Received 29 November 2013; revised 3 January 2014; accepted 23 January 2014

ABSTRACT

The 3d³4p, 3d³5p and 3d²4s4p odd configurations of the V II spectrum have been reanalysed and three 3d²4s4p triplets are assigned higher energies than previously proposed. We have determined the fine structure parameters, the largest and next largest eigenvector percentages of levels, their calculated Landé g_J-factors and predicted positions for missing experimental levels up to 100,000 cm⁻¹ for the 3d²4s4p configuration. Furthermore for the first time a hyperfine structure (HFS) parametric treatment, involving levels of these two configurations has been carried out. The deduced single-electron HFS parameter values are successfully checked with those obtained by means of ab initio calculations.

Keywords:

Fine Structure, Hyperfine Structure, Energy Levels, Ab Initio Calculations, V II Spectrum

1. Introduction

The first analysis of the V II spectrum was done by Meggers and Moore [1] who found most of the predicted terms of the 3d³4p configuration and several low terms of the 3d²4s4p configuration. Some years later this spectrum was reobserved by the Madrid Group [2] [3] who extended these observations to the vacuum ultraviolet and infrared regions. This permitted to revise the previous assignments and to establish new levels. This V⁺ ion fine structure (fs) analysis continued to be improved with the passing years [4] - [7] . Recently we carried out fs studies of some singly ionised atoms, e.g. Ta II [8] or Nb II [9] , in an effort to complete previous works and to eliminate erroneous level assignments. This time we propose a similar work concerning V II, which presents high interest for astrophysical investigations since it is very useful in the study of the history of nucleosynthesis, chemically peculiar stars and the sun.

2. Fine and Hyperfine Structure Analysis

As mentioned in our past work, the complex configuration-interaction between the odd-parity levels, especially between 3d4s²4p, 3d²4s4p and 3d³4p of V II, would make this work difficult. To overcome this, we use the extensively tested method of [10] - [12] that is suitable for systems of many mutually interacting Rydberg configurations, with the fs Hamiltonian of [13] - [17] . For this work on V II, we used the following as the configuration basis set: 3d4s²4p, 3d4s²5p, 3d²4s4p, 3d²4s5p, 3d³4p, 3d³5p. Since there are many interaction integrals in this basis, by constraining the radial integrals to physically reasonable ratios [10] , which also required a few assumptions from Hartree-Fock calculations, the fit procedure became manageable. Due to the relatively low atomic number of vanadium, LS coupling is preferably used. We fit the odd parity levels, up to 88 × 10³ cm⁻¹, that have been observed experimentally [7] . The configurations 3d²4s4p and 3d³4p have their entire fs parameter set adjusted, while the majority of 3d³5p is adjusted. As the levels of the other three configurations are not yet observed experimentally, the fs parameters of states in 3d4s²4p, 3d4s²5p and 3d²4s5p, that are expected to perturb the states of the three lowest configurations, could not be fit efficiently. Therefore, all parameters in the configurations 3d4s²4p, 3d4s²5p and 3d²4s5p are fixed to the weighted values from ab intio calculations. Our fs least squares fit of 186 energy levels from [7] used 27 free parameters, for a total of 128 parameters. This fit accurately reproduced the experimental data, with a standard deviation of 70 cm⁻¹, with the exception of three triplets in the 3d²4s4p configuration: ³ F , ¹P; ³D, ³ F , ¹P; ³ F and ³ F , ¹P; ³G whose level positions should be located higher than given in [7] .

In Table 1, we show the observed and calculated energy levels, percentages of leading eigenvector components with their LS-term symbols, along with observed and calculated g_J-factors. The fitted fs parameters are listed in Table 2 and Table 3 along with, from the Cowan calculations [18] , their weighted ab initio values. The

Slater integral ratio, , determined as in [8] [9] , for the 3d²4s4p con-

figuration, provides the weighting for all parameters except spin-orbit parameters ζ_nd and ζ_np. Not listed in Table 2 and Table 3 are those fs parameters which are fixed to zero as they are theoretically expected to be small. In Table 4 we propose our predicted positions for the energy levels of these three erroneous triplets. We furthermore present in Table 4, for the 3d²4s4p configuration, predicted energy level values for all missing experimental values up to 100,000 cm⁻¹ to suggest further experimental investigations.

A many-body parametrisation calculation for the HFS analysis allows us to exploit the similarities between configuration interaction effects seen in spin-orbit and hyperfine splitting. Using Equations (4) and (5) in [19] for the A and B HFS constants, the radial parameters, , a_i and b_i are determined by fits to experimental values.

The first published magnetic dipole HFS A constants of ⁵¹V II, consisting of 24 even levels and 31 odd levels, are determined in 2011 by Armstrong, Rosner and Holt applying fast-ion-beam laser-fluorescence spectroscopy [20] , which we use for our HFS fit. Our HFS fit does not require us to make extra assumptions for the parameter values as the number of HFS A-values makes the fit overdetermined. We can see from the well-known equation

(1)

(in MHz) that our values for p- and d-electron HFS many-body parameters are valid; where we use

for ⁵¹V and show the computed expectation values in Table 5.

Previously, it was found [9] , even more so for p-electrons, that weighting the parameters by a ratio of spin-orbit constants obtained thanks to fine structure study and ab initio calculations, i.e. to multiply the second member of Equation (1) by: ζ_nl(fs)/ζ_nl(ab initio) improves the agreement of the calculated HFS values to the experimental values. For d-electrons this ratio is generally close to one and is sometimes superfluous to insert it in (1).

To check the value of the most influential HFS-deduced parameter

(2)

for open s-shell configurations which shows that is directly proportional to s-electron density

B: 3d²4s4p configuration; C: 3d³4p configuration; D: 3d³5p configuration.

F.V.: fitted value; C.C.: Cowan code; ^a: fixed to fitted value; ^f: fixed to weighted ab initio value (C.C.).

we suggest to compare the ratio: obtained by using fitted experimental HFS V II data (Table 5 of Part I of this work and Table 5 of part II) to the same ratio using the computed data given in Table 6, obtained

thanks to use of the pseudo relativistic Hartree-Fock code [21] . We can see that these two ratios are very similar

to the fourth decimal place and we can conclude that the value is very satisfactory. In Table 5,

comparing the ab initio and fitted HFS one-electron parameters for d-, p- and s-electrons we have achieved good agreement further verifying the veracity of the fitting model. Moreover we confirm through our calculations the well-founded basis of the experimental data of Armstrong, Rosner and Holt [20] , gathered in Table 7 where one can note that experimental values are close to calculated ones except for the level whose energy is equal to 54718 cm⁻¹.

^aTotals are contribution from all s-orbitals weighted by their occupation numbers.

3. Conclusion

We studied the spectrum of V II which permits to point out the incorrect positions of three triplets. We give refined fine structure parameters and leading eigenvectors percentages of levels and for the first time the calculated magnetic Landé g_I-factors, which are very useful for missing level assignments. Taking advantage of recent experimental work on hyperfine structure of this ion [20] we were also able to determine for the first time the predominant single-electron HFS parameter values, confirmed using ab initio calculations. It would be interesting to extend this study experimentally to the missing levels of 3d²4s4p to compare with our predicted positions. Further experimental work on the unknown levels of 3d³4p, 3d³5p, 3d4s²4p, 4d5s²5p and 3d²4s5p configurations would also be useful, since the situation is already formulated for future investigations of all existing levels of this basis.

References