line profile with a box function sufficiently approximates any ringing artifacts and distortion due to the instrumental profile (Davis et al. 2001). This is only usually necessary in modeling line profiles observed under lower resolving powers, which is readily seen from Figure 2, where the peaks without any component transitions on the sides were ringing artifacts significantly above the noise level.

As indicated in Section 3, a number of parameters are required to model the spectral line of a transition between fine structure energy levels of Co II. The J quantum numbers for both levels of each transition were known prior to this work, hence the following eight parameters required fitting in order to determine the HFS constants:

• 1.  
  A and B HFS constants of the two fine structure levels of the transition (four parameters).
• 2.  
  Component line Gaussian width G_w .
• 3.  
  Component line Lorentzian width L_w .
• 4.  
  Total intensity.
• 5.  
  Center of gravity wavenumber.

The hfs_fit program written during this work was the central tool of the analysis. Model lines were produced from the parameters, and the square of the residuals was minimized by optimizing the parameters using a simulated annealing algorithm. Optimizations were classified as complete when the residuals were of the order of the noise.

hfs_fit is capable of inspection of parameter dependence; comparisons between model and observed profiles are shown visually under parameter variation. Any parameter can be held constant for an optimization. The probability distributions for parameter values at each iteration within an optimization are Gaussian, their widths can be modified, as necessary. These features support the algorithm in reaching the global minimum by allowing supervision and the choice of suitable initial parameter values.

Of the eight parameters computationally optimized, line profiles were predominantly most sensitive to the A constants and G_w . The center of gravity wavenumber and total intensity only affect the position and area of the profiles, respectively; they do not affect the profile shape. Accurate initial values of the center of gravity wavenumber and total intensity were calculated from the wavenumber and area of the observed line data, or specified by inspection. L_w was about 10 mK and approximately constant within each spectrum, in comparison, G_w varied from 20–100 mK from the IR to visible and 120–220 mK across the UV, where the majority of the stronger Co II lines lie. Fits to all profiles showed much smaller sensitivity to B constants compared to that of the A constants, values of B are also in general one order smaller than the A constants (e.g., see Pickering 1996), so the initial values for B in the fitting process were always set to 0 mK.

Even with the above stated constraints, optimizations were not possible given arbitrary initial values of A constants and G_w . Furthermore, without any constraint on these three parameters, profiles such as the ones shown in Figures 1(a)–(c) would give a wide range of optimal values, which limited the accuracy of A constant measurements. In the worst cases, the optimal A constants could vary by more than 20 mK, and an example is shown in Figure 3. For this type of line profile, a redistribution of components (i.e., having different A constants to those shown), combined with slight changes in their width G_w and total intensity, could produce the same observed profile, this is seen from Figure 4.

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When performing HFS analysis on FT spectra with line profiles summarized in Figure 1, it is crucial to have further constraints on the A constants and G_w , and these can indeed be obtained as the analysis progresses.

Knowing one of the two A constants during the analysis of a line profile greatly reduces the range of possible optimal values for the other A and G_w . When the unknown A constant is accurately determined this way, it could then serve as a new reference A constant for other transitions involving its corresponding fine structure energy level. This process could be iterated from a set of known A constants used as references to increasingly include a larger number of energy levels, and their A constants could thus be sequentially determined.

Due to the iterative nature of this analysis method, values and uncertainties of subsequently determined A constants depend on the initial set of reference A constants. Highly accurate measurements, e.g., by laser or Fabry Pérot techniques, of A and B constants are not available for Co II. This Co II HFS analysis therefore began with the assumption of no previously known A or B constants.

Profiles offering the most constraints on A constants and G_w were selected for the set of initial reference A constants. They showed pronounced spectral features and their A constants could be determined with uncertainties ranging from ±1 mK up to ±3 mK, without any prior information on the A values or G_w . Figure 5 is an example of such profiles. Here, the highest intensity hyperfine component transition was clearly separated from others, which also allowed accurate determination of G_w . A total of 18 lines with similar profiles were chosen in the 36,000–42,000 cm⁻¹ spectral range from two UV spectra containing the largest number of Co II lines. The A constants of the 18 lines were then determined by fitting and checking for agreement among other lines that involved them. Any line involving a level with J = 0 naturally possesses tighter constraints on the values of A and G_w , since the A of the J = 0 level is zero. However, their profiles did not show pronounced spectral features and their unknown G_w hindered the determination of their A constants at this stage.

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This initial analysis not only produced a set of reference A constants, but also covered a broad spectral range within the two UV spectra, which provided an essential constraint on G_w . For a Maxwellian gas, the relationship between G_w and the wavenumber is linear. The G_w obtained from all fits in a particular UV spectrum are plotted against their wavenumbers in Figure 6. This relationship had a Pearson correlation coefficient of 0.8 for all spectra analyzed, which was also observed by Pickering (1996) in the analysis of Co I HFS. When combined with the strategy of using reference A constants, the unknown A constant of a fit became the only unknown parameter. This enabled a large number of further fits, which were then typically for symmetric profiles (e.g., Figure 3) or lower S/N profiles showing slight asymmetry. The most significant deviations, such as the five fits with G_w more than 3σ above the expectation seen in Figure 6, were suspected to be due to self-absorption.

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To determine an A constant and later use it as a reference to go on to find other A constants, lines of its associated fine structure level must show agreement across multiple spectra. In fulfilling this criterion for each A constant, any effects from self-absorption or blending with other lines could become apparent. When profiles failed to be fitted using expected G_w or A constants due to these effects, they were omitted from the estimation of associated HFS constants, this is further discussed in Section 4.5.

When an A constant was measured from multiple profiles observed in multiple spectra, the weighted sample mean and standard deviation were estimated. The weighting was determined by the S/N and the clarity of spectral features, where the latter was quantified by the derivative of the normalized model line profile. Specifically, the unnormalized weights w_i were calculated as

$$\begin{eqnarray}&&{w}_{i}={\rm{SNR}}\times {\left(\displaystyle \sum _{{\rm{n}}}\left|\displaystyle \frac{{I}_{n+1}-{I}_{n}}{{\rm{\Delta }}\sigma }\right|\right)}^{3},\end{eqnarray} \tag{ 3 }$$

where I_n is the normalized intensity of the model line profile and n is the index in the wavenumber axis. The wavenumber axis of each line profile was scaled to be from 0 to 1, representing the spectral range of the line profile. Δσ is the scaled resolution, and so profiles with fewer data points were also weighted less. The power of 3 on the summation places more weight on spectral features rather than S/N, because profiles with more asymmetry and features constrained the fitted A constants and G_w better. The S/N was weighted less because it did not affect the line profile unless it was very low, and in most cases such lines could not be used in the analysis. If the S/Ns of the profiles shown in Figure 1 were the same, (d) would be weighted greatest with 0.97, which is expected as this profile constrains A and G_w far better than other examples in the figure. Even if profiles (a), (b), and (c) were to have 10 times the S/N of (d), the weights from (a) to (d) would be 0.02, 0.11, 0.11, and 0.76 respectively, still in favor of profile (d). The electric quadrupole B constants were also estimated using this weighted method.

Most lines did not show pronounced spectral features (e.g., Figures 1(a)–(c), 2, and 3), and so their A constants were less well constrained. Furthermore, many levels had only such profiles and less than five lines suitable for fitting. The variability of optimal A values from the fits was therefore the indicator for uncertainty, and so the weighted estimation was then unused. Instead, the estimation of the A constant was done on a worst-fit basis, i.e., fits were carried out at all mean values and also at extremes of possible values of the reference A constant and G_w (from Figure 6) based on their uncertainties. The B constants could not be estimated this way, due to profile insensitivity to B.

Even with the G_w and reference A known, there were occasional two-value ambiguities for the A constant of interest. This problem was encountered by Lawler et al. (2018) and was similarly solved by checking agreement among multiple lines of the level with the A constant.

Spectral lines affected by self-absorption can appear broadened, flattened, or self-reversed (Cowan & Dieke 1948). Effects of self-absorption in a fine structure line were indicated by comparing its observed profiles in spectra with varying lamp conditions, particularly those with differing carrier gases (either Ar or Ne), while also examining G_w trends of each spectrum and the associated A constants determined from other lines. Blends could be similarly identified.

Self-absorption was modeled using a parameter S that incorporated the absorption coefficient of the transition, adopted from the DECOMP program by J. W. Brault (private communication; Abrams 1989):

$$\begin{eqnarray}&&I(\sigma )={I}_{0}(\sigma )\ {e}^{-{{SI}}_{0}(\sigma )},\end{eqnarray} \tag{ 4 }$$

where I(σ) and I₀(σ) are the line profiles after and prior to self-absorption respectively. Using one S parameter to account for self-absorption in profiles with anomalous G_w did not shift G_w down to expected values. Introducing an S parameter to each hyperfine component transition produced the expected G_w and A constants, but these fits had too many parameters for results to be meaningful. For blends, unknown HFS constants and line intensities introduce the same issue. Therefore, profiles with anomalous G_w and blends were omitted from the estimation of associated A constants.

All measured magnetic dipole interaction constants, A, are reported in Tables 1 and 2 for even and odd levels, respectively. The configuration, term, J value, and energy of each level are listed in the first four columns, which were reported in Pickering et al. (1998). The fifth and sixth column list the A constants and their uncertainties in mK (1 mK = 0.001 cm⁻¹). The number of line profiles used to determine A values and their uncertainties, N, is indicated in the seventh column. Previous measurements are given in the last column where available.

Table 1. HFS Magnetic Interaction A Constants of the Even-parity Energy Levels of Co II

Configuration	Term	J	Energy	A	Uncertainty	N	Previous Work
			(cm−1)	(mK)	(mK)		(mK)
3d8	$a{}^{3}{\rm{F}}$	4	0.000	13	5	2
		3	950.324	17	5	2
		2	1597.197	33	4	1
$3{d}^{7}{(}^{4}{\rm{F}})4{\rm{s}}$	$a{}^{5}{\rm{F}}$	5	3350.494	34.4	1.0	7	33.8 ± 0.8 a
		4	4028.988	29.5	1.8	9	29.9 ± 0.8 a
		3	4560.789	25.4	2.2	6	25.0 ± 0.8 a
		2	4950.062	17.1	2.0	8	17.6 ± 0.8 a
		1	5204.698	−8.4	1.2	5	−9.2 ± 1.0 a

Notes. Energy level values were taken from Pickering et al. (1998), Pickering (1998a), and Pickering (1998b) with a correction of 6.7 parts in 10⁸ applied to place the levels on the revised wavenumber scale as recommended by Nave & Sansonetti (2011). N is the number of observed profiles used to determine the value and uncertainty of the A constants.

^a Lawler et al. (2018).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 2. HFS Magnetic Interaction A Constants of the Odd-parity Energy Levels of Co II

Configuration	Term	J	Energy	A	Uncertainty	N	Previous Work
			(cm−1)	(mK)	(mK)		(mK)
$3{d}^{7}{(}^{4}{\rm{F}})4{\rm{p}}$	z5F	5	45,197.711	12.1	1.5	5	10.9 ± 0.8 a
		4	45,378.754	8.5	1.5	6	9.3 ± 0.8 a
		3	45,972.036	13.8	3.1	4	11.5 ± 0.8 a
		2	46,452.700	18.6	1.0	4
		1	46,786.409	45.6	1.0	1
$3{d}^{7}{(}^{4}{\rm{F}})4{\rm{p}}$	z5D	4	46,320.832	8.8	1.7	3	8.8 ± 0.8 a , 9 ± 2 b
		3	47,039.105	8.1	2.2	5	7.8 ± 0.8 a , 8 ± 20 b
		2	47,537.365	8.0	2.6	3	9.0 ± 0.8 a
		1	47,848.781	2	4	3	5.5 ± 0.7 a , 5 ± 5 b
		0	47,995.594	0	0

Notes. Energy level values were taken from Pickering et al. (1998), Pickering (1998a), and Pickering (1998b) with a correction of 6.7 parts in 10⁸ applied to place the levels on the revised wavenumber scale as recommended by Nave & Sansonetti (2011). N is the number of observed profiles used to determine the value and uncertainty of the A constants.

^a Lawler et al. (2018). ^b Bergemann et al. (2010).

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Agreement was found within joint uncertainties between previously published values by Bergemann et al. (2010), Lawler et al. (2018), and the new A constants of this work. The uncertainties of each A constants have been carefully estimated, allowing their contribution to any resulting astrophysical chemical abundance determination uncertainties to be calculated. In the vast majority of cases, the small uncertainties of the A constants are expected to have a negligible effect on the resulting abundance uncertainties, given typical observational uncertainties.

As expected, the A constants were typically larger for levels involving an unpaired s valence electron, and they became smaller as the principal and angular momentum quantum numbers of the valence electron increased. One A constant was found for the 5d levels, which was for the $3{{d}}^{7}{(}^{4}{\rm{F}})5{d}\ {}^{5}{{\rm{G}}}_{2}$ level. No HFS analysis was carried out for the 5g levels. Other 5d levels and all 5g levels are excluded from the tables.

The electric quadrupole interaction constants, B, were unable to be estimated accurately. Line profiles were mostly sensitive to the difference in the B constants of the profiles, rather than in the particular B values themselves. The resulting uncertainties from the weighted calculations were always similar to, if not larger than, the estimated means. Hence, they are not reported in the table.

However, lines involving the $3{d}^{7}{(}^{4}{\rm{P}})4{s}\ \ {a}{}^{5}{{\rm{P}}}_{2}$ level consistently indicated a positive value for its B constant. The 14 profiles investigated for this level gave estimates of B as 5 ± 4 mK.